Theorem frgpuplem 19814

Description: Any assignment of the generators to target elements can be extended (uniquely) to a homomorphism from a free monoid to an arbitrary other monoid. (Contributed by Mario Carneiro, 2-Oct-2015.)

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)

Assertion

Ref	Expression
frgpuplem	⊢ ((𝜑 ∧ 𝐴 ∼ 𝐶) → (𝐻 Σg (𝑇 ∘ 𝐴)) = (𝐻 Σg (𝑇 ∘ 𝐶)))

Distinct variable groups:   𝑦,𝑧,𝐴   𝑦,𝐹,𝑧   𝑦,𝑁,𝑧   𝑦,𝐵,𝑧   𝜑,𝑦,𝑧   𝑦,𝐼,𝑧

Allowed substitution hints:   𝐶(𝑦,𝑧)   ∼ (𝑦,𝑧)   𝑇(𝑦,𝑧)   𝐻(𝑦,𝑧)   𝑉(𝑦,𝑧)   𝑊(𝑦,𝑧)

Proof of Theorem frgpuplem

Dummy variables 𝑎 𝑏 𝑢 𝑣 𝑛 𝑟 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		frgpup.w	. . . . . . 7 ⊢ 𝑊 = ( I ‘Word (𝐼 × 2o))
2		frgpup.r	. . . . . . 7 ⊢ ∼ = ( ~FG ‘𝐼)
3	1, 2	efgval 19759	. . . . . 6 ⊢ ∼ = ∩ {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥 ∈ 𝑊 ∀𝑛 ∈ (0...(♯‘𝑥))∀𝑎 ∈ 𝐼 ∀𝑏 ∈ 2o 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩))}
4		coeq2 5883	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝑣 → (𝑇 ∘ 𝑢) = (𝑇 ∘ 𝑣))
5	4	oveq2d 7464	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑣 → (𝐻 Σg (𝑇 ∘ 𝑢)) = (𝐻 Σg (𝑇 ∘ 𝑣)))
6		eqid 2740	. . . . . . . . . . . 12 ⊢ {⟨𝑢, 𝑣⟩ ∣ (𝐻 Σg (𝑇 ∘ 𝑢)) = (𝐻 Σg (𝑇 ∘ 𝑣))} = {⟨𝑢, 𝑣⟩ ∣ (𝐻 Σg (𝑇 ∘ 𝑢)) = (𝐻 Σg (𝑇 ∘ 𝑣))}
7	5, 6	eqer 8799	. . . . . . . . . . 11 ⊢ {⟨𝑢, 𝑣⟩ ∣ (𝐻 Σg (𝑇 ∘ 𝑢)) = (𝐻 Σg (𝑇 ∘ 𝑣))} Er V
8	7	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → {⟨𝑢, 𝑣⟩ ∣ (𝐻 Σg (𝑇 ∘ 𝑢)) = (𝐻 Σg (𝑇 ∘ 𝑣))} Er V)
9		ssv 4033	. . . . . . . . . . 11 ⊢ 𝑊 ⊆ V
10	9	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 𝑊 ⊆ V)
11	8, 10	erinxp 8849	. . . . . . . . 9 ⊢ (𝜑 → ({⟨𝑢, 𝑣⟩ ∣ (𝐻 Σg (𝑇 ∘ 𝑢)) = (𝐻 Σg (𝑇 ∘ 𝑣))} ∩ (𝑊 × 𝑊)) Er 𝑊)
12		df-xp 5706	. . . . . . . . . . . . 13 ⊢ (𝑊 × 𝑊) = {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝑊 ∧ 𝑣 ∈ 𝑊)}
13	12	ineq1i 4237	. . . . . . . . . . . 12 ⊢ ((𝑊 × 𝑊) ∩ {⟨𝑢, 𝑣⟩ ∣ (𝐻 Σg (𝑇 ∘ 𝑢)) = (𝐻 Σg (𝑇 ∘ 𝑣))}) = ({⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝑊 ∧ 𝑣 ∈ 𝑊)} ∩ {⟨𝑢, 𝑣⟩ ∣ (𝐻 Σg (𝑇 ∘ 𝑢)) = (𝐻 Σg (𝑇 ∘ 𝑣))})
14		incom 4230	. . . . . . . . . . . 12 ⊢ ((𝑊 × 𝑊) ∩ {⟨𝑢, 𝑣⟩ ∣ (𝐻 Σg (𝑇 ∘ 𝑢)) = (𝐻 Σg (𝑇 ∘ 𝑣))}) = ({⟨𝑢, 𝑣⟩ ∣ (𝐻 Σg (𝑇 ∘ 𝑢)) = (𝐻 Σg (𝑇 ∘ 𝑣))} ∩ (𝑊 × 𝑊))
15		inopab 5853	. . . . . . . . . . . 12 ⊢ ({⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝑊 ∧ 𝑣 ∈ 𝑊)} ∩ {⟨𝑢, 𝑣⟩ ∣ (𝐻 Σg (𝑇 ∘ 𝑢)) = (𝐻 Σg (𝑇 ∘ 𝑣))}) = {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝑊 ∧ 𝑣 ∈ 𝑊) ∧ (𝐻 Σg (𝑇 ∘ 𝑢)) = (𝐻 Σg (𝑇 ∘ 𝑣)))}
16	13, 14, 15	3eqtr3i 2776	. . . . . . . . . . 11 ⊢ ({⟨𝑢, 𝑣⟩ ∣ (𝐻 Σg (𝑇 ∘ 𝑢)) = (𝐻 Σg (𝑇 ∘ 𝑣))} ∩ (𝑊 × 𝑊)) = {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝑊 ∧ 𝑣 ∈ 𝑊) ∧ (𝐻 Σg (𝑇 ∘ 𝑢)) = (𝐻 Σg (𝑇 ∘ 𝑣)))}
17		vex 3492	. . . . . . . . . . . . . 14 ⊢ 𝑢 ∈ V
18		vex 3492	. . . . . . . . . . . . . 14 ⊢ 𝑣 ∈ V
19	17, 18	prss 4845	. . . . . . . . . . . . 13 ⊢ ((𝑢 ∈ 𝑊 ∧ 𝑣 ∈ 𝑊) ↔ {𝑢, 𝑣} ⊆ 𝑊)
20	19	anbi1i 623	. . . . . . . . . . . 12 ⊢ (((𝑢 ∈ 𝑊 ∧ 𝑣 ∈ 𝑊) ∧ (𝐻 Σg (𝑇 ∘ 𝑢)) = (𝐻 Σg (𝑇 ∘ 𝑣))) ↔ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇 ∘ 𝑢)) = (𝐻 Σg (𝑇 ∘ 𝑣))))
21	20	opabbii 5233	. . . . . . . . . . 11 ⊢ {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝑊 ∧ 𝑣 ∈ 𝑊) ∧ (𝐻 Σg (𝑇 ∘ 𝑢)) = (𝐻 Σg (𝑇 ∘ 𝑣)))} = {⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇 ∘ 𝑢)) = (𝐻 Σg (𝑇 ∘ 𝑣)))}
22	16, 21	eqtri 2768	. . . . . . . . . 10 ⊢ ({⟨𝑢, 𝑣⟩ ∣ (𝐻 Σg (𝑇 ∘ 𝑢)) = (𝐻 Σg (𝑇 ∘ 𝑣))} ∩ (𝑊 × 𝑊)) = {⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇 ∘ 𝑢)) = (𝐻 Σg (𝑇 ∘ 𝑣)))}
23		ereq1 8770	. . . . . . . . . 10 ⊢ (({⟨𝑢, 𝑣⟩ ∣ (𝐻 Σg (𝑇 ∘ 𝑢)) = (𝐻 Σg (𝑇 ∘ 𝑣))} ∩ (𝑊 × 𝑊)) = {⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇 ∘ 𝑢)) = (𝐻 Σg (𝑇 ∘ 𝑣)))} → (({⟨𝑢, 𝑣⟩ ∣ (𝐻 Σg (𝑇 ∘ 𝑢)) = (𝐻 Σg (𝑇 ∘ 𝑣))} ∩ (𝑊 × 𝑊)) Er 𝑊 ↔ {⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇 ∘ 𝑢)) = (𝐻 Σg (𝑇 ∘ 𝑣)))} Er 𝑊))
24	22, 23	ax-mp 5	. . . . . . . . 9 ⊢ (({⟨𝑢, 𝑣⟩ ∣ (𝐻 Σg (𝑇 ∘ 𝑢)) = (𝐻 Σg (𝑇 ∘ 𝑣))} ∩ (𝑊 × 𝑊)) Er 𝑊 ↔ {⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇 ∘ 𝑢)) = (𝐻 Σg (𝑇 ∘ 𝑣)))} Er 𝑊)
25	11, 24	sylib 218	. . . . . . . 8 ⊢ (𝜑 → {⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇 ∘ 𝑢)) = (𝐻 Σg (𝑇 ∘ 𝑣)))} Er 𝑊)
