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Theorem gacan 19335
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 19322 . . . . . . . 8 ( ∈ (𝐺 GrpAct 𝑌) → 𝐺 ∈ Grp)
21adantr 480 . . . . . . 7 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → 𝐺 ∈ Grp)
3 simpr1 1193 . . . . . . 7 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → 𝐴𝑋)
4 galcan.1 . . . . . . . 8 𝑋 = (Base‘𝐺)
5 eqid 2734 . . . . . . . 8 (+g𝐺) = (+g𝐺)
6 eqid 2734 . . . . . . . 8 (0g𝐺) = (0g𝐺)
7 gacan.2 . . . . . . . 8 𝑁 = (invg𝐺)
84, 5, 6, 7grprinv 19020 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (𝐴(+g𝐺)(𝑁𝐴)) = (0g𝐺))
92, 3, 8syl2anc 584 . . . . . 6 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → (𝐴(+g𝐺)(𝑁𝐴)) = (0g𝐺))
109oveq1d 7445 . . . . 5 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → ((𝐴(+g𝐺)(𝑁𝐴)) 𝐶) = ((0g𝐺) 𝐶))
11 simpl 482 . . . . . 6 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → ∈ (𝐺 GrpAct 𝑌))
124, 7grpinvcl 19017 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (𝑁𝐴) ∈ 𝑋)
132, 3, 12syl2anc 584 . . . . . 6 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → (𝑁𝐴) ∈ 𝑋)
14 simpr3 1195 . . . . . 6 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → 𝐶𝑌)
154, 5gaass 19327 . . . . . 6 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋 ∧ (𝑁𝐴) ∈ 𝑋𝐶𝑌)) → ((𝐴(+g𝐺)(𝑁𝐴)) 𝐶) = (𝐴 ((𝑁𝐴) 𝐶)))
1611, 3, 13, 14, 15syl13anc 1371 . . . . 5 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → ((𝐴(+g𝐺)(𝑁𝐴)) 𝐶) = (𝐴 ((𝑁𝐴) 𝐶)))
176gagrpid 19324 . . . . . 6 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐶𝑌) → ((0g𝐺) 𝐶) = 𝐶)
1811, 14, 17syl2anc 584 . . . . 5 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → ((0g𝐺) 𝐶) = 𝐶)
1910, 16, 183eqtr3d 2782 . . . 4 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → (𝐴 ((𝑁𝐴) 𝐶)) = 𝐶)
2019eqeq2d 2745 . . 3 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → ((𝐴 𝐵) = (𝐴 ((𝑁𝐴) 𝐶)) ↔ (𝐴 𝐵) = 𝐶))
21 simpr2 1194 . . . 4 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → 𝐵𝑌)
224gaf 19325 . . . . . 6 ( ∈ (𝐺 GrpAct 𝑌) → :(𝑋 × 𝑌)⟶𝑌)
2322adantr 480 . . . . 5 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → :(𝑋 × 𝑌)⟶𝑌)
2423, 13, 14fovcdmd 7604 . . . 4 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → ((𝑁𝐴) 𝐶) ∈ 𝑌)
254galcan 19334 . . . 4 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌 ∧ ((𝑁𝐴) 𝐶) ∈ 𝑌)) → ((𝐴 𝐵) = (𝐴 ((𝑁𝐴) 𝐶)) ↔ 𝐵 = ((𝑁𝐴) 𝐶)))
2611, 3, 21, 24, 25syl13anc 1371 . . 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 2741 . 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 1086   = wceq 1536  wcel 2105   × cxp 5686  wf 6558  cfv 6562  (class class class)co 7430  Basecbs 17244  +gcplusg 17297  0gc0g 17485  Grpcgrp 18963  invgcminusg 18964   GrpAct cga 19319
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-map 8866  df-0g 17487  df-mgm 18665  df-sgrp 18744  df-mnd 18760  df-grp 18966  df-minusg 18967  df-ga 19320
This theorem is referenced by:  gapm  19336  gaorber  19338  gastacl  19339  gastacos  19340
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