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Theorem caofass 7659
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 3201 . . . . 5 (𝜑 → ∀𝑥𝑆𝑦𝑆𝑧𝑆 ((𝑥𝑅𝑦)𝑇𝑧) = (𝑥𝑂(𝑦𝑃𝑧)))
32adantr 482 . . . 4 ((𝜑𝑤𝐴) → ∀𝑥𝑆𝑦𝑆𝑧𝑆 ((𝑥𝑅𝑦)𝑇𝑧) = (𝑥𝑂(𝑦𝑃𝑧)))
4 caofref.2 . . . . . 6 (𝜑𝐹:𝐴𝑆)
54ffvelcdmda 7040 . . . . 5 ((𝜑𝑤𝐴) → (𝐹𝑤) ∈ 𝑆)
6 caofcom.3 . . . . . 6 (𝜑𝐺:𝐴𝑆)
76ffvelcdmda 7040 . . . . 5 ((𝜑𝑤𝐴) → (𝐺𝑤) ∈ 𝑆)
8 caofass.4 . . . . . 6 (𝜑𝐻:𝐴𝑆)
98ffvelcdmda 7040 . . . . 5 ((𝜑𝑤𝐴) → (𝐻𝑤) ∈ 𝑆)
10 oveq1 7369 . . . . . . . 8 (𝑥 = (𝐹𝑤) → (𝑥𝑅𝑦) = ((𝐹𝑤)𝑅𝑦))
1110oveq1d 7377 . . . . . . 7 (𝑥 = (𝐹𝑤) → ((𝑥𝑅𝑦)𝑇𝑧) = (((𝐹𝑤)𝑅𝑦)𝑇𝑧))
12 oveq1 7369 . . . . . . 7 (𝑥 = (𝐹𝑤) → (𝑥𝑂(𝑦𝑃𝑧)) = ((𝐹𝑤)𝑂(𝑦𝑃𝑧)))
1311, 12eqeq12d 2753 . . . . . 6 (𝑥 = (𝐹𝑤) → (((𝑥𝑅𝑦)𝑇𝑧) = (𝑥𝑂(𝑦𝑃𝑧)) ↔ (((𝐹𝑤)𝑅𝑦)𝑇𝑧) = ((𝐹𝑤)𝑂(𝑦𝑃𝑧))))
14 oveq2 7370 . . . . . . . 8 (𝑦 = (𝐺𝑤) → ((𝐹𝑤)𝑅𝑦) = ((𝐹𝑤)𝑅(𝐺𝑤)))
1514oveq1d 7377 . . . . . . 7 (𝑦 = (𝐺𝑤) → (((𝐹𝑤)𝑅𝑦)𝑇𝑧) = (((𝐹𝑤)𝑅(𝐺𝑤))𝑇𝑧))
16 oveq1 7369 . . . . . . . 8 (𝑦 = (𝐺𝑤) → (𝑦𝑃𝑧) = ((𝐺𝑤)𝑃𝑧))
1716oveq2d 7378 . . . . . . 7 (𝑦 = (𝐺𝑤) → ((𝐹𝑤)𝑂(𝑦𝑃𝑧)) = ((𝐹𝑤)𝑂((𝐺𝑤)𝑃𝑧)))
1815, 17eqeq12d 2753 . . . . . 6 (𝑦 = (𝐺𝑤) → ((((𝐹𝑤)𝑅𝑦)𝑇𝑧) = ((𝐹𝑤)𝑂(𝑦𝑃𝑧)) ↔ (((𝐹𝑤)𝑅(𝐺𝑤))𝑇𝑧) = ((𝐹𝑤)𝑂((𝐺𝑤)𝑃𝑧))))
19 oveq2 7370 . . . . . . 7 (𝑧 = (𝐻𝑤) → (((𝐹𝑤)𝑅(𝐺𝑤))𝑇𝑧) = (((𝐹𝑤)𝑅(𝐺𝑤))𝑇(𝐻𝑤)))
20 oveq2 7370 . . . . . . . 8 (𝑧 = (𝐻𝑤) → ((𝐺𝑤)𝑃𝑧) = ((𝐺𝑤)𝑃(𝐻𝑤)))
2120oveq2d 7378 . . . . . . 7 (𝑧 = (𝐻𝑤) → ((𝐹𝑤)𝑂((𝐺𝑤)𝑃𝑧)) = ((𝐹𝑤)𝑂((𝐺𝑤)𝑃(𝐻𝑤))))
