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

Theorem gaass 18426
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 2821 . . . . . . 7 (0g𝐺) = (0g𝐺)
41, 2, 3isga 18420 . . . . . 6 ( ∈ (𝐺 GrpAct 𝑌) ↔ ((𝐺 ∈ Grp ∧ 𝑌 ∈ V) ∧ ( :(𝑋 × 𝑌)⟶𝑌 ∧ ∀𝑥𝑌 (((0g𝐺) 𝑥) = 𝑥 ∧ ∀𝑦𝑋𝑧𝑋 ((𝑦 + 𝑧) 𝑥) = (𝑦 (𝑧 𝑥))))))
54simprbi 499 . . . . 5 ( ∈ (𝐺 GrpAct 𝑌) → ( :(𝑋 × 𝑌)⟶𝑌 ∧ ∀𝑥𝑌 (((0g𝐺) 𝑥) = 𝑥 ∧ ∀𝑦𝑋𝑧𝑋 ((𝑦 + 𝑧) 𝑥) = (𝑦 (𝑧 𝑥)))))
6 simpr 487 . . . . . 6 ((((0g𝐺) 𝑥) = 𝑥 ∧ ∀𝑦𝑋𝑧𝑋 ((𝑦 + 𝑧) 𝑥) = (𝑦 (𝑧 𝑥))) → ∀𝑦𝑋𝑧𝑋 ((𝑦 + 𝑧) 𝑥) = (𝑦 (𝑧 𝑥)))
76ralimi 3160 . . . . 5 (∀𝑥𝑌 (((0g𝐺) 𝑥) = 𝑥 ∧ ∀𝑦𝑋𝑧𝑋 ((𝑦 + 𝑧) 𝑥) = (𝑦 (𝑧 𝑥))) → ∀𝑥𝑌𝑦𝑋𝑧𝑋 ((𝑦 + 𝑧) 𝑥) = (𝑦 (𝑧 𝑥)))
85, 7simpl2im 506 . . . 4 ( ∈ (𝐺 GrpAct 𝑌) → ∀𝑥𝑌𝑦𝑋𝑧𝑋 ((𝑦 + 𝑧) 𝑥) = (𝑦 (𝑧 𝑥)))
9 oveq2 7163 . . . . . 6 (𝑥 = 𝐶 → ((𝑦 + 𝑧) 𝑥) = ((𝑦 + 𝑧) 𝐶))
10 oveq2 7163 . . . . . . 7 (𝑥 = 𝐶 → (𝑧 𝑥) = (𝑧 𝐶))
1110oveq2d 7171 . . . . . 6 (𝑥 = 𝐶 → (𝑦 (𝑧 𝑥)) = (𝑦 (𝑧 𝐶)))
129, 11eqeq12d 2837 . . . . 5 (𝑥 = 𝐶 → (((𝑦 + 𝑧) 𝑥) = (𝑦 (𝑧 𝑥)) ↔ ((𝑦 + 𝑧) 𝐶) = (𝑦 (𝑧 𝐶))))
13 oveq1 7162 . . . . . . 7 (𝑦 = 𝐴 → (𝑦 + 𝑧) = (𝐴 + 𝑧))
1413oveq1d 7170 . . . . . 6 (𝑦 = 𝐴 → ((𝑦 + 𝑧) 𝐶) = ((𝐴 + 𝑧) 𝐶))
15 oveq1 7162 . . . . . 6 (𝑦 = 𝐴 → (𝑦 (𝑧 𝐶)) = (𝐴 (𝑧 𝐶)))
1614, 15eqeq12d 2837 . . . . 5 (𝑦 = 𝐴 → (((𝑦 + 𝑧) 𝐶) = (𝑦 (𝑧 𝐶)) ↔ ((𝐴 + 𝑧) 𝐶) = (𝐴 (𝑧 𝐶))))
17 oveq2 7163 . . . . . . 7 (𝑧 = 𝐵 → (𝐴 + 𝑧) = (𝐴 + 𝐵))
1817oveq1d 7170 . . . . . 6 (𝑧 = 𝐵 → ((𝐴 + 𝑧) 𝐶) = ((𝐴 + 𝐵) 𝐶))
19 oveq1 7162 . . . . . . 7 (𝑧 = 𝐵 → (𝑧 𝐶) = (𝐵 𝐶))
2019oveq2d 7171 . . . . . 6 (𝑧 = 𝐵 → (𝐴 (𝑧 𝐶)) = (𝐴 (𝐵 𝐶)))
2118, 20eqeq12d 2837 . . . . 5 (𝑧 = 𝐵 → (((𝐴 + 𝑧) 𝐶) = (𝐴 (𝑧 𝐶)) ↔ ((𝐴 + 𝐵) 𝐶) = (𝐴 (𝐵 𝐶))))
2212, 16, 21rspc3v 3635 . . . 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 1123 . 2 ((𝐴𝑋𝐵𝑋𝐶𝑌) → ( ∈ (𝐺 GrpAct 𝑌) → ((𝐴 + 𝐵) 𝐶) = (𝐴 (𝐵 𝐶))))
2524impcom 410 1 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑋𝐶𝑌)) → ((𝐴 + 𝐵) 𝐶) = (𝐴 (𝐵 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398  w3a 1083   = wceq 1533  wcel 2110  wral 3138  Vcvv 3494   × cxp 5552  wf 6350  cfv 6354  (class class class)co 7155  Basecbs 16482  +gcplusg 16564  0gc0g 16712  Grpcgrp 18102   GrpAct cga 18418
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-br 5066  df-opab 5128  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-fv 6362  df-ov 7158  df-oprab 7159  df-mpo 7160  df-map 8407  df-ga 18419
This theorem is referenced by:  gass  18430  gasubg  18431  galcan  18433  gacan  18434  gaorber  18437  gastacl  18438  gastacos  18439  galactghm  18531
  Copyright terms: Public domain W3C validator