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Theorem gacan 18088
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 18075 . . . . . . . 8 ( ∈ (𝐺 GrpAct 𝑌) → 𝐺 ∈ Grp)
21adantr 474 . . . . . . 7 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → 𝐺 ∈ Grp)
3 simpr1 1252 . . . . . . 7 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → 𝐴𝑋)
4 galcan.1 . . . . . . . 8 𝑋 = (Base‘𝐺)
5 eqid 2825 . . . . . . . 8 (+g𝐺) = (+g𝐺)
6 eqid 2825 . . . . . . . 8 (0g𝐺) = (0g𝐺)
7 gacan.2 . . . . . . . 8 𝑁 = (invg𝐺)
84, 5, 6, 7grprinv 17823 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (𝐴(+g𝐺)(𝑁𝐴)) = (0g𝐺))
92, 3, 8syl2anc 579 . . . . . 6 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → (𝐴(+g𝐺)(𝑁𝐴)) = (0g𝐺))
109oveq1d 6920 . . . . 5 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → ((𝐴(+g𝐺)(𝑁𝐴)) 𝐶) = ((0g𝐺) 𝐶))
11 simpl 476 . . . . . 6 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → ∈ (𝐺 GrpAct 𝑌))
124, 7grpinvcl 17821 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (𝑁𝐴) ∈ 𝑋)
132, 3, 12syl2anc 579 . . . . . 6 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → (𝑁𝐴) ∈ 𝑋)
14 simpr3 1256 . . . . . 6 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → 𝐶𝑌)
154, 5gaass 18080 . . . . . 6 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋 ∧ (𝑁𝐴) ∈ 𝑋𝐶𝑌)) → ((𝐴(+g𝐺)(𝑁𝐴)) 𝐶) = (𝐴 ((𝑁𝐴) 𝐶)))
1611, 3, 13, 14, 15syl13anc 1495 . . . . 5 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → ((𝐴(+g𝐺)(𝑁𝐴)) 𝐶) = (𝐴 ((𝑁𝐴) 𝐶)))
176gagrpid 18077 . . . . . 6 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐶𝑌) → ((0g𝐺) 𝐶) = 𝐶)
1811, 14, 17syl2anc 579 . . . . 5 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → ((0g𝐺) 𝐶) = 𝐶)
1910, 16, 183eqtr3d 2869 . . . 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 1254 . . . 4 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → 𝐵𝑌)
224gaf 18078 . . . . . 6 ( ∈ (𝐺 GrpAct 𝑌) → :(𝑋 × 𝑌)⟶𝑌)
2322adantr 474 . . . . 5 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → :(𝑋 × 𝑌)⟶𝑌)
2423, 13, 14fovrnd 7066 . . . 4 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → ((𝑁𝐴) 𝐶) ∈ 𝑌)
254galcan 18087 . . . 4 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌 ∧ ((𝑁𝐴) 𝐶) ∈ 𝑌)) → ((𝐴 𝐵) = (𝐴 ((𝑁𝐴) 𝐶)) ↔ 𝐵 = ((𝑁𝐴) 𝐶)))
2611, 3, 21, 24, 25syl13anc 1495 . . 3 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → ((𝐴 𝐵) = (𝐴 ((𝑁𝐴) 𝐶)) ↔ 𝐵 = ((𝑁𝐴) 𝐶)))
2720, 26bitr3d 273 . 2 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → ((𝐴 𝐵) = 𝐶𝐵 = ((𝑁𝐴) 𝐶)))
28 eqcom 2832 . 2 (𝐵 = ((𝑁𝐴) 𝐶) ↔ ((𝑁𝐴) 𝐶) = 𝐵)
2927, 28syl6bb 279 1 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → ((𝐴 𝐵) = 𝐶 ↔ ((𝑁𝐴) 𝐶) = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386  w3a 1111   = wceq 1656  wcel 2164   × cxp 5340  wf 6119  cfv 6123  (class class class)co 6905  Basecbs 16222  +gcplusg 16305  0gc0g 16453  Grpcgrp 17776  invgcminusg 17777   GrpAct cga 18072
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-riota 6866  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-map 8124  df-0g 16455  df-mgm 17595  df-sgrp 17637  df-mnd 17648  df-grp 17779  df-minusg 17780  df-ga 18073
This theorem is referenced by:  gapm  18089  gaorber  18091  gastacl  18092  gastacos  18093
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