Theorem gafo 19338

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 19337	. 2 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → ⊕ :(𝑋 × 𝑌)⟶𝑌)
3		gagrp 19334	. . . . . 6 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → 𝐺 ∈ Grp)
4	3	adantr 480	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑥 ∈ 𝑌) → 𝐺 ∈ Grp)
5		eqid 2740	. . . . . 6 ⊢ (0g‘𝐺) = (0g‘𝐺)
6	1, 5	grpidcl 19007	. . . . 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 19336	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑥 ∈ 𝑌) → ((0g‘𝐺) ⊕ 𝑥) = 𝑥)
10	9	eqcomd 2746	. . . 4 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑥 ∈ 𝑌) → 𝑥 = ((0g‘𝐺) ⊕ 𝑥))
11		rspceov 7499	. . . 4 ⊢ (((0g‘𝐺) ∈ 𝑋 ∧ 𝑥 ∈ 𝑌 ∧ 𝑥 = ((0g‘𝐺) ⊕ 𝑥)) → ∃𝑦 ∈ 𝑋 ∃𝑧 ∈ 𝑌 𝑥 = (𝑦 ⊕ 𝑧))
12	7, 8, 10, 11	syl3anc 1371	. . 3 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑥 ∈ 𝑌) → ∃𝑦 ∈ 𝑋 ∃𝑧 ∈ 𝑌 𝑥 = (𝑦 ⊕ 𝑧))
13	12	ralrimiva 3152	. 2 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → ∀𝑥 ∈ 𝑌 ∃𝑦 ∈ 𝑋 ∃𝑧 ∈ 𝑌 𝑥 = (𝑦 ⊕ 𝑧))
14		foov 7626	. 2 ⊢ ( ⊕ :(𝑋 × 𝑌)–onto→𝑌 ↔ ( ⊕ :(𝑋 × 𝑌)⟶𝑌 ∧ ∀𝑥 ∈ 𝑌 ∃𝑦 ∈ 𝑋 ∃𝑧 ∈ 𝑌 𝑥 = (𝑦 ⊕ 𝑧)))
15	2, 13, 14	sylanbrc 582	1 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → ⊕ :(𝑋 × 𝑌)–onto→𝑌)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ∀wral 3067  ∃wrex 3076   × cxp 5698  ⟶wf 6571  –onto→wfo 6573  ‘cfv 6575  (class class class)co 7450  Basecbs 17260  0_gc0g 17501  Grpcgrp 18975   GrpAct cga 19331

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7772

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-iota 6527  df-fun 6577  df-fn 6578  df-f 6579  df-fo 6581  df-fv 6583  df-riota 7406  df-ov 7453  df-oprab 7454  df-mpo 7455  df-map 8888  df-0g 17503  df-mgm 18680  df-sgrp 18759  df-mnd 18775  df-grp 18978  df-ga 19332

This theorem is referenced by: (None)