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Theorem gacan 19323
Description: Group inverses cancel in a group action. (Contributed by Jeff Hankins, 11-Aug-2009.) (Revised by Mario Carneiro, 13-Jan-2015.)
Hypotheses
Ref Expression
galcan.1 𝑋 = (Base‘𝐺)
gacan.2 𝑁 = (invg𝐺)
Assertion
Ref Expression
gacan (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → ((𝐴 𝐵) = 𝐶 ↔ ((𝑁𝐴) 𝐶) = 𝐵))

Proof of Theorem gacan
StepHypRef Expression
1 gagrp 19310 . . . . . . . 8 ( ∈ (𝐺 GrpAct 𝑌) → 𝐺 ∈ Grp)
21adantr 480 . . . . . . 7 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → 𝐺 ∈ Grp)
3 simpr1 1195 . . . . . . 7 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → 𝐴𝑋)
4 galcan.1 . . . . . . . 8 𝑋 = (Base‘𝐺)
5 eqid 2737 . . . . . . . 8 (+g𝐺) = (+g𝐺)
6 eqid 2737 . . . . . . . 8 (0g𝐺) = (0g𝐺)
7 gacan.2 . . . . . . . 8 𝑁 = (invg𝐺)
84, 5, 6, 7grprinv 19008 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (𝐴(+g𝐺)(𝑁𝐴)) = (0g𝐺))
92, 3, 8syl2anc 584 . . . . . 6 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → (𝐴(+g𝐺)(𝑁𝐴)) = (0g𝐺))
109oveq1d 7446 . . . . 5 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → ((𝐴(+g𝐺)(𝑁𝐴)) 𝐶) = ((0g𝐺) 𝐶))
11 simpl 482 . . . . . 6 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → ∈ (𝐺 GrpAct 𝑌))
124, 7grpinvcl 19005 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (𝑁𝐴) ∈ 𝑋)
132, 3, 12syl2anc 584 . . . . . 6 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → (𝑁𝐴) ∈ 𝑋)
14 simpr3 1197 . . . . . 6 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → 𝐶𝑌)
154, 5gaass 19315 . . . . . 6 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋 ∧ (𝑁𝐴) ∈ 𝑋𝐶𝑌)) → ((𝐴(+g𝐺)(𝑁𝐴)) 𝐶) = (𝐴 ((𝑁𝐴) 𝐶)))
1611, 3, 13, 14, 15syl13anc 1374 . . . . 5 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → ((𝐴(+g𝐺)(𝑁𝐴)) 𝐶) = (𝐴 ((𝑁𝐴) 𝐶)))
176gagrpid 19312 . . . . . 6 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐶𝑌) → ((0g𝐺) 𝐶) = 𝐶)
1811, 14, 17syl2anc 584 . . . . 5 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → ((0g𝐺) 𝐶) = 𝐶)
1910, 16, 183eqtr3d 2785 . . . 4 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → (𝐴 ((𝑁𝐴) 𝐶)) = 𝐶)
2019eqeq2d 2748 . . 3 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → ((𝐴 𝐵) = (𝐴 ((𝑁𝐴) 𝐶)) ↔ (𝐴 𝐵) = 𝐶))
21 simpr2 1196 . . . 4 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → 𝐵𝑌)
224gaf 19313 . . . . . 6 ( ∈ (𝐺 GrpAct 𝑌) → :(𝑋 × 𝑌)⟶𝑌)
2322adantr 480 . . . . 5 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → :(𝑋 × 𝑌)⟶𝑌)
2423, 13, 14fovcdmd 7605 . . . 4 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → ((𝑁𝐴) 𝐶) ∈ 𝑌)
254galcan 19322 . . . 4 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌 ∧ ((𝑁𝐴) 𝐶) ∈ 𝑌)) → ((𝐴 𝐵) = (𝐴 ((𝑁𝐴) 𝐶)) ↔ 𝐵 = ((𝑁𝐴) 𝐶)))
2611, 3, 21, 24, 25syl13anc 1374 . . 3 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → ((𝐴 𝐵) = (𝐴 ((𝑁𝐴) 𝐶)) ↔ 𝐵 = ((𝑁𝐴) 𝐶)))
2720, 26bitr3d 281 . 2 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → ((𝐴 𝐵) = 𝐶𝐵 = ((𝑁𝐴) 𝐶)))
28 eqcom 2744 . 2 (𝐵 = ((𝑁𝐴) 𝐶) ↔ ((𝑁𝐴) 𝐶) = 𝐵)
2927, 28bitrdi 287 1 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → ((𝐴 𝐵) = 𝐶 ↔ ((𝑁𝐴) 𝐶) = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108   × cxp 5683  wf 6557  cfv 6561  (class class class)co 7431  Basecbs 17247  +gcplusg 17297  0gc0g 17484  Grpcgrp 18951  invgcminusg 18952   GrpAct cga 19307
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8868  df-0g 17486  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-grp 18954  df-minusg 18955  df-ga 19308
This theorem is referenced by:  gapm  19324  gaorber  19326  gastacl  19327  gastacos  19328
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