Theorem phrel 30744

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 30743	. . 3 ⊢ (𝑥 ∈ CPreHilOLD → 𝑥 ∈ NrmCVec)
2	1	ssriv 3950	. 2 ⊢ CPreHilOLD ⊆ NrmCVec
3		nvrel 30531	. 2 ⊢ Rel NrmCVec
4		relss 5744	. 2 ⊢ (CPreHilOLD ⊆ NrmCVec → (Rel NrmCVec → Rel CPreHilOLD))
5	2, 3, 4	mp2 9	1 ⊢ Rel CPreHilOLD

Syntax hints:   ⊆ wss 3914  Rel wrel 5643  NrmCVeccnv 30513  CPreHil_OLDccphlo 30741

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-11 2158  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-opab 5170  df-xp 5644  df-rel 5645  df-oprab 7391  df-nv 30521  df-ph 30742

This theorem is referenced by:  phop  30747