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Theorem symggen 18359

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 3433	. . . 4 ⊢ (𝐷 ∈ 𝑉 → 𝐷 ∈ V)
2		symgtrf.g	. . . . . . 7 ⊢ 𝐺 = (SymGrp‘𝐷)
3	2	symggrp 18289	. . . . . 6 ⊢ (𝐷 ∈ V → 𝐺 ∈ Grp)
4		grpmnd 17898	. . . . . 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 17833	. . . . 5 ⊢ (𝐺 ∈ Mnd → (SubMnd‘𝐺) ∈ (ACS‘𝐵))
8		acsmre 16781	. . . . 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 18358	. . . . 5 ⊢ 𝑇 ⊆ 𝐵
13	12	a1i 11	. . . 4 ⊢ (𝐷 ∈ 𝑉 → 𝑇 ⊆ 𝐵)
14		2onn 8067	. . . . . 6 ⊢ 2_o ∈ ω
15		nnfi 8506	. . . . . 6 ⊢ (2_o ∈ ω → 2_o ∈ Fin)
16	14, 15	ax-mp 5	. . . . 5 ⊢ 2_o ∈ Fin
17		eqid 2778	. . . . . . . . 9 ⊢ (pmTrsp‘𝐷) = (pmTrsp‘𝐷)
18	17, 11	pmtrfb 18354	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑇 ↔ (𝐷 ∈ V ∧ 𝑥:𝐷–1-1-onto→𝐷 ∧ dom (𝑥 ∖ I ) ≈ 2_o))
19	18	simp3bi 1127	. . . . . . 7 ⊢ (𝑥 ∈ 𝑇 → dom (𝑥 ∖ I ) ≈ 2_o)
20		enfi 8529	. . . . . . 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 474	. . . . 5 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑥 ∈ 𝑇) → (dom (𝑥 ∖ I ) ∈ Fin ↔ 2_o ∈ Fin))
23	16, 22	mpbiri 250	. . . 4 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑥 ∈ 𝑇) → dom (𝑥 ∖ I ) ∈ Fin)
24	13, 23	ssrabdv 3940	. . 3 ⊢ (𝐷 ∈ 𝑉 → 𝑇 ⊆ {𝑥 ∈ 𝐵 ∣ dom (𝑥 ∖ I ) ∈ Fin})
25	2, 6	symgfisg 18357	. . . 4 ⊢ (𝐷 ∈ 𝑉 → {𝑥 ∈ 𝐵 ∣ dom (𝑥 ∖ I ) ∈ Fin} ∈ (SubGrp‘𝐺))
26		subgsubm 18085	. . . 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 16749	. . 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 1351	. 2 ⊢ (𝐷 ∈ 𝑉 → (𝐾‘𝑇) ⊆ {𝑥 ∈ 𝐵 ∣ dom (𝑥 ∖ I ) ∈ Fin})
31		vex 3418	. . . . . . 7 ⊢ 𝑥 ∈ V
32	31	a1i 11	. . . . . 6 ⊢ (dom (𝑥 ∖ I ) ∈ Fin → 𝑥 ∈ V)
33		finnum 9171	. . . . . 6 ⊢ (dom (𝑥 ∖ I ) ∈ Fin → dom (𝑥 ∖ I ) ∈ dom card)
34		domfi 8534	. . . . . . . 8 ⊢ ((dom (𝑥 ∖ I ) ∈ Fin ∧ dom (𝑦 ∖ I ) ≼ dom (𝑥 ∖ I )) → dom (𝑦 ∖ I ) ∈ Fin)
35	2, 6	symgbasf1o 18272	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ 𝐵 → 𝑦:𝐷–1-1-onto→𝐷)
36	35	adantl 474	. . . . . . . . . . . . . 14 ⊢ ((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) → 𝑦:𝐷–1-1-onto→𝐷)
37		f1ofn 6445	. . . . . . . . . . . . . 14 ⊢ (𝑦:𝐷–1-1-onto→𝐷 → 𝑦 Fn 𝐷)
38		fnnfpeq0 6763	. . . . . . . . . . . . . 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 16400	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ 𝐵 → 𝐷 ∈ V)
41	40	adantl 474	. . . . . . . . . . . . . . . 16 ⊢ ((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) → 𝐷 ∈ V)
42	2	symgid 18290	. . . . . . . . . . . . . . . 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 16740	. . . . . . . . . . . . . . . . 17 ⊢ (((SubMnd‘𝐺) ∈ (Moore‘𝐵) ∧ 𝑇 ⊆ 𝐵) → (𝐾‘𝑇) ∈ (SubMnd‘𝐺))
46	44, 12, 45	sylancl 577	. . . . . . . . . . . . . . . 16 ⊢ ((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) → (𝐾‘𝑇) ∈ (SubMnd‘𝐺))
47		eqid 2778	. . . . . . . . . . . . . . . . 17 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
48	47	subm0cl 17820	. . . . . . . . . . . . . . . 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 2866	. . . . . . . . . . . . . 14 ⊢ ((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) → ( I ↾ 𝐷) ∈ (𝐾‘𝑇))
