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Theorem ofcc 30498
Description: Left operation by a constant on a mixed operation with a constant. (Contributed by Thierry Arnoux, 31-Jan-2017.)
Hypotheses
Ref Expression
ofcc.1 (𝜑𝐴𝑉)
ofcc.2 (𝜑𝐵𝑊)
ofcc.3 (𝜑𝐶𝑋)
Assertion
Ref Expression
ofcc (𝜑 → ((𝐴 × {𝐵})∘𝑓/𝑐𝑅𝐶) = (𝐴 × {(𝐵𝑅𝐶)}))

Proof of Theorem ofcc
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 ofcc.2 . . . 4 (𝜑𝐵𝑊)
2 fnconstg 6254 . . . 4 (𝐵𝑊 → (𝐴 × {𝐵}) Fn 𝐴)
31, 2syl 17 . . 3 (𝜑 → (𝐴 × {𝐵}) Fn 𝐴)
4 ofcc.1 . . 3 (𝜑𝐴𝑉)
5 ofcc.3 . . 3 (𝜑𝐶𝑋)
6 fvconst2g 6632 . . . 4 ((𝐵𝑊𝑥𝐴) → ((𝐴 × {𝐵})‘𝑥) = 𝐵)
71, 6sylan 489 . . 3 ((𝜑𝑥𝐴) → ((𝐴 × {𝐵})‘𝑥) = 𝐵)
83, 4, 5, 7ofcfval 30490 . 2 (𝜑 → ((𝐴 × {𝐵})∘𝑓/𝑐𝑅𝐶) = (𝑥𝐴 ↦ (𝐵𝑅𝐶)))
9 fconstmpt 5320 . 2 (𝐴 × {(𝐵𝑅𝐶)}) = (𝑥𝐴 ↦ (𝐵𝑅𝐶))
108, 9syl6eqr 2812 1 (𝜑 → ((𝐴 × {𝐵})∘𝑓/𝑐𝑅𝐶) = (𝐴 × {(𝐵𝑅𝐶)}))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  wcel 2139  {csn 4321  cmpt 4881   × cxp 5264   Fn wfn 6044  cfv 6049  (class class class)co 6814  𝑓/𝑐cofc 30487
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 6817  df-oprab 6818  df-mpt2 6819  df-ofc 30488
This theorem is referenced by: (None)
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