Theorem phrel 29156

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 29155	. . 3 ⊢ (𝑥 ∈ CPreHilOLD → 𝑥 ∈ NrmCVec)
2	1	ssriv 3929	. 2 ⊢ CPreHilOLD ⊆ NrmCVec
3		nvrel 28943	. 2 ⊢ Rel NrmCVec
4		relss 5690	. 2 ⊢ (CPreHilOLD ⊆ NrmCVec → (Rel NrmCVec → Rel CPreHilOLD))
5	2, 3, 4	mp2 9	1 ⊢ Rel CPreHilOLD

Syntax hints:   ⊆ wss 3891  Rel wrel 5593  NrmCVeccnv 28925  CPreHil_OLDccphlo 29153

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-11 2157  ax-ext 2710  ax-sep 5226  ax-nul 5233  ax-pr 5355

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-sb 2071  df-clab 2717  df-cleq 2731  df-clel 2817  df-rab 3074  df-v 3432  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-sn 4567  df-pr 4569  df-op 4573  df-opab 5141  df-xp 5594  df-rel 5595  df-oprab 7272  df-nv 28933  df-ph 29154

This theorem is referenced by:  phop  29159