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Theorem symggen 18993

Description: The span of the transpositions is the subgroup that moves finitely many points. (Contributed by Stefan O'Rear, 28-Aug-2015.)

**Hypotheses**
Ref	Expression
symgtrf.t	⊢ 𝑇 = ran (pmTrsp‘𝐷)
symgtrf.g	⊢ 𝐺 = (SymGrp‘𝐷)
symgtrf.b	⊢ 𝐵 = (Base‘𝐺)
symggen.k	⊢ 𝐾 = (mrCls‘(SubMnd‘𝐺))

**Assertion**
Ref	Expression
symggen	⊢ (𝐷 ∈ 𝑉 → (𝐾‘𝑇) = {𝑥 ∈ 𝐵 ∣ dom (𝑥 ∖ I ) ∈ Fin})

Distinct variable groups: 𝑥,𝐵 𝑥,𝑇 𝑥,𝐾 𝑥,𝐷 𝑥,𝐺 𝑥,𝑉

Proof of Theorem symggen Dummy variables 𝑢 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3440	. . . 4 ⊢ (𝐷 ∈ 𝑉 → 𝐷 ∈ V)
2		symgtrf.g	. . . . . . 7 ⊢ 𝐺 = (SymGrp‘𝐷)
3	2	symggrp 18923	. . . . . 6 ⊢ (𝐷 ∈ V → 𝐺 ∈ Grp)
4	3	grpmndd 18504	. . . . 5 ⊢ (𝐷 ∈ V → 𝐺 ∈ Mnd)
5		symgtrf.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐺)
6	5	submacs 18380	. . . . 5 ⊢ (𝐺 ∈ Mnd → (SubMnd‘𝐺) ∈ (ACS‘𝐵))
7		acsmre 17278	. . . . 5 ⊢ ((SubMnd‘𝐺) ∈ (ACS‘𝐵) → (SubMnd‘𝐺) ∈ (Moore‘𝐵))
8	4, 6, 7	3syl 18	. . . 4 ⊢ (𝐷 ∈ V → (SubMnd‘𝐺) ∈ (Moore‘𝐵))
9	1, 8	syl 17	. . 3 ⊢ (𝐷 ∈ 𝑉 → (SubMnd‘𝐺) ∈ (Moore‘𝐵))
10		symgtrf.t	. . . . . 6 ⊢ 𝑇 = ran (pmTrsp‘𝐷)
11	10, 2, 5	symgtrf 18992	. . . . 5 ⊢ 𝑇 ⊆ 𝐵
12	11	a1i 11	. . . 4 ⊢ (𝐷 ∈ 𝑉 → 𝑇 ⊆ 𝐵)
13		2onn 8433	. . . . . 6 ⊢ 2_o ∈ ω
14		nnfi 8912	. . . . . 6 ⊢ (2_o ∈ ω → 2_o ∈ Fin)
15	13, 14	ax-mp 5	. . . . 5 ⊢ 2_o ∈ Fin
16		eqid 2738	. . . . . . . . 9 ⊢ (pmTrsp‘𝐷) = (pmTrsp‘𝐷)
17	16, 10	pmtrfb 18988	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑇 ↔ (𝐷 ∈ V ∧ 𝑥:𝐷–1-1-onto→𝐷 ∧ dom (𝑥 ∖ I ) ≈ 2_o))
18	17	simp3bi 1145	. . . . . . 7 ⊢ (𝑥 ∈ 𝑇 → dom (𝑥 ∖ I ) ≈ 2_o)
19		enfi 8933	. . . . . . 7 ⊢ (dom (𝑥 ∖ I ) ≈ 2_o → (dom (𝑥 ∖ I ) ∈ Fin ↔ 2_o ∈ Fin))
20	18, 19	syl 17	. . . . . 6 ⊢ (𝑥 ∈ 𝑇 → (dom (𝑥 ∖ I ) ∈ Fin ↔ 2_o ∈ Fin))
21	20	adantl 481	. . . . 5 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑥 ∈ 𝑇) → (dom (𝑥 ∖ I ) ∈ Fin ↔ 2_o ∈ Fin))
22	15, 21	mpbiri 257	. . . 4 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑥 ∈ 𝑇) → dom (𝑥 ∖ I ) ∈ Fin)
23	12, 22	ssrabdv 4003	. . 3 ⊢ (𝐷 ∈ 𝑉 → 𝑇 ⊆ {𝑥 ∈ 𝐵 ∣ dom (𝑥 ∖ I ) ∈ Fin})
24	2, 5	symgfisg 18991	. . . 4 ⊢ (𝐷 ∈ 𝑉 → {𝑥 ∈ 𝐵 ∣ dom (𝑥 ∖ I ) ∈ Fin} ∈ (SubGrp‘𝐺))
25		subgsubm 18692	. . . 4 ⊢ ({𝑥 ∈ 𝐵 ∣ dom (𝑥 ∖ I ) ∈ Fin} ∈ (SubGrp‘𝐺) → {𝑥 ∈ 𝐵 ∣ dom (𝑥 ∖ I ) ∈ Fin} ∈ (SubMnd‘𝐺))
26	24, 25	syl 17	. . 3 ⊢ (𝐷 ∈ 𝑉 → {𝑥 ∈ 𝐵 ∣ dom (𝑥 ∖ I ) ∈ Fin} ∈ (SubMnd‘𝐺))
27		symggen.k	. . . 4 ⊢ 𝐾 = (mrCls‘(SubMnd‘𝐺))
28	27	mrcsscl 17246	. . 3 ⊢ (((SubMnd‘𝐺) ∈ (Moore‘𝐵) ∧ 𝑇 ⊆ {𝑥 ∈ 𝐵 ∣ dom (𝑥 ∖ I ) ∈ Fin} ∧ {𝑥 ∈ 𝐵 ∣ dom (𝑥 ∖ I ) ∈ Fin} ∈ (SubMnd‘𝐺)) → (𝐾‘𝑇) ⊆ {𝑥 ∈ 𝐵 ∣ dom (𝑥 ∖ I ) ∈ Fin})
29	9, 23, 26, 28	syl3anc 1369	. 2 ⊢ (𝐷 ∈ 𝑉 → (𝐾‘𝑇) ⊆ {𝑥 ∈ 𝐵 ∣ dom (𝑥 ∖ I ) ∈ Fin})
30		vex 3426	. . . . . . 7 ⊢ 𝑥 ∈ V
31	30	a1i 11	. . . . . 6 ⊢ (dom (𝑥 ∖ I ) ∈ Fin → 𝑥 ∈ V)
32		finnum 9637	. . . . . 6 ⊢ (dom (𝑥 ∖ I ) ∈ Fin → dom (𝑥 ∖ I ) ∈ dom card)
33		domfi 8935	. . . . . . . 8 ⊢ ((dom (𝑥 ∖ I ) ∈ Fin ∧ dom (𝑦 ∖ I ) ≼ dom (𝑥 ∖ I )) → dom (𝑦 ∖ I ) ∈ Fin)
34	2, 5	symgbasf1o 18897	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ 𝐵 → 𝑦:𝐷–1-1-onto→𝐷)
35	34	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) → 𝑦:𝐷–1-1-onto→𝐷)
36		f1ofn 6701	. . . . . . . . . . . . . 14 ⊢ (𝑦:𝐷–1-1-onto→𝐷 → 𝑦 Fn 𝐷)
37		fnnfpeq0 7032	. . . . . . . . . . . . . 14 ⊢ (𝑦 Fn 𝐷 → (dom (𝑦 ∖ I ) = ∅ ↔ 𝑦 = ( I ↾ 𝐷)))
38	35, 36, 37	3syl 18	. . . . . . . . . . . . 13 ⊢ ((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) → (dom (𝑦 ∖ I ) = ∅ ↔ 𝑦 = ( I ↾ 𝐷)))
