Theorem gaorber 19291

Description: The orbit equivalence relation is an equivalence relation on the target set of the group action. (Contributed by NM, 11-Aug-2009.) (Revised by Mario Carneiro, 13-Jan-2015.)

Hypotheses

Ref	Expression
gaorb.1	⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑌 ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)}
gaorber.2	⊢ 𝑋 = (Base‘𝐺)

Assertion

Ref	Expression
gaorber	⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → ∼ Er 𝑌)

Distinct variable groups:   𝑥,𝑔,𝑦, ⊕   𝑔,𝑋,𝑥,𝑦   𝑥,𝑌,𝑦

Allowed substitution hints:   ∼ (𝑥,𝑦,𝑔)   𝐺(𝑥,𝑦,𝑔)   𝑌(𝑔)

Proof of Theorem gaorber

Dummy variables ℎ 𝑓 𝑘 𝑢 𝑣 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		gaorb.1	. . . 4 ⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑌 ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)}
2	1	relopabiv 5799	. . 3 ⊢ Rel ∼
3	2	a1i 11	. 2 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → Rel ∼ )
4		simpr 484	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑢 ∼ 𝑣) → 𝑢 ∼ 𝑣)
5	1	gaorb 19290	. . . . 5 ⊢ (𝑢 ∼ 𝑣 ↔ (𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌 ∧ ∃ℎ ∈ 𝑋 (ℎ ⊕ 𝑢) = 𝑣))
6	4, 5	sylib 218	. . . 4 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑢 ∼ 𝑣) → (𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌 ∧ ∃ℎ ∈ 𝑋 (ℎ ⊕ 𝑢) = 𝑣))
7	6	simp2d 1143	. . 3 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑢 ∼ 𝑣) → 𝑣 ∈ 𝑌)
8	6	simp1d 1142	. . 3 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑢 ∼ 𝑣) → 𝑢 ∈ 𝑌)
9	6	simp3d 1144	. . . 4 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑢 ∼ 𝑣) → ∃ℎ ∈ 𝑋 (ℎ ⊕ 𝑢) = 𝑣)
10		simpll 766	. . . . . . 7 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑢 ∼ 𝑣) ∧ ℎ ∈ 𝑋) → ⊕ ∈ (𝐺 GrpAct 𝑌))
11		simpr 484	. . . . . . 7 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑢 ∼ 𝑣) ∧ ℎ ∈ 𝑋) → ℎ ∈ 𝑋)
