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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 28004 | . 2 ⊢ NrmCVec ⊆ (CVecOLD × V) | |
2 | relxp 5361 | . 2 ⊢ Rel (CVecOLD × V) | |
3 | relss 5442 | . 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 3415 ⊆ wss 3799 × cxp 5341 Rel wrel 5348 CVecOLDcvc 27969 NrmCVeccnv 27995 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-sep 5006 ax-nul 5014 ax-pr 5128 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-rab 3127 df-v 3417 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-nul 4146 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-opab 4937 df-xp 5349 df-rel 5350 df-oprab 6910 df-nv 28003 |
This theorem is referenced by: nvop2 28019 nvop 28087 phrel 28226 bnrel 28279 |
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