Theorem frgpup3lem 18013

Description: The evaluation map has the intended behavior on the generators. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by Mario Carneiro, 28-Feb-2016.)

Hypotheses

Ref	Expression
frgpup.b	⊢ 𝐵 = (Base‘𝐻)
frgpup.n	⊢ 𝑁 = (invg‘𝐻)
frgpup.t	⊢ 𝑇 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2𝑜 ↦ if(𝑧 = ∅, (𝐹‘𝑦), (𝑁‘(𝐹‘𝑦))))
frgpup.h	⊢ (𝜑 → 𝐻 ∈ Grp)
frgpup.i	⊢ (𝜑 → 𝐼 ∈ 𝑉)
frgpup.a	⊢ (𝜑 → 𝐹:𝐼⟶𝐵)
frgpup.w	⊢ 𝑊 = ( I ‘Word (𝐼 × 2𝑜))
frgpup.r	⊢ ∼ = ( ~FG ‘𝐼)
frgpup.g	⊢ 𝐺 = (freeGrp‘𝐼)
frgpup.x	⊢ 𝑋 = (Base‘𝐺)
frgpup.e	⊢ 𝐸 = ran (𝑔 ∈ 𝑊 ↦ ⟨[𝑔] ∼ , (𝐻 Σg (𝑇 ∘ 𝑔))⟩)
frgpup.u	⊢ 𝑈 = (varFGrp‘𝐼)
frgpup3.k	⊢ (𝜑 → 𝐾 ∈ (𝐺 GrpHom 𝐻))
frgpup3.e	⊢ (𝜑 → (𝐾 ∘ 𝑈) = 𝐹)

Assertion

Ref	Expression
frgpup3lem	⊢ (𝜑 → 𝐾 = 𝐸)

Distinct variable groups:   𝑦,𝑔,𝑧   𝑔,𝐻   𝑦,𝐹,𝑧   𝑦,𝑁,𝑧   𝐵,𝑔,𝑦,𝑧   𝑇,𝑔   ∼ ,𝑔   𝜑,𝑔,𝑦,𝑧   𝑦,𝐼,𝑧   𝑔,𝑊

Allowed substitution hints:   ∼ (𝑦,𝑧)   𝑇(𝑦,𝑧)   𝑈(𝑦,𝑧,𝑔)   𝐸(𝑦,𝑧,𝑔)   𝐹(𝑔)   𝐺(𝑦,𝑧,𝑔)   𝐻(𝑦,𝑧)   𝐼(𝑔)   𝐾(𝑦,𝑧,𝑔)   𝑁(𝑔)   𝑉(𝑦,𝑧,𝑔)   𝑊(𝑦,𝑧)   𝑋(𝑦,𝑧,𝑔)

Proof of Theorem frgpup3lem

Dummy variables 𝑎 𝑡 𝑛 𝑖 𝑗 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		frgpup3.k	. . 3 ⊢ (𝜑 → 𝐾 ∈ (𝐺 GrpHom 𝐻))
2		frgpup.x	. . . 4 ⊢ 𝑋 = (Base‘𝐺)
3		frgpup.b	. . . 4 ⊢ 𝐵 = (Base‘𝐻)
4	2, 3	ghmf 17487	. . 3 ⊢ (𝐾 ∈ (𝐺 GrpHom 𝐻) → 𝐾:𝑋⟶𝐵)
5		ffn 5958	. . 3 ⊢ (𝐾:𝑋⟶𝐵 → 𝐾 Fn 𝑋)
6	1, 4, 5	3syl 18	. 2 ⊢ (𝜑 → 𝐾 Fn 𝑋)
7		frgpup.n	. . . 4 ⊢ 𝑁 = (invg‘𝐻)
8		frgpup.t	. . . 4 ⊢ 𝑇 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2𝑜 ↦ if(𝑧 = ∅, (𝐹‘𝑦), (𝑁‘(𝐹‘𝑦))))
9		frgpup.h	. . . 4 ⊢ (𝜑 → 𝐻 ∈ Grp)
10		frgpup.i	. . . 4 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
11		frgpup.a	. . . 4 ⊢ (𝜑 → 𝐹:𝐼⟶𝐵)
12		frgpup.w	. . . 4 ⊢ 𝑊 = ( I ‘Word (𝐼 × 2𝑜))
13		frgpup.r	. . . 4 ⊢ ∼ = ( ~FG ‘𝐼)
14		frgpup.g	. . . 4 ⊢ 𝐺 = (freeGrp‘𝐼)
15		frgpup.e	. . . 4 ⊢ 𝐸 = ran (𝑔 ∈ 𝑊 ↦ ⟨[𝑔] ∼ , (𝐻 Σg (𝑇 ∘ 𝑔))⟩)
16	3, 7, 8, 9, 10, 11, 12, 13, 14, 2, 15	frgpup1 18011	. . 3 ⊢ (𝜑 → 𝐸 ∈ (𝐺 GrpHom 𝐻))
17	2, 3	ghmf 17487	. . 3 ⊢ (𝐸 ∈ (𝐺 GrpHom 𝐻) → 𝐸:𝑋⟶𝐵)
18		ffn 5958	. . 3 ⊢ (𝐸:𝑋⟶𝐵 → 𝐸 Fn 𝑋)
19	16, 17, 18	3syl 18	. 2 ⊢ (𝜑 → 𝐸 Fn 𝑋)
20		eqid 2610	. . . . . . . . 9 ⊢ (freeMnd‘(𝐼 × 2𝑜)) = (freeMnd‘(𝐼 × 2𝑜))
21	14, 20, 13	frgpval 17994	. . . . . . . 8 ⊢ (𝐼 ∈ 𝑉 → 𝐺 = ((freeMnd‘(𝐼 × 2𝑜)) /s ∼ ))
22	10, 21	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐺 = ((freeMnd‘(𝐼 × 2𝑜)) /s ∼ ))
23		2on 7455	. . . . . . . . . . 11 ⊢ 2𝑜 ∈ On
24		xpexg 6858	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ 𝑉 ∧ 2𝑜 ∈ On) → (𝐼 × 2𝑜) ∈ V)
25	10, 23, 24	sylancl 693	. . . . . . . . . 10 ⊢ (𝜑 → (𝐼 × 2𝑜) ∈ V)
26		wrdexg 13170	. . . . . . . . . 10 ⊢ ((𝐼 × 2𝑜) ∈ V → Word (𝐼 × 2𝑜) ∈ V)
27		fvi 6165	. . . . . . . . . 10 ⊢ (Word (𝐼 × 2𝑜) ∈ V → ( I ‘Word (𝐼 × 2𝑜)) = Word (𝐼 × 2𝑜))
28	25, 26, 27	3syl 18	. . . . . . . . 9 ⊢ (𝜑 → ( I ‘Word (𝐼 × 2𝑜)) = Word (𝐼 × 2𝑜))
29	12, 28	syl5eq 2656	. . . . . . . 8 ⊢ (𝜑 → 𝑊 = Word (𝐼 × 2𝑜))
30		eqid 2610	. . . . . . . . . 10 ⊢ (Base‘(freeMnd‘(𝐼 × 2𝑜))) = (Base‘(freeMnd‘(𝐼 × 2𝑜)))