39	2, 5	elbasfv 16846	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ 𝐵 → 𝐷 ∈ V)
40	39	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) → 𝐷 ∈ V)
41	2	symgid 18924	. . . . . . . . . . . . . . . 16 ⊢ (𝐷 ∈ V → ( I ↾ 𝐷) = (0_g‘𝐺))
42	40, 41	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) → ( I ↾ 𝐷) = (0_g‘𝐺))
43	40, 8	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) → (SubMnd‘𝐺) ∈ (Moore‘𝐵))
44	27	mrccl 17237	. . . . . . . . . . . . . . . . 17 ⊢ (((SubMnd‘𝐺) ∈ (Moore‘𝐵) ∧ 𝑇 ⊆ 𝐵) → (𝐾‘𝑇) ∈ (SubMnd‘𝐺))
45	43, 11, 44	sylancl 585	. . . . . . . . . . . . . . . 16 ⊢ ((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) → (𝐾‘𝑇) ∈ (SubMnd‘𝐺))
46		eqid 2738	. . . . . . . . . . . . . . . . 17 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
47	46	subm0cl 18365	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾‘𝑇) ∈ (SubMnd‘𝐺) → (0_g‘𝐺) ∈ (𝐾‘𝑇))
48	45, 47	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) → (0_g‘𝐺) ∈ (𝐾‘𝑇))
49	42, 48	eqeltrd 2839	. . . . . . . . . . . . . 14 ⊢ ((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) → ( I ↾ 𝐷) ∈ (𝐾‘𝑇))
50		eleq1a 2834	. . . . . . . . . . . . . 14 ⊢ (( I ↾ 𝐷) ∈ (𝐾‘𝑇) → (𝑦 = ( I ↾ 𝐷) → 𝑦 ∈ (𝐾‘𝑇)))
51	49, 50	syl 17	. . . . . . . . . . . . 13 ⊢ ((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) → (𝑦 = ( I ↾ 𝐷) → 𝑦 ∈ (𝐾‘𝑇)))
52	38, 51	sylbid 239	. . . . . . . . . . . 12 ⊢ ((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) → (dom (𝑦 ∖ I ) = ∅ → 𝑦 ∈ (𝐾‘𝑇)))
53	52	adantr 480	. . . . . . . . . . 11 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ ∀𝑧(dom (𝑧 ∖ I ) ≺ dom (𝑦 ∖ I ) → (𝑧 ∈ 𝐵 → 𝑧 ∈ (𝐾‘𝑇)))) → (dom (𝑦 ∖ I ) = ∅ → 𝑦 ∈ (𝐾‘𝑇)))
54		n0 4277	. . . . . . . . . . . 12 ⊢ (dom (𝑦 ∖ I ) ≠ ∅ ↔ ∃𝑢 𝑢 ∈ dom (𝑦 ∖ I ))
55	40	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → 𝐷 ∈ V)
56		simpr 484	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → 𝑢 ∈ dom (𝑦 ∖ I ))
57		f1omvdmvd 18966	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑦:𝐷–1-1-onto→𝐷 ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → (𝑦‘𝑢) ∈ (dom (𝑦 ∖ I ) ∖ {𝑢}))
58	35, 57	sylan 579	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → (𝑦‘𝑢) ∈ (dom (𝑦 ∖ I ) ∖ {𝑢}))
59	58	eldifad 3895	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → (𝑦‘𝑢) ∈ dom (𝑦 ∖ I ))
60	56, 59	prssd 4752	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → {𝑢, (𝑦‘𝑢)} ⊆ dom (𝑦 ∖ I ))
61		difss 4062	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 ∖ I ) ⊆ 𝑦
62		dmss 5800	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑦 ∖ I ) ⊆ 𝑦 → dom (𝑦 ∖ I ) ⊆ dom 𝑦)
63	61, 62	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ dom (𝑦 ∖ I ) ⊆ dom 𝑦
64		f1odm 6704	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦:𝐷–1-1-onto→𝐷 → dom 𝑦 = 𝐷)
65	35, 64	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) → dom 𝑦 = 𝐷)
66	63, 65	sseqtrid 3969	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) → dom (𝑦 ∖ I ) ⊆ 𝐷)
67	66	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → dom (𝑦 ∖ I ) ⊆ 𝐷)
68	60, 67	sstrd 3927	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → {𝑢, (𝑦‘𝑢)} ⊆ 𝐷)
69		vex 3426	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝑢 ∈ V
70		fvex 6769	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦‘𝑢) ∈ V
71	35	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → 𝑦:𝐷–1-1-onto→𝐷)
