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Theorem symggen 17890

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 3212	. . . 4 ⊢ (𝐷 ∈ 𝑉 → 𝐷 ∈ V)
2		symgtrf.g	. . . . . . 7 ⊢ 𝐺 = (SymGrp‘𝐷)
3	2	symggrp 17820	. . . . . 6 ⊢ (𝐷 ∈ V → 𝐺 ∈ Grp)
4		grpmnd 17429	. . . . . 6 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ Mnd)
5	3, 4	syl 17	. . . . 5 ⊢ (𝐷 ∈ V → 𝐺 ∈ Mnd)
6		symgtrf.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐺)
7	6	submacs 17365	. . . . 5 ⊢ (𝐺 ∈ Mnd → (SubMnd‘𝐺) ∈ (ACS‘𝐵))
8		acsmre 16313	. . . . 5 ⊢ ((SubMnd‘𝐺) ∈ (ACS‘𝐵) → (SubMnd‘𝐺) ∈ (Moore‘𝐵))
9	5, 7, 8	3syl 18	. . . 4 ⊢ (𝐷 ∈ V → (SubMnd‘𝐺) ∈ (Moore‘𝐵))
10	1, 9	syl 17	. . 3 ⊢ (𝐷 ∈ 𝑉 → (SubMnd‘𝐺) ∈ (Moore‘𝐵))
11		symgtrf.t	. . . . . 6 ⊢ 𝑇 = ran (pmTrsp‘𝐷)
12	11, 2, 6	symgtrf 17889	. . . . 5 ⊢ 𝑇 ⊆ 𝐵
13	12	a1i 11	. . . 4 ⊢ (𝐷 ∈ 𝑉 → 𝑇 ⊆ 𝐵)
14		2onn 7720	. . . . . 6 ⊢ 2_𝑜 ∈ ω
15		nnfi 8153	. . . . . 6 ⊢ (2_𝑜 ∈ ω → 2_𝑜 ∈ Fin)
16	14, 15	ax-mp 5	. . . . 5 ⊢ 2_𝑜 ∈ Fin
17		eqid 2622	. . . . . . . . 9 ⊢ (pmTrsp‘𝐷) = (pmTrsp‘𝐷)
18	17, 11	pmtrfb 17885	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑇 ↔ (𝐷 ∈ V ∧ 𝑥:𝐷–1-1-onto→𝐷 ∧ dom (𝑥 ∖ I ) ≈ 2_𝑜))
19	18	simp3bi 1078	. . . . . . 7 ⊢ (𝑥 ∈ 𝑇 → dom (𝑥 ∖ I ) ≈ 2_𝑜)
20		enfi 8176	. . . . . . 7 ⊢ (dom (𝑥 ∖ I ) ≈ 2_𝑜 → (dom (𝑥 ∖ I ) ∈ Fin ↔ 2_𝑜 ∈ Fin))
21	19, 20	syl 17	. . . . . 6 ⊢ (𝑥 ∈ 𝑇 → (dom (𝑥 ∖ I ) ∈ Fin ↔ 2_𝑜 ∈ Fin))
22	21	adantl 482	. . . . 5 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑥 ∈ 𝑇) → (dom (𝑥 ∖ I ) ∈ Fin ↔ 2_𝑜 ∈ Fin))
23	16, 22	mpbiri 248	. . . 4 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑥 ∈ 𝑇) → dom (𝑥 ∖ I ) ∈ Fin)
24	13, 23	ssrabdv 3681	. . 3 ⊢ (𝐷 ∈ 𝑉 → 𝑇 ⊆ {𝑥 ∈ 𝐵 ∣ dom (𝑥 ∖ I ) ∈ Fin})
25	2, 6	symgfisg 17888	. . . 4 ⊢ (𝐷 ∈ 𝑉 → {𝑥 ∈ 𝐵 ∣ dom (𝑥 ∖ I ) ∈ Fin} ∈ (SubGrp‘𝐺))
26		subgsubm 17616	. . . 4 ⊢ ({𝑥 ∈ 𝐵 ∣ dom (𝑥 ∖ I ) ∈ Fin} ∈ (SubGrp‘𝐺) → {𝑥 ∈ 𝐵 ∣ dom (𝑥 ∖ I ) ∈ Fin} ∈ (SubMnd‘𝐺))
27	25, 26	syl 17	. . 3 ⊢ (𝐷 ∈ 𝑉 → {𝑥 ∈ 𝐵 ∣ dom (𝑥 ∖ I ) ∈ Fin} ∈ (SubMnd‘𝐺))
28		symggen.k	. . . 4 ⊢ 𝐾 = (mrCls‘(SubMnd‘𝐺))
29	28	mrcsscl 16280	. . 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})
30	10, 24, 27, 29	syl3anc 1326	. 2 ⊢ (𝐷 ∈ 𝑉 → (𝐾‘𝑇) ⊆ {𝑥 ∈ 𝐵 ∣ dom (𝑥 ∖ I ) ∈ Fin})
31		vex 3203	. . . . . . 7 ⊢ 𝑥 ∈ V
32	31	a1i 11	. . . . . 6 ⊢ (dom (𝑥 ∖ I ) ∈ Fin → 𝑥 ∈ V)
33		finnum 8774	. . . . . 6 ⊢ (dom (𝑥 ∖ I ) ∈ Fin → dom (𝑥 ∖ I ) ∈ dom card)
34		domfi 8181	. . . . . . . 8 ⊢ ((dom (𝑥 ∖ I ) ∈ Fin ∧ dom (𝑦 ∖ I ) ≼ dom (𝑥 ∖ I )) → dom (𝑦 ∖ I ) ∈ Fin)
35	2, 6	symgbasf1o 17803	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ 𝐵 → 𝑦:𝐷–1-1-onto→𝐷)
36	35	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) → 𝑦:𝐷–1-1-onto→𝐷)
37		f1ofn 6138	. . . . . . . . . . . . . 14 ⊢ (𝑦:𝐷–1-1-onto→𝐷 → 𝑦 Fn 𝐷)
38		fnnfpeq0 6444	. . . . . . . . . . . . . 14 ⊢ (𝑦 Fn 𝐷 → (dom (𝑦 ∖ I ) = ∅ ↔ 𝑦 = ( I ↾ 𝐷)))
39	36, 37, 38	3syl 18	. . . . . . . . . . . . 13 ⊢ ((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) → (dom (𝑦 ∖ I ) = ∅ ↔ 𝑦 = ( I ↾ 𝐷)))
40	2, 6	elbasfv 15920	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ 𝐵 → 𝐷 ∈ V)
