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Theorem hodmval 23232
 Description: Value of the difference of two Hilbert space operators. (Contributed by NM, 9-Nov-2000.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.)
Assertion
Ref Expression
hodmval
Distinct variable groups:   ,   ,

Proof of Theorem hodmval
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ax-hilex 22494 . . 3
21, 1elmap 7034 . 2
31, 1elmap 7034 . 2
4 fveq1 5719 . . . . 5
54oveq1d 6088 . . . 4
65mpteq2dv 4288 . . 3
7 fveq1 5719 . . . . 5
87oveq2d 6089 . . . 4
98mpteq2dv 4288 . . 3
10 df-hodif 23227 . . 3
111mptex 5958 . . 3
126, 9, 10, 11ovmpt2 6201 . 2
132, 3, 12syl2anbr 467 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   wceq 1652   wcel 1725   cmpt 4258  wf 5442  cfv 5446  (class class class)co 6073   cmap 7010  chil 22414   cmv 22420   chod 22435 This theorem is referenced by:  hodval  23237  hosubcli  23264 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-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-hilex 22494 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-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-map 7012  df-hodif 23227
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