72	71, 36	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → 𝑦 Fn 𝐷)
73	66	sselda 3917	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → 𝑢 ∈ 𝐷)
74		fnelnfp 7031	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑦 Fn 𝐷 ∧ 𝑢 ∈ 𝐷) → (𝑢 ∈ dom (𝑦 ∖ I ) ↔ (𝑦‘𝑢) ≠ 𝑢))
75	72, 73, 74	syl2anc 583	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → (𝑢 ∈ dom (𝑦 ∖ I ) ↔ (𝑦‘𝑢) ≠ 𝑢))
76	56, 75	mpbid 231	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → (𝑦‘𝑢) ≠ 𝑢)
77	76	necomd 2998	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → 𝑢 ≠ (𝑦‘𝑢))
78		pr2nelem 9691	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑢 ∈ V ∧ (𝑦‘𝑢) ∈ V ∧ 𝑢 ≠ (𝑦‘𝑢)) → {𝑢, (𝑦‘𝑢)} ≈ 2_o)
79	69, 70, 77, 78	mp3an12i 1463	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → {𝑢, (𝑦‘𝑢)} ≈ 2_o)
80	16, 10	pmtrrn 18980	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐷 ∈ V ∧ {𝑢, (𝑦‘𝑢)} ⊆ 𝐷 ∧ {𝑢, (𝑦‘𝑢)} ≈ 2_o) → ((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∈ 𝑇)
81	55, 68, 79, 80	syl3anc 1369	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → ((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∈ 𝑇)
82	11, 81	sselid 3915	. . . . . . . . . . . . . . . . . . 19 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → ((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∈ 𝐵)
83		simplr 765	. . . . . . . . . . . . . . . . . . 19 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → 𝑦 ∈ 𝐵)
84		eqid 2738	. . . . . . . . . . . . . . . . . . . 20 ⊢ (+_g‘𝐺) = (+_g‘𝐺)
85	2, 5, 84	symgov 18906	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+_g‘𝐺)𝑦) = (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦))
86	82, 83, 85	syl2anc 583	. . . . . . . . . . . . . . . . . 18 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+_g‘𝐺)𝑦) = (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦))
87	86	oveq2d 7271	. . . . . . . . . . . . . . . . 17 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+_g‘𝐺)(((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+_g‘𝐺)𝑦)) = (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+_g‘𝐺)(((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦)))
88	40, 3	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) → 𝐺 ∈ Grp)
89	88	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → 𝐺 ∈ Grp)
90	5, 84	grpcl 18500	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐺 ∈ Grp ∧ ((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+_g‘𝐺)𝑦) ∈ 𝐵)
91	89, 82, 83, 90	syl3anc 1369	. . . . . . . . . . . . . . . . . . 19 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+_g‘𝐺)𝑦) ∈ 𝐵)
92	86, 91	eqeltrrd 2840	. . . . . . . . . . . . . . . . . 18 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦) ∈ 𝐵)
93	2, 5, 84	symgov 18906	. . . . . . . . . . . . . . . . . 18 ⊢ ((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∈ 𝐵 ∧ (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦) ∈ 𝐵) → (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+_g‘𝐺)(((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦)) = (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦)))
94	82, 92, 93	syl2anc 583	. . . . . . . . . . . . . . . . 17 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+_g‘𝐺)(((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦)) = (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦)))
