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Theorem caofass 7191
Description: Transfer an associative 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)
caofass.4 (𝜑𝐻:𝐴𝑆)
caofass.5 ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥𝑅𝑦)𝑇𝑧) = (𝑥𝑂(𝑦𝑃𝑧)))
Assertion
Ref Expression
caofass (𝜑 → ((𝐹𝑓 𝑅𝐺) ∘𝑓 𝑇𝐻) = (𝐹𝑓 𝑂(𝐺𝑓 𝑃𝐻)))
Distinct variable groups:   𝑥,𝑦,𝑧,𝐹   𝑥,𝐺,𝑦,𝑧   𝑥,𝐻,𝑦,𝑧   𝑥,𝑂,𝑦,𝑧   𝑥,𝑃,𝑦,𝑧   𝜑,𝑥,𝑦,𝑧   𝑥,𝑅,𝑦,𝑧   𝑥,𝑆,𝑦,𝑧   𝑥,𝑇,𝑦,𝑧
Allowed substitution hints:   𝐴(𝑥,𝑦,𝑧)   𝑉(𝑥,𝑦,𝑧)

Proof of Theorem caofass
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 caofass.5 . . . . . 6 ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥𝑅𝑦)𝑇𝑧) = (𝑥𝑂(𝑦𝑃𝑧)))
21ralrimivvva 3181 . . . . 5 (𝜑 → ∀𝑥𝑆𝑦𝑆𝑧𝑆 ((𝑥𝑅𝑦)𝑇𝑧) = (𝑥𝑂(𝑦𝑃𝑧)))
32adantr 474 . . . 4 ((𝜑𝑤𝐴) → ∀𝑥𝑆𝑦𝑆𝑧𝑆 ((𝑥𝑅𝑦)𝑇𝑧) = (𝑥𝑂(𝑦𝑃𝑧)))
4 caofref.2 . . . . . 6 (𝜑𝐹:𝐴𝑆)
54ffvelrnda 6608 . . . . 5 ((𝜑𝑤𝐴) → (𝐹𝑤) ∈ 𝑆)
6 caofcom.3 . . . . . 6 (𝜑𝐺:𝐴𝑆)
76ffvelrnda 6608 . . . . 5 ((𝜑𝑤𝐴) → (𝐺𝑤) ∈ 𝑆)
8 caofass.4 . . . . . 6 (𝜑𝐻:𝐴𝑆)
98ffvelrnda 6608 . . . . 5 ((𝜑𝑤𝐴) → (𝐻𝑤) ∈ 𝑆)
10 oveq1 6912 . . . . . . . 8 (𝑥 = (𝐹𝑤) → (𝑥𝑅𝑦) = ((𝐹𝑤)𝑅𝑦))
1110oveq1d 6920 . . . . . . 7 (𝑥 = (𝐹𝑤) → ((𝑥𝑅𝑦)𝑇𝑧) = (((𝐹𝑤)𝑅𝑦)𝑇𝑧))
12 oveq1 6912 . . . . . . 7 (𝑥 = (𝐹𝑤) → (𝑥𝑂(𝑦𝑃𝑧)) = ((𝐹𝑤)𝑂(𝑦𝑃𝑧)))
1311, 12eqeq12d 2840 . . . . . 6 (𝑥 = (𝐹𝑤) → (((𝑥𝑅𝑦)𝑇𝑧) = (𝑥𝑂(𝑦𝑃𝑧)) ↔ (((𝐹𝑤)𝑅𝑦)𝑇𝑧) = ((𝐹𝑤)𝑂(𝑦𝑃𝑧))))
14 oveq2 6913 . . . . . . . 8 (𝑦 = (𝐺𝑤) → ((𝐹𝑤)𝑅𝑦) = ((𝐹𝑤)𝑅(𝐺𝑤)))
1514oveq1d 6920 . . . . . . 7 (𝑦 = (𝐺𝑤) → (((𝐹𝑤)𝑅𝑦)𝑇𝑧) = (((𝐹𝑤)𝑅(𝐺𝑤))𝑇𝑧))
16 oveq1 6912 . . . . . . . 8 (𝑦 = (𝐺𝑤) → (𝑦𝑃𝑧) = ((𝐺𝑤)𝑃𝑧))
1716oveq2d 6921 . . . . . . 7 (𝑦 = (𝐺𝑤) → ((𝐹𝑤)𝑂(𝑦𝑃𝑧)) = ((𝐹𝑤)𝑂((𝐺𝑤)𝑃𝑧)))
1815, 17eqeq12d 2840 . . . . . 6 (𝑦 = (𝐺𝑤) → ((((𝐹𝑤)𝑅𝑦)𝑇𝑧) = ((𝐹𝑤)𝑂(𝑦𝑃𝑧)) ↔ (((𝐹𝑤)𝑅(𝐺𝑤))𝑇𝑧) = ((𝐹𝑤)𝑂((𝐺𝑤)𝑃𝑧))))
19 oveq2 6913 . . . . . . 7 (𝑧 = (𝐻𝑤) → (((𝐹𝑤)𝑅(𝐺𝑤))𝑇𝑧) = (((𝐹𝑤)𝑅(𝐺𝑤))𝑇(𝐻𝑤)))
