Theorem gapm 19346

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 19335	. . . 4 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → ⊕ :(𝑋 × 𝑌)⟶𝑌)
4	3	ad2antrr 736	. . 3 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝑥 ∈ 𝑌) → ⊕ :(𝑋 × 𝑌)⟶𝑌)
5		simplr 778	. . 3 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝑥 ∈ 𝑌) → 𝐴 ∈ 𝑋)
6		simpr 488	. . 3 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝑥 ∈ 𝑌) → 𝑥 ∈ 𝑌)
7	4, 5, 6	fovcdmd 7568	. 2 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝑥 ∈ 𝑌) → (𝐴 ⊕ 𝑥) ∈ 𝑌)
8	3	ad2antrr 736	. . 3 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → ⊕ :(𝑋 × 𝑌)⟶𝑌)
9		gagrp 19332	. . . . 5 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → 𝐺 ∈ Grp)
10	9	ad2antrr 736	. . . 4 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → 𝐺 ∈ Grp)
11		simplr 778	. . . 4 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → 𝐴 ∈ 𝑋)
12		eqid 2762	. . . . 5 ⊢ (invg‘𝐺) = (invg‘𝐺)
13	2, 12	grpinvcl 19029	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ((invg‘𝐺)‘𝐴) ∈ 𝑋)
14	10, 11, 13	syl2anc 593	. . 3 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → ((invg‘𝐺)‘𝐴) ∈ 𝑋)
15		simpr 488	. . 3 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → 𝑦 ∈ 𝑌)
16	8, 14, 15	fovcdmd 7568	. 2 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → (((invg‘𝐺)‘𝐴) ⊕ 𝑦) ∈ 𝑌)
17		simpll 776	. . . . 5 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → ⊕ ∈ (𝐺 GrpAct 𝑌))
18		simplr 778	. . . . 5 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → 𝐴 ∈ 𝑋)
19		simprl 780	. . . . 5 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → 𝑥 ∈ 𝑌)
20		simprr 782	. . . . 5 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → 𝑦 ∈ 𝑌)
21	2, 12	gacan 19345	. . . . 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 1391	. . . 4 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → ((𝐴 ⊕ 𝑥) = 𝑦 ↔ (((invg‘𝐺)‘𝐴) ⊕ 𝑦) = 𝑥))
23	22	bicomd 225	. . 3 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → ((((invg‘𝐺)‘𝐴) ⊕ 𝑦) = 𝑥 ↔ (𝐴 ⊕ 𝑥) = 𝑦))
24		eqcom 2769	. . 3 ⊢ (𝑥 = (((invg‘𝐺)‘𝐴) ⊕ 𝑦) ↔ (((invg‘𝐺)‘𝐴) ⊕ 𝑦) = 𝑥)
25		eqcom 2769	. . 3 ⊢ (𝑦 = (𝐴 ⊕ 𝑥) ↔ (𝐴 ⊕ 𝑥) = 𝑦)
26	23, 24, 25	3bitr4g 316	. 2 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → (𝑥 = (((invg‘𝐺)‘𝐴) ⊕ 𝑦) ↔ 𝑦 = (𝐴 ⊕ 𝑥)))
27	1, 7, 16, 26	f1o2d 7650	1 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑋) → 𝐹:𝑌–1-1-onto→𝑌)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1560   ∈ wcel 2142   ↦ cmpt 5181   × cxp 5645  ⟶wf 6517  –1-1-onto→wf1o 6520  ‘cfv 6521  (class class class)co 7396  Basecbs 17245  Grpcgrp 18975  inv_gcminusg 18976   GrpAct cga 19329

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-riota 7353  df-ov 7399  df-oprab 7400  df-mpo 7401  df-map 8810  df-0g 17470  df-mgm 18674  df-sgrp 18753  df-mnd 18769  df-grp 18978  df-minusg 18979  df-ga 19330

This theorem is referenced by:  galactghm  19444