12	8	adantr 480	. . . . . . 7 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑢 ∼ 𝑣) ∧ ℎ ∈ 𝑋) → 𝑢 ∈ 𝑌)
13	7	adantr 480	. . . . . . 7 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑢 ∼ 𝑣) ∧ ℎ ∈ 𝑋) → 𝑣 ∈ 𝑌)
14		gaorber.2	. . . . . . . 8 ⊢ 𝑋 = (Base‘𝐺)
15		eqid 2735	. . . . . . . 8 ⊢ (invg‘𝐺) = (invg‘𝐺)
16	14, 15	gacan 19288	. . . . . . 7 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (ℎ ∈ 𝑋 ∧ 𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌)) → ((ℎ ⊕ 𝑢) = 𝑣 ↔ (((invg‘𝐺)‘ℎ) ⊕ 𝑣) = 𝑢))
17	10, 11, 12, 13, 16	syl13anc 1374	. . . . . 6 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑢 ∼ 𝑣) ∧ ℎ ∈ 𝑋) → ((ℎ ⊕ 𝑢) = 𝑣 ↔ (((invg‘𝐺)‘ℎ) ⊕ 𝑣) = 𝑢))
18		gagrp 19275	. . . . . . . . 9 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → 𝐺 ∈ Grp)
19	18	adantr 480	. . . . . . . 8 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑢 ∼ 𝑣) → 𝐺 ∈ Grp)
20	14, 15	grpinvcl 18970	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ ℎ ∈ 𝑋) → ((invg‘𝐺)‘ℎ) ∈ 𝑋)
21	19, 20	sylan 580	. . . . . . 7 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑢 ∼ 𝑣) ∧ ℎ ∈ 𝑋) → ((invg‘𝐺)‘ℎ) ∈ 𝑋)
22		oveq1 7412	. . . . . . . . . 10 ⊢ (𝑘 = ((invg‘𝐺)‘ℎ) → (𝑘 ⊕ 𝑣) = (((invg‘𝐺)‘ℎ) ⊕ 𝑣))
23	22	eqeq1d 2737	. . . . . . . . 9 ⊢ (𝑘 = ((invg‘𝐺)‘ℎ) → ((𝑘 ⊕ 𝑣) = 𝑢 ↔ (((invg‘𝐺)‘ℎ) ⊕ 𝑣) = 𝑢))
24	23	rspcev 3601	. . . . . . . 8 ⊢ ((((invg‘𝐺)‘ℎ) ∈ 𝑋 ∧ (((invg‘𝐺)‘ℎ) ⊕ 𝑣) = 𝑢) → ∃𝑘 ∈ 𝑋 (𝑘 ⊕ 𝑣) = 𝑢)
25	24	ex 412	. . . . . . 7 ⊢ (((invg‘𝐺)‘ℎ) ∈ 𝑋 → ((((invg‘𝐺)‘ℎ) ⊕ 𝑣) = 𝑢 → ∃𝑘 ∈ 𝑋 (𝑘 ⊕ 𝑣) = 𝑢))
26	21, 25	syl 17	. . . . . 6 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑢 ∼ 𝑣) ∧ ℎ ∈ 𝑋) → ((((invg‘𝐺)‘ℎ) ⊕ 𝑣) = 𝑢 → ∃𝑘 ∈ 𝑋 (𝑘 ⊕ 𝑣) = 𝑢))
27	17, 26	sylbid 240	. . . . 5 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑢 ∼ 𝑣) ∧ ℎ ∈ 𝑋) → ((ℎ ⊕ 𝑢) = 𝑣 → ∃𝑘 ∈ 𝑋 (𝑘 ⊕ 𝑣) = 𝑢))