26		simplrl 776	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → 𝑥 ∈ 𝑊)
27		fviss 6999	. . . . . . . . . . . . . . 15 ⊢ ( I ‘Word (𝐼 × 2o)) ⊆ Word (𝐼 × 2o)
28	1, 27	eqsstri 4043	. . . . . . . . . . . . . 14 ⊢ 𝑊 ⊆ Word (𝐼 × 2o)
29	28, 26	sselid 4006	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → 𝑥 ∈ Word (𝐼 × 2o))
30		opelxpi 5737	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o) → ⟨𝑎, 𝑏⟩ ∈ (𝐼 × 2o))
31	30	adantl 481	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → ⟨𝑎, 𝑏⟩ ∈ (𝐼 × 2o))
32		simprl 770	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → 𝑎 ∈ 𝐼)
33		2oconcl 8559	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 ∈ 2o → (1o ∖ 𝑏) ∈ 2o)
34	33	ad2antll 728	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (1o ∖ 𝑏) ∈ 2o)
35	32, 34	opelxpd 5739	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → ⟨𝑎, (1o ∖ 𝑏)⟩ ∈ (𝐼 × 2o))
36	31, 35	s2cld 14920	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩ ∈ Word (𝐼 × 2o))
37		splcl 14800	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ Word (𝐼 × 2o) ∧ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩ ∈ Word (𝐼 × 2o)) → (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩) ∈ Word (𝐼 × 2o))
38	29, 36, 37	syl2anc 583	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩) ∈ Word (𝐼 × 2o))
39	1	efgrcl 19757	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝑊 → (𝐼 ∈ V ∧ 𝑊 = Word (𝐼 × 2o)))
40	26, 39	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (𝐼 ∈ V ∧ 𝑊 = Word (𝐼 × 2o)))
41	40	simprd 495	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → 𝑊 = Word (𝐼 × 2o))
42	38, 41	eleqtrrd 2847	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩) ∈ 𝑊)
43		pfxcl 14725	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ Word (𝐼 × 2o) → (𝑥 prefix 𝑛) ∈ Word (𝐼 × 2o))
44	29, 43	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (𝑥 prefix 𝑛) ∈ Word (𝐼 × 2o))
45		frgpup.b	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐵 = (Base‘𝐻)
46		frgpup.n	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑁 = (invg‘𝐻)
47		frgpup.t	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑇 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ if(𝑧 = ∅, (𝐹‘𝑦), (𝑁‘(𝐹‘𝑦))))
48		frgpup.h	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐻 ∈ Grp)
49		frgpup.i	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
50		frgpup.a	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐹:𝐼⟶𝐵)
51	45, 46, 47, 48, 49, 50	frgpuptf 19812	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑇:(𝐼 × 2o)⟶𝐵)
52	51	ad2antrr 725	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → 𝑇:(𝐼 × 2o)⟶𝐵)
53		ccatco 14884	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 prefix 𝑛) ∈ Word (𝐼 × 2o) ∧ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩ ∈ Word (𝐼 × 2o) ∧ 𝑇:(𝐼 × 2o)⟶𝐵) → (𝑇 ∘ ((𝑥 prefix 𝑛) ++ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩)) = ((𝑇 ∘ (𝑥 prefix 𝑛)) ++ (𝑇 ∘ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩)))
54	44, 36, 52, 53	syl3anc 1371	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (𝑇 ∘ ((𝑥 prefix 𝑛) ++ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩)) = ((𝑇 ∘ (𝑥 prefix 𝑛)) ++ (𝑇 ∘ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩)))
55	54	oveq2d 7464	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (𝐻 Σg (𝑇 ∘ ((𝑥 prefix 𝑛) ++ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩))) = (𝐻 Σg ((𝑇 ∘ (𝑥 prefix 𝑛)) ++ (𝑇 ∘ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩))))
56	48	ad2antrr 725	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → 𝐻 ∈ Grp)
57	56	grpmndd 18986	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → 𝐻 ∈ Mnd)
58		wrdco 14880	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 prefix 𝑛) ∈ Word (𝐼 × 2o) ∧ 𝑇:(𝐼 × 2o)⟶𝐵) → (𝑇 ∘ (𝑥 prefix 𝑛)) ∈ Word 𝐵)
59	44, 52, 58	syl2anc 583	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (𝑇 ∘ (𝑥 prefix 𝑛)) ∈ Word 𝐵)
