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

Theorem caofass 7082
 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 3121 . . . . 5 (𝜑 → ∀𝑥𝑆𝑦𝑆𝑧𝑆 ((𝑥𝑅𝑦)𝑇𝑧) = (𝑥𝑂(𝑦𝑃𝑧)))
32adantr 466 . . . 4 ((𝜑𝑤𝐴) → ∀𝑥𝑆𝑦𝑆𝑧𝑆 ((𝑥𝑅𝑦)𝑇𝑧) = (𝑥𝑂(𝑦𝑃𝑧)))
4 caofref.2 . . . . . 6 (𝜑𝐹:𝐴𝑆)
54ffvelrnda 6504 . . . . 5 ((𝜑𝑤𝐴) → (𝐹𝑤) ∈ 𝑆)
6 caofcom.3 . . . . . 6 (𝜑𝐺:𝐴𝑆)
76ffvelrnda 6504 . . . . 5 ((𝜑𝑤𝐴) → (𝐺𝑤) ∈ 𝑆)
8 caofass.4 . . . . . 6 (𝜑𝐻:𝐴𝑆)
98ffvelrnda 6504 . . . . 5 ((𝜑𝑤𝐴) → (𝐻𝑤) ∈ 𝑆)
10 oveq1 6803 . . . . . . . 8 (𝑥 = (𝐹𝑤) → (𝑥𝑅𝑦) = ((𝐹𝑤)𝑅𝑦))
1110oveq1d 6811 . . . . . . 7 (𝑥 = (𝐹𝑤) → ((𝑥𝑅𝑦)𝑇𝑧) = (((𝐹𝑤)𝑅𝑦)𝑇𝑧))
12 oveq1 6803 . . . . . . 7 (𝑥 = (𝐹𝑤) → (𝑥𝑂(𝑦𝑃𝑧)) = ((𝐹𝑤)𝑂(𝑦𝑃𝑧)))
1311, 12eqeq12d 2786 . . . . . 6 (𝑥 = (𝐹𝑤) → (((𝑥𝑅𝑦)𝑇𝑧) = (𝑥𝑂(𝑦𝑃𝑧)) ↔ (((𝐹𝑤)𝑅𝑦)𝑇𝑧) = ((𝐹𝑤)𝑂(𝑦𝑃𝑧))))
14 oveq2 6804 . . . . . . . 8 (𝑦 = (𝐺𝑤) → ((𝐹𝑤)𝑅𝑦) = ((𝐹𝑤)𝑅(𝐺𝑤)))
1514oveq1d 6811 . . . . . . 7 (𝑦 = (𝐺𝑤) → (((𝐹𝑤)𝑅𝑦)𝑇𝑧) = (((𝐹𝑤)𝑅(𝐺𝑤))𝑇𝑧))
16 oveq1 6803 . . . . . . . 8 (𝑦 = (𝐺𝑤) → (𝑦𝑃𝑧) = ((𝐺𝑤)𝑃𝑧))
1716oveq2d 6812 . . . . . . 7 (𝑦 = (𝐺𝑤) → ((𝐹𝑤)𝑂(𝑦𝑃𝑧)) = ((𝐹𝑤)𝑂((𝐺𝑤)𝑃𝑧)))
1815, 17eqeq12d 2786 . . . . . 6 (𝑦 = (𝐺𝑤) → ((((𝐹𝑤)𝑅𝑦)𝑇𝑧) = ((𝐹𝑤)𝑂(𝑦𝑃𝑧)) ↔ (((𝐹𝑤)𝑅(𝐺𝑤))𝑇𝑧) = ((𝐹𝑤)𝑂((𝐺𝑤)𝑃𝑧))))
19 oveq2 6804 . . . . . . 7 (𝑧 = (𝐻𝑤) → (((𝐹𝑤)𝑅(𝐺𝑤))𝑇𝑧) = (((𝐹𝑤)𝑅(𝐺𝑤))𝑇(𝐻𝑤)))
20 oveq2 6804 . . . . . . . 8 (𝑧 = (𝐻𝑤) → ((𝐺𝑤)𝑃𝑧) = ((𝐺𝑤)𝑃(𝐻𝑤)))
2120oveq2d 6812 . . . . . . 7 (𝑧 = (𝐻𝑤) → ((𝐹𝑤)𝑂((𝐺𝑤)𝑃𝑧)) = ((𝐹𝑤)𝑂((𝐺𝑤)𝑃(𝐻𝑤))))