95		coass 6158	. . . . . . . . . . . . . . . . . 18 ⊢ ((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ ((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})) ∘ 𝑦) = (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦))
96	16, 10	pmtrfinv 18984	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∈ 𝑇 → (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ ((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})) = ( I ↾ 𝐷))
97	81, 96	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ ((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})) = ( I ↾ 𝐷))
98	97	coeq1d 5759	. . . . . . . . . . . . . . . . . . 19 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → ((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ ((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})) ∘ 𝑦) = (( I ↾ 𝐷) ∘ 𝑦))
99		f1of 6700	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦:𝐷–1-1-onto→𝐷 → 𝑦:𝐷⟶𝐷)
100		fcoi2 6633	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦:𝐷⟶𝐷 → (( I ↾ 𝐷) ∘ 𝑦) = 𝑦)
101	71, 99, 100	3syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → (( I ↾ 𝐷) ∘ 𝑦) = 𝑦)
102	98, 101	eqtrd 2778	. . . . . . . . . . . . . . . . . 18 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → ((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ ((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})) ∘ 𝑦) = 𝑦)
103	95, 102	eqtr3id 2793	. . . . . . . . . . . . . . . . 17 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦)) = 𝑦)
104	87, 94, 103	3eqtrd 2782	. . . . . . . . . . . . . . . 16 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+_g‘𝐺)(((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+_g‘𝐺)𝑦)) = 𝑦)
105	104	adantlr 711	. . . . . . . . . . . . . . 15 ⊢ ((((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ ∀𝑧(dom (𝑧 ∖ I ) ≺ dom (𝑦 ∖ I ) → (𝑧 ∈ 𝐵 → 𝑧 ∈ (𝐾‘𝑇)))) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+_g‘𝐺)(((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+_g‘𝐺)𝑦)) = 𝑦)
106	45	ad2antrr 722	. . . . . . . . . . . . . . . 16 ⊢ ((((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ ∀𝑧(dom (𝑧 ∖ I ) ≺ dom (𝑦 ∖ I ) → (𝑧 ∈ 𝐵 → 𝑧 ∈ (𝐾‘𝑇)))) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → (𝐾‘𝑇) ∈ (SubMnd‘𝐺))
107	27	mrcssid 17243	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((SubMnd‘𝐺) ∈ (Moore‘𝐵) ∧ 𝑇 ⊆ 𝐵) → 𝑇 ⊆ (𝐾‘𝑇))
108	43, 11, 107	sylancl 585	. . . . . . . . . . . . . . . . . . 19 ⊢ ((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) → 𝑇 ⊆ (𝐾‘𝑇))
109	108	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → 𝑇 ⊆ (𝐾‘𝑇))
110	109, 81	sseldd 3918	. . . . . . . . . . . . . . . . 17 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → ((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∈ (𝐾‘𝑇))
111	110	adantlr 711	. . . . . . . . . . . . . . . 16 ⊢ ((((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ ∀𝑧(dom (𝑧 ∖ I ) ≺ dom (𝑦 ∖ I ) → (𝑧 ∈ 𝐵 → 𝑧 ∈ (𝐾‘𝑇)))) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → ((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∈ (𝐾‘𝑇))
112	86	difeq1d 4052	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → ((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+_g‘𝐺)𝑦) ∖ I ) = ((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦) ∖ I ))
113	112	dmeqd 5803	. . . . . . . . . . . . . . . . . . 19 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → dom ((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+_g‘𝐺)𝑦) ∖ I ) = dom ((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦) ∖ I ))
