Theorem phrel 30068

Description: The class of all complex inner product spaces is a relation. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.)

Assertion

Ref	Expression
phrel	⊢ Rel CPreHilOLD

Proof of Theorem phrel

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		phnv 30067	. . 3 ⊢ (𝑥 ∈ CPreHilOLD → 𝑥 ∈ NrmCVec)
2	1	ssriv 3987	. 2 ⊢ CPreHilOLD ⊆ NrmCVec
3		nvrel 29855	. 2 ⊢ Rel NrmCVec
4		relss 5782	. 2 ⊢ (CPreHilOLD ⊆ NrmCVec → (Rel NrmCVec → Rel CPreHilOLD))
5	2, 3, 4	mp2 9	1 ⊢ Rel CPreHilOLD

Syntax hints:   ⊆ wss 3949  Rel wrel 5682  NrmCVeccnv 29837  CPreHil_OLDccphlo 30065

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-11 2155  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-opab 5212  df-xp 5683  df-rel 5684  df-oprab 7413  df-nv 29845  df-ph 30066

This theorem is referenced by:  phop  30071