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Theorem caofass 7418
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 3180 . . . . 5 (𝜑 → ∀𝑥𝑆𝑦𝑆𝑧𝑆 ((𝑥𝑅𝑦)𝑇𝑧) = (𝑥𝑂(𝑦𝑃𝑧)))
32adantr 484 . . . 4 ((𝜑𝑤𝐴) → ∀𝑥𝑆𝑦𝑆𝑧𝑆 ((𝑥𝑅𝑦)𝑇𝑧) = (𝑥𝑂(𝑦𝑃𝑧)))
4 caofref.2 . . . . . 6 (𝜑𝐹:𝐴𝑆)
54ffvelrnda 6824 . . . . 5 ((𝜑𝑤𝐴) → (𝐹𝑤) ∈ 𝑆)
6 caofcom.3 . . . . . 6 (𝜑𝐺:𝐴𝑆)
76ffvelrnda 6824 . . . . 5 ((𝜑𝑤𝐴) → (𝐺𝑤) ∈ 𝑆)
8 caofass.4 . . . . . 6 (𝜑𝐻:𝐴𝑆)
98ffvelrnda 6824 . . . . 5 ((𝜑𝑤𝐴) → (𝐻𝑤) ∈ 𝑆)
10 oveq1 7137 . . . . . . . 8 (𝑥 = (𝐹𝑤) → (𝑥𝑅𝑦) = ((𝐹𝑤)𝑅𝑦))
1110oveq1d 7145 . . . . . . 7 (𝑥 = (𝐹𝑤) → ((𝑥𝑅𝑦)𝑇𝑧) = (((𝐹𝑤)𝑅𝑦)𝑇𝑧))
12 oveq1 7137 . . . . . . 7 (𝑥 = (𝐹𝑤) → (𝑥𝑂(𝑦𝑃𝑧)) = ((𝐹𝑤)𝑂(𝑦𝑃𝑧)))
1311, 12eqeq12d 2837 . . . . . 6 (𝑥 = (𝐹𝑤) → (((𝑥𝑅𝑦)𝑇𝑧) = (𝑥𝑂(𝑦𝑃𝑧)) ↔ (((𝐹𝑤)𝑅𝑦)𝑇𝑧) = ((𝐹𝑤)𝑂(𝑦𝑃𝑧))))
14 oveq2 7138 . . . . . . . 8 (𝑦 = (𝐺𝑤) → ((𝐹𝑤)𝑅𝑦) = ((𝐹𝑤)𝑅(𝐺𝑤)))
1514oveq1d 7145 . . . . . . 7 (𝑦 = (𝐺𝑤) → (((𝐹𝑤)𝑅𝑦)𝑇𝑧) = (((𝐹𝑤)𝑅(𝐺𝑤))𝑇𝑧))
16 oveq1 7137 . . . . . . . 8 (𝑦 = (𝐺𝑤) → (𝑦𝑃𝑧) = ((𝐺𝑤)𝑃𝑧))
1716oveq2d 7146 . . . . . . 7 (𝑦 = (𝐺𝑤) → ((𝐹𝑤)𝑂(𝑦𝑃𝑧)) = ((𝐹𝑤)𝑂((𝐺𝑤)𝑃𝑧)))
1815, 17eqeq12d 2837 . . . . . 6 (𝑦 = (𝐺𝑤) → ((((𝐹𝑤)𝑅𝑦)𝑇𝑧) = ((𝐹𝑤)𝑂(𝑦𝑃𝑧)) ↔ (((𝐹𝑤)𝑅(𝐺𝑤))𝑇𝑧) = ((𝐹𝑤)𝑂((𝐺𝑤)𝑃𝑧))))
19 oveq2 7138 . . . . . . 7 (𝑧 = (𝐻𝑤) → (((𝐹𝑤)𝑅(𝐺𝑤))𝑇𝑧) = (((𝐹𝑤)𝑅(𝐺𝑤))𝑇(𝐻𝑤)))
20 oveq2 7138 . . . . . . . 8 (𝑧 = (𝐻𝑤) → ((𝐺𝑤)𝑃𝑧) = ((𝐺𝑤)𝑃(𝐻𝑤)))
2120oveq2d 7146 . . . . . . 7 (𝑧 = (𝐻𝑤) → ((𝐹𝑤)𝑂((𝐺𝑤)𝑃𝑧)) = ((𝐹𝑤)𝑂((𝐺𝑤)𝑃(𝐻𝑤))))
2219, 21eqeq12d 2837 . . . . . 6 (𝑧 = (𝐻𝑤) → ((((𝐹𝑤)𝑅(𝐺𝑤))𝑇𝑧) = ((𝐹𝑤)𝑂((𝐺𝑤)𝑃𝑧)) ↔ (((𝐹𝑤)𝑅(𝐺𝑤))𝑇(𝐻𝑤)) = ((𝐹𝑤)𝑂((𝐺𝑤)𝑃(𝐻𝑤)))))
