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Theorem gacan 17719
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 17706 . . . . . . . 8 ( ∈ (𝐺 GrpAct 𝑌) → 𝐺 ∈ Grp)
21adantr 481 . . . . . . 7 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → 𝐺 ∈ Grp)
3 simpr1 1065 . . . . . . 7 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → 𝐴𝑋)
4 galcan.1 . . . . . . . 8 𝑋 = (Base‘𝐺)
5 eqid 2620 . . . . . . . 8 (+g𝐺) = (+g𝐺)
6 eqid 2620 . . . . . . . 8 (0g𝐺) = (0g𝐺)
7 gacan.2 . . . . . . . 8 𝑁 = (invg𝐺)
84, 5, 6, 7grprinv 17450 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (𝐴(+g𝐺)(𝑁𝐴)) = (0g𝐺))
92, 3, 8syl2anc 692 . . . . . 6 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → (𝐴(+g𝐺)(𝑁𝐴)) = (0g𝐺))
109oveq1d 6650 . . . . 5 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → ((𝐴(+g𝐺)(𝑁𝐴)) 𝐶) = ((0g𝐺) 𝐶))
11 simpl 473 . . . . . 6 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → ∈ (𝐺 GrpAct 𝑌))
124, 7grpinvcl 17448 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (𝑁𝐴) ∈ 𝑋)
132, 3, 12syl2anc 692 . . . . . 6 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → (𝑁𝐴) ∈ 𝑋)
14 simpr3 1067 . . . . . 6 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → 𝐶𝑌)
154, 5gaass 17711 . . . . . 6 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋 ∧ (𝑁𝐴) ∈ 𝑋𝐶𝑌)) → ((𝐴(+g𝐺)(𝑁𝐴)) 𝐶) = (𝐴 ((𝑁𝐴) 𝐶)))
1611, 3, 13, 14, 15syl13anc 1326 . . . . 5 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → ((𝐴(+g𝐺)(𝑁𝐴)) 𝐶) = (𝐴 ((𝑁𝐴) 𝐶)))
176gagrpid 17708 . . . . . 6 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐶𝑌) → ((0g𝐺) 𝐶) = 𝐶)
1811, 14, 17syl2anc 692 . . . . 5 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → ((0g𝐺) 𝐶) = 𝐶)
1910, 16, 183eqtr3d 2662 . . . 4 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → (𝐴 ((𝑁𝐴) 𝐶)) = 𝐶)
2019eqeq2d 2630 . . 3 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → ((𝐴 𝐵) = (𝐴 ((𝑁𝐴) 𝐶)) ↔ (𝐴 𝐵) = 𝐶))
21 simpr2 1066 . . . 4 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → 𝐵𝑌)
224gaf 17709 . . . . . 6 ( ∈ (𝐺 GrpAct 𝑌) → :(𝑋 × 𝑌)⟶𝑌)
2322adantr 481 . . . . 5 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → :(𝑋 × 𝑌)⟶𝑌)
2423, 13, 14fovrnd 6791 . . . 4 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → ((𝑁𝐴) 𝐶) ∈ 𝑌)
254galcan 17718 . . . 4 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌 ∧ ((𝑁𝐴) 𝐶) ∈ 𝑌)) → ((𝐴 𝐵) = (𝐴 ((𝑁𝐴) 𝐶)) ↔ 𝐵 = ((𝑁𝐴) 𝐶)))
2611, 3, 21, 24, 25syl13anc 1326 . . 3 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → ((𝐴 𝐵) = (𝐴 ((𝑁𝐴) 𝐶)) ↔ 𝐵 = ((𝑁𝐴) 𝐶)))
2720, 26bitr3d 270 . 2 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → ((𝐴 𝐵) = 𝐶𝐵 = ((𝑁𝐴) 𝐶)))
28 eqcom 2627 . 2 (𝐵 = ((𝑁𝐴) 𝐶) ↔ ((𝑁𝐴) 𝐶) = 𝐵)
2927, 28syl6bb 276 1 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → ((𝐴 𝐵) = 𝐶 ↔ ((𝑁𝐴) 𝐶) = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1481  wcel 1988   × cxp 5102  wf 5872  cfv 5876  (class class class)co 6635  Basecbs 15838  +gcplusg 15922  0gc0g 16081  Grpcgrp 17403  invgcminusg 17404   GrpAct cga 17703
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-map 7844  df-0g 16083  df-mgm 17223  df-sgrp 17265  df-mnd 17276  df-grp 17406  df-minusg 17407  df-ga 17704
This theorem is referenced by:  gapm  17720  gaorber  17722  gastacl  17723  gastacos  17724
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