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Mirrors > Home > MPE Home > Th. List > phrel | Structured version Visualization version GIF version |
Description: The class of all complex inner product spaces is a relation. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.) |
Ref | Expression |
---|---|
phrel | ⊢ Rel CPreHilOLD |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | phnv 28570 | . . 3 ⊢ (𝑥 ∈ CPreHilOLD → 𝑥 ∈ NrmCVec) | |
2 | 1 | ssriv 3954 | . 2 ⊢ CPreHilOLD ⊆ NrmCVec |
3 | nvrel 28358 | . 2 ⊢ Rel NrmCVec | |
4 | relss 5637 | . 2 ⊢ (CPreHilOLD ⊆ NrmCVec → (Rel NrmCVec → Rel CPreHilOLD)) | |
5 | 2, 3, 4 | mp2 9 | 1 ⊢ Rel CPreHilOLD |
Colors of variables: wff setvar class |
Syntax hints: ⊆ wss 3919 Rel wrel 5541 NrmCVeccnv 28340 CPreHilOLDccphlo 28568 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2792 ax-sep 5184 ax-nul 5191 ax-pr 5311 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2799 df-cleq 2813 df-clel 2891 df-nfc 2959 df-rab 3142 df-v 3483 df-dif 3922 df-un 3924 df-in 3926 df-ss 3935 df-nul 4275 df-if 4449 df-sn 4549 df-pr 4551 df-op 4555 df-opab 5110 df-xp 5542 df-rel 5543 df-oprab 7141 df-nv 28348 df-ph 28569 |
This theorem is referenced by: phop 28574 |
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