Theorem gaorber 19522

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 5797	. . 3 ⊢ Rel ∼
3	2	a1i 11	. 2 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → Rel ∼ )
4		simpr 490	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑢 ∼ 𝑣) → 𝑢 ∼ 𝑣)
5	1	gaorb 19521	. . . . 5 ⊢ (𝑢 ∼ 𝑣 ↔ (𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌 ∧ ∃ℎ ∈ 𝑋 (ℎ ⊕ 𝑢) = 𝑣))
6	4, 5	sylib 221	. . . 4 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑢 ∼ 𝑣) → (𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌 ∧ ∃ℎ ∈ 𝑋 (ℎ ⊕ 𝑢) = 𝑣))
7	6	simp2d 1161	. . 3 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑢 ∼ 𝑣) → 𝑣 ∈ 𝑌)
8	6	simp1d 1160	. . 3 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑢 ∼ 𝑣) → 𝑢 ∈ 𝑌)
9	6	simp3d 1162	. . . 4 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑢 ∼ 𝑣) → ∃ℎ ∈ 𝑋 (ℎ ⊕ 𝑢) = 𝑣)
10		simpll 779	. . . . . . 7 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑢 ∼ 𝑣) ∧ ℎ ∈ 𝑋) → ⊕ ∈ (𝐺 GrpAct 𝑌))
11		simpr 490	. . . . . . 7 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑢 ∼ 𝑣) ∧ ℎ ∈ 𝑋) → ℎ ∈ 𝑋)
12	8	adantr 486	. . . . . . 7 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑢 ∼ 𝑣) ∧ ℎ ∈ 𝑋) → 𝑢 ∈ 𝑌)
13	7	adantr 486	. . . . . . 7 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑢 ∼ 𝑣) ∧ ℎ ∈ 𝑋) → 𝑣 ∈ 𝑌)
14		gaorber.2	. . . . . . . 8 ⊢ 𝑋 = (Base‘𝐺)
15		eqid 2760	. . . . . . . 8 ⊢ (invg‘𝐺) = (invg‘𝐺)
16	14, 15	gacan 19519	. . . . . . 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 1399	. . . . . 6 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑢 ∼ 𝑣) ∧ ℎ ∈ 𝑋) → ((ℎ ⊕ 𝑢) = 𝑣 ↔ (((invg‘𝐺)‘ℎ) ⊕ 𝑣) = 𝑢))
18		gagrp 19506	. . . . . . . . 9 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → 𝐺 ∈ Grp)
19	18	adantr 486	. . . . . . . 8 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑢 ∼ 𝑣) → 𝐺 ∈ Grp)
20	14, 15	grpinvcl 19198	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ ℎ ∈ 𝑋) → ((invg‘𝐺)‘ℎ) ∈ 𝑋)
21	19, 20	sylan 592	. . . . . . 7 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑢 ∼ 𝑣) ∧ ℎ ∈ 𝑋) → ((invg‘𝐺)‘ℎ) ∈ 𝑋)
22		oveq1 7427	. . . . . . . . . 10 ⊢ (𝑘 = ((invg‘𝐺)‘ℎ) → (𝑘 ⊕ 𝑣) = (((invg‘𝐺)‘ℎ) ⊕ 𝑣))
23	22	eqeq1d 2762	. . . . . . . . 9 ⊢ (𝑘 = ((invg‘𝐺)‘ℎ) → ((𝑘 ⊕ 𝑣) = 𝑢 ↔ (((invg‘𝐺)‘ℎ) ⊕ 𝑣) = 𝑢))
24	23	rspcev 3576	. . . . . . . 8 ⊢ ((((invg‘𝐺)‘ℎ) ∈ 𝑋 ∧ (((invg‘𝐺)‘ℎ) ⊕ 𝑣) = 𝑢) → ∃𝑘 ∈ 𝑋 (𝑘 ⊕ 𝑣) = 𝑢)
