Theorem gapm 19532

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 19521	. . . 4 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → ⊕ :(𝑋 × 𝑌)⟶𝑌)
4	3	ad2antrr 739	. . 3 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝑥 ∈ 𝑌) → ⊕ :(𝑋 × 𝑌)⟶𝑌)
5		simplr 781	. . 3 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝑥 ∈ 𝑌) → 𝐴 ∈ 𝑋)
6		simpr 490	. . 3 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝑥 ∈ 𝑌) → 𝑥 ∈ 𝑌)
7	4, 5, 6	fovcdmd 7600	. 2 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝑥 ∈ 𝑌) → (𝐴 ⊕ 𝑥) ∈ 𝑌)
8	3	ad2antrr 739	. . 3 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → ⊕ :(𝑋 × 𝑌)⟶𝑌)
9		gagrp 19518	. . . . 5 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → 𝐺 ∈ Grp)
10	9	ad2antrr 739	. . . 4 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → 𝐺 ∈ Grp)
11		simplr 781	. . . 4 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → 𝐴 ∈ 𝑋)
12		eqid 2763	. . . . 5 ⊢ (invg‘𝐺) = (invg‘𝐺)
13	2, 12	grpinvcl 19210	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ((invg‘𝐺)‘𝐴) ∈ 𝑋)
14	10, 11, 13	syl2anc 596	. . 3 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → ((invg‘𝐺)‘𝐴) ∈ 𝑋)
15		simpr 490	. . 3 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → 𝑦 ∈ 𝑌)
16	8, 14, 15	fovcdmd 7600	. 2 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → (((invg‘𝐺)‘𝐴) ⊕ 𝑦) ∈ 𝑌)
17		simpll 779	. . . . 5 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → ⊕ ∈ (𝐺 GrpAct 𝑌))
18		simplr 781	. . . . 5 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → 𝐴 ∈ 𝑋)
19		simprl 783	. . . . 5 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → 𝑥 ∈ 𝑌)
20		simprr 785	. . . . 5 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → 𝑦 ∈ 𝑌)
21	2, 12	gacan 19531	. . . . 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 1399	. . . 4 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → ((𝐴 ⊕ 𝑥) = 𝑦 ↔ (((invg‘𝐺)‘𝐴) ⊕ 𝑦) = 𝑥))
23	22	bicomd 226	. . 3 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → ((((invg‘𝐺)‘𝐴) ⊕ 𝑦) = 𝑥 ↔ (𝐴 ⊕ 𝑥) = 𝑦))
24		eqcom 2770	. . 3 ⊢ (𝑥 = (((invg‘𝐺)‘𝐴) ⊕ 𝑦) ↔ (((invg‘𝐺)‘𝐴) ⊕ 𝑦) = 𝑥)
25		eqcom 2770	. . 3 ⊢ (𝑦 = (𝐴 ⊕ 𝑥) ↔ (𝐴 ⊕ 𝑥) = 𝑦)
26	23, 24, 25	3bitr4g 317	. 2 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → (𝑥 = (((invg‘𝐺)‘𝐴) ⊕ 𝑦) ↔ 𝑦 = (𝐴 ⊕ 𝑥)))
27	1, 7, 16, 26	f1o2d 7682	1 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑋) → 𝐹:𝑌–1-1-onto→𝑌)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2146   ↦ cmpt 5188   × cxp 5652  ⟶wf 6542  –1-1-onto→wf1o 6545  ‘cfv 6546  (class class class)co 7427  Basecbs 17399  Grpcgrp 19156  inv_gcminusg 19157   GrpAct cga 19515

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-sep 5251  ax-nul 5262  ax-pow 5330  ax-pr 5394  ax-un 7758

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3455  df-sbc 3742  df-csb 3851  df-ss 3900  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-pw 4561  df-sn 4588  df-pr 4590  df-op 4838  df-uni 4870  df-br 5106  df-opab 5170  df-mpt 5189  df-id 5549  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6500  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-riota 7384  df-ov 7430  df-oprab 7431  df-mpo 7432  df-map 8860  df-0g 17624  df-mgm 18828  df-sgrp 18920  df-mnd 18936  df-grp 19159  df-minusg 19160  df-ga 19516

This theorem is used by:  galactghm  19630