Theorem gafo 17552

Description: A group action is onto its base set. (Contributed by Jeff Hankins, 10-Aug-2009.) (Revised by Mario Carneiro, 13-Jan-2015.)

Hypothesis

Ref	Expression
gaf.1	⊢ 𝑋 = (Base‘𝐺)

Assertion

Ref	Expression
gafo	⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → ⊕ :(𝑋 × 𝑌)–onto→𝑌)

Proof of Theorem gafo

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		gaf.1	. . 3 ⊢ 𝑋 = (Base‘𝐺)
2	1	gaf 17551	. 2 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → ⊕ :(𝑋 × 𝑌)⟶𝑌)
3		gagrp 17548	. . . . . 6 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → 𝐺 ∈ Grp)
4	3	adantr 480	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑥 ∈ 𝑌) → 𝐺 ∈ Grp)
5		eqid 2610	. . . . . 6 ⊢ (0g‘𝐺) = (0g‘𝐺)
6	1, 5	grpidcl 17273	. . . . 5 ⊢ (𝐺 ∈ Grp → (0g‘𝐺) ∈ 𝑋)
7	4, 6	syl 17	. . . 4 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑥 ∈ 𝑌) → (0g‘𝐺) ∈ 𝑋)
8		simpr 476	. . . 4 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑥 ∈ 𝑌) → 𝑥 ∈ 𝑌)
9	5	gagrpid 17550	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑥 ∈ 𝑌) → ((0g‘𝐺) ⊕ 𝑥) = 𝑥)
10	9	eqcomd 2616	. . . 4 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑥 ∈ 𝑌) → 𝑥 = ((0g‘𝐺) ⊕ 𝑥))
11		rspceov 6590	. . . 4 ⊢ (((0g‘𝐺) ∈ 𝑋 ∧ 𝑥 ∈ 𝑌 ∧ 𝑥 = ((0g‘𝐺) ⊕ 𝑥)) → ∃𝑦 ∈ 𝑋 ∃𝑧 ∈ 𝑌 𝑥 = (𝑦 ⊕ 𝑧))
12	7, 8, 10, 11	syl3anc 1318	. . 3 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑥 ∈ 𝑌) → ∃𝑦 ∈ 𝑋 ∃𝑧 ∈ 𝑌 𝑥 = (𝑦 ⊕ 𝑧))
13	12	ralrimiva 2949	. 2 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → ∀𝑥 ∈ 𝑌 ∃𝑦 ∈ 𝑋 ∃𝑧 ∈ 𝑌 𝑥 = (𝑦 ⊕ 𝑧))
14		foov 6706	. 2 ⊢ ( ⊕ :(𝑋 × 𝑌)–onto→𝑌 ↔ ( ⊕ :(𝑋 × 𝑌)⟶𝑌 ∧ ∀𝑥 ∈ 𝑌 ∃𝑦 ∈ 𝑋 ∃𝑧 ∈ 𝑌 𝑥 = (𝑦 ⊕ 𝑧)))
15	2, 13, 14	sylanbrc 695	1 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → ⊕ :(𝑋 × 𝑌)–onto→𝑌)

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897   × cxp 5036  ⟶wf 5800  –onto→wfo 5802  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  0_gc0g 15923  Grpcgrp 17245   GrpAct cga 17545

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fo 5810  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-map 7746  df-0g 15925  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-grp 17248  df-ga 17546

This theorem is referenced by: (None)