41	40	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ ((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) → 𝐷 ∈ V)
42	2	symgid 17821	. . . . . . . . . . . . . . . 16 ⊢ (𝐷 ∈ V → ( I ↾ 𝐷) = (0_g‘𝐺))
43	41, 42	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) → ( I ↾ 𝐷) = (0_g‘𝐺))
44	41, 9	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) → (SubMnd‘𝐺) ∈ (Moore‘𝐵))
45	28	mrccl 16271	. . . . . . . . . . . . . . . . 17 ⊢ (((SubMnd‘𝐺) ∈ (Moore‘𝐵) ∧ 𝑇 ⊆ 𝐵) → (𝐾‘𝑇) ∈ (SubMnd‘𝐺))
46	44, 12, 45	sylancl 694	. . . . . . . . . . . . . . . 16 ⊢ ((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) → (𝐾‘𝑇) ∈ (SubMnd‘𝐺))
47		eqid 2622	. . . . . . . . . . . . . . . . 17 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
48	47	subm0cl 17352	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾‘𝑇) ∈ (SubMnd‘𝐺) → (0_g‘𝐺) ∈ (𝐾‘𝑇))
49	46, 48	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) → (0_g‘𝐺) ∈ (𝐾‘𝑇))
50	43, 49	eqeltrd 2701	. . . . . . . . . . . . . 14 ⊢ ((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) → ( I ↾ 𝐷) ∈ (𝐾‘𝑇))
51		eleq1a 2696	. . . . . . . . . . . . . 14 ⊢ (( I ↾ 𝐷) ∈ (𝐾‘𝑇) → (𝑦 = ( I ↾ 𝐷) → 𝑦 ∈ (𝐾‘𝑇)))
52	50, 51	syl 17	. . . . . . . . . . . . 13 ⊢ ((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) → (𝑦 = ( I ↾ 𝐷) → 𝑦 ∈ (𝐾‘𝑇)))
53	39, 52	sylbid 230	. . . . . . . . . . . 12 ⊢ ((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) → (dom (𝑦 ∖ I ) = ∅ → 𝑦 ∈ (𝐾‘𝑇)))
54	53	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)))
55		n0 3931	. . . . . . . . . . . 12 ⊢ (dom (𝑦 ∖ I ) ≠ ∅ ↔ ∃𝑢 𝑢 ∈ dom (𝑦 ∖ I ))
56	41	adantr 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → 𝐷 ∈ V)
57		simpr 477	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → 𝑢 ∈ dom (𝑦 ∖ I ))
58		f1omvdmvd 17863	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑦:𝐷–1-1-onto→𝐷 ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → (𝑦‘𝑢) ∈ (dom (𝑦 ∖ I ) ∖ {𝑢}))
59	36, 58	sylan 488	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → (𝑦‘𝑢) ∈ (dom (𝑦 ∖ I ) ∖ {𝑢}))
60	59	eldifad 3586	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → (𝑦‘𝑢) ∈ dom (𝑦 ∖ I ))
61		prssi 4353	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑢 ∈ dom (𝑦 ∖ I ) ∧ (𝑦‘𝑢) ∈ dom (𝑦 ∖ I )) → {𝑢, (𝑦‘𝑢)} ⊆ dom (𝑦 ∖ I ))
62	57, 60, 61	syl2anc 693	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → {𝑢, (𝑦‘𝑢)} ⊆ dom (𝑦 ∖ I ))
63		difss 3737	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 ∖ I ) ⊆ 𝑦
64		dmss 5323	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑦 ∖ I ) ⊆ 𝑦 → dom (𝑦 ∖ I ) ⊆ dom 𝑦)
65	63, 64	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ dom (𝑦 ∖ I ) ⊆ dom 𝑦
66		f1odm 6141	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦:𝐷–1-1-onto→𝐷 → dom 𝑦 = 𝐷)
67	36, 66	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) → dom 𝑦 = 𝐷)
68	65, 67	syl5sseq 3653	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) → dom (𝑦 ∖ I ) ⊆ 𝐷)
69	68	adantr 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → dom (𝑦 ∖ I ) ⊆ 𝐷)
