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Theorem gacan 18424
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 18411 . . . . . . . 8 ( ∈ (𝐺 GrpAct 𝑌) → 𝐺 ∈ Grp)
21adantr 484 . . . . . . 7 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → 𝐺 ∈ Grp)
3 simpr1 1191 . . . . . . 7 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → 𝐴𝑋)
4 galcan.1 . . . . . . . 8 𝑋 = (Base‘𝐺)
5 eqid 2824 . . . . . . . 8 (+g𝐺) = (+g𝐺)
6 eqid 2824 . . . . . . . 8 (0g𝐺) = (0g𝐺)
7 gacan.2 . . . . . . . 8 𝑁 = (invg𝐺)
84, 5, 6, 7grprinv 18142 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (𝐴(+g𝐺)(𝑁𝐴)) = (0g𝐺))
92, 3, 8syl2anc 587 . . . . . 6 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → (𝐴(+g𝐺)(𝑁𝐴)) = (0g𝐺))
109oveq1d 7153 . . . . 5 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → ((𝐴(+g𝐺)(𝑁𝐴)) 𝐶) = ((0g𝐺) 𝐶))
11 simpl 486 . . . . . 6 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → ∈ (𝐺 GrpAct 𝑌))
124, 7grpinvcl 18140 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (𝑁𝐴) ∈ 𝑋)
132, 3, 12syl2anc 587 . . . . . 6 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → (𝑁𝐴) ∈ 𝑋)
14 simpr3 1193 . . . . . 6 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → 𝐶𝑌)
154, 5gaass 18416 . . . . . 6 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋 ∧ (𝑁𝐴) ∈ 𝑋𝐶𝑌)) → ((𝐴(+g𝐺)(𝑁𝐴)) 𝐶) = (𝐴 ((𝑁𝐴) 𝐶)))
1611, 3, 13, 14, 15syl13anc 1369 . . . . 5 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → ((𝐴(+g𝐺)(𝑁𝐴)) 𝐶) = (𝐴 ((𝑁𝐴) 𝐶)))
176gagrpid 18413 . . . . . 6 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐶𝑌) → ((0g𝐺) 𝐶) = 𝐶)
1811, 14, 17syl2anc 587 . . . . 5 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → ((0g𝐺) 𝐶) = 𝐶)
1910, 16, 183eqtr3d 2867 . . . 4 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → (𝐴 ((𝑁𝐴) 𝐶)) = 𝐶)
2019eqeq2d 2835 . . 3 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → ((𝐴 𝐵) = (𝐴 ((𝑁𝐴) 𝐶)) ↔ (𝐴 𝐵) = 𝐶))
21 simpr2 1192 . . . 4 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → 𝐵𝑌)
224gaf 18414 . . . . . 6 ( ∈ (𝐺 GrpAct 𝑌) → :(𝑋 × 𝑌)⟶𝑌)
2322adantr 484 . . . . 5 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → :(𝑋 × 𝑌)⟶𝑌)
2423, 13, 14fovrnd 7303 . . . 4 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → ((𝑁𝐴) 𝐶) ∈ 𝑌)
254galcan 18423 . . . 4 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌 ∧ ((𝑁𝐴) 𝐶) ∈ 𝑌)) → ((𝐴 𝐵) = (𝐴 ((𝑁𝐴) 𝐶)) ↔ 𝐵 = ((𝑁𝐴) 𝐶)))
2611, 3, 21, 24, 25syl13anc 1369 . . 3 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → ((𝐴 𝐵) = (𝐴 ((𝑁𝐴) 𝐶)) ↔ 𝐵 = ((𝑁𝐴) 𝐶)))
2720, 26bitr3d 284 . 2 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → ((𝐴 𝐵) = 𝐶𝐵 = ((𝑁𝐴) 𝐶)))
28 eqcom 2831 . 2 (𝐵 = ((𝑁𝐴) 𝐶) ↔ ((𝑁𝐴) 𝐶) = 𝐵)
2927, 28syl6bb 290 1 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → ((𝐴 𝐵) = 𝐶 ↔ ((𝑁𝐴) 𝐶) = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2115   × cxp 5534  wf 6332  cfv 6336  (class class class)co 7138  Basecbs 16472  +gcplusg 16554  0gc0g 16702  Grpcgrp 18092  invgcminusg 18093   GrpAct cga 18408
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5184  ax-nul 5191  ax-pow 5247  ax-pr 5311  ax-un 7444
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3014  df-ral 3137  df-rex 3138  df-reu 3139  df-rmo 3140  df-rab 3141  df-v 3481  df-sbc 3758  df-csb 3866  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4820  df-br 5048  df-opab 5110  df-mpt 5128  df-id 5441  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-fv 6344  df-riota 7096  df-ov 7141  df-oprab 7142  df-mpo 7143  df-map 8391  df-0g 16704  df-mgm 17841  df-sgrp 17890  df-mnd 17901  df-grp 18095  df-minusg 18096  df-ga 18409
This theorem is referenced by:  gapm  18425  gaorber  18427  gastacl  18428  gastacos  18429
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