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Theorem ofc1 7424
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 6560 . . 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 4193 . 2 (𝐴𝐴) = 𝐴
7 fvconst2g 6957 . . 3 ((𝐵𝑊𝑋𝐴) → ((𝐴 × {𝐵})‘𝑋) = 𝐵)
81, 7sylan 582 . 2 ((𝜑𝑋𝐴) → ((𝐴 × {𝐵})‘𝑋) = 𝐵)
9 ofc1.4 . 2 ((𝜑𝑋𝐴) → (𝐹𝑋) = 𝐶)
103, 4, 5, 5, 6, 8, 9ofval 7410 1 ((𝜑𝑋𝐴) → (((𝐴 × {𝐵}) ∘f 𝑅𝐹)‘𝑋) = (𝐵𝑅𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1531  wcel 2108  {csn 4559   × cxp 5546   Fn wfn 6343  cfv 6348  (class class class)co 7148  f cof 7399
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7151  df-oprab 7152  df-mpo 7153  df-of 7401
This theorem is referenced by:  ofnegsub  11628  pwsvscaval  16760  lmhmvsca  19809  psrvscaval  20164  mplvscaval  20220  coe1sclmulfv  20443  mamuvs1  21006  mamuvs2  21007  matvscacell  21037  mdetrsca  21204  mbfmulc2lem  24240  i1fmulclem  24295  itg1mulc  24297  itg2monolem1  24343  uc1pmon1p  24737  coemulc  24837  basellem9  25658  ofdivrec  40649
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