70	62, 69	sstrd 3613	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → {𝑢, (𝑦‘𝑢)} ⊆ 𝐷)
71	36	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → 𝑦:𝐷–1-1-onto→𝐷)
72	71, 37	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → 𝑦 Fn 𝐷)
73	68	sselda 3603	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → 𝑢 ∈ 𝐷)
74		fnelnfp 6443	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑦 Fn 𝐷 ∧ 𝑢 ∈ 𝐷) → (𝑢 ∈ dom (𝑦 ∖ I ) ↔ (𝑦‘𝑢) ≠ 𝑢))
75	72, 73, 74	syl2anc 693	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → (𝑢 ∈ dom (𝑦 ∖ I ) ↔ (𝑦‘𝑢) ≠ 𝑢))
76	57, 75	mpbid 222	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → (𝑦‘𝑢) ≠ 𝑢)
77	76	necomd 2849	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → 𝑢 ≠ (𝑦‘𝑢))
78		vex 3203	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝑢 ∈ V
79		fvex 6201	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦‘𝑢) ∈ V
80		pr2nelem 8827	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑢 ∈ V ∧ (𝑦‘𝑢) ∈ V ∧ 𝑢 ≠ (𝑦‘𝑢)) → {𝑢, (𝑦‘𝑢)} ≈ 2_𝑜)
81	78, 79, 80	mp3an12 1414	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑢 ≠ (𝑦‘𝑢) → {𝑢, (𝑦‘𝑢)} ≈ 2_𝑜)
82	77, 81	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → {𝑢, (𝑦‘𝑢)} ≈ 2_𝑜)
83	17, 11	pmtrrn 17877	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐷 ∈ V ∧ {𝑢, (𝑦‘𝑢)} ⊆ 𝐷 ∧ {𝑢, (𝑦‘𝑢)} ≈ 2_𝑜) → ((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∈ 𝑇)
84	56, 70, 82, 83	syl3anc 1326	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → ((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∈ 𝑇)
85	12, 84	sseldi 3601	. . . . . . . . . . . . . . . . . . 19 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → ((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∈ 𝐵)
86		simplr 792	. . . . . . . . . . . . . . . . . . 19 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → 𝑦 ∈ 𝐵)
87		eqid 2622	. . . . . . . . . . . . . . . . . . . 20 ⊢ (+_g‘𝐺) = (+_g‘𝐺)
88	2, 6, 87	symgov 17810	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+_g‘𝐺)𝑦) = (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦))
89	85, 86, 88	syl2anc 693	. . . . . . . . . . . . . . . . . 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))
90	89	oveq2d 6666	. . . . . . . . . . . . . . . . 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)))
91	41, 3	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) → 𝐺 ∈ Grp)
92	91	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → 𝐺 ∈ Grp)
93	6, 87	grpcl 17430	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐺 ∈ Grp ∧ ((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+_g‘𝐺)𝑦) ∈ 𝐵)
94	92, 85, 86, 93	syl3anc 1326	. . . . . . . . . . . . . . . . . . 19 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+_g‘𝐺)𝑦) ∈ 𝐵)
95	89, 94	eqeltrrd 2702	. . . . . . . . . . . . . . . . . 18 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦) ∈ 𝐵)
96	2, 6, 87	symgov 17810	. . . . . . . . . . . . . . . . . 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)))
97	85, 95, 96	syl2anc 693	. . . . . . . . . . . . . . . . 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)))