51		eleq1a 2861	. . . . . . . . . . . . . 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 232	. . . . . . . . . . . 12 ⊢ ((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) → (dom (𝑦 ∖ I ) = ∅ → 𝑦 ∈ (𝐾‘𝑇)))
54	53	adantr 473	. . . . . . . . . . 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 4196	. . . . . . . . . . . 12 ⊢ (dom (𝑦 ∖ I ) ≠ ∅ ↔ ∃𝑢 𝑢 ∈ dom (𝑦 ∖ I ))
56	41	adantr 473	. . . . . . . . . . . . . . . . . . . . 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 18332	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑦:𝐷–1-1-onto→𝐷 ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → (𝑦‘𝑢) ∈ (dom (𝑦 ∖ I ) ∖ {𝑢}))
59	36, 58	sylan 572	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → (𝑦‘𝑢) ∈ (dom (𝑦 ∖ I ) ∖ {𝑢}))
60	59	eldifad 3841	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → (𝑦‘𝑢) ∈ dom (𝑦 ∖ I ))
61	57, 60	prssd 4629	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → {𝑢, (𝑦‘𝑢)} ⊆ dom (𝑦 ∖ I ))
62		difss 3998	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 ∖ I ) ⊆ 𝑦
63		dmss 5621	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑦 ∖ I ) ⊆ 𝑦 → dom (𝑦 ∖ I ) ⊆ dom 𝑦)
64	62, 63	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ dom (𝑦 ∖ I ) ⊆ dom 𝑦
65		f1odm 6448	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦:𝐷–1-1-onto→𝐷 → dom 𝑦 = 𝐷)
66	36, 65	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) → dom 𝑦 = 𝐷)
67	64, 66	syl5sseq 3909	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) → dom (𝑦 ∖ I ) ⊆ 𝐷)
68	67	adantr 473	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → dom (𝑦 ∖ I ) ⊆ 𝐷)
69	61, 68	sstrd 3868	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → {𝑢, (𝑦‘𝑢)} ⊆ 𝐷)
70		vex 3418	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝑢 ∈ V
71		fvex 6512	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦‘𝑢) ∈ V
72	36	adantr 473	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → 𝑦:𝐷–1-1-onto→𝐷)
73	72, 37	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → 𝑦 Fn 𝐷)
74	67	sselda 3858	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → 𝑢 ∈ 𝐷)
75		fnelnfp 6762	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑦 Fn 𝐷 ∧ 𝑢 ∈ 𝐷) → (𝑢 ∈ dom (𝑦 ∖ I ) ↔ (𝑦‘𝑢) ≠ 𝑢))
76	73, 74, 75	syl2anc 576	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → (𝑢 ∈ dom (𝑦 ∖ I ) ↔ (𝑦‘𝑢) ≠ 𝑢))
77	57, 76	mpbid 224	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → (𝑦‘𝑢) ≠ 𝑢)
78	77	necomd 3022	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → 𝑢 ≠ (𝑦‘𝑢))
79		pr2nelem 9224	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑢 ∈ V ∧ (𝑦‘𝑢) ∈ V ∧ 𝑢 ≠ (𝑦‘𝑢)) → {𝑢, (𝑦‘𝑢)} ≈ 2_o)
80	70, 71, 78, 79	mp3an12i 1444	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → {𝑢, (𝑦‘𝑢)} ≈ 2_o)
81	17, 11	pmtrrn 18346	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐷 ∈ V ∧ {𝑢, (𝑦‘𝑢)} ⊆ 𝐷 ∧ {𝑢, (𝑦‘𝑢)} ≈ 2_o) → ((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∈ 𝑇)
82	56, 69, 80, 81	syl3anc 1351	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → ((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∈ 𝑇)
