MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  caofass Structured version   Visualization version   GIF version

Theorem caofass 7524
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 3114 . . . . 5 (𝜑 → ∀𝑥𝑆𝑦𝑆𝑧𝑆 ((𝑥𝑅𝑦)𝑇𝑧) = (𝑥𝑂(𝑦𝑃𝑧)))
32adantr 484 . . . 4 ((𝜑𝑤𝐴) → ∀𝑥𝑆𝑦𝑆𝑧𝑆 ((𝑥𝑅𝑦)𝑇𝑧) = (𝑥𝑂(𝑦𝑃𝑧)))
4 caofref.2 . . . . . 6 (𝜑𝐹:𝐴𝑆)
54ffvelrnda 6923 . . . . 5 ((𝜑𝑤𝐴) → (𝐹𝑤) ∈ 𝑆)
6 caofcom.3 . . . . . 6 (𝜑𝐺:𝐴𝑆)
76ffvelrnda 6923 . . . . 5 ((𝜑𝑤𝐴) → (𝐺𝑤) ∈ 𝑆)
8 caofass.4 . . . . . 6 (𝜑𝐻:𝐴𝑆)
98ffvelrnda 6923 . . . . 5 ((𝜑𝑤𝐴) → (𝐻𝑤) ∈ 𝑆)
10 oveq1 7239 . . . . . . . 8 (𝑥 = (𝐹𝑤) → (𝑥𝑅𝑦) = ((𝐹𝑤)𝑅𝑦))
1110oveq1d 7247 . . . . . . 7 (𝑥 = (𝐹𝑤) → ((𝑥𝑅𝑦)𝑇𝑧) = (((𝐹𝑤)𝑅𝑦)𝑇𝑧))
12 oveq1 7239 . . . . . . 7 (𝑥 = (𝐹𝑤) → (𝑥𝑂(𝑦𝑃𝑧)) = ((𝐹𝑤)𝑂(𝑦𝑃𝑧)))
1311, 12eqeq12d 2754 . . . . . 6 (𝑥 = (𝐹𝑤) → (((𝑥𝑅𝑦)𝑇𝑧) = (𝑥𝑂(𝑦𝑃𝑧)) ↔ (((𝐹𝑤)𝑅𝑦)𝑇𝑧) = ((𝐹𝑤)𝑂(𝑦𝑃𝑧))))
14 oveq2 7240 . . . . . . . 8 (𝑦 = (𝐺𝑤) → ((𝐹𝑤)𝑅𝑦) = ((𝐹𝑤)𝑅(𝐺𝑤)))
1514oveq1d 7247 . . . . . . 7 (𝑦 = (𝐺𝑤) → (((𝐹𝑤)𝑅𝑦)𝑇𝑧) = (((𝐹𝑤)𝑅(𝐺𝑤))𝑇𝑧))
16 oveq1 7239 . . . . . . . 8 (𝑦 = (𝐺𝑤) → (𝑦𝑃𝑧) = ((𝐺𝑤)𝑃𝑧))
1716oveq2d 7248 . . . . . . 7 (𝑦 = (𝐺𝑤) → ((𝐹𝑤)𝑂(𝑦𝑃𝑧)) = ((𝐹𝑤)𝑂((𝐺𝑤)𝑃𝑧)))
1815, 17eqeq12d 2754 . . . . . 6 (𝑦 = (𝐺𝑤) → ((((𝐹𝑤)𝑅𝑦)𝑇𝑧) = ((𝐹𝑤)𝑂(𝑦𝑃𝑧)) ↔ (((𝐹𝑤)𝑅(𝐺𝑤))𝑇𝑧) = ((𝐹𝑤)𝑂((𝐺𝑤)𝑃𝑧))))
19 oveq2 7240 . . . . . . 7 (𝑧 = (𝐻𝑤) → (((𝐹𝑤)𝑅(𝐺𝑤))𝑇𝑧) = (((𝐹𝑤)𝑅(𝐺𝑤))𝑇(𝐻𝑤)))
20 oveq2 7240 . . . . . . . 8 (𝑧 = (𝐻𝑤) → ((𝐺𝑤)𝑃𝑧) = ((𝐺𝑤)𝑃(𝐻𝑤)))
2120oveq2d 7248 . . . . . . 7 (𝑧 = (𝐻𝑤) → ((𝐹𝑤)𝑂((𝐺𝑤)𝑃𝑧)) = ((𝐹𝑤)𝑂((𝐺𝑤)𝑃(𝐻𝑤))))
2219, 21eqeq12d 2754 . . . . . 6 (𝑧 = (𝐻𝑤) → ((((𝐹𝑤)𝑅(𝐺𝑤))𝑇𝑧) = ((𝐹𝑤)𝑂((𝐺𝑤)𝑃𝑧)) ↔ (((𝐹𝑤)𝑅(𝐺𝑤))𝑇(𝐻𝑤)) = ((𝐹𝑤)𝑂((𝐺𝑤)𝑃(𝐻𝑤)))))
2313, 18, 22rspc3v 3563 . . . . 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 1373 . . . 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 5165 . 2 (𝜑 → (𝑤𝐴 ↦ (((𝐹𝑤)𝑅(𝐺𝑤))𝑇(𝐻𝑤))) = (𝑤𝐴 ↦ ((𝐹𝑤)𝑂((𝐺𝑤)𝑃(𝐻𝑤)))))