60		wrdco 14880	. . . . . . . . . . . . . . . . 17 ⊢ ((⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩ ∈ Word (𝐼 × 2o) ∧ 𝑇:(𝐼 × 2o)⟶𝐵) → (𝑇 ∘ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩) ∈ Word 𝐵)
61	36, 52, 60	syl2anc 583	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (𝑇 ∘ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩) ∈ Word 𝐵)
62		eqid 2740	. . . . . . . . . . . . . . . . 17 ⊢ (+g‘𝐻) = (+g‘𝐻)
63	45, 62	gsumccat 18876	. . . . . . . . . . . . . . . 16 ⊢ ((𝐻 ∈ Mnd ∧ (𝑇 ∘ (𝑥 prefix 𝑛)) ∈ Word 𝐵 ∧ (𝑇 ∘ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩) ∈ Word 𝐵) → (𝐻 Σg ((𝑇 ∘ (𝑥 prefix 𝑛)) ++ (𝑇 ∘ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩))) = ((𝐻 Σg (𝑇 ∘ (𝑥 prefix 𝑛)))(+g‘𝐻)(𝐻 Σg (𝑇 ∘ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩))))
64	57, 59, 61, 63	syl3anc 1371	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (𝐻 Σg ((𝑇 ∘ (𝑥 prefix 𝑛)) ++ (𝑇 ∘ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩))) = ((𝐻 Σg (𝑇 ∘ (𝑥 prefix 𝑛)))(+g‘𝐻)(𝐻 Σg (𝑇 ∘ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩))))
65	52, 31, 35	s2co 14969	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (𝑇 ∘ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩) = ⟨“(𝑇‘⟨𝑎, 𝑏⟩)(𝑇‘⟨𝑎, (1o ∖ 𝑏)⟩)”⟩)
66		df-ov 7451	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑎𝑇𝑏) = (𝑇‘⟨𝑎, 𝑏⟩)
67	66	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (𝑎𝑇𝑏) = (𝑇‘⟨𝑎, 𝑏⟩))
68	66	fveq2i 6923	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑁‘(𝑎𝑇𝑏)) = (𝑁‘(𝑇‘⟨𝑎, 𝑏⟩))
69		df-ov 7451	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑎(𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)𝑏) = ((𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)‘⟨𝑎, 𝑏⟩)
70		eqid 2740	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩) = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)
71	70	efgmval 19754	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o) → (𝑎(𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)𝑏) = ⟨𝑎, (1o ∖ 𝑏)⟩)
72	69, 71	eqtr3id 2794	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o) → ((𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)‘⟨𝑎, 𝑏⟩) = ⟨𝑎, (1o ∖ 𝑏)⟩)
73	72	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → ((𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)‘⟨𝑎, 𝑏⟩) = ⟨𝑎, (1o ∖ 𝑏)⟩)
74	73	fveq2d 6924	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (𝑇‘((𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)‘⟨𝑎, 𝑏⟩)) = (𝑇‘⟨𝑎, (1o ∖ 𝑏)⟩))
75	45, 46, 47, 48, 49, 50, 70	frgpuptinv 19813	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ⟨𝑎, 𝑏⟩ ∈ (𝐼 × 2o)) → (𝑇‘((𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)‘⟨𝑎, 𝑏⟩)) = (𝑁‘(𝑇‘⟨𝑎, 𝑏⟩)))
76	30, 75	sylan2 592	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (𝑇‘((𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)‘⟨𝑎, 𝑏⟩)) = (𝑁‘(𝑇‘⟨𝑎, 𝑏⟩)))
77	76	adantlr 714	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (𝑇‘((𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)‘⟨𝑎, 𝑏⟩)) = (𝑁‘(𝑇‘⟨𝑎, 𝑏⟩)))
78	74, 77	eqtr3d 2782	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (𝑇‘⟨𝑎, (1o ∖ 𝑏)⟩) = (𝑁‘(𝑇‘⟨𝑎, 𝑏⟩)))
79	68, 78	eqtr4id 2799	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (𝑁‘(𝑎𝑇𝑏)) = (𝑇‘⟨𝑎, (1o ∖ 𝑏)⟩))
80	67, 79	s2eqd 14912	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → ⟨“(𝑎𝑇𝑏)(𝑁‘(𝑎𝑇𝑏))”⟩ = ⟨“(𝑇‘⟨𝑎, 𝑏⟩)(𝑇‘⟨𝑎, (1o ∖ 𝑏)⟩)”⟩)
81	65, 80	eqtr4d 2783	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (𝑇 ∘ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩) = ⟨“(𝑎𝑇𝑏)(𝑁‘(𝑎𝑇𝑏))”⟩)
82	81	oveq2d 7464	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (𝐻 Σg (𝑇 ∘ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩)) = (𝐻 Σg ⟨“(𝑎𝑇𝑏)(𝑁‘(𝑎𝑇𝑏))”⟩))
83		simprr 772	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → 𝑏 ∈ 2o)
84	52, 32, 83	fovcdmd 7622	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (𝑎𝑇𝑏) ∈ 𝐵)
85	45, 46	grpinvcl 19027	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐻 ∈ Grp ∧ (𝑎𝑇𝑏) ∈ 𝐵) → (𝑁‘(𝑎𝑇𝑏)) ∈ 𝐵)
86	56, 84, 85	syl2anc 583	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (𝑁‘(𝑎𝑇𝑏)) ∈ 𝐵)
