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Theorem ofc1 7725
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 6797 . . 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 4235 . 2 (𝐴𝐴) = 𝐴
7 fvconst2g 7222 . . 3 ((𝐵𝑊𝑋𝐴) → ((𝐴 × {𝐵})‘𝑋) = 𝐵)
81, 7sylan 580 . 2 ((𝜑𝑋𝐴) → ((𝐴 × {𝐵})‘𝑋) = 𝐵)
9 ofc1.4 . 2 ((𝜑𝑋𝐴) → (𝐹𝑋) = 𝐶)
103, 4, 5, 5, 6, 8, 9ofval 7708 1 ((𝜑𝑋𝐴) → (((𝐴 × {𝐵}) ∘f 𝑅𝐹)‘𝑋) = (𝐵𝑅𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  {csn 4631   × cxp 5687   Fn wfn 6558  cfv 6563  (class class class)co 7431  f cof 7695
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697
This theorem is referenced by:  ofnegsub  12262  pwsvscaval  17542  lmhmvsca  21062  psrvscaval  21988  mplvscaval  22054  coe1sclmulfv  22302  mamuvs1  22425  mamuvs2  22426  matvscacell  22458  mdetrsca  22625  mbfmulc2lem  25696  i1fmulclem  25752  itg1mulc  25754  itg2monolem1  25800  uc1pmon1p  26206  coemulc  26309  basellem9  27147  mhphf  42584  ofdivrec  44322
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