2219, 21eqeq12d 2753 . . . . . 6 (𝑧 = (𝐻𝑤) → ((((𝐹𝑤)𝑅(𝐺𝑤))𝑇𝑧) = ((𝐹𝑤)𝑂((𝐺𝑤)𝑃𝑧)) ↔ (((𝐹𝑤)𝑅(𝐺𝑤))𝑇(𝐻𝑤)) = ((𝐹𝑤)𝑂((𝐺𝑤)𝑃(𝐻𝑤)))))
2313, 18, 22rspc3v 3596 . . . . 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 1372 . . . 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 5210 . 2 (𝜑 → (𝑤𝐴 ↦ (((𝐹𝑤)𝑅(𝐺𝑤))𝑇(𝐻𝑤))) = (𝑤𝐴 ↦ ((𝐹𝑤)𝑂((𝐺𝑤)𝑃(𝐻𝑤)))))
27 caofref.1 . . 3 (𝜑𝐴𝑉)
28 ovexd 7397 . . 3 ((𝜑𝑤𝐴) → ((𝐹𝑤)𝑅(𝐺𝑤)) ∈ V)
294feqmptd 6915 . . . 4 (𝜑𝐹 = (𝑤𝐴 ↦ (𝐹𝑤)))
306feqmptd 6915 . . . 4 (𝜑𝐺 = (𝑤𝐴 ↦ (𝐺𝑤)))
3127, 5, 7, 29, 30offval2 7642 . . 3 (𝜑 → (𝐹f 𝑅𝐺) = (𝑤𝐴 ↦ ((𝐹𝑤)𝑅(𝐺𝑤))))
328feqmptd 6915 . . 3 (𝜑𝐻 = (𝑤𝐴 ↦ (𝐻𝑤)))
3327, 28, 9, 31, 32offval2 7642 . 2 (𝜑 → ((𝐹f 𝑅𝐺) ∘f 𝑇𝐻) = (𝑤𝐴 ↦ (((𝐹𝑤)𝑅(𝐺𝑤))𝑇(𝐻𝑤))))
34 ovexd 7397 . . 3 ((𝜑𝑤𝐴) → ((𝐺𝑤)𝑃(𝐻𝑤)) ∈ V)
3527, 7, 9, 30, 32offval2 7642 . . 3 (𝜑 → (𝐺f 𝑃𝐻) = (𝑤𝐴 ↦ ((𝐺𝑤)𝑃(𝐻𝑤))))
3627, 5, 34, 29, 35offval2 7642 . 2 (𝜑 → (𝐹f 𝑂(𝐺f 𝑃𝐻)) = (𝑤𝐴 ↦ ((𝐹𝑤)𝑂((𝐺𝑤)𝑃(𝐻𝑤)))))
3726, 33, 363eqtr4d 2787 1 (𝜑 → ((𝐹f 𝑅𝐺) ∘f 𝑇𝐻) = (𝐹f 𝑂(𝐺f 𝑃𝐻)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1088   = wceq 1542  wcel 2107  wral 3065  Vcvv 3448  cmpt 5193  wf 6497  cfv 6501  (class class class)co 7362  f cof 7620
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367  df-of 7622
This theorem is referenced by:  psrgrpOLD  21383  psrlmod  21386  mndvass  21757  itg2mulc  25128  plydivlem4  25672  dchrabl  26618  lfladdass  37564  lflvsass  37572  expgrowth  42689
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