31	20, 30	frmdbas 17212	. . . . . . . . 9 ⊢ ((𝐼 × 2𝑜) ∈ V → (Base‘(freeMnd‘(𝐼 × 2𝑜))) = Word (𝐼 × 2𝑜))
32	25, 31	syl 17	. . . . . . . 8 ⊢ (𝜑 → (Base‘(freeMnd‘(𝐼 × 2𝑜))) = Word (𝐼 × 2𝑜))
33	29, 32	eqtr4d 2647	. . . . . . 7 ⊢ (𝜑 → 𝑊 = (Base‘(freeMnd‘(𝐼 × 2𝑜))))
34		fvex 6113	. . . . . . . . 9 ⊢ ( ~FG ‘𝐼) ∈ V
35	13, 34	eqeltri 2684	. . . . . . . 8 ⊢ ∼ ∈ V
36	35	a1i 11	. . . . . . 7 ⊢ (𝜑 → ∼ ∈ V)
37		fvex 6113	. . . . . . . 8 ⊢ (freeMnd‘(𝐼 × 2𝑜)) ∈ V
38	37	a1i 11	. . . . . . 7 ⊢ (𝜑 → (freeMnd‘(𝐼 × 2𝑜)) ∈ V)
39	22, 33, 36, 38	qusbas 16028	. . . . . 6 ⊢ (𝜑 → (𝑊 / ∼ ) = (Base‘𝐺))
40	39, 2	syl6reqr 2663	. . . . 5 ⊢ (𝜑 → 𝑋 = (𝑊 / ∼ ))
41		eqimss 3620	. . . . 5 ⊢ (𝑋 = (𝑊 / ∼ ) → 𝑋 ⊆ (𝑊 / ∼ ))
42	40, 41	syl 17	. . . 4 ⊢ (𝜑 → 𝑋 ⊆ (𝑊 / ∼ ))
43	42	sselda 3568	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑋) → 𝑎 ∈ (𝑊 / ∼ ))
44		eqid 2610	. . . 4 ⊢ (𝑊 / ∼ ) = (𝑊 / ∼ )
45		fveq2 6103	. . . . 5 ⊢ ([𝑡] ∼ = 𝑎 → (𝐾‘[𝑡] ∼ ) = (𝐾‘𝑎))
46		fveq2 6103	. . . . 5 ⊢ ([𝑡] ∼ = 𝑎 → (𝐸‘[𝑡] ∼ ) = (𝐸‘𝑎))
47	45, 46	eqeq12d 2625	. . . 4 ⊢ ([𝑡] ∼ = 𝑎 → ((𝐾‘[𝑡] ∼ ) = (𝐸‘[𝑡] ∼ ) ↔ (𝐾‘𝑎) = (𝐸‘𝑎)))
48		simpr 476	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) → 𝑡 ∈ 𝑊)
49	29	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) → 𝑊 = Word (𝐼 × 2𝑜))
50	48, 49	eleqtrd 2690	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) → 𝑡 ∈ Word (𝐼 × 2𝑜))
51		wrdf 13165	. . . . . . . . . . . . 13 ⊢ (𝑡 ∈ Word (𝐼 × 2𝑜) → 𝑡:(0..^(#‘𝑡))⟶(𝐼 × 2𝑜))
52	50, 51	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) → 𝑡:(0..^(#‘𝑡))⟶(𝐼 × 2𝑜))
53	52	ffvelrnda 6267	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑊) ∧ 𝑛 ∈ (0..^(#‘𝑡))) → (𝑡‘𝑛) ∈ (𝐼 × 2𝑜))
54		elxp2 5056	. . . . . . . . . . 11 ⊢ ((𝑡‘𝑛) ∈ (𝐼 × 2𝑜) ↔ ∃𝑖 ∈ 𝐼 ∃𝑗 ∈ 2𝑜 (𝑡‘𝑛) = ⟨𝑖, 𝑗⟩)
55	53, 54	sylib 207	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑊) ∧ 𝑛 ∈ (0..^(#‘𝑡))) → ∃𝑖 ∈ 𝐼 ∃𝑗 ∈ 2𝑜 (𝑡‘𝑛) = ⟨𝑖, 𝑗⟩)
56		fveq2 6103	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑖 → (𝐹‘𝑦) = (𝐹‘𝑖))
57	56	fveq2d 6107	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑖 → (𝑁‘(𝐹‘𝑦)) = (𝑁‘(𝐹‘𝑖)))
58	56, 57	ifeq12d 4056	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑖 → if(𝑧 = ∅, (𝐹‘𝑦), (𝑁‘(𝐹‘𝑦))) = if(𝑧 = ∅, (𝐹‘𝑖), (𝑁‘(𝐹‘𝑖))))
59		eqeq1 2614	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = 𝑗 → (𝑧 = ∅ ↔ 𝑗 = ∅))
60	59	ifbid 4058	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑗 → if(𝑧 = ∅, (𝐹‘𝑖), (𝑁‘(𝐹‘𝑖))) = if(𝑗 = ∅, (𝐹‘𝑖), (𝑁‘(𝐹‘𝑖))))
61		fvex 6113	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹‘𝑖) ∈ V
62		fvex 6113	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁‘(𝐹‘𝑖)) ∈ V
63	61, 62	ifex 4106	. . . . . . . . . . . . . . . 16 ⊢ if(𝑗 = ∅, (𝐹‘𝑖), (𝑁‘(𝐹‘𝑖))) ∈ V
64	58, 60, 8, 63	ovmpt2 6694	. . . . . . . . . . . . . . 15 ⊢ ((𝑖 ∈ 𝐼 ∧ 𝑗 ∈ 2𝑜) → (𝑖𝑇𝑗) = if(𝑗 = ∅, (𝐹‘𝑖), (𝑁‘(𝐹‘𝑖))))
65	64	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝐼 ∧ 𝑗 ∈ 2𝑜)) → (𝑖𝑇𝑗) = if(𝑗 = ∅, (𝐹‘𝑖), (𝑁‘(𝐹‘𝑖))))
