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Theorem ofc1 7085
Description: Left operation by a constant. (Contributed by Mario Carneiro, 24-Jul-2014.)
Hypotheses
Ref Expression
ofc1.1 (𝜑𝐴𝑉)
ofc1.2 (𝜑𝐵𝑊)
ofc1.3 (𝜑𝐹 Fn 𝐴)
ofc1.4 ((𝜑𝑋𝐴) → (𝐹𝑋) = 𝐶)
Assertion
Ref Expression
ofc1 ((𝜑𝑋𝐴) → (((𝐴 × {𝐵}) ∘𝑓 𝑅𝐹)‘𝑋) = (𝐵𝑅𝐶))

Proof of Theorem ofc1
StepHypRef Expression
1 ofc1.2 . . 3 (𝜑𝐵𝑊)
2 fnconstg 6254 . . 3 (𝐵𝑊 → (𝐴 × {𝐵}) Fn 𝐴)
31, 2syl 17 . 2 (𝜑 → (𝐴 × {𝐵}) Fn 𝐴)
4 ofc1.3 . 2 (𝜑𝐹 Fn 𝐴)
5 ofc1.1 . 2 (𝜑𝐴𝑉)
6 inidm 3965 . 2 (𝐴𝐴) = 𝐴
7 fvconst2g 6631 . . 3 ((𝐵𝑊𝑋𝐴) → ((𝐴 × {𝐵})‘𝑋) = 𝐵)
81, 7sylan 489 . 2 ((𝜑𝑋𝐴) → ((𝐴 × {𝐵})‘𝑋) = 𝐵)
9 ofc1.4 . 2 ((𝜑𝑋𝐴) → (𝐹𝑋) = 𝐶)
103, 4, 5, 5, 6, 8, 9ofval 7071 1 ((𝜑𝑋𝐴) → (((𝐴 × {𝐵}) ∘𝑓 𝑅𝐹)‘𝑋) = (𝐵𝑅𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  {csn 4321   × cxp 5264   Fn wfn 6044  cfv 6049  (class class class)co 6813  𝑓 cof 7060
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-of 7062
This theorem is referenced by:  ofnegsub  11210  pwsvscaval  16357  lmhmvsca  19247  psrvscaval  19594  mplvscaval  19650  coe1sclmulfv  19855  mamuvs1  20413  mamuvs2  20414  matvscacell  20444  mdetrsca  20611  mbfmulc2lem  23613  i1fmulclem  23668  itg1mulc  23670  itg2monolem1  23716  uc1pmon1p  24110  coemulc  24210  basellem9  25014  ofdivrec  39027
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