87	45, 62	gsumws2 18877	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐻 ∈ Mnd ∧ (𝑎𝑇𝑏) ∈ 𝐵 ∧ (𝑁‘(𝑎𝑇𝑏)) ∈ 𝐵) → (𝐻 Σg ⟨“(𝑎𝑇𝑏)(𝑁‘(𝑎𝑇𝑏))”⟩) = ((𝑎𝑇𝑏)(+g‘𝐻)(𝑁‘(𝑎𝑇𝑏))))
88	57, 84, 86, 87	syl3anc 1371	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (𝐻 Σg ⟨“(𝑎𝑇𝑏)(𝑁‘(𝑎𝑇𝑏))”⟩) = ((𝑎𝑇𝑏)(+g‘𝐻)(𝑁‘(𝑎𝑇𝑏))))
89		eqid 2740	. . . . . . . . . . . . . . . . . . . 20 ⊢ (0g‘𝐻) = (0g‘𝐻)
90	45, 62, 89, 46	grprinv 19030	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐻 ∈ Grp ∧ (𝑎𝑇𝑏) ∈ 𝐵) → ((𝑎𝑇𝑏)(+g‘𝐻)(𝑁‘(𝑎𝑇𝑏))) = (0g‘𝐻))
91	56, 84, 90	syl2anc 583	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → ((𝑎𝑇𝑏)(+g‘𝐻)(𝑁‘(𝑎𝑇𝑏))) = (0g‘𝐻))
92	82, 88, 91	3eqtrd 2784	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (𝐻 Σg (𝑇 ∘ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩)) = (0g‘𝐻))
93	92	oveq2d 7464	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → ((𝐻 Σg (𝑇 ∘ (𝑥 prefix 𝑛)))(+g‘𝐻)(𝐻 Σg (𝑇 ∘ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩))) = ((𝐻 Σg (𝑇 ∘ (𝑥 prefix 𝑛)))(+g‘𝐻)(0g‘𝐻)))
94	45	gsumwcl 18874	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐻 ∈ Mnd ∧ (𝑇 ∘ (𝑥 prefix 𝑛)) ∈ Word 𝐵) → (𝐻 Σg (𝑇 ∘ (𝑥 prefix 𝑛))) ∈ 𝐵)
95	57, 59, 94	syl2anc 583	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (𝐻 Σg (𝑇 ∘ (𝑥 prefix 𝑛))) ∈ 𝐵)
96	45, 62, 89	grprid 19008	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐻 ∈ Grp ∧ (𝐻 Σg (𝑇 ∘ (𝑥 prefix 𝑛))) ∈ 𝐵) → ((𝐻 Σg (𝑇 ∘ (𝑥 prefix 𝑛)))(+g‘𝐻)(0g‘𝐻)) = (𝐻 Σg (𝑇 ∘ (𝑥 prefix 𝑛))))
97	56, 95, 96	syl2anc 583	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → ((𝐻 Σg (𝑇 ∘ (𝑥 prefix 𝑛)))(+g‘𝐻)(0g‘𝐻)) = (𝐻 Σg (𝑇 ∘ (𝑥 prefix 𝑛))))
98	93, 97	eqtrd 2780	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → ((𝐻 Σg (𝑇 ∘ (𝑥 prefix 𝑛)))(+g‘𝐻)(𝐻 Σg (𝑇 ∘ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩))) = (𝐻 Σg (𝑇 ∘ (𝑥 prefix 𝑛))))
99	55, 64, 98	3eqtrrd 2785	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (𝐻 Σg (𝑇 ∘ (𝑥 prefix 𝑛))) = (𝐻 Σg (𝑇 ∘ ((𝑥 prefix 𝑛) ++ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩))))
100	99	oveq1d 7463	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → ((𝐻 Σg (𝑇 ∘ (𝑥 prefix 𝑛)))(+g‘𝐻)(𝐻 Σg (𝑇 ∘ (𝑥 substr ⟨𝑛, (♯‘𝑥)⟩)))) = ((𝐻 Σg (𝑇 ∘ ((𝑥 prefix 𝑛) ++ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩)))(+g‘𝐻)(𝐻 Σg (𝑇 ∘ (𝑥 substr ⟨𝑛, (♯‘𝑥)⟩)))))
101		swrdcl 14693	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ Word (𝐼 × 2o) → (𝑥 substr ⟨𝑛, (♯‘𝑥)⟩) ∈ Word (𝐼 × 2o))
102	29, 101	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (𝑥 substr ⟨𝑛, (♯‘𝑥)⟩) ∈ Word (𝐼 × 2o))
103		wrdco 14880	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 substr ⟨𝑛, (♯‘𝑥)⟩) ∈ Word (𝐼 × 2o) ∧ 𝑇:(𝐼 × 2o)⟶𝐵) → (𝑇 ∘ (𝑥 substr ⟨𝑛, (♯‘𝑥)⟩)) ∈ Word 𝐵)
104	102, 52, 103	syl2anc 583	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (𝑇 ∘ (𝑥 substr ⟨𝑛, (♯‘𝑥)⟩)) ∈ Word 𝐵)
105	45, 62	gsumccat 18876	. . . . . . . . . . . . . 14 ⊢ ((𝐻 ∈ Mnd ∧ (𝑇 ∘ (𝑥 prefix 𝑛)) ∈ Word 𝐵 ∧ (𝑇 ∘ (𝑥 substr ⟨𝑛, (♯‘𝑥)⟩)) ∈ Word 𝐵) → (𝐻 Σg ((𝑇 ∘ (𝑥 prefix 𝑛)) ++ (𝑇 ∘ (𝑥 substr ⟨𝑛, (♯‘𝑥)⟩)))) = ((𝐻 Σg (𝑇 ∘ (𝑥 prefix 𝑛)))(+g‘𝐻)(𝐻 Σg (𝑇 ∘ (𝑥 substr ⟨𝑛, (♯‘𝑥)⟩)))))
106	57, 59, 104, 105	syl3anc 1371	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (𝐻 Σg ((𝑇 ∘ (𝑥 prefix 𝑛)) ++ (𝑇 ∘ (𝑥 substr ⟨𝑛, (♯‘𝑥)⟩)))) = ((𝐻 Σg (𝑇 ∘ (𝑥 prefix 𝑛)))(+g‘𝐻)(𝐻 Σg (𝑇 ∘ (𝑥 substr ⟨𝑛, (♯‘𝑥)⟩)))))
107		ccatcl 14622	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 prefix 𝑛) ∈ Word (𝐼 × 2o) ∧ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩ ∈ Word (𝐼 × 2o)) → ((𝑥 prefix 𝑛) ++ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩) ∈ Word (𝐼 × 2o))
108	44, 36, 107	syl2anc 583	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → ((𝑥 prefix 𝑛) ++ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩) ∈ Word (𝐼 × 2o))
109		wrdco 14880	. . . . . . . . . . . . . . 15 ⊢ ((((𝑥 prefix 𝑛) ++ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩) ∈ Word (𝐼 × 2o) ∧ 𝑇:(𝐼 × 2o)⟶𝐵) → (𝑇 ∘ ((𝑥 prefix 𝑛) ++ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩)) ∈ Word 𝐵)
110	108, 52, 109	syl2anc 583	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (𝑇 ∘ ((𝑥 prefix 𝑛) ++ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩)) ∈ Word 𝐵)
