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Theorem caofcom 7721
Description: Transfer a commutative law to the function operation. (Contributed by Mario Carneiro, 26-Jul-2014.)
Hypotheses
Ref Expression
caofref.1 (𝜑𝐴𝑉)
caofref.2 (𝜑𝐹:𝐴𝑆)
caofcom.3 (𝜑𝐺:𝐴𝑆)
caofcom.4 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝑅𝑦) = (𝑦𝑅𝑥))
Assertion
Ref Expression
caofcom (𝜑 → (𝐹f 𝑅𝐺) = (𝐺f 𝑅𝐹))
Distinct variable groups:   𝑥,𝑦,𝐹   𝑥,𝐺,𝑦   𝜑,𝑥,𝑦   𝑥,𝑅,𝑦   𝑥,𝑆,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem caofcom
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 caofref.2 . . . . . 6 (𝜑𝐹:𝐴𝑆)
21ffvelcdmda 7093 . . . . 5 ((𝜑𝑤𝐴) → (𝐹𝑤) ∈ 𝑆)
3 caofcom.3 . . . . . 6 (𝜑𝐺:𝐴𝑆)
43ffvelcdmda 7093 . . . . 5 ((𝜑𝑤𝐴) → (𝐺𝑤) ∈ 𝑆)
52, 4jca 510 . . . 4 ((𝜑𝑤𝐴) → ((𝐹𝑤) ∈ 𝑆 ∧ (𝐺𝑤) ∈ 𝑆))
6 caofcom.4 . . . . 5 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝑅𝑦) = (𝑦𝑅𝑥))
76caovcomg 7616 . . . 4 ((𝜑 ∧ ((𝐹𝑤) ∈ 𝑆 ∧ (𝐺𝑤) ∈ 𝑆)) → ((𝐹𝑤)𝑅(𝐺𝑤)) = ((𝐺𝑤)𝑅(𝐹𝑤)))
85, 7syldan 589 . . 3 ((𝜑𝑤𝐴) → ((𝐹𝑤)𝑅(𝐺𝑤)) = ((𝐺𝑤)𝑅(𝐹𝑤)))
98mpteq2dva 5249 . 2 (𝜑 → (𝑤𝐴 ↦ ((𝐹𝑤)𝑅(𝐺𝑤))) = (𝑤𝐴 ↦ ((𝐺𝑤)𝑅(𝐹𝑤))))
10 caofref.1 . . 3 (𝜑𝐴𝑉)
111feqmptd 6966 . . 3 (𝜑𝐹 = (𝑤𝐴 ↦ (𝐹𝑤)))
123feqmptd 6966 . . 3 (𝜑𝐺 = (𝑤𝐴 ↦ (𝐺𝑤)))
1310, 2, 4, 11, 12offval2 7705 . 2 (𝜑 → (𝐹f 𝑅𝐺) = (𝑤𝐴 ↦ ((𝐹𝑤)𝑅(𝐺𝑤))))
1410, 4, 2, 12, 11offval2 7705 . 2 (𝜑 → (𝐺f 𝑅𝐹) = (𝑤𝐴 ↦ ((𝐺𝑤)𝑅(𝐹𝑤))))
159, 13, 143eqtr4d 2775 1 (𝜑 → (𝐹f 𝑅𝐺) = (𝐺f 𝑅𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  cmpt 5232  wf 6545  cfv 6549  (class class class)co 7419  f cof 7683
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-of 7685
This theorem is referenced by:  plydivlem4  26276  quotcan  26289  dchrabl  27232  plymulx0  34310  lfladdcom  38674  expgrowth  43914  amgmwlem  48421
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