98		coass 5654	. . . . . . . . . . . . . . . . . 18 ⊢ ((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ ((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})) ∘ 𝑦) = (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦))
99	17, 11	pmtrfinv 17881	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∈ 𝑇 → (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ ((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})) = ( I ↾ 𝐷))
100	84, 99	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))
101	100	coeq1d 5283	. . . . . . . . . . . . . . . . . . 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))
102		f1of 6137	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦:𝐷–1-1-onto→𝐷 → 𝑦:𝐷⟶𝐷)
103		fcoi2 6079	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦:𝐷⟶𝐷 → (( I ↾ 𝐷) ∘ 𝑦) = 𝑦)
104	71, 102, 103	3syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → (( I ↾ 𝐷) ∘ 𝑦) = 𝑦)
105	101, 104	eqtrd 2656	. . . . . . . . . . . . . . . . . 18 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → ((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ ((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})) ∘ 𝑦) = 𝑦)
106	98, 105	syl5eqr 2670	. . . . . . . . . . . . . . . . 17 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦)) = 𝑦)
107	90, 97, 106	3eqtrd 2660	. . . . . . . . . . . . . . . 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)
108	107	adantlr 751	. . . . . . . . . . . . . . 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)
109	46	ad2antrr 762	. . . . . . . . . . . . . . . 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))
110	28	mrcssid 16277	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((SubMnd‘𝐺) ∈ (Moore‘𝐵) ∧ 𝑇 ⊆ 𝐵) → 𝑇 ⊆ (𝐾‘𝑇))
111	44, 12, 110	sylancl 694	. . . . . . . . . . . . . . . . . . 19 ⊢ ((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) → 𝑇 ⊆ (𝐾‘𝑇))
112	111	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → 𝑇 ⊆ (𝐾‘𝑇))
113	112, 84	sseldd 3604	. . . . . . . . . . . . . . . . 17 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → ((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∈ (𝐾‘𝑇))
114	113	adantlr 751	. . . . . . . . . . . . . . . 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))
115	89	difeq1d 3727	. . . . . . . . . . . . . . . . . . . 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 ))
116	115	dmeqd 5326	. . . . . . . . . . . . . . . . . . 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 ))
117		simpll 790	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → dom (𝑦 ∖ I ) ∈ Fin)
118		mvdco 17865	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ dom ((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦) ∖ I ) ⊆ (dom (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∖ I ) ∪ dom (𝑦 ∖ I ))
119	17	pmtrmvd 17876	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐷 ∈ V ∧ {𝑢, (𝑦‘𝑢)} ⊆ 𝐷 ∧ {𝑢, (𝑦‘𝑢)} ≈ 2_𝑜) → dom (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∖ I ) = {𝑢, (𝑦‘𝑢)})
