Theorem phrel 30751

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 30750	. . 3 ⊢ (𝑥 ∈ CPreHilOLD → 𝑥 ∈ NrmCVec)
2	1	ssriv 3953	. 2 ⊢ CPreHilOLD ⊆ NrmCVec
3		nvrel 30538	. 2 ⊢ Rel NrmCVec
4		relss 5747	. 2 ⊢ (CPreHilOLD ⊆ NrmCVec → (Rel NrmCVec → Rel CPreHilOLD))
5	2, 3, 4	mp2 9	1 ⊢ Rel CPreHilOLD

Syntax hints:   ⊆ wss 3917  Rel wrel 5646  NrmCVeccnv 30520  CPreHil_OLDccphlo 30748

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 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

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 2709  df-cleq 2722  df-clel 2804  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-opab 5173  df-xp 5647  df-rel 5648  df-oprab 7394  df-nv 30528  df-ph 30749

This theorem is referenced by:  phop  30754