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Theorem gaass 17937
 Description: An "associative" property for group actions. (Contributed by Jeff Hankins, 11-Aug-2009.) (Revised by Mario Carneiro, 13-Jan-2015.)
Hypotheses
Ref Expression
gaass.1 𝑋 = (Base‘𝐺)
gaass.2 + = (+g𝐺)
Assertion
Ref Expression
gaass (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑋𝐶𝑌)) → ((𝐴 + 𝐵) 𝐶) = (𝐴 (𝐵 𝐶)))

Proof of Theorem gaass
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 gaass.1 . . . . . . . 8 𝑋 = (Base‘𝐺)
2 gaass.2 . . . . . . . 8 + = (+g𝐺)
3 eqid 2771 . . . . . . . 8 (0g𝐺) = (0g𝐺)
41, 2, 3isga 17931 . . . . . . 7 ( ∈ (𝐺 GrpAct 𝑌) ↔ ((𝐺 ∈ Grp ∧ 𝑌 ∈ V) ∧ ( :(𝑋 × 𝑌)⟶𝑌 ∧ ∀𝑥𝑌 (((0g𝐺) 𝑥) = 𝑥 ∧ ∀𝑦𝑋𝑧𝑋 ((𝑦 + 𝑧) 𝑥) = (𝑦 (𝑧 𝑥))))))
54simprbi 484 . . . . . 6 ( ∈ (𝐺 GrpAct 𝑌) → ( :(𝑋 × 𝑌)⟶𝑌 ∧ ∀𝑥𝑌 (((0g𝐺) 𝑥) = 𝑥 ∧ ∀𝑦𝑋𝑧𝑋 ((𝑦 + 𝑧) 𝑥) = (𝑦 (𝑧 𝑥)))))
65simprd 483 . . . . 5 ( ∈ (𝐺 GrpAct 𝑌) → ∀𝑥𝑌 (((0g𝐺) 𝑥) = 𝑥 ∧ ∀𝑦𝑋𝑧𝑋 ((𝑦 + 𝑧) 𝑥) = (𝑦 (𝑧 𝑥))))
7 simpr 471 . . . . . 6 ((((0g𝐺) 𝑥) = 𝑥 ∧ ∀𝑦𝑋𝑧𝑋 ((𝑦 + 𝑧) 𝑥) = (𝑦 (𝑧 𝑥))) → ∀𝑦𝑋𝑧𝑋 ((𝑦 + 𝑧) 𝑥) = (𝑦 (𝑧 𝑥)))
87ralimi 3101 . . . . 5 (∀𝑥𝑌 (((0g𝐺) 𝑥) = 𝑥 ∧ ∀𝑦𝑋𝑧𝑋 ((𝑦 + 𝑧) 𝑥) = (𝑦 (𝑧 𝑥))) → ∀𝑥𝑌𝑦𝑋𝑧𝑋 ((𝑦 + 𝑧) 𝑥) = (𝑦 (𝑧 𝑥)))
96, 8syl 17 . . . 4 ( ∈ (𝐺 GrpAct 𝑌) → ∀𝑥𝑌𝑦𝑋𝑧𝑋 ((𝑦 + 𝑧) 𝑥) = (𝑦 (𝑧 𝑥)))
10 oveq2 6801 . . . . . 6 (𝑥 = 𝐶 → ((𝑦 + 𝑧) 𝑥) = ((𝑦 + 𝑧) 𝐶))
11 oveq2 6801 . . . . . . 7 (𝑥 = 𝐶 → (𝑧 𝑥) = (𝑧 𝐶))
1211oveq2d 6809 . . . . . 6 (𝑥 = 𝐶 → (𝑦 (𝑧 𝑥)) = (𝑦 (𝑧 𝐶)))
1310, 12eqeq12d 2786 . . . . 5 (𝑥 = 𝐶 → (((𝑦 + 𝑧) 𝑥) = (𝑦 (𝑧 𝑥)) ↔ ((𝑦 + 𝑧) 𝐶) = (𝑦 (𝑧 𝐶))))
14 oveq1 6800 . . . . . . 7 (𝑦 = 𝐴 → (𝑦 + 𝑧) = (𝐴 + 𝑧))
1514oveq1d 6808 . . . . . 6 (𝑦 = 𝐴 → ((𝑦 + 𝑧) 𝐶) = ((𝐴 + 𝑧) 𝐶))
16 oveq1 6800 . . . . . 6 (𝑦 = 𝐴 → (𝑦 (𝑧 𝐶)) = (𝐴 (𝑧 𝐶)))
1715, 16eqeq12d 2786 . . . . 5 (𝑦 = 𝐴 → (((𝑦 + 𝑧) 𝐶) = (𝑦 (𝑧 𝐶)) ↔ ((𝐴 + 𝑧) 𝐶) = (𝐴 (𝑧 𝐶))))
18 oveq2 6801 . . . . . . 7 (𝑧 = 𝐵 → (𝐴 + 𝑧) = (𝐴 + 𝐵))
1918oveq1d 6808 . . . . . 6 (𝑧 = 𝐵 → ((𝐴 + 𝑧) 𝐶) = ((𝐴 + 𝐵) 𝐶))
20 oveq1 6800 . . . . . . 7 (𝑧 = 𝐵 → (𝑧 𝐶) = (𝐵 𝐶))
2120oveq2d 6809 . . . . . 6 (𝑧 = 𝐵 → (𝐴 (𝑧 𝐶)) = (𝐴 (𝐵 𝐶)))
2219, 21eqeq12d 2786 . . . . 5 (𝑧 = 𝐵 → (((𝐴 + 𝑧) 𝐶) = (𝐴 (𝑧 𝐶)) ↔ ((𝐴 + 𝐵) 𝐶) = (𝐴 (𝐵 𝐶))))
2313, 17, 22rspc3v 3475 . . . 4 ((𝐶𝑌𝐴𝑋𝐵𝑋) → (∀𝑥𝑌𝑦𝑋𝑧𝑋 ((𝑦 + 𝑧) 𝑥) = (𝑦 (𝑧 𝑥)) → ((𝐴 + 𝐵) 𝐶) = (𝐴 (𝐵 𝐶))))
249, 23syl5 34 . . 3 ((𝐶𝑌𝐴𝑋𝐵𝑋) → ( ∈ (𝐺 GrpAct 𝑌) → ((𝐴 + 𝐵) 𝐶) = (𝐴 (𝐵 𝐶))))
25243coml 1121 . 2 ((𝐴𝑋𝐵𝑋𝐶𝑌) → ( ∈ (𝐺 GrpAct 𝑌) → ((𝐴 + 𝐵) 𝐶) = (𝐴 (𝐵 𝐶))))
2625impcom 394 1 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑋𝐶𝑌)) → ((𝐴 + 𝐵) 𝐶) = (𝐴 (𝐵 𝐶)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 382   ∧ w3a 1071   = wceq 1631   ∈ wcel 2145  ∀wral 3061  Vcvv 3351   × cxp 5247  ⟶wf 6027  ‘cfv 6031  (class class class)co 6793  Basecbs 16064  +gcplusg 16149  0gc0g 16308  Grpcgrp 17630   GrpAct cga 17929 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-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096 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-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 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-map 8011  df-ga 17930 This theorem is referenced by:  gass  17941  gasubg  17942  galcan  17944  gacan  17945  gaorber  17948  gastacl  17949  gastacos  17950  galactghm  18030
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