20 oveq2 6913 . . . . . . . 8 (𝑧 = (𝐻𝑤) → ((𝐺𝑤)𝑃𝑧) = ((𝐺𝑤)𝑃(𝐻𝑤)))
2120oveq2d 6921 . . . . . . 7 (𝑧 = (𝐻𝑤) → ((𝐹𝑤)𝑂((𝐺𝑤)𝑃𝑧)) = ((𝐹𝑤)𝑂((𝐺𝑤)𝑃(𝐻𝑤))))
2219, 21eqeq12d 2840 . . . . . 6 (𝑧 = (𝐻𝑤) → ((((𝐹𝑤)𝑅(𝐺𝑤))𝑇𝑧) = ((𝐹𝑤)𝑂((𝐺𝑤)𝑃𝑧)) ↔ (((𝐹𝑤)𝑅(𝐺𝑤))𝑇(𝐻𝑤)) = ((𝐹𝑤)𝑂((𝐺𝑤)𝑃(𝐻𝑤)))))
2313, 18, 22rspc3v 3542 . . . . 5 (((𝐹𝑤) ∈ 𝑆 ∧ (𝐺𝑤) ∈ 𝑆 ∧ (𝐻𝑤) ∈ 𝑆) → (∀𝑥𝑆𝑦𝑆𝑧𝑆 ((𝑥𝑅𝑦)𝑇𝑧) = (𝑥𝑂(𝑦𝑃𝑧)) → (((𝐹𝑤)𝑅(𝐺𝑤))𝑇(𝐻𝑤)) = ((𝐹𝑤)𝑂((𝐺𝑤)𝑃(𝐻𝑤)))))
245, 7, 9, 23syl3anc 1494 . . . 4 ((𝜑𝑤𝐴) → (∀𝑥𝑆𝑦𝑆𝑧𝑆 ((𝑥𝑅𝑦)𝑇𝑧) = (𝑥𝑂(𝑦𝑃𝑧)) → (((𝐹𝑤)𝑅(𝐺𝑤))𝑇(𝐻𝑤)) = ((𝐹𝑤)𝑂((𝐺𝑤)𝑃(𝐻𝑤)))))
253, 24mpd 15 . . 3 ((𝜑𝑤𝐴) → (((𝐹𝑤)𝑅(𝐺𝑤))𝑇(𝐻𝑤)) = ((𝐹𝑤)𝑂((𝐺𝑤)𝑃(𝐻𝑤))))
2625mpteq2dva 4967 . 2 (𝜑 → (𝑤𝐴 ↦ (((𝐹𝑤)𝑅(𝐺𝑤))𝑇(𝐻𝑤))) = (𝑤𝐴 ↦ ((𝐹𝑤)𝑂((𝐺𝑤)𝑃(𝐻𝑤)))))
27 caofref.1 . . 3 (𝜑𝐴𝑉)
28 ovexd 6939 . . 3 ((𝜑𝑤𝐴) → ((𝐹𝑤)𝑅(𝐺𝑤)) ∈ V)
294feqmptd 6496 . . . 4 (𝜑𝐹 = (𝑤𝐴 ↦ (𝐹𝑤)))
306feqmptd 6496 . . . 4 (𝜑𝐺 = (𝑤𝐴 ↦ (𝐺𝑤)))
3127, 5, 7, 29, 30offval2 7174 . . 3 (𝜑 → (𝐹𝑓 𝑅𝐺) = (𝑤𝐴 ↦ ((𝐹𝑤)𝑅(𝐺𝑤))))
328feqmptd 6496 . . 3 (𝜑𝐻 = (𝑤𝐴 ↦ (𝐻𝑤)))
3327, 28, 9, 31, 32offval2 7174 . 2 (𝜑 → ((𝐹𝑓 𝑅𝐺) ∘𝑓 𝑇𝐻) = (𝑤𝐴 ↦ (((𝐹𝑤)𝑅(𝐺𝑤))𝑇(𝐻𝑤))))
34 ovexd 6939 . . 3 ((𝜑𝑤𝐴) → ((𝐺𝑤)𝑃(𝐻𝑤)) ∈ V)
3527, 7, 9, 30, 32offval2 7174 . . 3 (𝜑 → (𝐺𝑓 𝑃𝐻) = (𝑤𝐴 ↦ ((𝐺𝑤)𝑃(𝐻𝑤))))
3627, 5, 34, 29, 35offval2 7174 . 2 (𝜑 → (𝐹𝑓 𝑂(𝐺𝑓 𝑃𝐻)) = (𝑤𝐴 ↦ ((𝐹𝑤)𝑂((𝐺𝑤)𝑃(𝐻𝑤)))))
3726, 33, 363eqtr4d 2871 1 (𝜑 → ((𝐹𝑓 𝑅𝐺) ∘𝑓 𝑇𝐻) = (𝐹𝑓 𝑂(𝐺𝑓 𝑃𝐻)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386  w3a 1111   = wceq 1656  wcel 2164  wral 3117  Vcvv 3414  cmpt 4952  wf 6119  cfv 6123  (class class class)co 6905  𝑓 cof 7155
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-of 7157
This theorem is referenced by:  psrgrp  19759  psrlmod  19762  mndvass  20565  itg2mulc  23913  plydivlem4  24450  dchrabl  25392  lfladdass  35141  lflvsass  35149  expgrowth  39367
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