111	45, 62	gsumccat 18876	. . . . . . . . . . . . . 14 ⊢ ((𝐻 ∈ Mnd ∧ (𝑇 ∘ ((𝑥 prefix 𝑛) ++ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩)) ∈ Word 𝐵 ∧ (𝑇 ∘ (𝑥 substr ⟨𝑛, (♯‘𝑥)⟩)) ∈ Word 𝐵) → (𝐻 Σg ((𝑇 ∘ ((𝑥 prefix 𝑛) ++ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩)) ++ (𝑇 ∘ (𝑥 substr ⟨𝑛, (♯‘𝑥)⟩)))) = ((𝐻 Σg (𝑇 ∘ ((𝑥 prefix 𝑛) ++ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩)))(+g‘𝐻)(𝐻 Σg (𝑇 ∘ (𝑥 substr ⟨𝑛, (♯‘𝑥)⟩)))))
112	57, 110, 104, 111	syl3anc 1371	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (𝐻 Σg ((𝑇 ∘ ((𝑥 prefix 𝑛) ++ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩)) ++ (𝑇 ∘ (𝑥 substr ⟨𝑛, (♯‘𝑥)⟩)))) = ((𝐻 Σg (𝑇 ∘ ((𝑥 prefix 𝑛) ++ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩)))(+g‘𝐻)(𝐻 Σg (𝑇 ∘ (𝑥 substr ⟨𝑛, (♯‘𝑥)⟩)))))
113	100, 106, 112	3eqtr4d 2790	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (𝐻 Σg ((𝑇 ∘ (𝑥 prefix 𝑛)) ++ (𝑇 ∘ (𝑥 substr ⟨𝑛, (♯‘𝑥)⟩)))) = (𝐻 Σg ((𝑇 ∘ ((𝑥 prefix 𝑛) ++ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩)) ++ (𝑇 ∘ (𝑥 substr ⟨𝑛, (♯‘𝑥)⟩)))))
114		simplrr 777	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → 𝑛 ∈ (0...(♯‘𝑥)))
115		lencl 14581	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ Word (𝐼 × 2o) → (♯‘𝑥) ∈ ℕ0)
116	29, 115	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (♯‘𝑥) ∈ ℕ0)
117		nn0uz 12945	. . . . . . . . . . . . . . . . . . 19 ⊢ ℕ0 = (ℤ≥‘0)
118	116, 117	eleqtrdi 2854	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (♯‘𝑥) ∈ (ℤ≥‘0))
119		eluzfz2 13592	. . . . . . . . . . . . . . . . . 18 ⊢ ((♯‘𝑥) ∈ (ℤ≥‘0) → (♯‘𝑥) ∈ (0...(♯‘𝑥)))
120	118, 119	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (♯‘𝑥) ∈ (0...(♯‘𝑥)))
121		ccatpfx 14749	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ Word (𝐼 × 2o) ∧ 𝑛 ∈ (0...(♯‘𝑥)) ∧ (♯‘𝑥) ∈ (0...(♯‘𝑥))) → ((𝑥 prefix 𝑛) ++ (𝑥 substr ⟨𝑛, (♯‘𝑥)⟩)) = (𝑥 prefix (♯‘𝑥)))
122	29, 114, 120, 121	syl3anc 1371	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → ((𝑥 prefix 𝑛) ++ (𝑥 substr ⟨𝑛, (♯‘𝑥)⟩)) = (𝑥 prefix (♯‘𝑥)))
123		pfxid 14732	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ Word (𝐼 × 2o) → (𝑥 prefix (♯‘𝑥)) = 𝑥)
124	29, 123	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (𝑥 prefix (♯‘𝑥)) = 𝑥)
125	122, 124	eqtrd 2780	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → ((𝑥 prefix 𝑛) ++ (𝑥 substr ⟨𝑛, (♯‘𝑥)⟩)) = 𝑥)
126	125	coeq2d 5887	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (𝑇 ∘ ((𝑥 prefix 𝑛) ++ (𝑥 substr ⟨𝑛, (♯‘𝑥)⟩))) = (𝑇 ∘ 𝑥))
127		ccatco 14884	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 prefix 𝑛) ∈ Word (𝐼 × 2o) ∧ (𝑥 substr ⟨𝑛, (♯‘𝑥)⟩) ∈ Word (𝐼 × 2o) ∧ 𝑇:(𝐼 × 2o)⟶𝐵) → (𝑇 ∘ ((𝑥 prefix 𝑛) ++ (𝑥 substr ⟨𝑛, (♯‘𝑥)⟩))) = ((𝑇 ∘ (𝑥 prefix 𝑛)) ++ (𝑇 ∘ (𝑥 substr ⟨𝑛, (♯‘𝑥)⟩))))
128	44, 102, 52, 127	syl3anc 1371	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (𝑇 ∘ ((𝑥 prefix 𝑛) ++ (𝑥 substr ⟨𝑛, (♯‘𝑥)⟩))) = ((𝑇 ∘ (𝑥 prefix 𝑛)) ++ (𝑇 ∘ (𝑥 substr ⟨𝑛, (♯‘𝑥)⟩))))
129	126, 128	eqtr3d 2782	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (𝑇 ∘ 𝑥) = ((𝑇 ∘ (𝑥 prefix 𝑛)) ++ (𝑇 ∘ (𝑥 substr ⟨𝑛, (♯‘𝑥)⟩))))
130	129	oveq2d 7464	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (𝐻 Σg (𝑇 ∘ 𝑥)) = (𝐻 Σg ((𝑇 ∘ (𝑥 prefix 𝑛)) ++ (𝑇 ∘ (𝑥 substr ⟨𝑛, (♯‘𝑥)⟩)))))
131		splval 14799	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ 𝑊 ∧ (𝑛 ∈ (0...(♯‘𝑥)) ∧ 𝑛 ∈ (0...(♯‘𝑥)) ∧ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩ ∈ Word (𝐼 × 2o))) → (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩) = (((𝑥 prefix 𝑛) ++ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩) ++ (𝑥 substr ⟨𝑛, (♯‘𝑥)⟩)))
132	26, 114, 114, 36, 131	syl13anc 1372	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩) = (((𝑥 prefix 𝑛) ++ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩) ++ (𝑥 substr ⟨𝑛, (♯‘𝑥)⟩)))
133	132	coeq2d 5887	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (𝑇 ∘ (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩)) = (𝑇 ∘ (((𝑥 prefix 𝑛) ++ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩) ++ (𝑥 substr ⟨𝑛, (♯‘𝑥)⟩))))