83	12, 82	sseldi 3856	. . . . . . . . . . . . . . . . . . 19 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → ((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∈ 𝐵)
84		simplr 756	. . . . . . . . . . . . . . . . . . 19 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → 𝑦 ∈ 𝐵)
85		eqid 2778	. . . . . . . . . . . . . . . . . . . 20 ⊢ (+_g‘𝐺) = (+_g‘𝐺)
86	2, 6, 85	symgov 18279	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+_g‘𝐺)𝑦) = (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦))
87	83, 84, 86	syl2anc 576	. . . . . . . . . . . . . . . . . 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))
88	87	oveq2d 6992	. . . . . . . . . . . . . . . . 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)))
89	41, 3	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) → 𝐺 ∈ Grp)
90	89	adantr 473	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → 𝐺 ∈ Grp)
91	6, 85	grpcl 17899	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐺 ∈ Grp ∧ ((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+_g‘𝐺)𝑦) ∈ 𝐵)
92	90, 83, 84, 91	syl3anc 1351	. . . . . . . . . . . . . . . . . . 19 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+_g‘𝐺)𝑦) ∈ 𝐵)
93	87, 92	eqeltrrd 2867	. . . . . . . . . . . . . . . . . 18 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦) ∈ 𝐵)
94	2, 6, 85	symgov 18279	. . . . . . . . . . . . . . . . . 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)))
95	83, 93, 94	syl2anc 576	. . . . . . . . . . . . . . . . 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)))
96		coass 5957	. . . . . . . . . . . . . . . . . 18 ⊢ ((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ ((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})) ∘ 𝑦) = (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦))
97	17, 11	pmtrfinv 18350	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∈ 𝑇 → (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ ((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})) = ( I ↾ 𝐷))
98	82, 97	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))
99	98	coeq1d 5582	. . . . . . . . . . . . . . . . . . 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))
100		f1of 6444	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦:𝐷–1-1-onto→𝐷 → 𝑦:𝐷⟶𝐷)
101		fcoi2 6382	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦:𝐷⟶𝐷 → (( I ↾ 𝐷) ∘ 𝑦) = 𝑦)
102	72, 100, 101	3syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → (( I ↾ 𝐷) ∘ 𝑦) = 𝑦)
103	99, 102	eqtrd 2814	. . . . . . . . . . . . . . . . . 18 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → ((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ ((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})) ∘ 𝑦) = 𝑦)
104	96, 103	syl5eqr 2828	. . . . . . . . . . . . . . . . 17 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦)) = 𝑦)
105	88, 95, 104	3eqtrd 2818	. . . . . . . . . . . . . . . 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)
106	105	adantlr 702	. . . . . . . . . . . . . . 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)