66		elpri 4145	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ {∅, 1𝑜} → (𝑗 = ∅ ∨ 𝑗 = 1𝑜))
67		df2o3 7460	. . . . . . . . . . . . . . . . 17 ⊢ 2𝑜 = {∅, 1𝑜}
68	66, 67	eleq2s 2706	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ 2𝑜 → (𝑗 = ∅ ∨ 𝑗 = 1𝑜))
69		frgpup3.e	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝐾 ∘ 𝑈) = 𝐹)
70	69	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (𝐾 ∘ 𝑈) = 𝐹)
71	70	fveq1d 6105	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ((𝐾 ∘ 𝑈)‘𝑖) = (𝐹‘𝑖))
72		frgpup.u	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝑈 = (varFGrp‘𝐼)
73	13, 72, 14, 2	vrgpf 18004	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐼 ∈ 𝑉 → 𝑈:𝐼⟶𝑋)
74	10, 73	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝑈:𝐼⟶𝑋)
75		fvco3 6185	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑈:𝐼⟶𝑋 ∧ 𝑖 ∈ 𝐼) → ((𝐾 ∘ 𝑈)‘𝑖) = (𝐾‘(𝑈‘𝑖)))
76	74, 75	sylan 487	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ((𝐾 ∘ 𝑈)‘𝑖) = (𝐾‘(𝑈‘𝑖)))
77	71, 76	eqtr3d 2646	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (𝐹‘𝑖) = (𝐾‘(𝑈‘𝑖)))
78	77	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 = ∅) → (𝐹‘𝑖) = (𝐾‘(𝑈‘𝑖)))
79		iftrue 4042	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 = ∅ → if(𝑗 = ∅, (𝐹‘𝑖), (𝑁‘(𝐹‘𝑖))) = (𝐹‘𝑖))
80	79	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 = ∅) → if(𝑗 = ∅, (𝐹‘𝑖), (𝑁‘(𝐹‘𝑖))) = (𝐹‘𝑖))
81		simpr 476	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 = ∅) → 𝑗 = ∅)
82	81	opeq2d 4347	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 = ∅) → ⟨𝑖, 𝑗⟩ = ⟨𝑖, ∅⟩)
83	82	s1eqd 13234	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 = ∅) → ⟨“⟨𝑖, 𝑗⟩”⟩ = ⟨“⟨𝑖, ∅⟩”⟩)
84	83	eceq1d 7670	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 = ∅) → [⟨“⟨𝑖, 𝑗⟩”⟩] ∼ = [⟨“⟨𝑖, ∅⟩”⟩] ∼ )
85	13, 72	vrgpval 18003	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑖 ∈ 𝐼) → (𝑈‘𝑖) = [⟨“⟨𝑖, ∅⟩”⟩] ∼ )
86	10, 85	sylan 487	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (𝑈‘𝑖) = [⟨“⟨𝑖, ∅⟩”⟩] ∼ )
87	86	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 = ∅) → (𝑈‘𝑖) = [⟨“⟨𝑖, ∅⟩”⟩] ∼ )
88	84, 87	eqtr4d 2647	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 = ∅) → [⟨“⟨𝑖, 𝑗⟩”⟩] ∼ = (𝑈‘𝑖))
89	88	fveq2d 6107	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 = ∅) → (𝐾‘[⟨“⟨𝑖, 𝑗⟩”⟩] ∼ ) = (𝐾‘(𝑈‘𝑖)))
90	78, 80, 89	3eqtr4d 2654	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 = ∅) → if(𝑗 = ∅, (𝐹‘𝑖), (𝑁‘(𝐹‘𝑖))) = (𝐾‘[⟨“⟨𝑖, 𝑗⟩”⟩] ∼ ))
91	77	fveq2d 6107	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (𝑁‘(𝐹‘𝑖)) = (𝑁‘(𝐾‘(𝑈‘𝑖))))
92	1	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝐾 ∈ (𝐺 GrpHom 𝐻))
93	74	ffvelrnda 6267	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (𝑈‘𝑖) ∈ 𝑋)
94		eqid 2610	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (invg‘𝐺) = (invg‘𝐺)
95	2, 94, 7	ghminv 17490	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐾 ∈ (𝐺 GrpHom 𝐻) ∧ (𝑈‘𝑖) ∈ 𝑋) → (𝐾‘((invg‘𝐺)‘(𝑈‘𝑖))) = (𝑁‘(𝐾‘(𝑈‘𝑖))))
