Theorem gaorber 18829

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 5719	. . 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 18828	. . . . 5 ⊢ (𝑢 ∼ 𝑣 ↔ (𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌 ∧ ∃ℎ ∈ 𝑋 (ℎ ⊕ 𝑢) = 𝑣))
6	4, 5	sylib 217	. . . 4 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑢 ∼ 𝑣) → (𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌 ∧ ∃ℎ ∈ 𝑋 (ℎ ⊕ 𝑢) = 𝑣))
7	6	simp2d 1141	. . 3 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑢 ∼ 𝑣) → 𝑣 ∈ 𝑌)
8	6	simp1d 1140	. . 3 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑢 ∼ 𝑣) → 𝑢 ∈ 𝑌)
9	6	simp3d 1142	. . . 4 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑢 ∼ 𝑣) → ∃ℎ ∈ 𝑋 (ℎ ⊕ 𝑢) = 𝑣)
10		simpll 763	. . . . . . 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 2738	. . . . . . . 8 ⊢ (invg‘𝐺) = (invg‘𝐺)
16	14, 15	gacan 18826	. . . . . . 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 1370	. . . . . 6 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑢 ∼ 𝑣) ∧ ℎ ∈ 𝑋) → ((ℎ ⊕ 𝑢) = 𝑣 ↔ (((invg‘𝐺)‘ℎ) ⊕ 𝑣) = 𝑢))
18		gagrp 18813	. . . . . . . . 9 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → 𝐺 ∈ Grp)
19	18	adantr 480	. . . . . . . 8 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑢 ∼ 𝑣) → 𝐺 ∈ Grp)
20	14, 15	grpinvcl 18542	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ ℎ ∈ 𝑋) → ((invg‘𝐺)‘ℎ) ∈ 𝑋)
21	19, 20	sylan 579	. . . . . . 7 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑢 ∼ 𝑣) ∧ ℎ ∈ 𝑋) → ((invg‘𝐺)‘ℎ) ∈ 𝑋)
22		oveq1 7262	. . . . . . . . . 10 ⊢ (𝑘 = ((invg‘𝐺)‘ℎ) → (𝑘 ⊕ 𝑣) = (((invg‘𝐺)‘ℎ) ⊕ 𝑣))
23	22	eqeq1d 2740	. . . . . . . . 9 ⊢ (𝑘 = ((invg‘𝐺)‘ℎ) → ((𝑘 ⊕ 𝑣) = 𝑢 ↔ (((invg‘𝐺)‘ℎ) ⊕ 𝑣) = 𝑢))
24	23	rspcev 3552	. . . . . . . 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 239	. . . . 5 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑢 ∼ 𝑣) ∧ ℎ ∈ 𝑋) → ((ℎ ⊕ 𝑢) = 𝑣 → ∃𝑘 ∈ 𝑋 (𝑘 ⊕ 𝑣) = 𝑢))
28	27	rexlimdva 3212	. . . 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 18828	. . 3 ⊢ (𝑣 ∼ 𝑢 ↔ (𝑣 ∈ 𝑌 ∧ 𝑢 ∈ 𝑌 ∧ ∃𝑘 ∈ 𝑋 (𝑘 ⊕ 𝑣) = 𝑢))
31	7, 8, 29, 30	syl3anbrc 1341	. 2 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑢 ∼ 𝑣) → 𝑣 ∼ 𝑢)
32	8	adantrr 713	. . 3 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢 ∼ 𝑣 ∧ 𝑣 ∼ 𝑤)) → 𝑢 ∈ 𝑌)
33		simprr 769	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢 ∼ 𝑣 ∧ 𝑣 ∼ 𝑤)) → 𝑣 ∼ 𝑤)
34	1	gaorb 18828	. . . . 5 ⊢ (𝑣 ∼ 𝑤 ↔ (𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌 ∧ ∃𝑘 ∈ 𝑋 (𝑘 ⊕ 𝑣) = 𝑤))
