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Theorem dvhvscaval 31834
 Description: The scalar product operation for the constructed full vector space H. (Contributed by NM, 20-Nov-2013.)
Hypothesis
Ref Expression
dvhvscaval.s
Assertion
Ref Expression
dvhvscaval
Distinct variable groups:   ,,   ,,
Allowed substitution hints:   (,)   (,)   (,)

Proof of Theorem dvhvscaval
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq1 5719 . . 3
2 coeq1 5022 . . 3
31, 2opeq12d 3984 . 2
4 fveq2 5720 . . . 4
54fveq2d 5724 . . 3
6 fveq2 5720 . . . 4
76coeq2d 5027 . . 3
85, 7opeq12d 3984 . 2
9 dvhvscaval.s . . 3
109dvhvscacbv 31833 . 2
11 opex 4419 . 2
123, 8, 10, 11ovmpt2 6201 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   wceq 1652   wcel 1725  cop 3809   cxp 4868   ccom 4874  cfv 5446  (class class class)co 6073   cmpt2 6075  c1st 6339  c2nd 6340 This theorem is referenced by:  dvhvsca  31836  dvhopspN  31850 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078
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