Theorem gafo 19265

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 19264	. 2 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → ⊕ :(𝑋 × 𝑌)⟶𝑌)
3		gagrp 19261	. . . . . 6 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → 𝐺 ∈ Grp)
4	3	adantr 480	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑥 ∈ 𝑌) → 𝐺 ∈ Grp)
5		eqid 2737	. . . . . 6 ⊢ (0g‘𝐺) = (0g‘𝐺)
6	1, 5	grpidcl 18935	. . . . 5 ⊢ (𝐺 ∈ Grp → (0g‘𝐺) ∈ 𝑋)
7	4, 6	syl 17	. . . 4 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑥 ∈ 𝑌) → (0g‘𝐺) ∈ 𝑋)
8		simpr 484	. . . 4 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑥 ∈ 𝑌) → 𝑥 ∈ 𝑌)
9	5	gagrpid 19263	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑥 ∈ 𝑌) → ((0g‘𝐺) ⊕ 𝑥) = 𝑥)
10	9	eqcomd 2743	. . . 4 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑥 ∈ 𝑌) → 𝑥 = ((0g‘𝐺) ⊕ 𝑥))
11		rspceov 7410	. . . 4 ⊢ (((0g‘𝐺) ∈ 𝑋 ∧ 𝑥 ∈ 𝑌 ∧ 𝑥 = ((0g‘𝐺) ⊕ 𝑥)) → ∃𝑦 ∈ 𝑋 ∃𝑧 ∈ 𝑌 𝑥 = (𝑦 ⊕ 𝑧))
12	7, 8, 10, 11	syl3anc 1374	. . 3 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑥 ∈ 𝑌) → ∃𝑦 ∈ 𝑋 ∃𝑧 ∈ 𝑌 𝑥 = (𝑦 ⊕ 𝑧))
13	12	ralrimiva 3130	. 2 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → ∀𝑥 ∈ 𝑌 ∃𝑦 ∈ 𝑋 ∃𝑧 ∈ 𝑌 𝑥 = (𝑦 ⊕ 𝑧))
14		foov 7535	. 2 ⊢ ( ⊕ :(𝑋 × 𝑌)–onto→𝑌 ↔ ( ⊕ :(𝑋 × 𝑌)⟶𝑌 ∧ ∀𝑥 ∈ 𝑌 ∃𝑦 ∈ 𝑋 ∃𝑧 ∈ 𝑌 𝑥 = (𝑦 ⊕ 𝑧)))
15	2, 13, 14	sylanbrc 584	1 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → ⊕ :(𝑋 × 𝑌)–onto→𝑌)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062   × cxp 5623  ⟶wf 6489  –onto→wfo 6491  ‘cfv 6493  (class class class)co 7361  Basecbs 17173  0_gc0g 17396  Grpcgrp 18903   GrpAct cga 19258

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-fo 6499  df-fv 6501  df-riota 7318  df-ov 7364  df-oprab 7365  df-mpo 7366  df-map 8769  df-0g 17398  df-mgm 18602  df-sgrp 18681  df-mnd 18697  df-grp 18906  df-ga 19259

This theorem is referenced by: (None)