35	33, 34	sylib 217	. . . 4 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢 ∼ 𝑣 ∧ 𝑣 ∼ 𝑤)) → (𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌 ∧ ∃𝑘 ∈ 𝑋 (𝑘 ⊕ 𝑣) = 𝑤))
36	35	simp2d 1141	. . 3 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢 ∼ 𝑣 ∧ 𝑣 ∼ 𝑤)) → 𝑤 ∈ 𝑌)
37	9	adantrr 713	. . . 4 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢 ∼ 𝑣 ∧ 𝑣 ∼ 𝑤)) → ∃ℎ ∈ 𝑋 (ℎ ⊕ 𝑢) = 𝑣)
38	35	simp3d 1142	. . . 4 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢 ∼ 𝑣 ∧ 𝑣 ∼ 𝑤)) → ∃𝑘 ∈ 𝑋 (𝑘 ⊕ 𝑣) = 𝑤)
39		reeanv 3292	. . . . 5 ⊢ (∃ℎ ∈ 𝑋 ∃𝑘 ∈ 𝑋 ((ℎ ⊕ 𝑢) = 𝑣 ∧ (𝑘 ⊕ 𝑣) = 𝑤) ↔ (∃ℎ ∈ 𝑋 (ℎ ⊕ 𝑢) = 𝑣 ∧ ∃𝑘 ∈ 𝑋 (𝑘 ⊕ 𝑣) = 𝑤))
40	18	ad2antrr 722	. . . . . . . . 9 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢 ∼ 𝑣 ∧ 𝑣 ∼ 𝑤)) ∧ ((ℎ ∈ 𝑋 ∧ 𝑘 ∈ 𝑋) ∧ ((ℎ ⊕ 𝑢) = 𝑣 ∧ (𝑘 ⊕ 𝑣) = 𝑤))) → 𝐺 ∈ Grp)
41		simprlr 776	. . . . . . . . 9 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢 ∼ 𝑣 ∧ 𝑣 ∼ 𝑤)) ∧ ((ℎ ∈ 𝑋 ∧ 𝑘 ∈ 𝑋) ∧ ((ℎ ⊕ 𝑢) = 𝑣 ∧ (𝑘 ⊕ 𝑣) = 𝑤))) → 𝑘 ∈ 𝑋)
42		simprll 775	. . . . . . . . 9 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢 ∼ 𝑣 ∧ 𝑣 ∼ 𝑤)) ∧ ((ℎ ∈ 𝑋 ∧ 𝑘 ∈ 𝑋) ∧ ((ℎ ⊕ 𝑢) = 𝑣 ∧ (𝑘 ⊕ 𝑣) = 𝑤))) → ℎ ∈ 𝑋)
43		eqid 2738	. . . . . . . . . 10 ⊢ (+g‘𝐺) = (+g‘𝐺)
44	14, 43	grpcl 18500	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ 𝑘 ∈ 𝑋 ∧ ℎ ∈ 𝑋) → (𝑘(+g‘𝐺)ℎ) ∈ 𝑋)
45	40, 41, 42, 44	syl3anc 1369	. . . . . . . 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 763	. . . . . . . . . 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 18818	. . . . . . . . . 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 1370	. . . . . . . . 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 777	. . . . . . . . . 10 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢 ∼ 𝑣 ∧ 𝑣 ∼ 𝑤)) ∧ ((ℎ ∈ 𝑋 ∧ 𝑘 ∈ 𝑋) ∧ ((ℎ ⊕ 𝑢) = 𝑣 ∧ (𝑘 ⊕ 𝑣) = 𝑤))) → (ℎ ⊕ 𝑢) = 𝑣)
51	50	oveq2d 7271	. . . . . . . . 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 778	. . . . . . . . 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 2782	. . . . . . . 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 7262	. . . . . . . . . 10 ⊢ (𝑓 = (𝑘(+g‘𝐺)ℎ) → (𝑓 ⊕ 𝑢) = ((𝑘(+g‘𝐺)ℎ) ⊕ 𝑢))
55	54	eqeq1d 2740	. . . . . . . . 9 ⊢ (𝑓 = (𝑘(+g‘𝐺)ℎ) → ((𝑓 ⊕ 𝑢) = 𝑤 ↔ ((𝑘(+g‘𝐺)ℎ) ⊕ 𝑢) = 𝑤))
56	55	rspcev 3552	. . . . . . . 8 ⊢ (((𝑘(+g‘𝐺)ℎ) ∈ 𝑋 ∧ ((𝑘(+g‘𝐺)ℎ) ⊕ 𝑢) = 𝑤) → ∃𝑓 ∈ 𝑋 (𝑓 ⊕ 𝑢) = 𝑤)
57	45, 53, 56	syl2anc 583	. . . . . . 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 3222	. . . . 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	syl5bir 242	. . . 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 695	. . 3 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢 ∼ 𝑣 ∧ 𝑣 ∼ 𝑤)) → ∃𝑓 ∈ 𝑋 (𝑓 ⊕ 𝑢) = 𝑤)