107	46	ad2antrr 713	. . . . . . . . . . . . . . . 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))
108	28	mrcssid 16746	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((SubMnd‘𝐺) ∈ (Moore‘𝐵) ∧ 𝑇 ⊆ 𝐵) → 𝑇 ⊆ (𝐾‘𝑇))
109	44, 12, 108	sylancl 577	. . . . . . . . . . . . . . . . . . 19 ⊢ ((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) → 𝑇 ⊆ (𝐾‘𝑇))
110	109	adantr 473	. . . . . . . . . . . . . . . . . 18 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → 𝑇 ⊆ (𝐾‘𝑇))
111	110, 82	sseldd 3859	. . . . . . . . . . . . . . . . 17 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → ((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∈ (𝐾‘𝑇))
112	111	adantlr 702	. . . . . . . . . . . . . . . 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))
113	87	difeq1d 3988	. . . . . . . . . . . . . . . . . . . 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 ))
114	113	dmeqd 5624	. . . . . . . . . . . . . . . . . . 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 ))
115		simpll 754	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → dom (𝑦 ∖ I ) ∈ Fin)
116		mvdco 18334	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ dom ((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦) ∖ I ) ⊆ (dom (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∖ I ) ∪ dom (𝑦 ∖ I ))
117	17	pmtrmvd 18345	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐷 ∈ V ∧ {𝑢, (𝑦‘𝑢)} ⊆ 𝐷 ∧ {𝑢, (𝑦‘𝑢)} ≈ 2_o) → dom (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∖ I ) = {𝑢, (𝑦‘𝑢)})
118	56, 69, 80, 117	syl3anc 1351	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → dom (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∖ I ) = {𝑢, (𝑦‘𝑢)})
119	118, 61	eqsstrd 3895	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → dom (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∖ I ) ⊆ dom (𝑦 ∖ I ))
120		ssidd 3880	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → dom (𝑦 ∖ I ) ⊆ dom (𝑦 ∖ I ))
121	119, 120	unssd 4050	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → (dom (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∖ I ) ∪ dom (𝑦 ∖ I )) ⊆ dom (𝑦 ∖ I ))
122	116, 121	syl5ss 3869	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → dom ((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦) ∖ I ) ⊆ dom (𝑦 ∖ I ))
123		fvco2 6586	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑦 Fn 𝐷 ∧ 𝑢 ∈ 𝐷) → ((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦)‘𝑢) = (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})‘(𝑦‘𝑢)))
124	73, 74, 123	syl2anc 576	. . . . . . . . . . . . . . . . . . . . . . 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)))
125		prcom 4542	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ {𝑢, (𝑦‘𝑢)} = {(𝑦‘𝑢), 𝑢}
126	125	fveq2i 6502	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) = ((pmTrsp‘𝐷)‘{(𝑦‘𝑢), 𝑢})
127	126	fveq1i 6500	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})‘(𝑦‘𝑢)) = (((pmTrsp‘𝐷)‘{(𝑦‘𝑢), 𝑢})‘(𝑦‘𝑢))
128	68, 60	sseldd 3859	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → (𝑦‘𝑢) ∈ 𝐷)