2219, 21eqeq12d 2786 . . . . . 6 (𝑧 = (𝐻𝑤) → ((((𝐹𝑤)𝑅(𝐺𝑤))𝑇𝑧) = ((𝐹𝑤)𝑂((𝐺𝑤)𝑃𝑧)) ↔ (((𝐹𝑤)𝑅(𝐺𝑤))𝑇(𝐻𝑤)) = ((𝐹𝑤)𝑂((𝐺𝑤)𝑃(𝐻𝑤)))))
2313, 18, 22rspc3v 3475 . . . . 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 1476 . . . 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 4879 . 2 (𝜑 → (𝑤𝐴 ↦ (((𝐹𝑤)𝑅(𝐺𝑤))𝑇(𝐻𝑤))) = (𝑤𝐴 ↦ ((𝐹𝑤)𝑂((𝐺𝑤)𝑃(𝐻𝑤)))))
27 caofref.1 . . 3 (𝜑𝐴𝑉)
28 ovexd 6829 . . 3 ((𝜑𝑤𝐴) → ((𝐹𝑤)𝑅(𝐺𝑤)) ∈ V)
294feqmptd 6393 . . . 4 (𝜑𝐹 = (𝑤𝐴 ↦ (𝐹𝑤)))
306feqmptd 6393 . . . 4 (𝜑𝐺 = (𝑤𝐴 ↦ (𝐺𝑤)))
3127, 5, 7, 29, 30offval2 7065 . . 3 (𝜑 → (𝐹𝑓 𝑅𝐺) = (𝑤𝐴 ↦ ((𝐹𝑤)𝑅(𝐺𝑤))))
328feqmptd 6393 . . 3 (𝜑𝐻 = (𝑤𝐴 ↦ (𝐻𝑤)))
3327, 28, 9, 31, 32offval2 7065 . 2 (𝜑 → ((𝐹𝑓 𝑅𝐺) ∘𝑓 𝑇𝐻) = (𝑤𝐴 ↦ (((𝐹𝑤)𝑅(𝐺𝑤))𝑇(𝐻𝑤))))
34 ovexd 6829 . . 3 ((𝜑𝑤𝐴) → ((𝐺𝑤)𝑃(𝐻𝑤)) ∈ V)
3527, 7, 9, 30, 32offval2 7065 . . 3 (𝜑 → (𝐺𝑓 𝑃𝐻) = (𝑤𝐴 ↦ ((𝐺𝑤)𝑃(𝐻𝑤))))
3627, 5, 34, 29, 35offval2 7065 . 2 (𝜑 → (𝐹𝑓 𝑂(𝐺𝑓 𝑃𝐻)) = (𝑤𝐴 ↦ ((𝐹𝑤)𝑂((𝐺𝑤)𝑃(𝐻𝑤)))))
3726, 33, 363eqtr4d 2815 1 (𝜑 → ((𝐹𝑓 𝑅𝐺) ∘𝑓 𝑇𝐻) = (𝐹𝑓 𝑂(𝐺𝑓 𝑃𝐻)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 382   ∧ w3a 1071   = wceq 1631   ∈ wcel 2145  ∀wral 3061  Vcvv 3351   ↦ cmpt 4864  ⟶wf 6026  ‘cfv 6030  (class class class)co 6796   ∘𝑓 cof 7046 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-of 7048 This theorem is referenced by:  psrgrp  19613  psrlmod  19616  mndvass  20415  itg2mulc  23734  plydivlem4  24271  dchrabl  25200  lfladdass  34882  lflvsass  34890  expgrowth  39060
 Copyright terms: Public domain W3C validator