Theorem gafo 19424

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 19423	. 2 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → ⊕ :(𝑋 × 𝑌)⟶𝑌)
3		gagrp 19420	. . . . . 6 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → 𝐺 ∈ Grp)
4	3	adantr 486	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑥 ∈ 𝑌) → 𝐺 ∈ Grp)
5		eqid 2760	. . . . . 6 ⊢ (0g‘𝐺) = (0g‘𝐺)
6	1, 5	grpidcl 19090	. . . . 5 ⊢ (𝐺 ∈ Grp → (0g‘𝐺) ∈ 𝑋)
7	4, 6	syl 18	. . . 4 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑥 ∈ 𝑌) → (0g‘𝐺) ∈ 𝑋)
8		simpr 490	. . . 4 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑥 ∈ 𝑌) → 𝑥 ∈ 𝑌)
9	5	gagrpid 19422	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑥 ∈ 𝑌) → ((0g‘𝐺) ⊕ 𝑥) = 𝑥)
10	9	eqcomd 2766	. . . 4 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑥 ∈ 𝑌) → 𝑥 = ((0g‘𝐺) ⊕ 𝑥))
11		rspceov 7463	. . . 4 ⊢ (((0g‘𝐺) ∈ 𝑋 ∧ 𝑥 ∈ 𝑌 ∧ 𝑥 = ((0g‘𝐺) ⊕ 𝑥)) → ∃𝑦 ∈ 𝑋 ∃𝑧 ∈ 𝑌 𝑥 = (𝑦 ⊕ 𝑧))
12	7, 8, 10, 11	syl3anc 1398	. . 3 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑥 ∈ 𝑌) → ∃𝑦 ∈ 𝑋 ∃𝑧 ∈ 𝑌 𝑥 = (𝑦 ⊕ 𝑧))
13	12	ralrimiva 3154	. 2 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → ∀𝑥 ∈ 𝑌 ∃𝑦 ∈ 𝑋 ∃𝑧 ∈ 𝑌 𝑥 = (𝑦 ⊕ 𝑧))
14		foov 7589	. 2 ⊢ ( ⊕ :(𝑋 × 𝑌)–onto→𝑌 ↔ ( ⊕ :(𝑋 × 𝑌)⟶𝑌 ∧ ∀𝑥 ∈ 𝑌 ∃𝑦 ∈ 𝑋 ∃𝑧 ∈ 𝑌 𝑥 = (𝑦 ⊕ 𝑧)))
15	2, 13, 14	sylanbrc 595	1 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → ⊕ :(𝑋 × 𝑌)–onto→𝑌)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3076  ∃wrex 3086   × cxp 5653  ⟶wf 6529  –onto→wfo 6531  ‘cfv 6533  (class class class)co 7414  Basecbs 17302  0_gc0g 17525  Grpcgrp 19058   GrpAct cga 19417

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7737

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 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-fo 6539  df-fv 6541  df-riota 7371  df-ov 7417  df-oprab 7418  df-mpo 7419  df-map 8829  df-0g 17527  df-mgm 18731  df-sgrp 18822  df-mnd 18838  df-grp 19061  df-ga 19418

This theorem is used by: (None)