Theorem gapm 19571

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⟶1-1onto𝑌)

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 19560	. . . 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 7625	. 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 19557	. . . . 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 19248	. . . 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 7625	. 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 19570	. . . . 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 1400	. . . 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 7707	1 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑋) → 𝐹:𝑌⟶1-1onto𝑌)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147   ↦ cmpt 5192   × cxp 5656  ⟶wf 6562  ⟶1-1ontowf1o 6565  ‘cfv 6566  (class class class)co 7451  Basecbs 17431  Grpcgrp 19194  inv_gcminusg 19195   GrpAct cga 19554

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-map 8887  df-0g 17662  df-mgm 18866  df-sgrp 18958  df-mnd 18974  df-grp 19197  df-minusg 19198  df-ga 19555

This theorem is used by:  galactghm  19669