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Theorem galcan 17783
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 6698 . . 3 ((𝐴 𝐵) = (𝐴 𝐶) → (((invg𝐺)‘𝐴) (𝐴 𝐵)) = (((invg𝐺)‘𝐴) (𝐴 𝐶)))
2 simpl 472 . . . . . . . 8 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → ∈ (𝐺 GrpAct 𝑌))
3 gagrp 17771 . . . . . . . 8 ( ∈ (𝐺 GrpAct 𝑌) → 𝐺 ∈ Grp)
42, 3syl 17 . . . . . . 7 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → 𝐺 ∈ Grp)
5 simpr1 1087 . . . . . . 7 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → 𝐴𝑋)
6 galcan.1 . . . . . . . 8 𝑋 = (Base‘𝐺)
7 eqid 2651 . . . . . . . 8 (+g𝐺) = (+g𝐺)
8 eqid 2651 . . . . . . . 8 (0g𝐺) = (0g𝐺)
9 eqid 2651 . . . . . . . 8 (invg𝐺) = (invg𝐺)
106, 7, 8, 9grplinv 17515 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (((invg𝐺)‘𝐴)(+g𝐺)𝐴) = (0g𝐺))
114, 5, 10syl2anc 694 . . . . . 6 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → (((invg𝐺)‘𝐴)(+g𝐺)𝐴) = (0g𝐺))
1211oveq1d 6705 . . . . 5 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → ((((invg𝐺)‘𝐴)(+g𝐺)𝐴) 𝐵) = ((0g𝐺) 𝐵))
136, 9grpinvcl 17514 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → ((invg𝐺)‘𝐴) ∈ 𝑋)
144, 5, 13syl2anc 694 . . . . . 6 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → ((invg𝐺)‘𝐴) ∈ 𝑋)
15 simpr2 1088 . . . . . 6 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → 𝐵𝑌)
166, 7gaass 17776 . . . . . 6 (( ∈ (𝐺 GrpAct 𝑌) ∧ (((invg𝐺)‘𝐴) ∈ 𝑋𝐴𝑋𝐵𝑌)) → ((((invg𝐺)‘𝐴)(+g𝐺)𝐴) 𝐵) = (((invg𝐺)‘𝐴) (𝐴 𝐵)))
172, 14, 5, 15, 16syl13anc 1368 . . . . 5 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → ((((invg𝐺)‘𝐴)(+g𝐺)𝐴) 𝐵) = (((invg𝐺)‘𝐴) (𝐴 𝐵)))
188gagrpid 17773 . . . . . 6 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐵𝑌) → ((0g𝐺) 𝐵) = 𝐵)
192, 15, 18syl2anc 694 . . . . 5 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → ((0g𝐺) 𝐵) = 𝐵)
2012, 17, 193eqtr3d 2693 . . . 4 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → (((invg𝐺)‘𝐴) (𝐴 𝐵)) = 𝐵)
2111oveq1d 6705 . . . . 5 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → ((((invg𝐺)‘𝐴)(+g𝐺)𝐴) 𝐶) = ((0g𝐺) 𝐶))
22 simpr3 1089 . . . . . 6 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → 𝐶𝑌)
236, 7gaass 17776 . . . . . 6 (( ∈ (𝐺 GrpAct 𝑌) ∧ (((invg𝐺)‘𝐴) ∈ 𝑋𝐴𝑋𝐶𝑌)) → ((((invg𝐺)‘𝐴)(+g𝐺)𝐴) 𝐶) = (((invg𝐺)‘𝐴) (𝐴 𝐶)))
242, 14, 5, 22, 23syl13anc 1368 . . . . 5 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → ((((invg𝐺)‘𝐴)(+g𝐺)𝐴) 𝐶) = (((invg𝐺)‘𝐴) (𝐴 𝐶)))
258gagrpid 17773 . . . . . 6 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐶𝑌) → ((0g𝐺) 𝐶) = 𝐶)
262, 22, 25syl2anc 694 . . . . 5 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → ((0g𝐺) 𝐶) = 𝐶)
2721, 24, 263eqtr3d 2693 . . . 4 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → (((invg𝐺)‘𝐴) (𝐴 𝐶)) = 𝐶)
2820, 27eqeq12d 2666 . . 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 6698 . 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 383  w3a 1054   = wceq 1523  wcel 2030  cfv 5926  (class class class)co 6690  Basecbs 15904  +gcplusg 15988  0gc0g 16147  Grpcgrp 17469  invgcminusg 17470   GrpAct cga 17768
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-map 7901  df-0g 16149  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-grp 17472  df-minusg 17473  df-ga 17769
This theorem is referenced by:  gacan  17784
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