120	56, 70, 82, 119	syl3anc 1326	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → dom (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∖ I ) = {𝑢, (𝑦‘𝑢)})
121	120, 62	eqsstrd 3639	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → dom (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∖ I ) ⊆ dom (𝑦 ∖ I ))
122		ssid 3624	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ dom (𝑦 ∖ I ) ⊆ dom (𝑦 ∖ I )
123	122	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → dom (𝑦 ∖ I ) ⊆ dom (𝑦 ∖ I ))
124	121, 123	unssd 3789	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → (dom (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∖ I ) ∪ dom (𝑦 ∖ I )) ⊆ dom (𝑦 ∖ I ))
125	118, 124	syl5ss 3614	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → dom ((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦) ∖ I ) ⊆ dom (𝑦 ∖ I ))
126		fvco2 6273	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑦 Fn 𝐷 ∧ 𝑢 ∈ 𝐷) → ((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦)‘𝑢) = (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})‘(𝑦‘𝑢)))
127	72, 73, 126	syl2anc 693	. . . . . . . . . . . . . . . . . . . . . . 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)))
128		prcom 4267	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ {𝑢, (𝑦‘𝑢)} = {(𝑦‘𝑢), 𝑢}
129	128	fveq2i 6194	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) = ((pmTrsp‘𝐷)‘{(𝑦‘𝑢), 𝑢})
130	129	fveq1i 6192	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})‘(𝑦‘𝑢)) = (((pmTrsp‘𝐷)‘{(𝑦‘𝑢), 𝑢})‘(𝑦‘𝑢))
131	69, 60	sseldd 3604	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → (𝑦‘𝑢) ∈ 𝐷)
132	17	pmtrprfv 17873	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐷 ∈ V ∧ ((𝑦‘𝑢) ∈ 𝐷 ∧ 𝑢 ∈ 𝐷 ∧ (𝑦‘𝑢) ≠ 𝑢)) → (((pmTrsp‘𝐷)‘{(𝑦‘𝑢), 𝑢})‘(𝑦‘𝑢)) = 𝑢)
133	56, 131, 73, 76, 132	syl13anc 1328	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → (((pmTrsp‘𝐷)‘{(𝑦‘𝑢), 𝑢})‘(𝑦‘𝑢)) = 𝑢)
134	130, 133	syl5eq 2668	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})‘(𝑦‘𝑢)) = 𝑢)
135	127, 134	eqtrd 2656	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → ((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦)‘𝑢) = 𝑢)
136	2, 6	symgbasf1o 17803	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦) ∈ 𝐵 → (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦):𝐷–1-1-onto→𝐷)
137		f1ofn 6138	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦):𝐷–1-1-onto→𝐷 → (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦) Fn 𝐷)
138	95, 136, 137	3syl 18	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦) Fn 𝐷)
139		fnelnfp 6443	. . . . . . . . . . . . . . . . . . . . . . . 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))
140	139	necon2bbid 2837	. . . . . . . . . . . . . . . . . . . . . . 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 )))
141	138, 73, 140	syl2anc 693	. . . . . . . . . . . . . . . . . . . . . 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 )))
142	135, 141	mpbid 222	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → ¬ 𝑢 ∈ dom ((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦) ∖ I ))
