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Theorem symggen 19436

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 3452	. . . 4 ⊢ (𝐷 ∈ 𝑉 → 𝐷 ∈ V)
2		symgtrf.g	. . . . . . 7 ⊢ 𝐺 = (SymGrp‘𝐷)
3	2	symggrp 19366	. . . . . 6 ⊢ (𝐷 ∈ V → 𝐺 ∈ Grp)
4	3	grpmndd 18913	. . . . 5 ⊢ (𝐷 ∈ V → 𝐺 ∈ Mnd)
5		symgtrf.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐺)
6	5	submacs 18786	. . . . 5 ⊢ (𝐺 ∈ Mnd → (SubMnd‘𝐺) ∈ (ACS‘𝐵))
7		acsmre 17609	. . . . 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 19435	. . . . 5 ⊢ 𝑇 ⊆ 𝐵
12	11	a1i 11	. . . 4 ⊢ (𝐷 ∈ 𝑉 → 𝑇 ⊆ 𝐵)
13		2onn 8568	. . . . . 6 ⊢ 2_o ∈ ω
14		nnfi 9092	. . . . . 6 ⊢ (2_o ∈ ω → 2_o ∈ Fin)
15	13, 14	ax-mp 5	. . . . 5 ⊢ 2_o ∈ Fin
16		eqid 2739	. . . . . . . . 9 ⊢ (pmTrsp‘𝐷) = (pmTrsp‘𝐷)
17	16, 10	pmtrfb 19431	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑇 ↔ (𝐷 ∈ V ∧ 𝑥:𝐷–1-1-onto→𝐷 ∧ dom (𝑥 ∖ I ) ≈ 2_o))
18	17	simp3bi 1153	. . . . . . 7 ⊢ (𝑥 ∈ 𝑇 → dom (𝑥 ∖ I ) ≈ 2_o)
19		enfi 9111	. . . . . . 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 482	. . . . 5 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑥 ∈ 𝑇) → (dom (𝑥 ∖ I ) ∈ Fin ↔ 2_o ∈ Fin))
22	15, 21	mpbiri 259	. . . 4 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑥 ∈ 𝑇) → dom (𝑥 ∖ I ) ∈ Fin)
23	12, 22	ssrabdv 4004	. . 3 ⊢ (𝐷 ∈ 𝑉 → 𝑇 ⊆ {𝑥 ∈ 𝐵 ∣ dom (𝑥 ∖ I ) ∈ Fin})
24	2, 5	symgfisg 19434	. . . 4 ⊢ (𝐷 ∈ 𝑉 → {𝑥 ∈ 𝐵 ∣ dom (𝑥 ∖ I ) ∈ Fin} ∈ (SubGrp‘𝐺))
25		subgsubm 19115	. . . 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 17577	. . 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 1379	. 2 ⊢ (𝐷 ∈ 𝑉 → (𝐾‘𝑇) ⊆ {𝑥 ∈ 𝐵 ∣ dom (𝑥 ∖ I ) ∈ Fin})
30		vex 3435	. . . . . . 7 ⊢ 𝑥 ∈ V
31	30	a1i 11	. . . . . 6 ⊢ (dom (𝑥 ∖ I ) ∈ Fin → 𝑥 ∈ V)
32		finnum 9863	. . . . . 6 ⊢ (dom (𝑥 ∖ I ) ∈ Fin → dom (𝑥 ∖ I ) ∈ dom card)
33		domfi 9113	. . . . . . . 8 ⊢ ((dom (𝑥 ∖ I ) ∈ Fin ∧ dom (𝑦 ∖ I ) ≼ dom (𝑥 ∖ I )) → dom (𝑦 ∖ I ) ∈ Fin)
34	2, 5	symgbasf1o 19341	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ 𝐵 → 𝑦:𝐷–1-1-onto→𝐷)
35	34	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) → 𝑦:𝐷–1-1-onto→𝐷)
36		f1ofn 6768	. . . . . . . . . . . . . 14 ⊢ (𝑦:𝐷–1-1-onto→𝐷 → 𝑦 Fn 𝐷)
37		fnnfpeq0 7122	. . . . . . . . . . . . . 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 17176	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ 𝐵 → 𝐷 ∈ V)
40	39	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ ((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) → 𝐷 ∈ V)
41	2	symgid 19367	. . . . . . . . . . . . . . . 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 17568	. . . . . . . . . . . . . . . . 17 ⊢ (((SubMnd‘𝐺) ∈ (Moore‘𝐵) ∧ 𝑇 ⊆ 𝐵) → (𝐾‘𝑇) ∈ (SubMnd‘𝐺))
45	43, 11, 44	sylancl 592	. . . . . . . . . . . . . . . 16 ⊢ ((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) → (𝐾‘𝑇) ∈ (SubMnd‘𝐺))
46		eqid 2739	. . . . . . . . . . . . . . . . 17 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
47	46	subm0cl 18770	. . . . . . . . . . . . . . . 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 241	. . . . . . . . . . . 12 ⊢ ((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) → (dom (𝑦 ∖ I ) = ∅ → 𝑦 ∈ (𝐾‘𝑇)))
53	52	adantr 481	. . . . . . . . . . 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 4281	. . . . . . . . . . . 12 ⊢ (dom (𝑦 ∖ I ) ≠ ∅ ↔ ∃𝑢 𝑢 ∈ dom (𝑦 ∖ I ))
55	40	adantr 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → 𝐷 ∈ V)