96	92, 93, 95	syl2anc 691	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (𝐾‘((invg‘𝐺)‘(𝑈‘𝑖))) = (𝑁‘(𝐾‘(𝑈‘𝑖))))
97	91, 96	eqtr4d 2647	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (𝑁‘(𝐹‘𝑖)) = (𝐾‘((invg‘𝐺)‘(𝑈‘𝑖))))
98	97	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 = 1𝑜) → (𝑁‘(𝐹‘𝑖)) = (𝐾‘((invg‘𝐺)‘(𝑈‘𝑖))))
99		1n0 7462	. . . . . . . . . . . . . . . . . . . 20 ⊢ 1𝑜 ≠ ∅
100		simpr 476	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 = 1𝑜) → 𝑗 = 1𝑜)
101	100	neeq1d 2841	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 = 1𝑜) → (𝑗 ≠ ∅ ↔ 1𝑜 ≠ ∅))
102	99, 101	mpbiri 247	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 = 1𝑜) → 𝑗 ≠ ∅)
103		ifnefalse 4048	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 ≠ ∅ → if(𝑗 = ∅, (𝐹‘𝑖), (𝑁‘(𝐹‘𝑖))) = (𝑁‘(𝐹‘𝑖)))
104	102, 103	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 = 1𝑜) → if(𝑗 = ∅, (𝐹‘𝑖), (𝑁‘(𝐹‘𝑖))) = (𝑁‘(𝐹‘𝑖)))
105	100	opeq2d 4347	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 = 1𝑜) → ⟨𝑖, 𝑗⟩ = ⟨𝑖, 1𝑜⟩)
106	105	s1eqd 13234	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 = 1𝑜) → ⟨“⟨𝑖, 𝑗⟩”⟩ = ⟨“⟨𝑖, 1𝑜⟩”⟩)
107	106	eceq1d 7670	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 = 1𝑜) → [⟨“⟨𝑖, 𝑗⟩”⟩] ∼ = [⟨“⟨𝑖, 1𝑜⟩”⟩] ∼ )
108	13, 72, 14, 94	vrgpinv 18005	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑖 ∈ 𝐼) → ((invg‘𝐺)‘(𝑈‘𝑖)) = [⟨“⟨𝑖, 1𝑜⟩”⟩] ∼ )
109	10, 108	sylan 487	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ((invg‘𝐺)‘(𝑈‘𝑖)) = [⟨“⟨𝑖, 1𝑜⟩”⟩] ∼ )
110	109	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 = 1𝑜) → ((invg‘𝐺)‘(𝑈‘𝑖)) = [⟨“⟨𝑖, 1𝑜⟩”⟩] ∼ )
111	107, 110	eqtr4d 2647	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 = 1𝑜) → [⟨“⟨𝑖, 𝑗⟩”⟩] ∼ = ((invg‘𝐺)‘(𝑈‘𝑖)))
112	111	fveq2d 6107	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 = 1𝑜) → (𝐾‘[⟨“⟨𝑖, 𝑗⟩”⟩] ∼ ) = (𝐾‘((invg‘𝐺)‘(𝑈‘𝑖))))
113	98, 104, 112	3eqtr4d 2654	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 = 1𝑜) → if(𝑗 = ∅, (𝐹‘𝑖), (𝑁‘(𝐹‘𝑖))) = (𝐾‘[⟨“⟨𝑖, 𝑗⟩”⟩] ∼ ))
114	90, 113	jaodan 822	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ (𝑗 = ∅ ∨ 𝑗 = 1𝑜)) → if(𝑗 = ∅, (𝐹‘𝑖), (𝑁‘(𝐹‘𝑖))) = (𝐾‘[⟨“⟨𝑖, 𝑗⟩”⟩] ∼ ))
115	68, 114	sylan2 490	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ 2𝑜) → if(𝑗 = ∅, (𝐹‘𝑖), (𝑁‘(𝐹‘𝑖))) = (𝐾‘[⟨“⟨𝑖, 𝑗⟩”⟩] ∼ ))
116	115	anasss 677	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝐼 ∧ 𝑗 ∈ 2𝑜)) → if(𝑗 = ∅, (𝐹‘𝑖), (𝑁‘(𝐹‘𝑖))) = (𝐾‘[⟨“⟨𝑖, 𝑗⟩”⟩] ∼ ))
117	65, 116	eqtrd 2644	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝐼 ∧ 𝑗 ∈ 2𝑜)) → (𝑖𝑇𝑗) = (𝐾‘[⟨“⟨𝑖, 𝑗⟩”⟩] ∼ ))
118		fveq2 6103	. . . . . . . . . . . . . . 15 ⊢ ((𝑡‘𝑛) = ⟨𝑖, 𝑗⟩ → (𝑇‘(𝑡‘𝑛)) = (𝑇‘⟨𝑖, 𝑗⟩))
119		df-ov 6552	. . . . . . . . . . . . . . 15 ⊢ (𝑖𝑇𝑗) = (𝑇‘⟨𝑖, 𝑗⟩)
120	118, 119	syl6eqr 2662	. . . . . . . . . . . . . 14 ⊢ ((𝑡‘𝑛) = ⟨𝑖, 𝑗⟩ → (𝑇‘(𝑡‘𝑛)) = (𝑖𝑇𝑗))
121		s1eq 13233	. . . . . . . . . . . . . . . 16 ⊢ ((𝑡‘𝑛) = ⟨𝑖, 𝑗⟩ → ⟨“(𝑡‘𝑛)”⟩ = ⟨“⟨𝑖, 𝑗⟩”⟩)
122	121	eceq1d 7670	. . . . . . . . . . . . . . 15 ⊢ ((𝑡‘𝑛) = ⟨𝑖, 𝑗⟩ → [⟨“(𝑡‘𝑛)”⟩] ∼ = [⟨“⟨𝑖, 𝑗⟩”⟩] ∼ )
