Theorem phrel 31422

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 31421	. . 3 ⊢ (𝑥 ∈ CPreHilOLD → 𝑥 ∈ NrmCVec)
2	1	ssriv 3912	. 2 ⊢ CPreHilOLD ⊆ NrmCVec
3		nvrel 31209	. 2 ⊢ Rel NrmCVec
4		relss 5757	. 2 ⊢ (CPreHilOLD ⊆ NrmCVec → (Rel NrmCVec → Rel CPreHilOLD))
5	2, 3, 4	mp2 9	1 ⊢ Rel CPreHilOLD

This proof depends on syntax axioms:   ⊆ wss 3895  Rel wrel 5655  NrmCVeccnv 31191  CPreHil_OLDccphlo 31419

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-11 2193  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-opab 5167  df-xp 5656  df-rel 5657  df-oprab 7424  df-nv 31199  df-ph 31420

This theorem is used by:  phop  31425