56		simpr 485	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → 𝑢 ∈ dom (𝑦 ∖ I ))
57		f1omvdmvd 19409	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑦:𝐷–1-1-onto→𝐷 ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → (𝑦‘𝑢) ∈ (dom (𝑦 ∖ I ) ∖ {𝑢}))
58	35, 57	sylan 586	. . . . . . . . . . . . . . . . . . . . . . . 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 4753	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → {𝑢, (𝑦‘𝑢)} ⊆ dom (𝑦 ∖ I ))
61		difss 4066	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 ∖ I ) ⊆ 𝑦
62		dmss 5844	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑦 ∖ I ) ⊆ 𝑦 → dom (𝑦 ∖ I ) ⊆ dom 𝑦)
63	61, 62	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ dom (𝑦 ∖ I ) ⊆ dom 𝑦
64		f1odm 6771	. . . . . . . . . . . . . . . . . . . . . . . . 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 3957	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) → dom (𝑦 ∖ I ) ⊆ 𝐷)
67	66	adantr 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → dom (𝑦 ∖ I ) ⊆ 𝐷)
68	60, 67	sstrd 3925	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → {𝑢, (𝑦‘𝑢)} ⊆ 𝐷)
69		vex 3435	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝑢 ∈ V
70		fvex 6840	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦‘𝑢) ∈ V
71	35	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . 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 3915	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → 𝑢 ∈ 𝐷)
74		fnelnfp 7121	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑦 Fn 𝐷 ∧ 𝑢 ∈ 𝐷) → (𝑢 ∈ dom (𝑦 ∖ I ) ↔ (𝑦‘𝑢) ≠ 𝑢))
75	72, 73, 74	syl2anc 590	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → (𝑢 ∈ dom (𝑦 ∖ I ) ↔ (𝑦‘𝑢) ≠ 𝑢))
76	56, 75	mpbid 233	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → (𝑦‘𝑢) ≠ 𝑢)
77	76	necomd 2989	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → 𝑢 ≠ (𝑦‘𝑢))
78		enpr2 9917	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑢 ∈ V ∧ (𝑦‘𝑢) ∈ V ∧ 𝑢 ≠ (𝑦‘𝑢)) → {𝑢, (𝑦‘𝑢)} ≈ 2_o)
79	69, 70, 77, 78	mp3an12i 1473	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → {𝑢, (𝑦‘𝑢)} ≈ 2_o)
80	16, 10	pmtrrn 19423	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐷 ∈ V ∧ {𝑢, (𝑦‘𝑢)} ⊆ 𝐷 ∧ {𝑢, (𝑦‘𝑢)} ≈ 2_o) → ((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∈ 𝑇)
81	55, 68, 79, 80	syl3anc 1379	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → ((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∈ 𝑇)
82	11, 81	sselid 3913	. . . . . . . . . . . . . . . . . . 19 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → ((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∈ 𝐵)
83		simplr 774	. . . . . . . . . . . . . . . . . . 19 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → 𝑦 ∈ 𝐵)
84		eqid 2739	. . . . . . . . . . . . . . . . . . . 20 ⊢ (+_g‘𝐺) = (+_g‘𝐺)
85	2, 5, 84	symgov 19350	. . . . . . . . . . . . . . . . . . 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 590	. . . . . . . . . . . . . . . . . 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 7372	. . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → 𝐺 ∈ Grp)