123	122	fveq2d 6107	. . . . . . . . . . . . . 14 ⊢ ((𝑡‘𝑛) = ⟨𝑖, 𝑗⟩ → (𝐾‘[⟨“(𝑡‘𝑛)”⟩] ∼ ) = (𝐾‘[⟨“⟨𝑖, 𝑗⟩”⟩] ∼ ))
124	120, 123	eqeq12d 2625	. . . . . . . . . . . . 13 ⊢ ((𝑡‘𝑛) = ⟨𝑖, 𝑗⟩ → ((𝑇‘(𝑡‘𝑛)) = (𝐾‘[⟨“(𝑡‘𝑛)”⟩] ∼ ) ↔ (𝑖𝑇𝑗) = (𝐾‘[⟨“⟨𝑖, 𝑗⟩”⟩] ∼ )))
125	117, 124	syl5ibrcom 236	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝐼 ∧ 𝑗 ∈ 2𝑜)) → ((𝑡‘𝑛) = ⟨𝑖, 𝑗⟩ → (𝑇‘(𝑡‘𝑛)) = (𝐾‘[⟨“(𝑡‘𝑛)”⟩] ∼ )))
126	125	rexlimdvva 3020	. . . . . . . . . . 11 ⊢ (𝜑 → (∃𝑖 ∈ 𝐼 ∃𝑗 ∈ 2𝑜 (𝑡‘𝑛) = ⟨𝑖, 𝑗⟩ → (𝑇‘(𝑡‘𝑛)) = (𝐾‘[⟨“(𝑡‘𝑛)”⟩] ∼ )))
127	126	ad2antrr 758	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑊) ∧ 𝑛 ∈ (0..^(#‘𝑡))) → (∃𝑖 ∈ 𝐼 ∃𝑗 ∈ 2𝑜 (𝑡‘𝑛) = ⟨𝑖, 𝑗⟩ → (𝑇‘(𝑡‘𝑛)) = (𝐾‘[⟨“(𝑡‘𝑛)”⟩] ∼ )))
128	55, 127	mpd 15	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑊) ∧ 𝑛 ∈ (0..^(#‘𝑡))) → (𝑇‘(𝑡‘𝑛)) = (𝐾‘[⟨“(𝑡‘𝑛)”⟩] ∼ ))
129	128	mpteq2dva 4672	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) → (𝑛 ∈ (0..^(#‘𝑡)) ↦ (𝑇‘(𝑡‘𝑛))) = (𝑛 ∈ (0..^(#‘𝑡)) ↦ (𝐾‘[⟨“(𝑡‘𝑛)”⟩] ∼ )))
130	3, 7, 8, 9, 10, 11	frgpuptf 18006	. . . . . . . . . 10 ⊢ (𝜑 → 𝑇:(𝐼 × 2𝑜)⟶𝐵)
131	130	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) → 𝑇:(𝐼 × 2𝑜)⟶𝐵)
132		fcompt 6306	. . . . . . . . 9 ⊢ ((𝑇:(𝐼 × 2𝑜)⟶𝐵 ∧ 𝑡:(0..^(#‘𝑡))⟶(𝐼 × 2𝑜)) → (𝑇 ∘ 𝑡) = (𝑛 ∈ (0..^(#‘𝑡)) ↦ (𝑇‘(𝑡‘𝑛))))
133	131, 52, 132	syl2anc 691	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) → (𝑇 ∘ 𝑡) = (𝑛 ∈ (0..^(#‘𝑡)) ↦ (𝑇‘(𝑡‘𝑛))))
134	53	s1cld 13236	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑊) ∧ 𝑛 ∈ (0..^(#‘𝑡))) → ⟨“(𝑡‘𝑛)”⟩ ∈ Word (𝐼 × 2𝑜))
135	29	ad2antrr 758	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑊) ∧ 𝑛 ∈ (0..^(#‘𝑡))) → 𝑊 = Word (𝐼 × 2𝑜))
136	134, 135	eleqtrrd 2691	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑊) ∧ 𝑛 ∈ (0..^(#‘𝑡))) → ⟨“(𝑡‘𝑛)”⟩ ∈ 𝑊)
137	14, 13, 12, 2	frgpeccl 17997	. . . . . . . . . 10 ⊢ (⟨“(𝑡‘𝑛)”⟩ ∈ 𝑊 → [⟨“(𝑡‘𝑛)”⟩] ∼ ∈ 𝑋)
138	136, 137	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑊) ∧ 𝑛 ∈ (0..^(#‘𝑡))) → [⟨“(𝑡‘𝑛)”⟩] ∼ ∈ 𝑋)
139	52	feqmptd 6159	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) → 𝑡 = (𝑛 ∈ (0..^(#‘𝑡)) ↦ (𝑡‘𝑛)))
140	10	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) → 𝐼 ∈ 𝑉)
141	140, 23, 24	sylancl 693	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) → (𝐼 × 2𝑜) ∈ V)
142		eqid 2610	. . . . . . . . . . . . 13 ⊢ (varFMnd‘(𝐼 × 2𝑜)) = (varFMnd‘(𝐼 × 2𝑜))
143	142	vrmdfval 17216	. . . . . . . . . . . 12 ⊢ ((𝐼 × 2𝑜) ∈ V → (varFMnd‘(𝐼 × 2𝑜)) = (𝑤 ∈ (𝐼 × 2𝑜) ↦ ⟨“𝑤”⟩))
144	141, 143	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) → (varFMnd‘(𝐼 × 2𝑜)) = (𝑤 ∈ (𝐼 × 2𝑜) ↦ ⟨“𝑤”⟩))
145		s1eq 13233	. . . . . . . . . . 11 ⊢ (𝑤 = (𝑡‘𝑛) → ⟨“𝑤”⟩ = ⟨“(𝑡‘𝑛)”⟩)
146	53, 139, 144, 145	fmptco 6303	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) → ((varFMnd‘(𝐼 × 2𝑜)) ∘ 𝑡) = (𝑛 ∈ (0..^(#‘𝑡)) ↦ ⟨“(𝑡‘𝑛)”⟩))
147		eqidd 2611	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) → (𝑤 ∈ 𝑊 ↦ [𝑤] ∼ ) = (𝑤 ∈ 𝑊 ↦ [𝑤] ∼ ))
148		eceq1 7669	. . . . . . . . . 10 ⊢ (𝑤 = ⟨“(𝑡‘𝑛)”⟩ → [𝑤] ∼ = [⟨“(𝑡‘𝑛)”⟩] ∼ )