134		ccatco 14884	. . . . . . . . . . . . . . 15 ⊢ ((((𝑥 prefix 𝑛) ++ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩) ∈ Word (𝐼 × 2o) ∧ (𝑥 substr ⟨𝑛, (♯‘𝑥)⟩) ∈ Word (𝐼 × 2o) ∧ 𝑇:(𝐼 × 2o)⟶𝐵) → (𝑇 ∘ (((𝑥 prefix 𝑛) ++ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩) ++ (𝑥 substr ⟨𝑛, (♯‘𝑥)⟩))) = ((𝑇 ∘ ((𝑥 prefix 𝑛) ++ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩)) ++ (𝑇 ∘ (𝑥 substr ⟨𝑛, (♯‘𝑥)⟩))))
135	108, 102, 52, 134	syl3anc 1371	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (𝑇 ∘ (((𝑥 prefix 𝑛) ++ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩) ++ (𝑥 substr ⟨𝑛, (♯‘𝑥)⟩))) = ((𝑇 ∘ ((𝑥 prefix 𝑛) ++ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩)) ++ (𝑇 ∘ (𝑥 substr ⟨𝑛, (♯‘𝑥)⟩))))
136	133, 135	eqtrd 2780	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (𝑇 ∘ (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩)) = ((𝑇 ∘ ((𝑥 prefix 𝑛) ++ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩)) ++ (𝑇 ∘ (𝑥 substr ⟨𝑛, (♯‘𝑥)⟩))))
137	136	oveq2d 7464	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (𝐻 Σg (𝑇 ∘ (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩))) = (𝐻 Σg ((𝑇 ∘ ((𝑥 prefix 𝑛) ++ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩)) ++ (𝑇 ∘ (𝑥 substr ⟨𝑛, (♯‘𝑥)⟩)))))
138	113, 130, 137	3eqtr4d 2790	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (𝐻 Σg (𝑇 ∘ 𝑥)) = (𝐻 Σg (𝑇 ∘ (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩))))
139		vex 3492	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
140		ovex 7481	. . . . . . . . . . . 12 ⊢ (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩) ∈ V
141		eleq1 2832	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = 𝑥 → (𝑢 ∈ 𝑊 ↔ 𝑥 ∈ 𝑊))
142		eleq1 2832	. . . . . . . . . . . . . . 15 ⊢ (𝑣 = (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩) → (𝑣 ∈ 𝑊 ↔ (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩) ∈ 𝑊))
143	141, 142	bi2anan9 637	. . . . . . . . . . . . . 14 ⊢ ((𝑢 = 𝑥 ∧ 𝑣 = (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩)) → ((𝑢 ∈ 𝑊 ∧ 𝑣 ∈ 𝑊) ↔ (𝑥 ∈ 𝑊 ∧ (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩) ∈ 𝑊)))
144	19, 143	bitr3id 285	. . . . . . . . . . . . 13 ⊢ ((𝑢 = 𝑥 ∧ 𝑣 = (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩)) → ({𝑢, 𝑣} ⊆ 𝑊 ↔ (𝑥 ∈ 𝑊 ∧ (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩) ∈ 𝑊)))
145		coeq2 5883	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = 𝑥 → (𝑇 ∘ 𝑢) = (𝑇 ∘ 𝑥))
146	145	oveq2d 7464	. . . . . . . . . . . . . 14 ⊢ (𝑢 = 𝑥 → (𝐻 Σg (𝑇 ∘ 𝑢)) = (𝐻 Σg (𝑇 ∘ 𝑥)))
147		coeq2 5883	. . . . . . . . . . . . . . 15 ⊢ (𝑣 = (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩) → (𝑇 ∘ 𝑣) = (𝑇 ∘ (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩)))
148	147	oveq2d 7464	. . . . . . . . . . . . . 14 ⊢ (𝑣 = (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩) → (𝐻 Σg (𝑇 ∘ 𝑣)) = (𝐻 Σg (𝑇 ∘ (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩))))
149	146, 148	eqeqan12d 2754	. . . . . . . . . . . . 13 ⊢ ((𝑢 = 𝑥 ∧ 𝑣 = (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩)) → ((𝐻 Σg (𝑇 ∘ 𝑢)) = (𝐻 Σg (𝑇 ∘ 𝑣)) ↔ (𝐻 Σg (𝑇 ∘ 𝑥)) = (𝐻 Σg (𝑇 ∘ (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩)))))
150	144, 149	anbi12d 631	. . . . . . . . . . . 12 ⊢ ((𝑢 = 𝑥 ∧ 𝑣 = (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩)) → (({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇 ∘ 𝑢)) = (𝐻 Σg (𝑇 ∘ 𝑣))) ↔ ((𝑥 ∈ 𝑊 ∧ (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩) ∈ 𝑊) ∧ (𝐻 Σg (𝑇 ∘ 𝑥)) = (𝐻 Σg (𝑇 ∘ (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩))))))
151		eqid 2740	. . . . . . . . . . . 12 ⊢ {⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇 ∘ 𝑢)) = (𝐻 Σg (𝑇 ∘ 𝑣)))} = {⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇 ∘ 𝑢)) = (𝐻 Σg (𝑇 ∘ 𝑣)))}
152	139, 140, 150, 151	braba 5556	. . . . . . . . . . 11 ⊢ (𝑥{⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇 ∘ 𝑢)) = (𝐻 Σg (𝑇 ∘ 𝑣)))} (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩) ↔ ((𝑥 ∈ 𝑊 ∧ (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩) ∈ 𝑊) ∧ (𝐻 Σg (𝑇 ∘ 𝑥)) = (𝐻 Σg (𝑇 ∘ (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩)))))