28	27	rexlimdva 3141	. . . 4 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑢 ∼ 𝑣) → (∃ℎ ∈ 𝑋 (ℎ ⊕ 𝑢) = 𝑣 → ∃𝑘 ∈ 𝑋 (𝑘 ⊕ 𝑣) = 𝑢))
29	9, 28	mpd 15	. . 3 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑢 ∼ 𝑣) → ∃𝑘 ∈ 𝑋 (𝑘 ⊕ 𝑣) = 𝑢)
30	1	gaorb 19290	. . 3 ⊢ (𝑣 ∼ 𝑢 ↔ (𝑣 ∈ 𝑌 ∧ 𝑢 ∈ 𝑌 ∧ ∃𝑘 ∈ 𝑋 (𝑘 ⊕ 𝑣) = 𝑢))
31	7, 8, 29, 30	syl3anbrc 1344	. 2 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑢 ∼ 𝑣) → 𝑣 ∼ 𝑢)
32	8	adantrr 717	. . 3 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢 ∼ 𝑣 ∧ 𝑣 ∼ 𝑤)) → 𝑢 ∈ 𝑌)
33		simprr 772	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢 ∼ 𝑣 ∧ 𝑣 ∼ 𝑤)) → 𝑣 ∼ 𝑤)
34	1	gaorb 19290	. . . . 5 ⊢ (𝑣 ∼ 𝑤 ↔ (𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌 ∧ ∃𝑘 ∈ 𝑋 (𝑘 ⊕ 𝑣) = 𝑤))
35	33, 34	sylib 218	. . . 4 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢 ∼ 𝑣 ∧ 𝑣 ∼ 𝑤)) → (𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌 ∧ ∃𝑘 ∈ 𝑋 (𝑘 ⊕ 𝑣) = 𝑤))
36	35	simp2d 1143	. . 3 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢 ∼ 𝑣 ∧ 𝑣 ∼ 𝑤)) → 𝑤 ∈ 𝑌)
37	9	adantrr 717	. . . 4 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢 ∼ 𝑣 ∧ 𝑣 ∼ 𝑤)) → ∃ℎ ∈ 𝑋 (ℎ ⊕ 𝑢) = 𝑣)
38	35	simp3d 1144	. . . 4 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢 ∼ 𝑣 ∧ 𝑣 ∼ 𝑤)) → ∃𝑘 ∈ 𝑋 (𝑘 ⊕ 𝑣) = 𝑤)
39		reeanv 3213	. . . . 5 ⊢ (∃ℎ ∈ 𝑋 ∃𝑘 ∈ 𝑋 ((ℎ ⊕ 𝑢) = 𝑣 ∧ (𝑘 ⊕ 𝑣) = 𝑤) ↔ (∃ℎ ∈ 𝑋 (ℎ ⊕ 𝑢) = 𝑣 ∧ ∃𝑘 ∈ 𝑋 (𝑘 ⊕ 𝑣) = 𝑤))
40	18	ad2antrr 726	. . . . . . . . 9 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢 ∼ 𝑣 ∧ 𝑣 ∼ 𝑤)) ∧ ((ℎ ∈ 𝑋 ∧ 𝑘 ∈ 𝑋) ∧ ((ℎ ⊕ 𝑢) = 𝑣 ∧ (𝑘 ⊕ 𝑣) = 𝑤))) → 𝐺 ∈ Grp)
41		simprlr 779	. . . . . . . . 9 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢 ∼ 𝑣 ∧ 𝑣 ∼ 𝑤)) ∧ ((ℎ ∈ 𝑋 ∧ 𝑘 ∈ 𝑋) ∧ ((ℎ ⊕ 𝑢) = 𝑣 ∧ (𝑘 ⊕ 𝑣) = 𝑤))) → 𝑘 ∈ 𝑋)
42		simprll 778	. . . . . . . . 9 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢 ∼ 𝑣 ∧ 𝑣 ∼ 𝑤)) ∧ ((ℎ ∈ 𝑋 ∧ 𝑘 ∈ 𝑋) ∧ ((ℎ ⊕ 𝑢) = 𝑣 ∧ (𝑘 ⊕ 𝑣) = 𝑤))) → ℎ ∈ 𝑋)
43		eqid 2735	. . . . . . . . . 10 ⊢ (+g‘𝐺) = (+g‘𝐺)