149	136, 146, 147, 148	fmptco 6303	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) → ((𝑤 ∈ 𝑊 ↦ [𝑤] ∼ ) ∘ ((varFMnd‘(𝐼 × 2𝑜)) ∘ 𝑡)) = (𝑛 ∈ (0..^(#‘𝑡)) ↦ [⟨“(𝑡‘𝑛)”⟩] ∼ ))
150	1	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) → 𝐾 ∈ (𝐺 GrpHom 𝐻))
151	150, 4	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) → 𝐾:𝑋⟶𝐵)
152	151	feqmptd 6159	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) → 𝐾 = (𝑤 ∈ 𝑋 ↦ (𝐾‘𝑤)))
153		fveq2 6103	. . . . . . . . 9 ⊢ (𝑤 = [⟨“(𝑡‘𝑛)”⟩] ∼ → (𝐾‘𝑤) = (𝐾‘[⟨“(𝑡‘𝑛)”⟩] ∼ ))
154	138, 149, 152, 153	fmptco 6303	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) → (𝐾 ∘ ((𝑤 ∈ 𝑊 ↦ [𝑤] ∼ ) ∘ ((varFMnd‘(𝐼 × 2𝑜)) ∘ 𝑡))) = (𝑛 ∈ (0..^(#‘𝑡)) ↦ (𝐾‘[⟨“(𝑡‘𝑛)”⟩] ∼ )))
155	129, 133, 154	3eqtr4d 2654	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) → (𝑇 ∘ 𝑡) = (𝐾 ∘ ((𝑤 ∈ 𝑊 ↦ [𝑤] ∼ ) ∘ ((varFMnd‘(𝐼 × 2𝑜)) ∘ 𝑡))))
156	155	oveq2d 6565	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) → (𝐻 Σg (𝑇 ∘ 𝑡)) = (𝐻 Σg (𝐾 ∘ ((𝑤 ∈ 𝑊 ↦ [𝑤] ∼ ) ∘ ((varFMnd‘(𝐼 × 2𝑜)) ∘ 𝑡)))))
157	3, 7, 8, 9, 10, 11, 12, 13, 14, 2, 15	frgpupval 18010	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) → (𝐸‘[𝑡] ∼ ) = (𝐻 Σg (𝑇 ∘ 𝑡)))
158		ghmmhm 17493	. . . . . . . 8 ⊢ (𝐾 ∈ (𝐺 GrpHom 𝐻) → 𝐾 ∈ (𝐺 MndHom 𝐻))
159	150, 158	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) → 𝐾 ∈ (𝐺 MndHom 𝐻))
160	142	vrmdf 17218	. . . . . . . . . . 11 ⊢ ((𝐼 × 2𝑜) ∈ V → (varFMnd‘(𝐼 × 2𝑜)):(𝐼 × 2𝑜)⟶Word (𝐼 × 2𝑜))
161	141, 160	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) → (varFMnd‘(𝐼 × 2𝑜)):(𝐼 × 2𝑜)⟶Word (𝐼 × 2𝑜))
162	49	feq3d 5945	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) → ((varFMnd‘(𝐼 × 2𝑜)):(𝐼 × 2𝑜)⟶𝑊 ↔ (varFMnd‘(𝐼 × 2𝑜)):(𝐼 × 2𝑜)⟶Word (𝐼 × 2𝑜)))
163	161, 162	mpbird 246	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) → (varFMnd‘(𝐼 × 2𝑜)):(𝐼 × 2𝑜)⟶𝑊)
164		wrdco 13428	. . . . . . . . 9 ⊢ ((𝑡 ∈ Word (𝐼 × 2𝑜) ∧ (varFMnd‘(𝐼 × 2𝑜)):(𝐼 × 2𝑜)⟶𝑊) → ((varFMnd‘(𝐼 × 2𝑜)) ∘ 𝑡) ∈ Word 𝑊)
165	50, 163, 164	syl2anc 691	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) → ((varFMnd‘(𝐼 × 2𝑜)) ∘ 𝑡) ∈ Word 𝑊)
166	33	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) → 𝑊 = (Base‘(freeMnd‘(𝐼 × 2𝑜))))
167	166	mpteq1d 4666	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) → (𝑤 ∈ 𝑊 ↦ [𝑤] ∼ ) = (𝑤 ∈ (Base‘(freeMnd‘(𝐼 × 2𝑜))) ↦ [𝑤] ∼ ))
168		eqid 2610	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ (Base‘(freeMnd‘(𝐼 × 2𝑜))) ↦ [𝑤] ∼ ) = (𝑤 ∈ (Base‘(freeMnd‘(𝐼 × 2𝑜))) ↦ [𝑤] ∼ )
169	20, 30, 14, 13, 168	frgpmhm 18001	. . . . . . . . . . . 12 ⊢ (𝐼 ∈ 𝑉 → (𝑤 ∈ (Base‘(freeMnd‘(𝐼 × 2𝑜))) ↦ [𝑤] ∼ ) ∈ ((freeMnd‘(𝐼 × 2𝑜)) MndHom 𝐺))
170	140, 169	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) → (𝑤 ∈ (Base‘(freeMnd‘(𝐼 × 2𝑜))) ↦ [𝑤] ∼ ) ∈ ((freeMnd‘(𝐼 × 2𝑜)) MndHom 𝐺))
171	167, 170	eqeltrd 2688	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) → (𝑤 ∈ 𝑊 ↦ [𝑤] ∼ ) ∈ ((freeMnd‘(𝐼 × 2𝑜)) MndHom 𝐺))