90	5, 84	grpcl 18908	. . . . . . . . . . . . . . . . . . . 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 1379	. . . . . . . . . . . . . . . . . . 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 19350	. . . . . . . . . . . . . . . . . 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 590	. . . . . . . . . . . . . . . . 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 6217	. . . . . . . . . . . . . . . . . 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 19427	. . . . . . . . . . . . . . . . . . . . 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 5803	. . . . . . . . . . . . . . . . . . 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 6767	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦:𝐷–1-1-onto→𝐷 → 𝑦:𝐷⟶𝐷)
100		fcoi2 6702	. . . . . . . . . . . . . . . . . . . 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 2774	. . . . . . . . . . . . . . . . . 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 2788	. . . . . . . . . . . . . . . . 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 2778	. . . . . . . . . . . . . . . 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 721	. . . . . . . . . . . . . . 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 732	. . . . . . . . . . . . . . . 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 17574	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((SubMnd‘𝐺) ∈ (Moore‘𝐵) ∧ 𝑇 ⊆ 𝐵) → 𝑇 ⊆ (𝐾‘𝑇))
108	43, 11, 107	sylancl 592	. . . . . . . . . . . . . . . . . . 19 ⊢ ((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) → 𝑇 ⊆ (𝐾‘𝑇))
109	108	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → 𝑇 ⊆ (𝐾‘𝑇))
110	109, 81	sseldd 3916	. . . . . . . . . . . . . . . . 17 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → ((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∈ (𝐾‘𝑇))
111	110	adantlr 721	. . . . . . . . . . . . . . . 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 4056	. . . . . . . . . . . . . . . . . . . 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 5847	. . . . . . . . . . . . . . . . . . 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 772	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → dom (𝑦 ∖ I ) ∈ Fin)
115		mvdco 19411	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ dom ((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦) ∖ I ) ⊆ (dom (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∖ I ) ∪ dom (𝑦 ∖ I ))
116	16	pmtrmvd 19422	. . . . . . . . . . . . . . . . . . . . . . . . 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 1379	. . . . . . . . . . . . . . . . . . . . . . . 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 3949	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → dom (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∖ I ) ⊆ dom (𝑦 ∖ I ))
119		ssidd 3938	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → dom (𝑦 ∖ I ) ⊆ dom (𝑦 ∖ I ))
120	118, 119	unssd 4121	. . . . . . . . . . . . . . . . . . . . . 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 3926	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → dom ((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦) ∖ I ) ⊆ dom (𝑦 ∖ I ))
122		fvco2 6924	. . . . . . . . . . . . . . . . . . . . . . . 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 590	. . . . . . . . . . . . . . . . . . . . . . 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 4664	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ {𝑢, (𝑦‘𝑢)} = {(𝑦‘𝑢), 𝑢}
125	124	fveq2i 6830	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) = ((pmTrsp‘𝐷)‘{(𝑦‘𝑢), 𝑢})
126	125	fveq1i 6828	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})‘(𝑦‘𝑢)) = (((pmTrsp‘𝐷)‘{(𝑦‘𝑢), 𝑢})‘(𝑦‘𝑢))
