Theorem phrel 8405

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 8404	. . 3 |- (x e. CPreHil -> x e. NrmCVec)
2	1	ssriv 2059	. 2 |- CPreHil (_ NrmCVec
3		nvrel 8159	. 2 |- Rel NrmCVec
4		relss 3236	. 2 |- (CPreHil (_ NrmCVec -> (Rel NrmCVec -> Rel CPreHil))
5	2, 3, 4	mp2 43	1 |- Rel CPreHil

Syntax hints:   (_ wss 2037  Rel wrel 3165  NrmCVeccnv 8141  CPreHilcphl 8402

This theorem is referenced by:  phop 8408

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-br 2610  df-opab 2657  df-xp 3174  df-rel 3175  df-cnv 3176  df-dm 3178  df-rn 3179  df-oprab 3951  df-nv 8149  df-ph 8403

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