27 caofref.1 . . 3 (𝜑𝐴𝑉)
28 ovexd 7267 . . 3 ((𝜑𝑤𝐴) → ((𝐹𝑤)𝑅(𝐺𝑤)) ∈ V)
294feqmptd 6799 . . . 4 (𝜑𝐹 = (𝑤𝐴 ↦ (𝐹𝑤)))
306feqmptd 6799 . . . 4 (𝜑𝐺 = (𝑤𝐴 ↦ (𝐺𝑤)))
3127, 5, 7, 29, 30offval2 7507 . . 3 (𝜑 → (𝐹f 𝑅𝐺) = (𝑤𝐴 ↦ ((𝐹𝑤)𝑅(𝐺𝑤))))
328feqmptd 6799 . . 3 (𝜑𝐻 = (𝑤𝐴 ↦ (𝐻𝑤)))
3327, 28, 9, 31, 32offval2 7507 . 2 (𝜑 → ((𝐹f 𝑅𝐺) ∘f 𝑇𝐻) = (𝑤𝐴 ↦ (((𝐹𝑤)𝑅(𝐺𝑤))𝑇(𝐻𝑤))))
34 ovexd 7267 . . 3 ((𝜑𝑤𝐴) → ((𝐺𝑤)𝑃(𝐻𝑤)) ∈ V)
3527, 7, 9, 30, 32offval2 7507 . . 3 (𝜑 → (𝐺f 𝑃𝐻) = (𝑤𝐴 ↦ ((𝐺𝑤)𝑃(𝐻𝑤))))
3627, 5, 34, 29, 35offval2 7507 . 2 (𝜑 → (𝐹f 𝑂(𝐺f 𝑃𝐻)) = (𝑤𝐴 ↦ ((𝐹𝑤)𝑂((𝐺𝑤)𝑃(𝐻𝑤)))))
3726, 33, 363eqtr4d 2788 1 (𝜑 → ((𝐹f 𝑅𝐺) ∘f 𝑇𝐻) = (𝐹f 𝑂(𝐺f 𝑃𝐻)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1089   = wceq 1543  wcel 2111  wral 3062  Vcvv 3421  cmpt 5150  wf 6394  cfv 6398  (class class class)co 7232  f cof 7486
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2159  ax-12 2176  ax-ext 2709  ax-rep 5194  ax-sep 5207  ax-nul 5214  ax-pr 5337
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2072  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2887  df-ne 2942  df-ral 3067  df-rex 3068  df-reu 3069  df-rab 3071  df-v 3423  df-sbc 3710  df-csb 3827  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4253  df-if 4455  df-sn 4557  df-pr 4559  df-op 4563  df-uni 4835  df-iun 4921  df-br 5069  df-opab 5131  df-mpt 5151  df-id 5470  df-xp 5572  df-rel 5573  df-cnv 5574  df-co 5575  df-dm 5576  df-rn 5577  df-res 5578  df-ima 5579  df-iota 6356  df-fun 6400  df-fn 6401  df-f 6402  df-f1 6403  df-fo 6404  df-f1o 6405  df-fv 6406  df-ov 7235  df-oprab 7236  df-mpo 7237  df-of 7488
This theorem is referenced by:  psrgrp  20950  psrlmod  20953  mndvass  21318  itg2mulc  24672  plydivlem4  25216  dchrabl  26162  lfladdass  36854  lflvsass  36862  expgrowth  41659
  Copyright terms: Public domain W3C validator