172	30, 2	mhmf 17163	. . . . . . . . . 10 ⊢ ((𝑤 ∈ 𝑊 ↦ [𝑤] ∼ ) ∈ ((freeMnd‘(𝐼 × 2𝑜)) MndHom 𝐺) → (𝑤 ∈ 𝑊 ↦ [𝑤] ∼ ):(Base‘(freeMnd‘(𝐼 × 2𝑜)))⟶𝑋)
173	171, 172	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) → (𝑤 ∈ 𝑊 ↦ [𝑤] ∼ ):(Base‘(freeMnd‘(𝐼 × 2𝑜)))⟶𝑋)
174	166	feq2d 5944	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) → ((𝑤 ∈ 𝑊 ↦ [𝑤] ∼ ):𝑊⟶𝑋 ↔ (𝑤 ∈ 𝑊 ↦ [𝑤] ∼ ):(Base‘(freeMnd‘(𝐼 × 2𝑜)))⟶𝑋))
175	173, 174	mpbird 246	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) → (𝑤 ∈ 𝑊 ↦ [𝑤] ∼ ):𝑊⟶𝑋)
176		wrdco 13428	. . . . . . . 8 ⊢ ((((varFMnd‘(𝐼 × 2𝑜)) ∘ 𝑡) ∈ Word 𝑊 ∧ (𝑤 ∈ 𝑊 ↦ [𝑤] ∼ ):𝑊⟶𝑋) → ((𝑤 ∈ 𝑊 ↦ [𝑤] ∼ ) ∘ ((varFMnd‘(𝐼 × 2𝑜)) ∘ 𝑡)) ∈ Word 𝑋)
177	165, 175, 176	syl2anc 691	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) → ((𝑤 ∈ 𝑊 ↦ [𝑤] ∼ ) ∘ ((varFMnd‘(𝐼 × 2𝑜)) ∘ 𝑡)) ∈ Word 𝑋)
178	2	gsumwmhm 17205	. . . . . . 7 ⊢ ((𝐾 ∈ (𝐺 MndHom 𝐻) ∧ ((𝑤 ∈ 𝑊 ↦ [𝑤] ∼ ) ∘ ((varFMnd‘(𝐼 × 2𝑜)) ∘ 𝑡)) ∈ Word 𝑋) → (𝐾‘(𝐺 Σg ((𝑤 ∈ 𝑊 ↦ [𝑤] ∼ ) ∘ ((varFMnd‘(𝐼 × 2𝑜)) ∘ 𝑡)))) = (𝐻 Σg (𝐾 ∘ ((𝑤 ∈ 𝑊 ↦ [𝑤] ∼ ) ∘ ((varFMnd‘(𝐼 × 2𝑜)) ∘ 𝑡)))))
179	159, 177, 178	syl2anc 691	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) → (𝐾‘(𝐺 Σg ((𝑤 ∈ 𝑊 ↦ [𝑤] ∼ ) ∘ ((varFMnd‘(𝐼 × 2𝑜)) ∘ 𝑡)))) = (𝐻 Σg (𝐾 ∘ ((𝑤 ∈ 𝑊 ↦ [𝑤] ∼ ) ∘ ((varFMnd‘(𝐼 × 2𝑜)) ∘ 𝑡)))))
180	156, 157, 179	3eqtr4d 2654	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) → (𝐸‘[𝑡] ∼ ) = (𝐾‘(𝐺 Σg ((𝑤 ∈ 𝑊 ↦ [𝑤] ∼ ) ∘ ((varFMnd‘(𝐼 × 2𝑜)) ∘ 𝑡)))))
181	20, 142	frmdgsum 17222	. . . . . . . . 9 ⊢ (((𝐼 × 2𝑜) ∈ V ∧ 𝑡 ∈ Word (𝐼 × 2𝑜)) → ((freeMnd‘(𝐼 × 2𝑜)) Σg ((varFMnd‘(𝐼 × 2𝑜)) ∘ 𝑡)) = 𝑡)
182	141, 50, 181	syl2anc 691	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) → ((freeMnd‘(𝐼 × 2𝑜)) Σg ((varFMnd‘(𝐼 × 2𝑜)) ∘ 𝑡)) = 𝑡)
183	182	fveq2d 6107	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) → ((𝑤 ∈ 𝑊 ↦ [𝑤] ∼ )‘((freeMnd‘(𝐼 × 2𝑜)) Σg ((varFMnd‘(𝐼 × 2𝑜)) ∘ 𝑡))) = ((𝑤 ∈ 𝑊 ↦ [𝑤] ∼ )‘𝑡))
184		wrdco 13428	. . . . . . . . . 10 ⊢ ((𝑡 ∈ Word (𝐼 × 2𝑜) ∧ (varFMnd‘(𝐼 × 2𝑜)):(𝐼 × 2𝑜)⟶Word (𝐼 × 2𝑜)) → ((varFMnd‘(𝐼 × 2𝑜)) ∘ 𝑡) ∈ Word Word (𝐼 × 2𝑜))
185	50, 161, 184	syl2anc 691	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) → ((varFMnd‘(𝐼 × 2𝑜)) ∘ 𝑡) ∈ Word Word (𝐼 × 2𝑜))
186	32	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) → (Base‘(freeMnd‘(𝐼 × 2𝑜))) = Word (𝐼 × 2𝑜))
187		wrdeq 13182	. . . . . . . . . 10 ⊢ ((Base‘(freeMnd‘(𝐼 × 2𝑜))) = Word (𝐼 × 2𝑜) → Word (Base‘(freeMnd‘(𝐼 × 2𝑜))) = Word Word (𝐼 × 2𝑜))
188	186, 187	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) → Word (Base‘(freeMnd‘(𝐼 × 2𝑜))) = Word Word (𝐼 × 2𝑜))
189	185, 188	eleqtrrd 2691	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) → ((varFMnd‘(𝐼 × 2𝑜)) ∘ 𝑡) ∈ Word (Base‘(freeMnd‘(𝐼 × 2𝑜))))
190	30	gsumwmhm 17205	. . . . . . . 8 ⊢ (((𝑤 ∈ 𝑊 ↦ [𝑤] ∼ ) ∈ ((freeMnd‘(𝐼 × 2𝑜)) MndHom 𝐺) ∧ ((varFMnd‘(𝐼 × 2𝑜)) ∘ 𝑡) ∈ Word (Base‘(freeMnd‘(𝐼 × 2𝑜)))) → ((𝑤 ∈ 𝑊 ↦ [𝑤] ∼ )‘((freeMnd‘(𝐼 × 2𝑜)) Σg ((varFMnd‘(𝐼 × 2𝑜)) ∘ 𝑡))) = (𝐺 Σg ((𝑤 ∈ 𝑊 ↦ [𝑤] ∼ ) ∘ ((varFMnd‘(𝐼 × 2𝑜)) ∘ 𝑡))))
