Theorem gapm 19324

Description: The action of a particular group element is a permutation of the base set. (Contributed by Jeff Hankins, 11-Aug-2009.) (Proof shortened by Mario Carneiro, 13-Jan-2015.)

Hypotheses

Ref	Expression
gapm.1	⊢ 𝑋 = (Base‘𝐺)
gapm.2	⊢ 𝐹 = (𝑥 ∈ 𝑌 ↦ (𝐴 ⊕ 𝑥))

Assertion

Ref	Expression
gapm	⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑋) → 𝐹:𝑌–1-1-onto→𝑌)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐺   𝑥, ⊕   𝑥,𝑋   𝑥,𝑌

Allowed substitution hint:   𝐹(𝑥)

Proof of Theorem gapm

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		gapm.2	. 2 ⊢ 𝐹 = (𝑥 ∈ 𝑌 ↦ (𝐴 ⊕ 𝑥))
2		gapm.1	. . . . 5 ⊢ 𝑋 = (Base‘𝐺)
3	2	gaf 19313	. . . 4 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → ⊕ :(𝑋 × 𝑌)⟶𝑌)
4	3	ad2antrr 726	. . 3 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝑥 ∈ 𝑌) → ⊕ :(𝑋 × 𝑌)⟶𝑌)
5		simplr 769	. . 3 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝑥 ∈ 𝑌) → 𝐴 ∈ 𝑋)
6		simpr 484	. . 3 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝑥 ∈ 𝑌) → 𝑥 ∈ 𝑌)
7	4, 5, 6	fovcdmd 7605	. 2 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝑥 ∈ 𝑌) → (𝐴 ⊕ 𝑥) ∈ 𝑌)
8	3	ad2antrr 726	. . 3 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → ⊕ :(𝑋 × 𝑌)⟶𝑌)
9		gagrp 19310	. . . . 5 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → 𝐺 ∈ Grp)
10	9	ad2antrr 726	. . . 4 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → 𝐺 ∈ Grp)
11		simplr 769	. . . 4 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → 𝐴 ∈ 𝑋)
12		eqid 2737	. . . . 5 ⊢ (invg‘𝐺) = (invg‘𝐺)
13	2, 12	grpinvcl 19005	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ((invg‘𝐺)‘𝐴) ∈ 𝑋)
14	10, 11, 13	syl2anc 584	. . 3 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → ((invg‘𝐺)‘𝐴) ∈ 𝑋)
15		simpr 484	. . 3 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → 𝑦 ∈ 𝑌)
16	8, 14, 15	fovcdmd 7605	. 2 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → (((invg‘𝐺)‘𝐴) ⊕ 𝑦) ∈ 𝑌)
17		simpll 767	. . . . 5 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → ⊕ ∈ (𝐺 GrpAct 𝑌))
18		simplr 769	. . . . 5 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → 𝐴 ∈ 𝑋)
19		simprl 771	. . . . 5 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → 𝑥 ∈ 𝑌)
20		simprr 773	. . . . 5 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → 𝑦 ∈ 𝑌)
21	2, 12	gacan 19323	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → ((𝐴 ⊕ 𝑥) = 𝑦 ↔ (((invg‘𝐺)‘𝐴) ⊕ 𝑦) = 𝑥))
22	17, 18, 19, 20, 21	syl13anc 1374	. . . 4 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → ((𝐴 ⊕ 𝑥) = 𝑦 ↔ (((invg‘𝐺)‘𝐴) ⊕ 𝑦) = 𝑥))
23	22	bicomd 223	. . 3 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → ((((invg‘𝐺)‘𝐴) ⊕ 𝑦) = 𝑥 ↔ (𝐴 ⊕ 𝑥) = 𝑦))
24		eqcom 2744	. . 3 ⊢ (𝑥 = (((invg‘𝐺)‘𝐴) ⊕ 𝑦) ↔ (((invg‘𝐺)‘𝐴) ⊕ 𝑦) = 𝑥)
25		eqcom 2744	. . 3 ⊢ (𝑦 = (𝐴 ⊕ 𝑥) ↔ (𝐴 ⊕ 𝑥) = 𝑦)
26	23, 24, 25	3bitr4g 314	. 2 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → (𝑥 = (((invg‘𝐺)‘𝐴) ⊕ 𝑦) ↔ 𝑦 = (𝐴 ⊕ 𝑥)))
27	1, 7, 16, 26	f1o2d 7687	1 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑋) → 𝐹:𝑌–1-1-onto→𝑌)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108   ↦ cmpt 5225   × cxp 5683  ⟶wf 6557  –1-1-onto→wf1o 6560  ‘cfv 6561  (class class class)co 7431  Basecbs 17247  Grpcgrp 18951  inv_gcminusg 18952   GrpAct cga 19307

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8868  df-0g 17486  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-grp 18954  df-minusg 18955  df-ga 19308

This theorem is referenced by:  galactghm  19422