153	26, 42, 138, 152	syl21anbrc 1344	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → 𝑥{⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇 ∘ 𝑢)) = (𝐻 Σg (𝑇 ∘ 𝑣)))} (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩))
154	153	ralrimivva 3208	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) → ∀𝑎 ∈ 𝐼 ∀𝑏 ∈ 2o 𝑥{⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇 ∘ 𝑢)) = (𝐻 Σg (𝑇 ∘ 𝑣)))} (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩))
155	154	ralrimivva 3208	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ 𝑊 ∀𝑛 ∈ (0...(♯‘𝑥))∀𝑎 ∈ 𝐼 ∀𝑏 ∈ 2o 𝑥{⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇 ∘ 𝑢)) = (𝐻 Σg (𝑇 ∘ 𝑣)))} (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩))
156	1	fvexi 6934	. . . . . . . . . 10 ⊢ 𝑊 ∈ V
157		erex 8787	. . . . . . . . . 10 ⊢ ({⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇 ∘ 𝑢)) = (𝐻 Σg (𝑇 ∘ 𝑣)))} Er 𝑊 → (𝑊 ∈ V → {⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇 ∘ 𝑢)) = (𝐻 Σg (𝑇 ∘ 𝑣)))} ∈ V))
158	25, 156, 157	mpisyl 21	. . . . . . . . 9 ⊢ (𝜑 → {⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇 ∘ 𝑢)) = (𝐻 Σg (𝑇 ∘ 𝑣)))} ∈ V)
159		ereq1 8770	. . . . . . . . . . 11 ⊢ (𝑟 = {⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇 ∘ 𝑢)) = (𝐻 Σg (𝑇 ∘ 𝑣)))} → (𝑟 Er 𝑊 ↔ {⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇 ∘ 𝑢)) = (𝐻 Σg (𝑇 ∘ 𝑣)))} Er 𝑊))
160		breq 5168	. . . . . . . . . . . . 13 ⊢ (𝑟 = {⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇 ∘ 𝑢)) = (𝐻 Σg (𝑇 ∘ 𝑣)))} → (𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩) ↔ 𝑥{⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇 ∘ 𝑢)) = (𝐻 Σg (𝑇 ∘ 𝑣)))} (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩)))
161	160	2ralbidv 3227	. . . . . . . . . . . 12 ⊢ (𝑟 = {⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇 ∘ 𝑢)) = (𝐻 Σg (𝑇 ∘ 𝑣)))} → (∀𝑎 ∈ 𝐼 ∀𝑏 ∈ 2o 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩) ↔ ∀𝑎 ∈ 𝐼 ∀𝑏 ∈ 2o 𝑥{⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇 ∘ 𝑢)) = (𝐻 Σg (𝑇 ∘ 𝑣)))} (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩)))
162	161	2ralbidv 3227	. . . . . . . . . . 11 ⊢ (𝑟 = {⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇 ∘ 𝑢)) = (𝐻 Σg (𝑇 ∘ 𝑣)))} → (∀𝑥 ∈ 𝑊 ∀𝑛 ∈ (0...(♯‘𝑥))∀𝑎 ∈ 𝐼 ∀𝑏 ∈ 2o 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩) ↔ ∀𝑥 ∈ 𝑊 ∀𝑛 ∈ (0...(♯‘𝑥))∀𝑎 ∈ 𝐼 ∀𝑏 ∈ 2o 𝑥{⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇 ∘ 𝑢)) = (𝐻 Σg (𝑇 ∘ 𝑣)))} (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩)))
163	159, 162	anbi12d 631	. . . . . . . . . 10 ⊢ (𝑟 = {⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇 ∘ 𝑢)) = (𝐻 Σg (𝑇 ∘ 𝑣)))} → ((𝑟 Er 𝑊 ∧ ∀𝑥 ∈ 𝑊 ∀𝑛 ∈ (0...(♯‘𝑥))∀𝑎 ∈ 𝐼 ∀𝑏 ∈ 2o 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩)) ↔ ({⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇 ∘ 𝑢)) = (𝐻 Σg (𝑇 ∘ 𝑣)))} Er 𝑊 ∧ ∀𝑥 ∈ 𝑊 ∀𝑛 ∈ (0...(♯‘𝑥))∀𝑎 ∈ 𝐼 ∀𝑏 ∈ 2o 𝑥{⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇 ∘ 𝑢)) = (𝐻 Σg (𝑇 ∘ 𝑣)))} (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩))))
164	163	elabg 3690	. . . . . . . . 9 ⊢ ({⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇 ∘ 𝑢)) = (𝐻 Σg (𝑇 ∘ 𝑣)))} ∈ V → ({⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇 ∘ 𝑢)) = (𝐻 Σg (𝑇 ∘ 𝑣)))} ∈ {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥 ∈ 𝑊 ∀𝑛 ∈ (0...(♯‘𝑥))∀𝑎 ∈ 𝐼 ∀𝑏 ∈ 2o 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩))} ↔ ({⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇 ∘ 𝑢)) = (𝐻 Σg (𝑇 ∘ 𝑣)))} Er 𝑊 ∧ ∀𝑥 ∈ 𝑊 ∀𝑛 ∈ (0...(♯‘𝑥))∀𝑎 ∈ 𝐼 ∀𝑏 ∈ 2o 𝑥{⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇 ∘ 𝑢)) = (𝐻 Σg (𝑇 ∘ 𝑣)))} (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩))))
165	158, 164	syl 17	. . . . . . . 8 ⊢ (𝜑 → ({⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇 ∘ 𝑢)) = (𝐻 Σg (𝑇 ∘ 𝑣)))} ∈ {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥 ∈ 𝑊 ∀𝑛 ∈ (0...(♯‘𝑥))∀𝑎 ∈ 𝐼 ∀𝑏 ∈ 2o 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩))} ↔ ({⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇 ∘ 𝑢)) = (𝐻 Σg (𝑇 ∘ 𝑣)))} Er 𝑊 ∧ ∀𝑥 ∈ 𝑊 ∀𝑛 ∈ (0...(♯‘𝑥))∀𝑎 ∈ 𝐼 ∀𝑏 ∈ 2o 𝑥{⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇 ∘ 𝑢)) = (𝐻 Σg (𝑇 ∘ 𝑣)))} (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩))))