44	14, 43	grpcl 18924	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ 𝑘 ∈ 𝑋 ∧ ℎ ∈ 𝑋) → (𝑘(+g‘𝐺)ℎ) ∈ 𝑋)
45	40, 41, 42, 44	syl3anc 1373	. . . . . . . 8 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢 ∼ 𝑣 ∧ 𝑣 ∼ 𝑤)) ∧ ((ℎ ∈ 𝑋 ∧ 𝑘 ∈ 𝑋) ∧ ((ℎ ⊕ 𝑢) = 𝑣 ∧ (𝑘 ⊕ 𝑣) = 𝑤))) → (𝑘(+g‘𝐺)ℎ) ∈ 𝑋)
46		simpll 766	. . . . . . . . . 10 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢 ∼ 𝑣 ∧ 𝑣 ∼ 𝑤)) ∧ ((ℎ ∈ 𝑋 ∧ 𝑘 ∈ 𝑋) ∧ ((ℎ ⊕ 𝑢) = 𝑣 ∧ (𝑘 ⊕ 𝑣) = 𝑤))) → ⊕ ∈ (𝐺 GrpAct 𝑌))
47	32	adantr 480	. . . . . . . . . 10 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢 ∼ 𝑣 ∧ 𝑣 ∼ 𝑤)) ∧ ((ℎ ∈ 𝑋 ∧ 𝑘 ∈ 𝑋) ∧ ((ℎ ⊕ 𝑢) = 𝑣 ∧ (𝑘 ⊕ 𝑣) = 𝑤))) → 𝑢 ∈ 𝑌)
48	14, 43	gaass 19280	. . . . . . . . . 10 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑘 ∈ 𝑋 ∧ ℎ ∈ 𝑋 ∧ 𝑢 ∈ 𝑌)) → ((𝑘(+g‘𝐺)ℎ) ⊕ 𝑢) = (𝑘 ⊕ (ℎ ⊕ 𝑢)))
49	46, 41, 42, 47, 48	syl13anc 1374	. . . . . . . . 9 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢 ∼ 𝑣 ∧ 𝑣 ∼ 𝑤)) ∧ ((ℎ ∈ 𝑋 ∧ 𝑘 ∈ 𝑋) ∧ ((ℎ ⊕ 𝑢) = 𝑣 ∧ (𝑘 ⊕ 𝑣) = 𝑤))) → ((𝑘(+g‘𝐺)ℎ) ⊕ 𝑢) = (𝑘 ⊕ (ℎ ⊕ 𝑢)))
50		simprrl 780	. . . . . . . . . 10 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢 ∼ 𝑣 ∧ 𝑣 ∼ 𝑤)) ∧ ((ℎ ∈ 𝑋 ∧ 𝑘 ∈ 𝑋) ∧ ((ℎ ⊕ 𝑢) = 𝑣 ∧ (𝑘 ⊕ 𝑣) = 𝑤))) → (ℎ ⊕ 𝑢) = 𝑣)
51	50	oveq2d 7421	. . . . . . . . 9 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢 ∼ 𝑣 ∧ 𝑣 ∼ 𝑤)) ∧ ((ℎ ∈ 𝑋 ∧ 𝑘 ∈ 𝑋) ∧ ((ℎ ⊕ 𝑢) = 𝑣 ∧ (𝑘 ⊕ 𝑣) = 𝑤))) → (𝑘 ⊕ (ℎ ⊕ 𝑢)) = (𝑘 ⊕ 𝑣))
52		simprrr 781	. . . . . . . . 9 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢 ∼ 𝑣 ∧ 𝑣 ∼ 𝑤)) ∧ ((ℎ ∈ 𝑋 ∧ 𝑘 ∈ 𝑋) ∧ ((ℎ ⊕ 𝑢) = 𝑣 ∧ (𝑘 ⊕ 𝑣) = 𝑤))) → (𝑘 ⊕ 𝑣) = 𝑤)
53	49, 51, 52	3eqtrd 2774	. . . . . . . 8 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢 ∼ 𝑣 ∧ 𝑣 ∼ 𝑤)) ∧ ((ℎ ∈ 𝑋 ∧ 𝑘 ∈ 𝑋) ∧ ((ℎ ⊕ 𝑢) = 𝑣 ∧ (𝑘 ⊕ 𝑣) = 𝑤))) → ((𝑘(+g‘𝐺)ℎ) ⊕ 𝑢) = 𝑤)
54		oveq1 7412	. . . . . . . . . 10 ⊢ (𝑓 = (𝑘(+g‘𝐺)ℎ) → (𝑓 ⊕ 𝑢) = ((𝑘(+g‘𝐺)ℎ) ⊕ 𝑢))