129	17	pmtrprfv 18342	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐷 ∈ V ∧ ((𝑦‘𝑢) ∈ 𝐷 ∧ 𝑢 ∈ 𝐷 ∧ (𝑦‘𝑢) ≠ 𝑢)) → (((pmTrsp‘𝐷)‘{(𝑦‘𝑢), 𝑢})‘(𝑦‘𝑢)) = 𝑢)
130	56, 128, 74, 77, 129	syl13anc 1352	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → (((pmTrsp‘𝐷)‘{(𝑦‘𝑢), 𝑢})‘(𝑦‘𝑢)) = 𝑢)
131	127, 130	syl5eq 2826	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})‘(𝑦‘𝑢)) = 𝑢)
132	124, 131	eqtrd 2814	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → ((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦)‘𝑢) = 𝑢)
133	2, 6	symgbasf1o 18272	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦) ∈ 𝐵 → (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦):𝐷–1-1-onto→𝐷)
134		f1ofn 6445	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦):𝐷–1-1-onto→𝐷 → (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦) Fn 𝐷)
135	93, 133, 134	3syl 18	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦) Fn 𝐷)
136		fnelnfp 6762	. . . . . . . . . . . . . . . . . . . . . . . 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))
137	136	necon2bbid 3010	. . . . . . . . . . . . . . . . . . . . . . 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 )))
138	135, 74, 137	syl2anc 576	. . . . . . . . . . . . . . . . . . . . . 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 )))
139	132, 138	mpbid 224	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → ¬ 𝑢 ∈ dom ((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦) ∖ I ))
140	122, 57, 139	ssnelpssd 3979	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → dom ((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦) ∖ I ) ⊊ dom (𝑦 ∖ I ))
141		php3 8499	. . . . . . . . . . . . . . . . . . . 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 ))
142	115, 140, 141	syl2anc 576	. . . . . . . . . . . . . . . . . . 19 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → dom ((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦) ∖ I ) ≺ dom (𝑦 ∖ I ))
143	114, 142	eqbrtrd 4951	. . . . . . . . . . . . . . . . . 18 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → dom ((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+_g‘𝐺)𝑦) ∖ I ) ≺ dom (𝑦 ∖ I ))
144	143	adantlr 702	. . . . . . . . . . . . . . . . 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 ))
145	92	adantlr 702	. . . . . . . . . . . . . . . . 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)
146		ovex 7008	. . . . . . . . . . . . . . . . . . 19 ⊢ (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+_g‘𝐺)𝑦) ∈ V
147		difeq1 3982	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑧 = (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+_g‘𝐺)𝑦) → (𝑧 ∖ I ) = ((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+_g‘𝐺)𝑦) ∖ I ))
148	147	dmeqd 5624	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 = (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+_g‘𝐺)𝑦) → dom (𝑧 ∖ I ) = dom ((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+_g‘𝐺)𝑦) ∖ I ))
149	148	breq1d 4939	. . . . . . . . . . . . . . . . . . . 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 )))
