In the theory of tensor products of normed spaces, ‘The Five Basic Lemmas” (see Section 13 in Defant and Floret’s book Defant and Floret (Tensor Norms and Operator Ideals. North-Holland Mathematics Studies, vol. 176. North-Holland Publishing Co., Amsterdam, 1993)) are rather simple results which turn out to be “basic for the understanding and use of tensor norms”. Namely, they are the Approximation Lemma, the Extension Lemma, the Embedding Lemma, the Density Lemma and the Local Technique Lemma. We present here the analogous results for the operator space setting and also exhibit some applications as example of their potential. Our presentation follows the lines of Defant and Floret (Tensor Norms and Operator Ideals. North-Holland Mathematics Studies, vol. 176. North-Holland Publishing Co., Amsterdam, 1993). Although the proofs are similar to the Banach space case, the operator space nature of our tensor products introduces some difficulties and we have to deal in most of the cases with additional hypotheses of local reflexivity. However, for the newly introduced family of o.s. tensor norms (called extended \(\lambda \) -o.s. tensor norms, see Definition 4.2.3) the conditions about local reflexivity can be avoided.