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Theorem gaass 18931
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 . . . . . . 7 𝑋 = (Base‘𝐺)
2 gaass.2 . . . . . . 7 + = (+g𝐺)
3 eqid 2733 . . . . . . 7 (0g𝐺) = (0g𝐺)
41, 2, 3isga 18925 . . . . . 6 ( ∈ (𝐺 GrpAct 𝑌) ↔ ((𝐺 ∈ Grp ∧ 𝑌 ∈ V) ∧ ( :(𝑋 × 𝑌)⟶𝑌 ∧ ∀𝑥𝑌 (((0g𝐺) 𝑥) = 𝑥 ∧ ∀𝑦𝑋𝑧𝑋 ((𝑦 + 𝑧) 𝑥) = (𝑦 (𝑧 𝑥))))))
54simprbi 496 . . . . 5 ( ∈ (𝐺 GrpAct 𝑌) → ( :(𝑋 × 𝑌)⟶𝑌 ∧ ∀𝑥𝑌 (((0g𝐺) 𝑥) = 𝑥 ∧ ∀𝑦𝑋𝑧𝑋 ((𝑦 + 𝑧) 𝑥) = (𝑦 (𝑧 𝑥)))))
6 simpr 484 . . . . . 6 ((((0g𝐺) 𝑥) = 𝑥 ∧ ∀𝑦𝑋𝑧𝑋 ((𝑦 + 𝑧) 𝑥) = (𝑦 (𝑧 𝑥))) → ∀𝑦𝑋𝑧𝑋 ((𝑦 + 𝑧) 𝑥) = (𝑦 (𝑧 𝑥)))
76ralimi 3080 . . . . 5 (∀𝑥𝑌 (((0g𝐺) 𝑥) = 𝑥 ∧ ∀𝑦𝑋𝑧𝑋 ((𝑦 + 𝑧) 𝑥) = (𝑦 (𝑧 𝑥))) → ∀𝑥𝑌𝑦𝑋𝑧𝑋 ((𝑦 + 𝑧) 𝑥) = (𝑦 (𝑧 𝑥)))
85, 7simpl2im 503 . . . 4 ( ∈ (𝐺 GrpAct 𝑌) → ∀𝑥𝑌𝑦𝑋𝑧𝑋 ((𝑦 + 𝑧) 𝑥) = (𝑦 (𝑧 𝑥)))
9 oveq2 7303 . . . . . 6 (𝑥 = 𝐶 → ((𝑦 + 𝑧) 𝑥) = ((𝑦 + 𝑧) 𝐶))
10 oveq2 7303 . . . . . . 7 (𝑥 = 𝐶 → (𝑧 𝑥) = (𝑧 𝐶))
1110oveq2d 7311 . . . . . 6 (𝑥 = 𝐶 → (𝑦 (𝑧 𝑥)) = (𝑦 (𝑧 𝐶)))
129, 11eqeq12d 2749 . . . . 5 (𝑥 = 𝐶 → (((𝑦 + 𝑧) 𝑥) = (𝑦 (𝑧 𝑥)) ↔ ((𝑦 + 𝑧) 𝐶) = (𝑦 (𝑧 𝐶))))
13 oveq1 7302 . . . . . . 7 (𝑦 = 𝐴 → (𝑦 + 𝑧) = (𝐴 + 𝑧))
1413oveq1d 7310 . . . . . 6 (𝑦 = 𝐴 → ((𝑦 + 𝑧) 𝐶) = ((𝐴 + 𝑧) 𝐶))
15 oveq1 7302 . . . . . 6 (𝑦 = 𝐴 → (𝑦 (𝑧 𝐶)) = (𝐴 (𝑧 𝐶)))
1614, 15eqeq12d 2749 . . . . 5 (𝑦 = 𝐴 → (((𝑦 + 𝑧) 𝐶) = (𝑦 (𝑧 𝐶)) ↔ ((𝐴 + 𝑧) 𝐶) = (𝐴 (𝑧 𝐶))))
17 oveq2 7303 . . . . . . 7 (𝑧 = 𝐵 → (𝐴 + 𝑧) = (𝐴 + 𝐵))
1817oveq1d 7310 . . . . . 6 (𝑧 = 𝐵 → ((𝐴 + 𝑧) 𝐶) = ((𝐴 + 𝐵) 𝐶))
19 oveq1 7302 . . . . . . 7 (𝑧 = 𝐵 → (𝑧 𝐶) = (𝐵 𝐶))
2019oveq2d 7311 . . . . . 6 (𝑧 = 𝐵 → (𝐴 (𝑧 𝐶)) = (𝐴 (𝐵 𝐶)))
2118, 20eqeq12d 2749 . . . . 5 (𝑧 = 𝐵 → (((𝐴 + 𝑧) 𝐶) = (𝐴 (𝑧 𝐶)) ↔ ((𝐴 + 𝐵) 𝐶) = (𝐴 (𝐵 𝐶))))
2212, 16, 21rspc3v 3575 . . . 4 ((𝐶𝑌𝐴𝑋𝐵𝑋) → (∀𝑥𝑌𝑦𝑋𝑧𝑋 ((𝑦 + 𝑧) 𝑥) = (𝑦 (𝑧 𝑥)) → ((𝐴 + 𝐵) 𝐶) = (𝐴 (𝐵 𝐶))))
238, 22syl5 34 . . 3 ((𝐶𝑌𝐴𝑋𝐵𝑋) → ( ∈ (𝐺 GrpAct 𝑌) → ((𝐴 + 𝐵) 𝐶) = (𝐴 (𝐵 𝐶))))
24233coml 1125 . 2 ((𝐴𝑋𝐵𝑋𝐶𝑌) → ( ∈ (𝐺 GrpAct 𝑌) → ((𝐴 + 𝐵) 𝐶) = (𝐴 (𝐵 𝐶))))
2524impcom 407 1 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑋𝐶𝑌)) → ((𝐴 + 𝐵) 𝐶) = (𝐴 (𝐵 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1085   = wceq 1537  wcel 2101  wral 3059  Vcvv 3434   × cxp 5589  wf 6443  cfv 6447  (class class class)co 7295  Basecbs 16940  +gcplusg 16990  0gc0g 17178  Grpcgrp 18605   GrpAct cga 18923
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2103  ax-9 2111  ax-10 2132  ax-11 2149  ax-12 2166  ax-ext 2704  ax-sep 5226  ax-nul 5233  ax-pow 5291  ax-pr 5355  ax-un 7608
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2063  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2884  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3224  df-v 3436  df-sbc 3719  df-csb 3835  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4260  df-if 4463  df-pw 4538  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4842  df-br 5078  df-opab 5140  df-id 5491  df-xp 5597  df-rel 5598  df-cnv 5599  df-co 5600  df-dm 5601  df-rn 5602  df-iota 6399  df-fun 6449  df-fn 6450  df-f 6451  df-fv 6455  df-ov 7298  df-oprab 7299  df-mpo 7300  df-map 8637  df-ga 18924
This theorem is referenced by:  gass  18935  gasubg  18936  galcan  18938  gacan  18939  gaorber  18942  gastacl  18943  gastacos  18944  galactghm  19040
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