150		eleq1 2853	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 = (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+_g‘𝐺)𝑦) → (𝑧 ∈ 𝐵 ↔ (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+_g‘𝐺)𝑦) ∈ 𝐵))
151		eleq1 2853	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 = (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+_g‘𝐺)𝑦) → (𝑧 ∈ (𝐾‘𝑇) ↔ (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+_g‘𝐺)𝑦) ∈ (𝐾‘𝑇)))
152	150, 151	imbi12d 337	. . . . . . . . . . . . . . . . . . . 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))))
153	149, 152	imbi12d 337	. . . . . . . . . . . . . . . . . . 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)))))
154	146, 153	spcv 3524	. . . . . . . . . . . . . . . . . 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))))
155	154	ad2antlr 714	. . . . . . . . . . . . . . . . 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))))
156	144, 145, 155	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))
157	85	submcl 17821	. . . . . . . . . . . . . . . 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))
158	107, 112, 156, 157	syl3anc 1351	. . . . . . . . . . . . . . 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))
159	106, 158	eqeltrrd 2867	. . . . . . . . . . . . . 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))
160	159	ex 405	. . . . . . . . . . . . 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)))
161	160	exlimdv 1892	. . . . . . . . . . . 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)))
162	55, 161	syl5bi 234	. . . . . . . . . . 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)))
163	54, 162	pm2.61dne 3054	. . . . . . . . . 10 ⊢ (((dom (𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵) ∧ ∀𝑧(dom (𝑧 ∖ I ) ≺ dom (𝑦 ∖ I ) → (𝑧 ∈ 𝐵 → 𝑧 ∈ (𝐾‘𝑇)))) → 𝑦 ∈ (𝐾‘𝑇))
164	163	exp31 412	. . . . . . . . 9 ⊢ (dom (𝑦 ∖ I ) ∈ Fin → (𝑦 ∈ 𝐵 → (∀𝑧(dom (𝑧 ∖ I ) ≺ dom (𝑦 ∖ I ) → (𝑧 ∈ 𝐵 → 𝑧 ∈ (𝐾‘𝑇))) → 𝑦 ∈ (𝐾‘𝑇))))
165	164	com23 86	. . . . . . . 8 ⊢ (dom (𝑦 ∖ I ) ∈ Fin → (∀𝑧(dom (𝑧 ∖ I ) ≺ dom (𝑦 ∖ I ) → (𝑧 ∈ 𝐵 → 𝑧 ∈ (𝐾‘𝑇))) → (𝑦 ∈ 𝐵 → 𝑦 ∈ (𝐾‘𝑇))))
166	34, 165	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))))
167	166	3impia 1097	. . . . . 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)))
168		eleq1w 2848	. . . . . . 7 ⊢ (𝑦 = 𝑧 → (𝑦 ∈ 𝐵 ↔ 𝑧 ∈ 𝐵))
169		eleq1w 2848	. . . . . . 7 ⊢ (𝑦 = 𝑧 → (𝑦 ∈ (𝐾‘𝑇) ↔ 𝑧 ∈ (𝐾‘𝑇)))
170	168, 169	imbi12d 337	. . . . . 6 ⊢ (𝑦 = 𝑧 → ((𝑦 ∈ 𝐵 → 𝑦 ∈ (𝐾‘𝑇)) ↔ (𝑧 ∈ 𝐵 → 𝑧 ∈ (𝐾‘𝑇))))
171		eleq1w 2848	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ 𝐵 ↔ 𝑥 ∈ 𝐵))
172		eleq1w 2848	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ (𝐾‘𝑇) ↔ 𝑥 ∈ (𝐾‘𝑇)))
173	171, 172	imbi12d 337	. . . . . 6 ⊢ (𝑦 = 𝑥 → ((𝑦 ∈ 𝐵 → 𝑦 ∈ (𝐾‘𝑇)) ↔ (𝑥 ∈ 𝐵 → 𝑥 ∈ (𝐾‘𝑇))))
174		difeq1 3982	. . . . . . 7 ⊢ (𝑦 = 𝑧 → (𝑦 ∖ I ) = (𝑧 ∖ I ))
175	174	dmeqd 5624	. . . . . 6 ⊢ (𝑦 = 𝑧 → dom (𝑦 ∖ I ) = dom (𝑧 ∖ I ))
176		difeq1 3982	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (𝑦 ∖ I ) = (𝑥 ∖ I ))
177	176	dmeqd 5624	. . . . . 6 ⊢ (𝑦 = 𝑥 → dom (𝑦 ∖ I ) = dom (𝑥 ∖ I ))
