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Theorem galcan 17658
Description: The action of a particular group element is left-cancelable. (Contributed by FL, 17-May-2010.) (Revised by Mario Carneiro, 13-Jan-2015.)
Hypothesis
Ref Expression
galcan.1 𝑋 = (Base‘𝐺)
Assertion
Ref Expression
galcan (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → ((𝐴 𝐵) = (𝐴 𝐶) ↔ 𝐵 = 𝐶))

Proof of Theorem galcan
StepHypRef Expression
1 oveq2 6612 . . 3 ((𝐴 𝐵) = (𝐴 𝐶) → (((invg𝐺)‘𝐴) (𝐴 𝐵)) = (((invg𝐺)‘𝐴) (𝐴 𝐶)))
2 simpl 473 . . . . . . . 8 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → ∈ (𝐺 GrpAct 𝑌))
3 gagrp 17646 . . . . . . . 8 ( ∈ (𝐺 GrpAct 𝑌) → 𝐺 ∈ Grp)
42, 3syl 17 . . . . . . 7 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → 𝐺 ∈ Grp)
5 simpr1 1065 . . . . . . 7 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → 𝐴𝑋)
6 galcan.1 . . . . . . . 8 𝑋 = (Base‘𝐺)
7 eqid 2621 . . . . . . . 8 (+g𝐺) = (+g𝐺)
8 eqid 2621 . . . . . . . 8 (0g𝐺) = (0g𝐺)
9 eqid 2621 . . . . . . . 8 (invg𝐺) = (invg𝐺)
106, 7, 8, 9grplinv 17389 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (((invg𝐺)‘𝐴)(+g𝐺)𝐴) = (0g𝐺))
114, 5, 10syl2anc 692 . . . . . 6 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → (((invg𝐺)‘𝐴)(+g𝐺)𝐴) = (0g𝐺))
1211oveq1d 6619 . . . . 5 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → ((((invg𝐺)‘𝐴)(+g𝐺)𝐴) 𝐵) = ((0g𝐺) 𝐵))
136, 9grpinvcl 17388 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → ((invg𝐺)‘𝐴) ∈ 𝑋)
144, 5, 13syl2anc 692 . . . . . 6 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → ((invg𝐺)‘𝐴) ∈ 𝑋)
15 simpr2 1066 . . . . . 6 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → 𝐵𝑌)
166, 7gaass 17651 . . . . . 6 (( ∈ (𝐺 GrpAct 𝑌) ∧ (((invg𝐺)‘𝐴) ∈ 𝑋𝐴𝑋𝐵𝑌)) → ((((invg𝐺)‘𝐴)(+g𝐺)𝐴) 𝐵) = (((invg𝐺)‘𝐴) (𝐴 𝐵)))
172, 14, 5, 15, 16syl13anc 1325 . . . . 5 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → ((((invg𝐺)‘𝐴)(+g𝐺)𝐴) 𝐵) = (((invg𝐺)‘𝐴) (𝐴 𝐵)))
188gagrpid 17648 . . . . . 6 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐵𝑌) → ((0g𝐺) 𝐵) = 𝐵)
192, 15, 18syl2anc 692 . . . . 5 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → ((0g𝐺) 𝐵) = 𝐵)
2012, 17, 193eqtr3d 2663 . . . 4 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → (((invg𝐺)‘𝐴) (𝐴 𝐵)) = 𝐵)
2111oveq1d 6619 . . . . 5 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → ((((invg𝐺)‘𝐴)(+g𝐺)𝐴) 𝐶) = ((0g𝐺) 𝐶))
22 simpr3 1067 . . . . . 6 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → 𝐶𝑌)
236, 7gaass 17651 . . . . . 6 (( ∈ (𝐺 GrpAct 𝑌) ∧ (((invg𝐺)‘𝐴) ∈ 𝑋𝐴𝑋𝐶𝑌)) → ((((invg𝐺)‘𝐴)(+g𝐺)𝐴) 𝐶) = (((invg𝐺)‘𝐴) (𝐴 𝐶)))
242, 14, 5, 22, 23syl13anc 1325 . . . . 5 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → ((((invg𝐺)‘𝐴)(+g𝐺)𝐴) 𝐶) = (((invg𝐺)‘𝐴) (𝐴 𝐶)))
258gagrpid 17648 . . . . . 6 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐶𝑌) → ((0g𝐺) 𝐶) = 𝐶)
262, 22, 25syl2anc 692 . . . . 5 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → ((0g𝐺) 𝐶) = 𝐶)
2721, 24, 263eqtr3d 2663 . . . 4 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → (((invg𝐺)‘𝐴) (𝐴 𝐶)) = 𝐶)
2820, 27eqeq12d 2636 . . 3 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → ((((invg𝐺)‘𝐴) (𝐴 𝐵)) = (((invg𝐺)‘𝐴) (𝐴 𝐶)) ↔ 𝐵 = 𝐶))
291, 28syl5ib 234 . 2 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → ((𝐴 𝐵) = (𝐴 𝐶) → 𝐵 = 𝐶))
30 oveq2 6612 . 2 (𝐵 = 𝐶 → (𝐴 𝐵) = (𝐴 𝐶))
3129, 30impbid1 215 1 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → ((𝐴 𝐵) = (𝐴 𝐶) ↔ 𝐵 = 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  cfv 5847  (class class class)co 6604  Basecbs 15781  +gcplusg 15862  0gc0g 16021  Grpcgrp 17343  invgcminusg 17344   GrpAct cga 17643
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-map 7804  df-0g 16023  df-mgm 17163  df-sgrp 17205  df-mnd 17216  df-grp 17346  df-minusg 17347  df-ga 17644
This theorem is referenced by:  gacan  17659
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