191	171, 189, 190	syl2anc 691	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) → ((𝑤 ∈ 𝑊 ↦ [𝑤] ∼ )‘((freeMnd‘(𝐼 × 2𝑜)) Σg ((varFMnd‘(𝐼 × 2𝑜)) ∘ 𝑡))) = (𝐺 Σg ((𝑤 ∈ 𝑊 ↦ [𝑤] ∼ ) ∘ ((varFMnd‘(𝐼 × 2𝑜)) ∘ 𝑡))))
192	12, 13	efger 17954	. . . . . . . . 9 ⊢ ∼ Er 𝑊
193	192	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) → ∼ Er 𝑊)
194		fvex 6113	. . . . . . . . . 10 ⊢ ( I ‘Word (𝐼 × 2𝑜)) ∈ V
195	12, 194	eqeltri 2684	. . . . . . . . 9 ⊢ 𝑊 ∈ V
196	195	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) → 𝑊 ∈ V)
197		eqid 2610	. . . . . . . 8 ⊢ (𝑤 ∈ 𝑊 ↦ [𝑤] ∼ ) = (𝑤 ∈ 𝑊 ↦ [𝑤] ∼ )
198	193, 196, 197	divsfval 16030	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) → ((𝑤 ∈ 𝑊 ↦ [𝑤] ∼ )‘𝑡) = [𝑡] ∼ )
199	183, 191, 198	3eqtr3d 2652	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) → (𝐺 Σg ((𝑤 ∈ 𝑊 ↦ [𝑤] ∼ ) ∘ ((varFMnd‘(𝐼 × 2𝑜)) ∘ 𝑡))) = [𝑡] ∼ )
200	199	fveq2d 6107	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) → (𝐾‘(𝐺 Σg ((𝑤 ∈ 𝑊 ↦ [𝑤] ∼ ) ∘ ((varFMnd‘(𝐼 × 2𝑜)) ∘ 𝑡)))) = (𝐾‘[𝑡] ∼ ))
201	180, 200	eqtr2d 2645	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) → (𝐾‘[𝑡] ∼ ) = (𝐸‘[𝑡] ∼ ))
202	44, 47, 201	ectocld 7701	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑊 / ∼ )) → (𝐾‘𝑎) = (𝐸‘𝑎))
203	43, 202	syldan 486	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑋) → (𝐾‘𝑎) = (𝐸‘𝑎))
204	6, 19, 203	eqfnfvd 6222	1 ⊢ (𝜑 → 𝐾 = 𝐸)

Syntax hints:   → wi 4   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∃wrex 2897  Vcvv 3173   ⊆ wss 3540  ∅c0 3874  ifcif 4036  {cpr 4127  ⟨cop 4131   ↦ cmpt 4643   I cid 4948   × cxp 5036  ran crn 5039   ∘ ccom 5042  Oncon0 5640   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  1_𝑜c1o 7440  2_𝑜c2o 7441   Er wer 7626  [cec 7627   / cqs 7628  0cc0 9815  ..^cfzo 12334  #chash 12979  Word cword 13146  ⟨“cs1 13149  Basecbs 15695   Σ_g cgsu 15924   /_s cqus 15988   MndHom cmhm 17156  freeMndcfrmd 17207  var_FMndcvrmd 17208  Grpcgrp 17245  inv_gcminusg 17246   GrpHom cghm 17480   ~_FG cefg 17942  freeGrpcfrgp 17943  var_FGrpcvrgp 17944

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-ot 4134  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-ec 7631  df-qs 7635  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-inf 8232  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-xnn0 11241  df-z 11255  df-dec 11370  df-uz 11564  df-fz 12198  df-fzo 12335  df-seq 12664  df-hash 12980  df-word 13154  df-lsw 13155  df-concat 13156  df-s1 13157  df-substr 13158  df-splice 13159  df-reverse 13160  df-s2 13444  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-mulr 15782  df-sca 15784  df-vsca 15785  df-ip 15786  df-tset 15787  df-ple 15788  df-ds 15791  df-0g 15925  df-gsum 15926  df-imas 15991  df-qus 15992  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-mhm 17158  df-submnd 17159  df-frmd 17209  df-vrmd 17210  df-grp 17248  df-minusg 17249  df-ghm 17481  df-efg 17945  df-frgp 17946  df-vrgp 17947

This theorem is referenced by:  frgpup3  18014