114		simpll 763	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → dom (𝑦 ∖ I ) ∈ Fin)
115		mvdco 18968	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ dom ((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦) ∖ I ) ⊆ (dom (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∖ I ) ∪ dom (𝑦 ∖ I ))
116	16	pmtrmvd 18979	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐷 ∈ V ∧ {𝑢, (𝑦‘𝑢)} ⊆ 𝐷 ∧ {𝑢, (𝑦‘𝑢)} ≈ 2_o) → dom (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∖ I ) = {𝑢, (𝑦‘𝑢)})
117	55, 68, 79, 116	syl3anc 1369	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → dom (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∖ I ) = {𝑢, (𝑦‘𝑢)})
118	117, 60	eqsstrd 3955	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → dom (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∖ I ) ⊆ dom (𝑦 ∖ I ))
119		ssidd 3940	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → dom (𝑦 ∖ I ) ⊆ dom (𝑦 ∖ I ))
120	118, 119	unssd 4116	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → (dom (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∖ I ) ∪ dom (𝑦 ∖ I )) ⊆ dom (𝑦 ∖ I ))
121	115, 120	sstrid 3928	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → dom ((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦) ∖ I ) ⊆ dom (𝑦 ∖ I ))
122		fvco2 6847	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑦 Fn 𝐷 ∧ 𝑢 ∈ 𝐷) → ((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦)‘𝑢) = (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})‘(𝑦‘𝑢)))
123	72, 73, 122	syl2anc 583	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → ((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦)‘𝑢) = (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})‘(𝑦‘𝑢)))
124		prcom 4665	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ {𝑢, (𝑦‘𝑢)} = {(𝑦‘𝑢), 𝑢}
125	124	fveq2i 6759	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) = ((pmTrsp‘𝐷)‘{(𝑦‘𝑢), 𝑢})
126	125	fveq1i 6757	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})‘(𝑦‘𝑢)) = (((pmTrsp‘𝐷)‘{(𝑦‘𝑢), 𝑢})‘(𝑦‘𝑢))
127	67, 59	sseldd 3918	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → (𝑦‘𝑢) ∈ 𝐷)
128	16	pmtrprfv 18976	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐷 ∈ V ∧ ((𝑦‘𝑢) ∈ 𝐷 ∧ 𝑢 ∈ 𝐷 ∧ (𝑦‘𝑢) ≠ 𝑢)) → (((pmTrsp‘𝐷)‘{(𝑦‘𝑢), 𝑢})‘(𝑦‘𝑢)) = 𝑢)
129	55, 127, 73, 76, 128	syl13anc 1370	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → (((pmTrsp‘𝐷)‘{(𝑦‘𝑢), 𝑢})‘(𝑦‘𝑢)) = 𝑢)
130	126, 129	eqtrid 2790	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})‘(𝑦‘𝑢)) = 𝑢)
131	123, 130	eqtrd 2778	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → ((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦)‘𝑢) = 𝑢)
132	2, 5	symgbasf1o 18897	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦) ∈ 𝐵 → (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦):𝐷–1-1-onto→𝐷)
133		f1ofn 6701	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦):𝐷–1-1-onto→𝐷 → (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦) Fn 𝐷)
134	92, 132, 133	3syl 18	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦) Fn 𝐷)
135		fnelnfp 7031	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦) Fn 𝐷 ∧ 𝑢 ∈ 𝐷) → (𝑢 ∈ dom ((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦) ∖ I ) ↔ ((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦)‘𝑢) ≠ 𝑢))