25	24	ex 418	. . . . . . 7 ⊢ (((invg‘𝐺)‘ℎ) ∈ 𝑋 → ((((invg‘𝐺)‘ℎ) ⊕ 𝑣) = 𝑢 → ∃𝑘 ∈ 𝑋 (𝑘 ⊕ 𝑣) = 𝑢))
26	21, 25	syl 18	. . . . . 6 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑢 ∼ 𝑣) ∧ ℎ ∈ 𝑋) → ((((invg‘𝐺)‘ℎ) ⊕ 𝑣) = 𝑢 → ∃𝑘 ∈ 𝑋 (𝑘 ⊕ 𝑣) = 𝑢))
27	17, 26	sylbid 243	. . . . 5 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑢 ∼ 𝑣) ∧ ℎ ∈ 𝑋) → ((ℎ ⊕ 𝑢) = 𝑣 → ∃𝑘 ∈ 𝑋 (𝑘 ⊕ 𝑣) = 𝑢))
28	27	rexlimdva 3163	. . . 4 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑢 ∼ 𝑣) → (∃ℎ ∈ 𝑋 (ℎ ⊕ 𝑢) = 𝑣 → ∃𝑘 ∈ 𝑋 (𝑘 ⊕ 𝑣) = 𝑢))
29	9, 28	mpd 16	. . 3 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑢 ∼ 𝑣) → ∃𝑘 ∈ 𝑋 (𝑘 ⊕ 𝑣) = 𝑢)
30	1	gaorb 19521	. . 3 ⊢ (𝑣 ∼ 𝑢 ↔ (𝑣 ∈ 𝑌 ∧ 𝑢 ∈ 𝑌 ∧ ∃𝑘 ∈ 𝑋 (𝑘 ⊕ 𝑣) = 𝑢))
31	7, 8, 29, 30	syl3anbrc 1362	. 2 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑢 ∼ 𝑣) → 𝑣 ∼ 𝑢)
32	8	adantrr 730	. . 3 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢 ∼ 𝑣 ∧ 𝑣 ∼ 𝑤)) → 𝑢 ∈ 𝑌)
33		simprr 785	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢 ∼ 𝑣 ∧ 𝑣 ∼ 𝑤)) → 𝑣 ∼ 𝑤)
34	1	gaorb 19521	. . . . 5 ⊢ (𝑣 ∼ 𝑤 ↔ (𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌 ∧ ∃𝑘 ∈ 𝑋 (𝑘 ⊕ 𝑣) = 𝑤))
35	33, 34	sylib 221	. . . 4 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢 ∼ 𝑣 ∧ 𝑣 ∼ 𝑤)) → (𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌 ∧ ∃𝑘 ∈ 𝑋 (𝑘 ⊕ 𝑣) = 𝑤))
36	35	simp2d 1161	. . 3 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢 ∼ 𝑣 ∧ 𝑣 ∼ 𝑤)) → 𝑤 ∈ 𝑌)
37	9	adantrr 730	. . . 4 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢 ∼ 𝑣 ∧ 𝑣 ∼ 𝑤)) → ∃ℎ ∈ 𝑋 (ℎ ⊕ 𝑢) = 𝑣)
38	35	simp3d 1162	. . . 4 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢 ∼ 𝑣 ∧ 𝑣 ∼ 𝑤)) → ∃𝑘 ∈ 𝑋 (𝑘 ⊕ 𝑣) = 𝑤)
39		reeanv 3234	. . . . 5 ⊢ (∃ℎ ∈ 𝑋 ∃𝑘 ∈ 𝑋 ((ℎ ⊕ 𝑢) = 𝑣 ∧ (𝑘 ⊕ 𝑣) = 𝑤) ↔ (∃ℎ ∈ 𝑋 (ℎ ⊕ 𝑢) = 𝑣 ∧ ∃𝑘 ∈ 𝑋 (𝑘 ⊕ 𝑣) = 𝑤))
40	18	ad2antrr 739	. . . . . . . . 9 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢 ∼ 𝑣 ∧ 𝑣 ∼ 𝑤)) ∧ ((ℎ ∈ 𝑋 ∧ 𝑘 ∈ 𝑋) ∧ ((ℎ ⊕ 𝑢) = 𝑣 ∧ (𝑘 ⊕ 𝑣) = 𝑤))) → 𝐺 ∈ Grp)
41		simprlr 792	. . . . . . . . 9 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢 ∼ 𝑣 ∧ 𝑣 ∼ 𝑤)) ∧ ((ℎ ∈ 𝑋 ∧ 𝑘 ∈ 𝑋) ∧ ((ℎ ⊕ 𝑢) = 𝑣 ∧ (𝑘 ⊕ 𝑣) = 𝑤))) → 𝑘 ∈ 𝑋)
42		simprll 791	. . . . . . . . 9 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢 ∼ 𝑣 ∧ 𝑣 ∼ 𝑤)) ∧ ((ℎ ∈ 𝑋 ∧ 𝑘 ∈ 𝑋) ∧ ((ℎ ⊕ 𝑢) = 𝑣 ∧ (𝑘 ⊕ 𝑣) = 𝑤))) → ℎ ∈ 𝑋)