127	67, 59	sseldd 3916	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → (𝑦‘𝑢) ∈ 𝐷)
128	16	pmtrprfv 19419	. . . . . . . . . . . . . . . . . . . . . . . . 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 1380	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → (((pmTrsp‘𝐷)‘{(𝑦‘𝑢), 𝑢})‘(𝑦‘𝑢)) = 𝑢)
130	126, 129	eqtrid 2786	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})‘(𝑦‘𝑢)) = 𝑢)
131	123, 130	eqtrd 2774	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → ((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦)‘𝑢) = 𝑢)
132	2, 5	symgbasf1o 19341	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦) ∈ 𝐵 → (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦):𝐷–1-1-onto→𝐷)
133		f1ofn 6768	. . . . . . . . . . . . . . . . . . . . . . . 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 7121	. . . . . . . . . . . . . . . . . . . . . . . 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 2977	. . . . . . . . . . . . . . . . . . . . . . 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 590	. . . . . . . . . . . . . . . . . . . . . 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 233	. . . . . . . . . . . . . . . . . . . . 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 4046	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → dom ((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦) ∖ I ) ⊊ dom (𝑦 ∖ I ))
140		php3 9133	. . . . . . . . . . . . . . . . . . . 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 590	. . . . . . . . . . . . . . . . . . 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 5094	. . . . . . . . . . . . . . . . . 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 721	. . . . . . . . . . . . . . . . 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 721	. . . . . . . . . . . . . . . . 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 7389	. . . . . . . . . . . . . . . . . . 19 ⊢ (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+_g‘𝐺)𝑦) ∈ V
146		difeq1 4050	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑧 = (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+_g‘𝐺)𝑦) → (𝑧 ∖ I ) = ((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+_g‘𝐺)𝑦) ∖ I ))
147	146	dmeqd 5847	. . . . . . . . . . . . . . . . . . . . 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 5082	. . . . . . . . . . . . . . . . . . . 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 2827	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 = (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+_g‘𝐺)𝑦) → (𝑧 ∈ 𝐵 ↔ (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+_g‘𝐺)𝑦) ∈ 𝐵))
150		eleq1 2827	. . . . . . . . . . . . . . . . . . . . 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 345	. . . . . . . . . . . . . . . . . . . 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 345	. . . . . . . . . . . . . . . . . . 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 3543	. . . . . . . . . . . . . . . . . 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 733	. . . . . . . . . . . . . . . . 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 18771	. . . . . . . . . . . . . . . 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 1379	. . . . . . . . . . . . . . 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 413	. . . . . . . . . . . . 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 1940	. . . . . . . . . . . 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	biimtrid 243	. . . . . . . . . . 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 3020	. . . . . . . . . 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 420	. . . . . . . . 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 1123	. . . . . 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 2822	. . . . . . 7 ⊢ (𝑦 = 𝑧 → (𝑦 ∈ 𝐵 ↔ 𝑧 ∈ 𝐵))