143	125, 57, 142	ssnelpssd 3719	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → dom ((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦) ∖ I ) ⊊ dom (𝑦 ∖ I ))
144		php3 8146	. . . . . . . . . . . . . . . . . . . 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 ))
145	117, 143, 144	syl2anc 693	. . . . . . . . . . . . . . . . . . 19 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → dom ((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦) ∖ I ) ≺ dom (𝑦 ∖ I ))
146	116, 145	eqbrtrd 4675	. . . . . . . . . . . . . . . . . 18 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → dom ((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+_g‘𝐺)𝑦) ∖ I ) ≺ dom (𝑦 ∖ I ))
147	146	adantlr 751	. . . . . . . . . . . . . . . . 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 ))
148	94	adantlr 751	. . . . . . . . . . . . . . . . 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)
149		ovex 6678	. . . . . . . . . . . . . . . . . . 19 ⊢ (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+_g‘𝐺)𝑦) ∈ V
150		difeq1 3721	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑧 = (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+_g‘𝐺)𝑦) → (𝑧 ∖ I ) = ((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+_g‘𝐺)𝑦) ∖ I ))
151	150	dmeqd 5326	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 = (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+_g‘𝐺)𝑦) → dom (𝑧 ∖ I ) = dom ((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+_g‘𝐺)𝑦) ∖ I ))
152	151	breq1d 4663	. . . . . . . . . . . . . . . . . . . 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 )))
153		eleq1 2689	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 = (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+_g‘𝐺)𝑦) → (𝑧 ∈ 𝐵 ↔ (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+_g‘𝐺)𝑦) ∈ 𝐵))
154		eleq1 2689	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 = (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+_g‘𝐺)𝑦) → (𝑧 ∈ (𝐾‘𝑇) ↔ (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+_g‘𝐺)𝑦) ∈ (𝐾‘𝑇)))
155	153, 154	imbi12d 334	. . . . . . . . . . . . . . . . . . . 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))))
156	152, 155	imbi12d 334	. . . . . . . . . . . . . . . . . . 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)))))
157	149, 156	spcv 3299	. . . . . . . . . . . . . . . . . 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))))
158	157	ad2antlr 763	. . . . . . . . . . . . . . . . 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))))
159	147, 148, 158	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))
160	87	submcl 17353	. . . . . . . . . . . . . . . 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))
161	109, 114, 159, 160	syl3anc 1326	. . . . . . . . . . . . . . 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))
162	108, 161	eqeltrrd 2702	. . . . . . . . . . . . . 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))
