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Mirrors > Home > MPE Home > Th. List > nvrel | Structured version Visualization version GIF version |
Description: The class of all normed complex vectors spaces is a relation. (Contributed by NM, 14-Nov-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nvrel | ⊢ Rel NrmCVec |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nvss 27992 | . 2 ⊢ NrmCVec ⊆ (CVecOLD × V) | |
2 | relxp 5360 | . 2 ⊢ Rel (CVecOLD × V) | |
3 | relss 5441 | . 2 ⊢ (NrmCVec ⊆ (CVecOLD × V) → (Rel (CVecOLD × V) → Rel NrmCVec)) | |
4 | 1, 2, 3 | mp2 9 | 1 ⊢ Rel NrmCVec |
Colors of variables: wff setvar class |
Syntax hints: Vcvv 3414 ⊆ wss 3798 × cxp 5340 Rel wrel 5347 CVecOLDcvc 27957 NrmCVeccnv 27983 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pr 5127 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-rab 3126 df-v 3416 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-opab 4936 df-xp 5348 df-rel 5349 df-oprab 6909 df-nv 27991 |
This theorem is referenced by: nvop2 28007 nvop 28075 phrel 28214 bnrel 28267 |
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