168		eleq1w 2822	. . . . . . 7 ⊢ (𝑦 = 𝑧 → (𝑦 ∈ (𝐾‘𝑇) ↔ 𝑧 ∈ (𝐾‘𝑇)))
169	167, 168	imbi12d 345	. . . . . 6 ⊢ (𝑦 = 𝑧 → ((𝑦 ∈ 𝐵 → 𝑦 ∈ (𝐾‘𝑇)) ↔ (𝑧 ∈ 𝐵 → 𝑧 ∈ (𝐾‘𝑇))))
170		eleq1w 2822	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ 𝐵 ↔ 𝑥 ∈ 𝐵))
171		eleq1w 2822	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ (𝐾‘𝑇) ↔ 𝑥 ∈ (𝐾‘𝑇)))
172	170, 171	imbi12d 345	. . . . . 6 ⊢ (𝑦 = 𝑥 → ((𝑦 ∈ 𝐵 → 𝑦 ∈ (𝐾‘𝑇)) ↔ (𝑥 ∈ 𝐵 → 𝑥 ∈ (𝐾‘𝑇))))
173		difeq1 4050	. . . . . . 7 ⊢ (𝑦 = 𝑧 → (𝑦 ∖ I ) = (𝑧 ∖ I ))
174	173	dmeqd 5847	. . . . . 6 ⊢ (𝑦 = 𝑧 → dom (𝑦 ∖ I ) = dom (𝑧 ∖ I ))
175		difeq1 4050	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (𝑦 ∖ I ) = (𝑥 ∖ I ))
176	175	dmeqd 5847	. . . . . 6 ⊢ (𝑦 = 𝑥 → dom (𝑦 ∖ I ) = dom (𝑥 ∖ I ))
177	31, 32, 166, 169, 172, 174, 176	indcardi 9954	. . . . 5 ⊢ (dom (𝑥 ∖ I ) ∈ Fin → (𝑥 ∈ 𝐵 → 𝑥 ∈ (𝐾‘𝑇)))
178	177	impcom 408	. . . 4 ⊢ ((𝑥 ∈ 𝐵 ∧ dom (𝑥 ∖ I ) ∈ Fin) → 𝑥 ∈ (𝐾‘𝑇))
179	178	3adant1 1136	. . 3 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵 ∧ dom (𝑥 ∖ I ) ∈ Fin) → 𝑥 ∈ (𝐾‘𝑇))
180	179	rabssdv 4005	. 2 ⊢ (𝐷 ∈ 𝑉 → {𝑥 ∈ 𝐵 ∣ dom (𝑥 ∖ I ) ∈ Fin} ⊆ (𝐾‘𝑇))
181	29, 180	eqssd 3932	1 ⊢ (𝐷 ∈ 𝑉 → (𝐾‘𝑇) = {𝑥 ∈ 𝐵 ∣ dom (𝑥 ∖ I ) ∈ Fin})

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∧ wa 396 ∀wal 1545 = wceq 1547 ∃wex 1786 ∈ wcel 2119 ≠ wne 2934 {crab 3391 Vcvv 3431 ∖ cdif 3880 ∪ cun 3881 ⊆ wss 3883 ⊊ wpss 3884 ∅c0 4261 {csn 4555 {cpr 4557 class class class wbr 5072 I cid 5512 dom cdm 5618 ran crn 5619 ↾ cres 5620 ∘ ccom 5622 Fn wfn 6480 ⟶wf 6481 –1-1-onto→wf1o 6484 ‘cfv 6485 (class class class)co 7356 ωcom 7806 2_oc2o 8389 ≈ cen 8880 ≼ cdom 8881 ≺ csdm 8882 Fincfn 8883 Basecbs 17170 +_gcplusg 17211 0_gc0g 17393 Moorecmre 17535 mrClscmrc 17536 ACScacs 17538 Mndcmnd 18693 SubMndcsubmnd 18741 Grpcgrp 18900 SubGrpcsubg 19087 SymGrpcsymg 19335 pmTrspcpmtr 19407

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5199 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-tp 4560 df-op 4562 df-uni 4839 df-int 4878 df-iun 4923 df-iin 4924 df-br 5073 df-opab 5135 df-mpt 5154 df-tr 5180 df-id 5513 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5571 df-se 5572 df-we 5573 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-pred 6252 df-ord 6313 df-on 6314 df-lim 6315 df-suc 6316 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-isom 6494 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-er 8633 df-map 8765 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-card 9854 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-5 12238 df-6 12239 df-7 12240 df-8 12241 df-9 12242 df-n0 12429 df-z 12516 df-uz 12780 df-fz 13453 df-struct 17108 df-sets 17125 df-slot 17143 df-ndx 17155 df-base 17171 df-ress 17192 df-plusg 17224 df-tset 17230 df-0g 17395 df-mre 17539 df-mrc 17540 df-acs 17542 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-submnd 18743 df-efmnd 18828 df-grp 18903 df-minusg 18904 df-subg 19090 df-symg 19336 df-pmtr 19408

This theorem is referenced by: symggen2 19437 psgneldm2 19470

W3C validator