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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 27053 | . . 3 ⊢ (𝑥 ∈ CPreHilOLD → 𝑥 ∈ NrmCVec) | |
2 | 1 | ssriv 3572 | . 2 ⊢ CPreHilOLD ⊆ NrmCVec |
3 | nvrel 26841 | . 2 ⊢ Rel NrmCVec | |
4 | relss 5129 | . 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 3540 Rel wrel 5043 NrmCVeccnv 26823 CPreHilOLDccphlo 27051 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-opab 4644 df-xp 5044 df-rel 5045 df-oprab 6553 df-nv 26831 df-ph 27052 |
This theorem is referenced by: phop 27057 |
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