136	135	necon2bbid 2986	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦) Fn 𝐷 ∧ 𝑢 ∈ 𝐷) → (((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦)‘𝑢) = 𝑢 ↔ ¬ 𝑢 ∈ dom ((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦) ∖ I )))
137	134, 73, 136	syl2anc 583	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → (((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦)‘𝑢) = 𝑢 ↔ ¬ 𝑢 ∈ dom ((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦) ∖ I )))
138	131, 137	mpbid 231	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → ¬ 𝑢 ∈ dom ((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦) ∖ I ))
139	121, 56, 138	ssnelpssd 4043	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → dom ((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦) ∖ I ) ⊊ dom (𝑦 ∖ I ))
140		php3 8899	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((dom (𝑦 ∖ I ) ∈ Fin ∧ dom ((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦) ∖ I ) ⊊ dom (𝑦 ∖ I )) → dom ((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦) ∖ I ) ≺ dom (𝑦 ∖ I ))
141	114, 139, 140	syl2anc 583	. . . . . . . . . . . . . . . . . . 19 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → dom ((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦) ∖ I ) ≺ dom (𝑦 ∖ I ))
142	113, 141	eqbrtrd 5092	. . . . . . . . . . . . . . . . . 18 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → dom ((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+_g‘𝐺)𝑦) ∖ I ) ≺ dom (𝑦 ∖ I ))
143	142	adantlr 711	. . . . . . . . . . . . . . . . 17 ⊢ ((((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ ∀𝑧(dom (𝑧 ∖ I ) ≺ dom (𝑦 ∖ I ) → (𝑧 ∈ 𝐵 → 𝑧 ∈ (𝐾‘𝑇)))) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → dom ((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+_g‘𝐺)𝑦) ∖ I ) ≺ dom (𝑦 ∖ I ))
144	91	adantlr 711	. . . . . . . . . . . . . . . . 17 ⊢ ((((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ ∀𝑧(dom (𝑧 ∖ I ) ≺ dom (𝑦 ∖ I ) → (𝑧 ∈ 𝐵 → 𝑧 ∈ (𝐾‘𝑇)))) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+_g‘𝐺)𝑦) ∈ 𝐵)
145		ovex 7288	. . . . . . . . . . . . . . . . . . 19 ⊢ (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+_g‘𝐺)𝑦) ∈ V
146		difeq1 4046	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑧 = (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+_g‘𝐺)𝑦) → (𝑧 ∖ I ) = ((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+_g‘𝐺)𝑦) ∖ I ))
147	146	dmeqd 5803	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 = (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+_g‘𝐺)𝑦) → dom (𝑧 ∖ I ) = dom ((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+_g‘𝐺)𝑦) ∖ I ))
148	147	breq1d 5080	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 = (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+_g‘𝐺)𝑦) → (dom (𝑧 ∖ I ) ≺ dom (𝑦 ∖ I ) ↔ dom ((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+_g‘𝐺)𝑦) ∖ I ) ≺ dom (𝑦 ∖ I )))
149		eleq1 2826	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 = (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+_g‘𝐺)𝑦) → (𝑧 ∈ 𝐵 ↔ (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+_g‘𝐺)𝑦) ∈ 𝐵))
150		eleq1 2826	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 = (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+_g‘𝐺)𝑦) → (𝑧 ∈ (𝐾‘𝑇) ↔ (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+_g‘𝐺)𝑦) ∈ (𝐾‘𝑇)))