163	162	ex 450	. . . . . . . . . . . . 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)))
164	163	exlimdv 1861	. . . . . . . . . . . 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)))
165	55, 164	syl5bi 232	. . . . . . . . . . 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)))
166	54, 165	pm2.61dne 2880	. . . . . . . . . 10 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ ∀𝑧(dom (𝑧 ∖ I ) ≺ dom (𝑦 ∖ I ) → (𝑧 ∈ 𝐵 → 𝑧 ∈ (𝐾‘𝑇)))) → 𝑦 ∈ (𝐾‘𝑇))
167	166	exp31 630	. . . . . . . . 9 ⊢ (dom (𝑦 ∖ I ) ∈ Fin → (𝑦 ∈ 𝐵 → (∀𝑧(dom (𝑧 ∖ I ) ≺ dom (𝑦 ∖ I ) → (𝑧 ∈ 𝐵 → 𝑧 ∈ (𝐾‘𝑇))) → 𝑦 ∈ (𝐾‘𝑇))))
168	167	com23 86	. . . . . . . 8 ⊢ (dom (𝑦 ∖ I ) ∈ Fin → (∀𝑧(dom (𝑧 ∖ I ) ≺ dom (𝑦 ∖ I ) → (𝑧 ∈ 𝐵 → 𝑧 ∈ (𝐾‘𝑇))) → (𝑦 ∈ 𝐵 → 𝑦 ∈ (𝐾‘𝑇))))
169	34, 168	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))))
170	169	3impia 1261	. . . . . 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)))
171		eleq1 2689	. . . . . . 7 ⊢ (𝑦 = 𝑧 → (𝑦 ∈ 𝐵 ↔ 𝑧 ∈ 𝐵))
172		eleq1 2689	. . . . . . 7 ⊢ (𝑦 = 𝑧 → (𝑦 ∈ (𝐾‘𝑇) ↔ 𝑧 ∈ (𝐾‘𝑇)))
173	171, 172	imbi12d 334	. . . . . 6 ⊢ (𝑦 = 𝑧 → ((𝑦 ∈ 𝐵 → 𝑦 ∈ (𝐾‘𝑇)) ↔ (𝑧 ∈ 𝐵 → 𝑧 ∈ (𝐾‘𝑇))))
174		eleq1 2689	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ 𝐵 ↔ 𝑥 ∈ 𝐵))
175		eleq1 2689	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ (𝐾‘𝑇) ↔ 𝑥 ∈ (𝐾‘𝑇)))
176	174, 175	imbi12d 334	. . . . . 6 ⊢ (𝑦 = 𝑥 → ((𝑦 ∈ 𝐵 → 𝑦 ∈ (𝐾‘𝑇)) ↔ (𝑥 ∈ 𝐵 → 𝑥 ∈ (𝐾‘𝑇))))
177		difeq1 3721	. . . . . . 7 ⊢ (𝑦 = 𝑧 → (𝑦 ∖ I ) = (𝑧 ∖ I ))
178	177	dmeqd 5326	. . . . . 6 ⊢ (𝑦 = 𝑧 → dom (𝑦 ∖ I ) = dom (𝑧 ∖ I ))
179		difeq1 3721	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (𝑦 ∖ I ) = (𝑥 ∖ I ))
180	179	dmeqd 5326	. . . . . 6 ⊢ (𝑦 = 𝑥 → dom (𝑦 ∖ I ) = dom (𝑥 ∖ I ))
181	32, 33, 170, 173, 176, 178, 180	indcardi 8864	. . . . 5 ⊢ (dom (𝑥 ∖ I ) ∈ Fin → (𝑥 ∈ 𝐵 → 𝑥 ∈ (𝐾‘𝑇)))
182	181	impcom 446	. . . 4 ⊢ ((𝑥 ∈ 𝐵 ∧ dom (𝑥 ∖ I ) ∈ Fin) → 𝑥 ∈ (𝐾‘𝑇))
183	182	3adant1 1079	. . 3 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵 ∧ dom (𝑥 ∖ I ) ∈ Fin) → 𝑥 ∈ (𝐾‘𝑇))
184	183	rabssdv 3682	. 2 ⊢ (𝐷 ∈ 𝑉 → {𝑥 ∈ 𝐵 ∣ dom (𝑥 ∖ I ) ∈ Fin} ⊆ (𝐾‘𝑇))
185	30, 184	eqssd 3620	1 ⊢ (𝐷 ∈ 𝑉 → (𝐾‘𝑇) = {𝑥 ∈ 𝐵 ∣ dom (𝑥 ∖ I ) ∈ Fin})

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 384 ∀wal 1481 = wceq 1483 ∃wex 1704 ∈ wcel 1990 ≠ wne 2794 {crab 2916 Vcvv 3200 ∖ cdif 3571 ∪ cun 3572 ⊆ wss 3574 ⊊ wpss 3575 ∅c0 3915 {csn 4177 {cpr 4179 class class class wbr 4653 I cid 5023 dom cdm 5114 ran crn 5115 ↾ cres 5116 ∘ ccom 5118 Fn wfn 5883 ⟶wf 5884 –1-1-onto→wf1o 5887 ‘cfv 5888 (class class class)co 6650 ωcom 7065 2_𝑜c2o 7554 ≈ cen 7952 ≼ cdom 7953 ≺ csdm 7954 Fincfn 7955 Basecbs 15857 +_gcplusg 15941 0_gc0g 16100 Moorecmre 16242 mrClscmrc 16243 ACScacs 16245 Mndcmnd 17294 SubMndcsubmnd 17334 Grpcgrp 17422 SubGrpcsubg 17588 SymGrpcsymg 17797 pmTrspcpmtr 17861

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-tset 15960 df-0g 16102 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-grp 17425 df-minusg 17426 df-subg 17591 df-symg 17798 df-pmtr 17862

This theorem is referenced by: symggen2 17891 psgneldm2 17924

W3C validator