55	54	eqeq1d 2737	. . . . . . . . 9 ⊢ (𝑓 = (𝑘(+g‘𝐺)ℎ) → ((𝑓 ⊕ 𝑢) = 𝑤 ↔ ((𝑘(+g‘𝐺)ℎ) ⊕ 𝑢) = 𝑤))
56	55	rspcev 3601	. . . . . . . 8 ⊢ (((𝑘(+g‘𝐺)ℎ) ∈ 𝑋 ∧ ((𝑘(+g‘𝐺)ℎ) ⊕ 𝑢) = 𝑤) → ∃𝑓 ∈ 𝑋 (𝑓 ⊕ 𝑢) = 𝑤)
57	45, 53, 56	syl2anc 584	. . . . . . 7 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢 ∼ 𝑣 ∧ 𝑣 ∼ 𝑤)) ∧ ((ℎ ∈ 𝑋 ∧ 𝑘 ∈ 𝑋) ∧ ((ℎ ⊕ 𝑢) = 𝑣 ∧ (𝑘 ⊕ 𝑣) = 𝑤))) → ∃𝑓 ∈ 𝑋 (𝑓 ⊕ 𝑢) = 𝑤)
58	57	expr 456	. . . . . 6 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢 ∼ 𝑣 ∧ 𝑣 ∼ 𝑤)) ∧ (ℎ ∈ 𝑋 ∧ 𝑘 ∈ 𝑋)) → (((ℎ ⊕ 𝑢) = 𝑣 ∧ (𝑘 ⊕ 𝑣) = 𝑤) → ∃𝑓 ∈ 𝑋 (𝑓 ⊕ 𝑢) = 𝑤))
59	58	rexlimdvva 3198	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢 ∼ 𝑣 ∧ 𝑣 ∼ 𝑤)) → (∃ℎ ∈ 𝑋 ∃𝑘 ∈ 𝑋 ((ℎ ⊕ 𝑢) = 𝑣 ∧ (𝑘 ⊕ 𝑣) = 𝑤) → ∃𝑓 ∈ 𝑋 (𝑓 ⊕ 𝑢) = 𝑤))
60	39, 59	biimtrrid 243	. . . 4 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢 ∼ 𝑣 ∧ 𝑣 ∼ 𝑤)) → ((∃ℎ ∈ 𝑋 (ℎ ⊕ 𝑢) = 𝑣 ∧ ∃𝑘 ∈ 𝑋 (𝑘 ⊕ 𝑣) = 𝑤) → ∃𝑓 ∈ 𝑋 (𝑓 ⊕ 𝑢) = 𝑤))
61	37, 38, 60	mp2and 699	. . 3 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢 ∼ 𝑣 ∧ 𝑣 ∼ 𝑤)) → ∃𝑓 ∈ 𝑋 (𝑓 ⊕ 𝑢) = 𝑤)
62	1	gaorb 19290	. . 3 ⊢ (𝑢 ∼ 𝑤 ↔ (𝑢 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌 ∧ ∃𝑓 ∈ 𝑋 (𝑓 ⊕ 𝑢) = 𝑤))
63	32, 36, 61, 62	syl3anbrc 1344	. 2 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢 ∼ 𝑣 ∧ 𝑣 ∼ 𝑤)) → 𝑢 ∼ 𝑤)
64	18	adantr 480	. . . . . . . 8 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑢 ∈ 𝑌) → 𝐺 ∈ Grp)
65		eqid 2735	. . . . . . . . 9 ⊢ (0g‘𝐺) = (0g‘𝐺)
66	14, 65	grpidcl 18948	. . . . . . . 8 ⊢ (𝐺 ∈ Grp → (0g‘𝐺) ∈ 𝑋)
67	64, 66	syl 17	. . . . . . 7 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑢 ∈ 𝑌) → (0g‘𝐺) ∈ 𝑋)
68	65	gagrpid 19277	. . . . . . 7 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑢 ∈ 𝑌) → ((0g‘𝐺) ⊕ 𝑢) = 𝑢)
69		oveq1 7412	. . . . . . . . 9 ⊢ (ℎ = (0g‘𝐺) → (ℎ ⊕ 𝑢) = ((0g‘𝐺) ⊕ 𝑢))
70	69	eqeq1d 2737	. . . . . . . 8 ⊢ (ℎ = (0g‘𝐺) → ((ℎ ⊕ 𝑢) = 𝑢 ↔ ((0g‘𝐺) ⊕ 𝑢) = 𝑢))
71	70	rspcev 3601	. . . . . . 7 ⊢ (((0g‘𝐺) ∈ 𝑋 ∧ ((0g‘𝐺) ⊕ 𝑢) = 𝑢) → ∃ℎ ∈ 𝑋 (ℎ ⊕ 𝑢) = 𝑢)