43		eqid 2760	. . . . . . . . . 10 ⊢ (+g‘𝐺) = (+g‘𝐺)
44	14, 43	grpcl 19152	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ 𝑘 ∈ 𝑋 ∧ ℎ ∈ 𝑋) → (𝑘(+g‘𝐺)ℎ) ∈ 𝑋)
45	40, 41, 42, 44	syl3anc 1398	. . . . . . . 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 779	. . . . . . . . . 10 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢 ∼ 𝑣 ∧ 𝑣 ∼ 𝑤)) ∧ ((ℎ ∈ 𝑋 ∧ 𝑘 ∈ 𝑋) ∧ ((ℎ ⊕ 𝑢) = 𝑣 ∧ (𝑘 ⊕ 𝑣) = 𝑤))) → ⊕ ∈ (𝐺 GrpAct 𝑌))
47	32	adantr 486	. . . . . . . . . 10 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢 ∼ 𝑣 ∧ 𝑣 ∼ 𝑤)) ∧ ((ℎ ∈ 𝑋 ∧ 𝑘 ∈ 𝑋) ∧ ((ℎ ⊕ 𝑢) = 𝑣 ∧ (𝑘 ⊕ 𝑣) = 𝑤))) → 𝑢 ∈ 𝑌)
48	14, 43	gaass 19511	. . . . . . . . . 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 1399	. . . . . . . . 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 793	. . . . . . . . . 10 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢 ∼ 𝑣 ∧ 𝑣 ∼ 𝑤)) ∧ ((ℎ ∈ 𝑋 ∧ 𝑘 ∈ 𝑋) ∧ ((ℎ ⊕ 𝑢) = 𝑣 ∧ (𝑘 ⊕ 𝑣) = 𝑤))) → (ℎ ⊕ 𝑢) = 𝑣)
51	50	oveq2d 7436	. . . . . . . . 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 794	. . . . . . . . 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 2799	. . . . . . . 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 7427	. . . . . . . . . 10 ⊢ (𝑓 = (𝑘(+g‘𝐺)ℎ) → (𝑓 ⊕ 𝑢) = ((𝑘(+g‘𝐺)ℎ) ⊕ 𝑢))
55	54	eqeq1d 2762	. . . . . . . . 9 ⊢ (𝑓 = (𝑘(+g‘𝐺)ℎ) → ((𝑓 ⊕ 𝑢) = 𝑤 ↔ ((𝑘(+g‘𝐺)ℎ) ⊕ 𝑢) = 𝑤))
56	55	rspcev 3576	. . . . . . . 8 ⊢ (((𝑘(+g‘𝐺)ℎ) ∈ 𝑋 ∧ ((𝑘(+g‘𝐺)ℎ) ⊕ 𝑢) = 𝑤) → ∃𝑓 ∈ 𝑋 (𝑓 ⊕ 𝑢) = 𝑤)
57	45, 53, 56	syl2anc 596	. . . . . . 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 462	. . . . . 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 3219	. . . . 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 246	. . . 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 712	. . 3 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢 ∼ 𝑣 ∧ 𝑣 ∼ 𝑤)) → ∃𝑓 ∈ 𝑋 (𝑓 ⊕ 𝑢) = 𝑤)
62	1	gaorb 19521	. . 3 ⊢ (𝑢 ∼ 𝑤 ↔ (𝑢 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌 ∧ ∃𝑓 ∈ 𝑋 (𝑓 ⊕ 𝑢) = 𝑤))
63	32, 36, 61, 62	syl3anbrc 1362	. 2 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢 ∼ 𝑣 ∧ 𝑣 ∼ 𝑤)) → 𝑢 ∼ 𝑤)
64	18	adantr 486	. . . . . . . 8 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑢 ∈ 𝑌) → 𝐺 ∈ Grp)
65		eqid 2760	. . . . . . . . 9 ⊢ (0g‘𝐺) = (0g‘𝐺)
66	14, 65	grpidcl 19176	. . . . . . . 8 ⊢ (𝐺 ∈ Grp → (0g‘𝐺) ∈ 𝑋)