178	32, 33, 167, 170, 173, 175, 177	indcardi 9261	. . . . 5 ⊢ (dom (𝑥 ∖ I ) ∈ Fin → (𝑥 ∈ 𝐵 → 𝑥 ∈ (𝐾‘𝑇)))
179	178	impcom 399	. . . 4 ⊢ ((𝑥 ∈ 𝐵 ∧ dom (𝑥 ∖ I ) ∈ Fin) → 𝑥 ∈ (𝐾‘𝑇))
180	179	3adant1 1110	. . 3 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵 ∧ dom (𝑥 ∖ I ) ∈ Fin) → 𝑥 ∈ (𝐾‘𝑇))
181	180	rabssdv 3941	. 2 ⊢ (𝐷 ∈ 𝑉 → {𝑥 ∈ 𝐵 ∣ dom (𝑥 ∖ I ) ∈ Fin} ⊆ (𝐾‘𝑇))
182	30, 181	eqssd 3875	1 ⊢ (𝐷 ∈ 𝑉 → (𝐾‘𝑇) = {𝑥 ∈ 𝐵 ∣ dom (𝑥 ∖ I ) ∈ Fin})

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ wa 387 ∀wal 1505 = wceq 1507 ∃wex 1742 ∈ wcel 2050 ≠ wne 2967 {crab 3092 Vcvv 3415 ∖ cdif 3826 ∪ cun 3827 ⊆ wss 3829 ⊊ wpss 3830 ∅c0 4178 {csn 4441 {cpr 4443 class class class wbr 4929 I cid 5311 dom cdm 5407 ran crn 5408 ↾ cres 5409 ∘ ccom 5411 Fn wfn 6183 ⟶wf 6184 –1-1-onto→wf1o 6187 ‘cfv 6188 (class class class)co 6976 ωcom 7396 2_oc2o 7899 ≈ cen 8303 ≼ cdom 8304 ≺ csdm 8305 Fincfn 8306 Basecbs 16339 +_gcplusg 16421 0_gc0g 16569 Moorecmre 16711 mrClscmrc 16712 ACScacs 16714 Mndcmnd 17762 SubMndcsubmnd 17802 Grpcgrp 17891 SubGrpcsubg 18057 SymGrpcsymg 18266 pmTrspcpmtr 18330

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-rep 5049 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 ax-cnex 10391 ax-resscn 10392 ax-1cn 10393 ax-icn 10394 ax-addcl 10395 ax-addrcl 10396 ax-mulcl 10397 ax-mulrcl 10398 ax-mulcom 10399 ax-addass 10400 ax-mulass 10401 ax-distr 10402 ax-i2m1 10403 ax-1ne0 10404 ax-1rid 10405 ax-rnegex 10406 ax-rrecex 10407 ax-cnre 10408 ax-pre-lttri 10409 ax-pre-lttrn 10410 ax-pre-ltadd 10411 ax-pre-mulgt0 10412

This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rmo 3096 df-rab 3097 df-v 3417 df-sbc 3682 df-csb 3787 df-dif 3832 df-un 3834 df-in 3836 df-ss 3843 df-pss 3845 df-nul 4179 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-tp 4446 df-op 4448 df-uni 4713 df-int 4750 df-iun 4794 df-iin 4795 df-br 4930 df-opab 4992 df-mpt 5009 df-tr 5031 df-id 5312 df-eprel 5317 df-po 5326 df-so 5327 df-fr 5366 df-se 5367 df-we 5368 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-pred 5986 df-ord 6032 df-on 6033 df-lim 6034 df-suc 6035 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-isom 6197 df-riota 6937 df-ov 6979 df-oprab 6980 df-mpo 6981 df-om 7397 df-1st 7501 df-2nd 7502 df-wrecs 7750 df-recs 7812 df-rdg 7850 df-1o 7905 df-2o 7906 df-oadd 7909 df-er 8089 df-map 8208 df-en 8307 df-dom 8308 df-sdom 8309 df-fin 8310 df-card 9162 df-pnf 10476 df-mnf 10477 df-xr 10478 df-ltxr 10479 df-le 10480 df-sub 10672 df-neg 10673 df-nn 11440 df-2 11503 df-3 11504 df-4 11505 df-5 11506 df-6 11507 df-7 11508 df-8 11509 df-9 11510 df-n0 11708 df-z 11794 df-uz 12059 df-fz 12709 df-struct 16341 df-ndx 16342 df-slot 16343 df-base 16345 df-sets 16346 df-ress 16347 df-plusg 16434 df-tset 16440 df-0g 16571 df-mre 16715 df-mrc 16716 df-acs 16718 df-mgm 17710 df-sgrp 17752 df-mnd 17763 df-submnd 17804 df-grp 17894 df-minusg 17895 df-subg 18060 df-symg 18267 df-pmtr 18331

This theorem is referenced by: symggen2 18360 psgneldm2 18394

W3C validator