Theorem phrel 9815

Description: The class of all complex inner product spaces is a relation.

Assertion

Ref	Expression
phrel	|- Rel CPreHil

Proof of Theorem phrel

Step	Hyp	Ref	Expression
1		phnv 9814	. . 3 |- (x e. CPreHil -> x e. NrmCVec)
2	1	ssriv 2621	. 2 |- CPreHil C_ NrmCVec
3		nvrel 9553	. 2 |- Rel NrmCVec
4		relss 4074	. 2 |- (CPreHil C_ NrmCVec -> (Rel NrmCVec -> Rel CPreHil))
5	2, 3, 4	mp2 54	1 |- Rel CPreHil

Syntax hints:   C_ wss 2593  Rel wrel 3991  NrmCVeccnv 9535  CPreHilcphl 9812

This theorem is referenced by:  phop 9818

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339  df-opab 3396  df-xp 4000  df-rel 4001  df-cnv 4002  df-dm 4004  df-rn 4005  df-oprab 4887  df-nv 9543  df-ph 9813

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