67	64, 66	syl 18	. . . . . . 7 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑢 ∈ 𝑌) → (0g‘𝐺) ∈ 𝑋)
68	65	gagrpid 19508	. . . . . . 7 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑢 ∈ 𝑌) → ((0g‘𝐺) ⊕ 𝑢) = 𝑢)
69		oveq1 7427	. . . . . . . . 9 ⊢ (ℎ = (0g‘𝐺) → (ℎ ⊕ 𝑢) = ((0g‘𝐺) ⊕ 𝑢))
70	69	eqeq1d 2762	. . . . . . . 8 ⊢ (ℎ = (0g‘𝐺) → ((ℎ ⊕ 𝑢) = 𝑢 ↔ ((0g‘𝐺) ⊕ 𝑢) = 𝑢))
71	70	rspcev 3576	. . . . . . 7 ⊢ (((0g‘𝐺) ∈ 𝑋 ∧ ((0g‘𝐺) ⊕ 𝑢) = 𝑢) → ∃ℎ ∈ 𝑋 (ℎ ⊕ 𝑢) = 𝑢)
72	67, 68, 71	syl2anc 596	. . . . . 6 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑢 ∈ 𝑌) → ∃ℎ ∈ 𝑋 (ℎ ⊕ 𝑢) = 𝑢)
73	72	ex 418	. . . . 5 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → (𝑢 ∈ 𝑌 → ∃ℎ ∈ 𝑋 (ℎ ⊕ 𝑢) = 𝑢))
74	73	pm4.71rd 572	. . . 4 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → (𝑢 ∈ 𝑌 ↔ (∃ℎ ∈ 𝑋 (ℎ ⊕ 𝑢) = 𝑢 ∧ 𝑢 ∈ 𝑌)))
75		df-3an 1105	. . . . 5 ⊢ ((𝑢 ∈ 𝑌 ∧ 𝑢 ∈ 𝑌 ∧ ∃ℎ ∈ 𝑋 (ℎ ⊕ 𝑢) = 𝑢) ↔ ((𝑢 ∈ 𝑌 ∧ 𝑢 ∈ 𝑌) ∧ ∃ℎ ∈ 𝑋 (ℎ ⊕ 𝑢) = 𝑢))
76		anidm 575	. . . . . 6 ⊢ ((𝑢 ∈ 𝑌 ∧ 𝑢 ∈ 𝑌) ↔ 𝑢 ∈ 𝑌)
77	76	anbi2ci 637	. . . . 5 ⊢ (((𝑢 ∈ 𝑌 ∧ 𝑢 ∈ 𝑌) ∧ ∃ℎ ∈ 𝑋 (ℎ ⊕ 𝑢) = 𝑢) ↔ (∃ℎ ∈ 𝑋 (ℎ ⊕ 𝑢) = 𝑢 ∧ 𝑢 ∈ 𝑌))
78	75, 77	bitri 278	. . . 4 ⊢ ((𝑢 ∈ 𝑌 ∧ 𝑢 ∈ 𝑌 ∧ ∃ℎ ∈ 𝑋 (ℎ ⊕ 𝑢) = 𝑢) ↔ (∃ℎ ∈ 𝑋 (ℎ ⊕ 𝑢) = 𝑢 ∧ 𝑢 ∈ 𝑌))
79	74, 78	bitr4di 292	. . 3 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → (𝑢 ∈ 𝑌 ↔ (𝑢 ∈ 𝑌 ∧ 𝑢 ∈ 𝑌 ∧ ∃ℎ ∈ 𝑋 (ℎ ⊕ 𝑢) = 𝑢)))
80	1	gaorb 19521	. . 3 ⊢ (𝑢 ∼ 𝑢 ↔ (𝑢 ∈ 𝑌 ∧ 𝑢 ∈ 𝑌 ∧ ∃ℎ ∈ 𝑋 (ℎ ⊕ 𝑢) = 𝑢))
81	79, 80	bitr4di 292	. 2 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → (𝑢 ∈ 𝑌 ↔ 𝑢 ∼ 𝑢))
82	3, 31, 63, 81	iserd 8744	1 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → ∼ Er 𝑌)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  ∃wrex 3086   ⊆ wss 3895  {cpr 4585   class class class wbr 5102  {copab 5166  Rel wrel 5655  ‘cfv 6538  (class class class)co 7420   Er wer 8714  Basecbs 17387  +_gcplusg 17428  0_gc0g 17610  Grpcgrp 19144  inv_gcminusg 19145   GrpAct cga 19503

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  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 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-er 8717  df-map 8849  df-0g 17612  df-mgm 18816  df-sgrp 18908  df-mnd 18924  df-grp 19147  df-minusg 19148  df-ga 19504

This theorem is used by:  sylow1lem3  19814  sylow1lem5  19816  sylow2alem1  19831  sylow2alem2  19832  sylow2a  19833  sylow3lem3  19843