62	1	gaorb 18828	. . 3 ⊢ (𝑢 ∼ 𝑤 ↔ (𝑢 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌 ∧ ∃𝑓 ∈ 𝑋 (𝑓 ⊕ 𝑢) = 𝑤))
63	32, 36, 61, 62	syl3anbrc 1341	. 2 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢 ∼ 𝑣 ∧ 𝑣 ∼ 𝑤)) → 𝑢 ∼ 𝑤)
64	18	adantr 480	. . . . . . . 8 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑢 ∈ 𝑌) → 𝐺 ∈ Grp)
65		eqid 2738	. . . . . . . . 9 ⊢ (0g‘𝐺) = (0g‘𝐺)
66	14, 65	grpidcl 18522	. . . . . . . 8 ⊢ (𝐺 ∈ Grp → (0g‘𝐺) ∈ 𝑋)
67	64, 66	syl 17	. . . . . . 7 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑢 ∈ 𝑌) → (0g‘𝐺) ∈ 𝑋)
68	65	gagrpid 18815	. . . . . . 7 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑢 ∈ 𝑌) → ((0g‘𝐺) ⊕ 𝑢) = 𝑢)
69		oveq1 7262	. . . . . . . . 9 ⊢ (ℎ = (0g‘𝐺) → (ℎ ⊕ 𝑢) = ((0g‘𝐺) ⊕ 𝑢))
70	69	eqeq1d 2740	. . . . . . . 8 ⊢ (ℎ = (0g‘𝐺) → ((ℎ ⊕ 𝑢) = 𝑢 ↔ ((0g‘𝐺) ⊕ 𝑢) = 𝑢))
71	70	rspcev 3552	. . . . . . 7 ⊢ (((0g‘𝐺) ∈ 𝑋 ∧ ((0g‘𝐺) ⊕ 𝑢) = 𝑢) → ∃ℎ ∈ 𝑋 (ℎ ⊕ 𝑢) = 𝑢)
72	67, 68, 71	syl2anc 583	. . . . . 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 1087	. . . . 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 624	. . . . 5 ⊢ (((𝑢 ∈ 𝑌 ∧ 𝑢 ∈ 𝑌) ∧ ∃ℎ ∈ 𝑋 (ℎ ⊕ 𝑢) = 𝑢) ↔ (∃ℎ ∈ 𝑋 (ℎ ⊕ 𝑢) = 𝑢 ∧ 𝑢 ∈ 𝑌))
78	75, 77	bitri 274	. . . 4 ⊢ ((𝑢 ∈ 𝑌 ∧ 𝑢 ∈ 𝑌 ∧ ∃ℎ ∈ 𝑋 (ℎ ⊕ 𝑢) = 𝑢) ↔ (∃ℎ ∈ 𝑋 (ℎ ⊕ 𝑢) = 𝑢 ∧ 𝑢 ∈ 𝑌))
79	74, 78	bitr4di 288	. . 3 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → (𝑢 ∈ 𝑌 ↔ (𝑢 ∈ 𝑌 ∧ 𝑢 ∈ 𝑌 ∧ ∃ℎ ∈ 𝑋 (ℎ ⊕ 𝑢) = 𝑢)))
80	1	gaorb 18828	. . 3 ⊢ (𝑢 ∼ 𝑢 ↔ (𝑢 ∈ 𝑌 ∧ 𝑢 ∈ 𝑌 ∧ ∃ℎ ∈ 𝑋 (ℎ ⊕ 𝑢) = 𝑢))
81	79, 80	bitr4di 288	. 2 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → (𝑢 ∈ 𝑌 ↔ 𝑢 ∼ 𝑢))
82	3, 31, 63, 81	iserd 8482	1 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → ∼ Er 𝑌)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  ∃wrex 3064   ⊆ wss 3883  {cpr 4560   class class class wbr 5070  {copab 5132  Rel wrel 5585  ‘cfv 6418  (class class class)co 7255   Er wer 8453  Basecbs 16840  +_gcplusg 16888  0_gc0g 17067  Grpcgrp 18492  inv_gcminusg 18493   GrpAct cga 18810

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-er 8456  df-map 8575  df-0g 17069  df-mgm 18241  df-sgrp 18290  df-mnd 18301  df-grp 18495  df-minusg 18496  df-ga 18811

This theorem is referenced by:  sylow1lem3  19120  sylow1lem5  19122  sylow2alem1  19137  sylow2alem2  19138  sylow2a  19139  sylow3lem3  19149