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Theorem ofc1 7421
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 4192 . 2 (𝐴𝐴) = 𝐴
7 fvconst2g 6956 . . 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 7407 1 ((𝜑𝑋𝐴) → (((𝐴 × {𝐵}) ∘f 𝑅𝐹)‘𝑋) = (𝐵𝑅𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  {csn 4557   × cxp 5546   Fn wfn 6343  cfv 6348  (class class class)co 7145  f cof 7396
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  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 7148  df-oprab 7149  df-mpo 7150  df-of 7398
This theorem is referenced by:  ofnegsub  11624  pwsvscaval  16756  lmhmvsca  19746  psrvscaval  20100  mplvscaval  20156  coe1sclmulfv  20379  mamuvs1  20942  mamuvs2  20943  matvscacell  20973  mdetrsca  21140  mbfmulc2lem  24175  i1fmulclem  24230  itg1mulc  24232  itg2monolem1  24278  uc1pmon1p  24672  coemulc  24772  basellem9  25593  ofdivrec  40535
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