2313, 18, 22rspc3v 3613 . . . . 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 1368 . . . 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 5134 . 2 (𝜑 → (𝑤𝐴 ↦ (((𝐹𝑤)𝑅(𝐺𝑤))𝑇(𝐻𝑤))) = (𝑤𝐴 ↦ ((𝐹𝑤)𝑂((𝐺𝑤)𝑃(𝐻𝑤)))))
27 caofref.1 . . 3 (𝜑𝐴𝑉)
28 ovexd 7165 . . 3 ((𝜑𝑤𝐴) → ((𝐹𝑤)𝑅(𝐺𝑤)) ∈ V)
294feqmptd 6706 . . . 4 (𝜑𝐹 = (𝑤𝐴 ↦ (𝐹𝑤)))
306feqmptd 6706 . . . 4 (𝜑𝐺 = (𝑤𝐴 ↦ (𝐺𝑤)))
3127, 5, 7, 29, 30offval2 7401 . . 3 (𝜑 → (𝐹f 𝑅𝐺) = (𝑤𝐴 ↦ ((𝐹𝑤)𝑅(𝐺𝑤))))
328feqmptd 6706 . . 3 (𝜑𝐻 = (𝑤𝐴 ↦ (𝐻𝑤)))
3327, 28, 9, 31, 32offval2 7401 . 2 (𝜑 → ((𝐹f 𝑅𝐺) ∘f 𝑇𝐻) = (𝑤𝐴 ↦ (((𝐹𝑤)𝑅(𝐺𝑤))𝑇(𝐻𝑤))))
34 ovexd 7165 . . 3 ((𝜑𝑤𝐴) → ((𝐺𝑤)𝑃(𝐻𝑤)) ∈ V)
3527, 7, 9, 30, 32offval2 7401 . . 3 (𝜑 → (𝐺f 𝑃𝐻) = (𝑤𝐴 ↦ ((𝐺𝑤)𝑃(𝐻𝑤))))
3627, 5, 34, 29, 35offval2 7401 . 2 (𝜑 → (𝐹f 𝑂(𝐺f 𝑃𝐻)) = (𝑤𝐴 ↦ ((𝐹𝑤)𝑂((𝐺𝑤)𝑃(𝐻𝑤)))))
3726, 33, 363eqtr4d 2866 1 (𝜑 → ((𝐹f 𝑅𝐺) ∘f 𝑇𝐻) = (𝐹f 𝑂(𝐺f 𝑃𝐻)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2115  wral 3126  Vcvv 3471  cmpt 5119  wf 6324  cfv 6328  (class class class)co 7130  f cof 7382
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-rep 5163  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-reu 3133  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7133  df-oprab 7134  df-mpo 7135  df-of 7384
This theorem is referenced by:  psrgrp  20154  psrlmod  20157  mndvass  20979  itg2mulc  24330  plydivlem4  24871  dchrabl  25817  lfladdass  36251  lflvsass  36259  expgrowth  40850
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