72	67, 68, 71	syl2anc 584	. . . . . 6 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑢 ∈ 𝑌) → ∃ℎ ∈ 𝑋 (ℎ ⊕ 𝑢) = 𝑢)
73	72	ex 412	. . . . 5 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → (𝑢 ∈ 𝑌 → ∃ℎ ∈ 𝑋 (ℎ ⊕ 𝑢) = 𝑢))
74	73	pm4.71rd 562	. . . 4 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → (𝑢 ∈ 𝑌 ↔ (∃ℎ ∈ 𝑋 (ℎ ⊕ 𝑢) = 𝑢 ∧ 𝑢 ∈ 𝑌)))
75		df-3an 1088	. . . . 5 ⊢ ((𝑢 ∈ 𝑌 ∧ 𝑢 ∈ 𝑌 ∧ ∃ℎ ∈ 𝑋 (ℎ ⊕ 𝑢) = 𝑢) ↔ ((𝑢 ∈ 𝑌 ∧ 𝑢 ∈ 𝑌) ∧ ∃ℎ ∈ 𝑋 (ℎ ⊕ 𝑢) = 𝑢))
76		anidm 564	. . . . . 6 ⊢ ((𝑢 ∈ 𝑌 ∧ 𝑢 ∈ 𝑌) ↔ 𝑢 ∈ 𝑌)
77	76	anbi2ci 625	. . . . 5 ⊢ (((𝑢 ∈ 𝑌 ∧ 𝑢 ∈ 𝑌) ∧ ∃ℎ ∈ 𝑋 (ℎ ⊕ 𝑢) = 𝑢) ↔ (∃ℎ ∈ 𝑋 (ℎ ⊕ 𝑢) = 𝑢 ∧ 𝑢 ∈ 𝑌))
78	75, 77	bitri 275	. . . 4 ⊢ ((𝑢 ∈ 𝑌 ∧ 𝑢 ∈ 𝑌 ∧ ∃ℎ ∈ 𝑋 (ℎ ⊕ 𝑢) = 𝑢) ↔ (∃ℎ ∈ 𝑋 (ℎ ⊕ 𝑢) = 𝑢 ∧ 𝑢 ∈ 𝑌))
79	74, 78	bitr4di 289	. . 3 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → (𝑢 ∈ 𝑌 ↔ (𝑢 ∈ 𝑌 ∧ 𝑢 ∈ 𝑌 ∧ ∃ℎ ∈ 𝑋 (ℎ ⊕ 𝑢) = 𝑢)))
80	1	gaorb 19290	. . 3 ⊢ (𝑢 ∼ 𝑢 ↔ (𝑢 ∈ 𝑌 ∧ 𝑢 ∈ 𝑌 ∧ ∃ℎ ∈ 𝑋 (ℎ ⊕ 𝑢) = 𝑢))
81	79, 80	bitr4di 289	. 2 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → (𝑢 ∈ 𝑌 ↔ 𝑢 ∼ 𝑢))
82	3, 31, 63, 81	iserd 8745	1 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → ∼ Er 𝑌)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2108  ∃wrex 3060   ⊆ wss 3926  {cpr 4603   class class class wbr 5119  {copab 5181  Rel wrel 5659  ‘cfv 6531  (class class class)co 7405   Er wer 8716  Basecbs 17228  +_gcplusg 17271  0_gc0g 17453  Grpcgrp 18916  inv_gcminusg 18917   GrpAct cga 19272

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-fv 6539  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-er 8719  df-map 8842  df-0g 17455  df-mgm 18618  df-sgrp 18697  df-mnd 18713  df-grp 18919  df-minusg 18920  df-ga 19273

This theorem is referenced by:  sylow1lem3  19581  sylow1lem5  19583  sylow2alem1  19598  sylow2alem2  19599  sylow2a  19600  sylow3lem3  19610