151	149, 150	imbi12d 344	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 = (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+_g‘𝐺)𝑦) → ((𝑧 ∈ 𝐵 → 𝑧 ∈ (𝐾‘𝑇)) ↔ ((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+_g‘𝐺)𝑦) ∈ 𝐵 → (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+_g‘𝐺)𝑦) ∈ (𝐾‘𝑇))))
152	148, 151	imbi12d 344	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+_g‘𝐺)𝑦) → ((dom (𝑧 ∖ I ) ≺ dom (𝑦 ∖ I ) → (𝑧 ∈ 𝐵 → 𝑧 ∈ (𝐾‘𝑇))) ↔ (dom ((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+_g‘𝐺)𝑦) ∖ I ) ≺ dom (𝑦 ∖ I ) → ((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+_g‘𝐺)𝑦) ∈ 𝐵 → (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+_g‘𝐺)𝑦) ∈ (𝐾‘𝑇)))))
153	145, 152	spcv 3534	. . . . . . . . . . . . . . . . . 18 ⊢ (∀𝑧(dom (𝑧 ∖ I ) ≺ dom (𝑦 ∖ I ) → (𝑧 ∈ 𝐵 → 𝑧 ∈ (𝐾‘𝑇))) → (dom ((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+_g‘𝐺)𝑦) ∖ I ) ≺ dom (𝑦 ∖ I ) → ((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+_g‘𝐺)𝑦) ∈ 𝐵 → (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+_g‘𝐺)𝑦) ∈ (𝐾‘𝑇))))
154	153	ad2antlr 723	. . . . . . . . . . . . . . . . 17 ⊢ ((((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ ∀𝑧(dom (𝑧 ∖ I ) ≺ dom (𝑦 ∖ I ) → (𝑧 ∈ 𝐵 → 𝑧 ∈ (𝐾‘𝑇)))) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → (dom ((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+_g‘𝐺)𝑦) ∖ I ) ≺ dom (𝑦 ∖ I ) → ((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+_g‘𝐺)𝑦) ∈ 𝐵 → (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+_g‘𝐺)𝑦) ∈ (𝐾‘𝑇))))
155	143, 144, 154	mp2d 49	. . . . . . . . . . . . . . . 16 ⊢ ((((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ ∀𝑧(dom (𝑧 ∖ I ) ≺ dom (𝑦 ∖ I ) → (𝑧 ∈ 𝐵 → 𝑧 ∈ (𝐾‘𝑇)))) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+_g‘𝐺)𝑦) ∈ (𝐾‘𝑇))
156	84	submcl 18366	. . . . . . . . . . . . . . . 16 ⊢ (((𝐾‘𝑇) ∈ (SubMnd‘𝐺) ∧ ((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∈ (𝐾‘𝑇) ∧ (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+_g‘𝐺)𝑦) ∈ (𝐾‘𝑇)) → (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+_g‘𝐺)(((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+_g‘𝐺)𝑦)) ∈ (𝐾‘𝑇))
157	106, 111, 155, 156	syl3anc 1369	. . . . . . . . . . . . . . 15 ⊢ ((((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ ∀𝑧(dom (𝑧 ∖ I ) ≺ dom (𝑦 ∖ I ) → (𝑧 ∈ 𝐵 → 𝑧 ∈ (𝐾‘𝑇)))) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+_g‘𝐺)(((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+_g‘𝐺)𝑦)) ∈ (𝐾‘𝑇))
158	105, 157	eqeltrrd 2840	. . . . . . . . . . . . . 14 ⊢ ((((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ ∀𝑧(dom (𝑧 ∖ I ) ≺ dom (𝑦 ∖ I ) → (𝑧 ∈ 𝐵 → 𝑧 ∈ (𝐾‘𝑇)))) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → 𝑦 ∈ (𝐾‘𝑇))
159	158	ex 412	. . . . . . . . . . . . 13 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ ∀𝑧(dom (𝑧 ∖ I ) ≺ dom (𝑦 ∖ I ) → (𝑧 ∈ 𝐵 → 𝑧 ∈ (𝐾‘𝑇)))) → (𝑢 ∈ dom (𝑦 ∖ I ) → 𝑦 ∈ (𝐾‘𝑇)))
160	159	exlimdv 1937	. . . . . . . . . . . 12 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ ∀𝑧(dom (𝑧 ∖ I ) ≺ dom (𝑦 ∖ I ) → (𝑧 ∈ 𝐵 → 𝑧 ∈ (𝐾‘𝑇)))) → (∃𝑢 𝑢 ∈ dom (𝑦 ∖ I ) → 𝑦 ∈ (𝐾‘𝑇)))
161	54, 160	syl5bi 241	. . . . . . . . . . 11 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ ∀𝑧(dom (𝑧 ∖ I ) ≺ dom (𝑦 ∖ I ) → (𝑧 ∈ 𝐵 → 𝑧 ∈ (𝐾‘𝑇)))) → (dom (𝑦 ∖ I ) ≠ ∅ → 𝑦 ∈ (𝐾‘𝑇)))
162	53, 161	pm2.61dne 3030	. . . . . . . . . 10 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ ∀𝑧(dom (𝑧 ∖ I ) ≺ dom (𝑦 ∖ I ) → (𝑧 ∈ 𝐵 → 𝑧 ∈ (𝐾‘𝑇)))) → 𝑦 ∈ (𝐾‘𝑇))
163	162	exp31 419	. . . . . . . . 9 ⊢ (dom (𝑦 ∖ I ) ∈ Fin → (𝑦 ∈ 𝐵 → (∀𝑧(dom (𝑧 ∖ I ) ≺ dom (𝑦 ∖ I ) → (𝑧 ∈ 𝐵 → 𝑧 ∈ (𝐾‘𝑇))) → 𝑦 ∈ (𝐾‘𝑇))))
164	163	com23 86	. . . . . . . 8 ⊢ (dom (𝑦 ∖ I ) ∈ Fin → (∀𝑧(dom (𝑧 ∖ I ) ≺ dom (𝑦 ∖ I ) → (𝑧 ∈ 𝐵 → 𝑧 ∈ (𝐾‘𝑇))) → (𝑦 ∈ 𝐵 → 𝑦 ∈ (𝐾‘𝑇))))
165	33, 164	syl 17	. . . . . . 7 ⊢ ((dom (𝑥 ∖ I ) ∈ Fin ∧ dom (𝑦 ∖ I ) ≼ dom (𝑥 ∖ I )) → (∀𝑧(dom (𝑧 ∖ I ) ≺ dom (𝑦 ∖ I ) → (𝑧 ∈ 𝐵 → 𝑧 ∈ (𝐾‘𝑇))) → (𝑦 ∈ 𝐵 → 𝑦 ∈ (𝐾‘𝑇))))
166	165	3impia 1115	. . . . . 6 ⊢ ((dom (𝑥 ∖ I ) ∈ Fin ∧ dom (𝑦 ∖ I ) ≼ dom (𝑥 ∖ I ) ∧ ∀𝑧(dom (𝑧 ∖ I ) ≺ dom (𝑦 ∖ I ) → (𝑧 ∈ 𝐵 → 𝑧 ∈ (𝐾‘𝑇)))) → (𝑦 ∈ 𝐵 → 𝑦 ∈ (𝐾‘𝑇)))
167		eleq1w 2821	. . . . . . 7 ⊢ (𝑦 = 𝑧 → (𝑦 ∈ 𝐵 ↔ 𝑧 ∈ 𝐵))
168		eleq1w 2821	. . . . . . 7 ⊢ (𝑦 = 𝑧 → (𝑦 ∈ (𝐾‘𝑇) ↔ 𝑧 ∈ (𝐾‘𝑇)))
169	167, 168	imbi12d 344	. . . . . 6 ⊢ (𝑦 = 𝑧 → ((𝑦 ∈ 𝐵 → 𝑦 ∈ (𝐾‘𝑇)) ↔ (𝑧 ∈ 𝐵 → 𝑧 ∈ (𝐾‘𝑇))))
170		eleq1w 2821	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ 𝐵 ↔ 𝑥 ∈ 𝐵))
171		eleq1w 2821	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ (𝐾‘𝑇) ↔ 𝑥 ∈ (𝐾‘𝑇)))
172	170, 171	imbi12d 344	. . . . . 6 ⊢ (𝑦 = 𝑥 → ((𝑦 ∈ 𝐵 → 𝑦 ∈ (𝐾‘𝑇)) ↔ (𝑥 ∈ 𝐵 → 𝑥 ∈ (𝐾‘𝑇))))
173		difeq1 4046	. . . . . . 7 ⊢ (𝑦 = 𝑧 → (𝑦 ∖ I ) = (𝑧 ∖ I ))
174	173	dmeqd 5803	. . . . . 6 ⊢ (𝑦 = 𝑧 → dom (𝑦 ∖ I ) = dom (𝑧 ∖ I ))
175		difeq1 4046	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (𝑦 ∖ I ) = (𝑥 ∖ I ))
176	175	dmeqd 5803	. . . . . 6 ⊢ (𝑦 = 𝑥 → dom (𝑦 ∖ I ) = dom (𝑥 ∖ I ))
177	31, 32, 166, 169, 172, 174, 176	indcardi 9728	. . . . 5 ⊢ (dom (𝑥 ∖ I ) ∈ Fin → (𝑥 ∈ 𝐵 → 𝑥 ∈ (𝐾‘𝑇)))
178	177	impcom 407	. . . 4 ⊢ ((𝑥 ∈ 𝐵 ∧ dom (𝑥 ∖ I ) ∈ Fin) → 𝑥 ∈ (𝐾‘𝑇))
179	178	3adant1 1128	. . 3 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵 ∧ dom (𝑥 ∖ I ) ∈ Fin) → 𝑥 ∈ (𝐾‘𝑇))
180	179	rabssdv 4004	. 2 ⊢ (𝐷 ∈ 𝑉 → {𝑥 ∈ 𝐵 ∣ dom (𝑥 ∖ I ) ∈ Fin} ⊆ (𝐾‘𝑇))
181	29, 180	eqssd 3934	1 ⊢ (𝐷 ∈ 𝑉 → (𝐾‘𝑇) = {𝑥 ∈ 𝐵 ∣ dom (𝑥 ∖ I ) ∈ Fin})

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∀wal 1537 = wceq 1539 ∃wex 1783 ∈ wcel 2108 ≠ wne 2942 {crab 3067 Vcvv 3422 ∖ cdif 3880 ∪ cun 3881 ⊆ wss 3883 ⊊ wpss 3884 ∅c0 4253 {csn 4558 {cpr 4560 class class class wbr 5070 I cid 5479 dom cdm 5580 ran crn 5581 ↾ cres 5582 ∘ ccom 5584 Fn wfn 6413 ⟶wf 6414 –1-1-onto→wf1o 6417 ‘cfv 6418 (class class class)co 7255 ωcom 7687 2_oc2o 8261 ≈ cen 8688 ≼ cdom 8689 ≺ csdm 8690 Fincfn 8691 Basecbs 16840 +_gcplusg 16888 0_gc0g 17067 Moorecmre 17208 mrClscmrc 17209 ACScacs 17211 Mndcmnd 18300 SubMndcsubmnd 18344 Grpcgrp 18492 SubGrpcsubg 18664 SymGrpcsymg 18889 pmTrspcpmtr 18964

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-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 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-nel 3049 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-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-iin 4924 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 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-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-2o 8268 df-er 8456 df-map 8575 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-z 12250 df-uz 12512 df-fz 13169 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-tset 16907 df-0g 17069 df-mre 17212 df-mrc 17213 df-acs 17215 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-submnd 18346 df-efmnd 18423 df-grp 18495 df-minusg 18496 df-subg 18667 df-symg 18890 df-pmtr 18965

This theorem is referenced by: symggen2 18994 psgneldm2 19027

W3C validator