166	25, 155, 165	mpbir2and 712	. . . . . . 7 ⊢ (𝜑 → {⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇 ∘ 𝑢)) = (𝐻 Σg (𝑇 ∘ 𝑣)))} ∈ {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥 ∈ 𝑊 ∀𝑛 ∈ (0...(♯‘𝑥))∀𝑎 ∈ 𝐼 ∀𝑏 ∈ 2o 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩))})
167		intss1 4987	. . . . . . 7 ⊢ ({⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇 ∘ 𝑢)) = (𝐻 Σg (𝑇 ∘ 𝑣)))} ∈ {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥 ∈ 𝑊 ∀𝑛 ∈ (0...(♯‘𝑥))∀𝑎 ∈ 𝐼 ∀𝑏 ∈ 2o 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩))} → ∩ {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥 ∈ 𝑊 ∀𝑛 ∈ (0...(♯‘𝑥))∀𝑎 ∈ 𝐼 ∀𝑏 ∈ 2o 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩))} ⊆ {⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇 ∘ 𝑢)) = (𝐻 Σg (𝑇 ∘ 𝑣)))})
168	166, 167	syl 17	. . . . . 6 ⊢ (𝜑 → ∩ {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥 ∈ 𝑊 ∀𝑛 ∈ (0...(♯‘𝑥))∀𝑎 ∈ 𝐼 ∀𝑏 ∈ 2o 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩))} ⊆ {⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇 ∘ 𝑢)) = (𝐻 Σg (𝑇 ∘ 𝑣)))})
169	3, 168	eqsstrid 4057	. . . . 5 ⊢ (𝜑 → ∼ ⊆ {⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇 ∘ 𝑢)) = (𝐻 Σg (𝑇 ∘ 𝑣)))})
170	169	ssbrd 5209	. . . 4 ⊢ (𝜑 → (𝐴 ∼ 𝐶 → 𝐴{⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇 ∘ 𝑢)) = (𝐻 Σg (𝑇 ∘ 𝑣)))}𝐶))
171	170	imp 406	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∼ 𝐶) → 𝐴{⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇 ∘ 𝑢)) = (𝐻 Σg (𝑇 ∘ 𝑣)))}𝐶)
172	1, 2	efger 19760	. . . . . 6 ⊢ ∼ Er 𝑊
173		errel 8772	. . . . . 6 ⊢ ( ∼ Er 𝑊 → Rel ∼ )
174	172, 173	mp1i 13	. . . . 5 ⊢ (𝜑 → Rel ∼ )
175		brrelex12 5752	. . . . 5 ⊢ ((Rel ∼ ∧ 𝐴 ∼ 𝐶) → (𝐴 ∈ V ∧ 𝐶 ∈ V))
176	174, 175	sylan 579	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∼ 𝐶) → (𝐴 ∈ V ∧ 𝐶 ∈ V))
177		preq12 4760	. . . . . . 7 ⊢ ((𝑢 = 𝐴 ∧ 𝑣 = 𝐶) → {𝑢, 𝑣} = {𝐴, 𝐶})
178	177	sseq1d 4040	. . . . . 6 ⊢ ((𝑢 = 𝐴 ∧ 𝑣 = 𝐶) → ({𝑢, 𝑣} ⊆ 𝑊 ↔ {𝐴, 𝐶} ⊆ 𝑊))
179		coeq2 5883	. . . . . . . 8 ⊢ (𝑢 = 𝐴 → (𝑇 ∘ 𝑢) = (𝑇 ∘ 𝐴))
180	179	oveq2d 7464	. . . . . . 7 ⊢ (𝑢 = 𝐴 → (𝐻 Σg (𝑇 ∘ 𝑢)) = (𝐻 Σg (𝑇 ∘ 𝐴)))
181		coeq2 5883	. . . . . . . 8 ⊢ (𝑣 = 𝐶 → (𝑇 ∘ 𝑣) = (𝑇 ∘ 𝐶))
182	181	oveq2d 7464	. . . . . . 7 ⊢ (𝑣 = 𝐶 → (𝐻 Σg (𝑇 ∘ 𝑣)) = (𝐻 Σg (𝑇 ∘ 𝐶)))
183	180, 182	eqeqan12d 2754	. . . . . 6 ⊢ ((𝑢 = 𝐴 ∧ 𝑣 = 𝐶) → ((𝐻 Σg (𝑇 ∘ 𝑢)) = (𝐻 Σg (𝑇 ∘ 𝑣)) ↔ (𝐻 Σg (𝑇 ∘ 𝐴)) = (𝐻 Σg (𝑇 ∘ 𝐶))))
184	178, 183	anbi12d 631	. . . . 5 ⊢ ((𝑢 = 𝐴 ∧ 𝑣 = 𝐶) → (({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇 ∘ 𝑢)) = (𝐻 Σg (𝑇 ∘ 𝑣))) ↔ ({𝐴, 𝐶} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇 ∘ 𝐴)) = (𝐻 Σg (𝑇 ∘ 𝐶)))))
185	184, 151	brabga 5553	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐶 ∈ V) → (𝐴{⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇 ∘ 𝑢)) = (𝐻 Σg (𝑇 ∘ 𝑣)))}𝐶 ↔ ({𝐴, 𝐶} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇 ∘ 𝐴)) = (𝐻 Σg (𝑇 ∘ 𝐶)))))
186	176, 185	syl 17	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∼ 𝐶) → (𝐴{⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇 ∘ 𝑢)) = (𝐻 Σg (𝑇 ∘ 𝑣)))}𝐶 ↔ ({𝐴, 𝐶} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇 ∘ 𝐴)) = (𝐻 Σg (𝑇 ∘ 𝐶)))))
187	171, 186	mpbid 232	. 2 ⊢ ((𝜑 ∧ 𝐴 ∼ 𝐶) → ({𝐴, 𝐶} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇 ∘ 𝐴)) = (𝐻 Σg (𝑇 ∘ 𝐶))))
188	187	simprd 495	1 ⊢ ((𝜑 ∧ 𝐴 ∼ 𝐶) → (𝐻 Σg (𝑇 ∘ 𝐴)) = (𝐻 Σg (𝑇 ∘ 𝐶)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108  {cab 2717  ∀wral 3067  Vcvv 3488   ∖ cdif 3973   ∩ cin 3975   ⊆ wss 3976  ∅c0 4352  ifcif 4548  {cpr 4650  ⟨cop 4654  ⟨cotp 4656  ∩ cint 4970   class class class wbr 5166  {copab 5228   I cid 5592   × cxp 5698   ∘ ccom 5704  Rel wrel 5705  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448   ∈ cmpo 7450  1_oc1o 8515  2_oc2o 8516   Er wer 8760  0cc0 11184  ℕ₀cn0 12553  ℤ_≥cuz 12903  ...cfz 13567  ♯chash 14379  Word cword 14562   ++ cconcat 14618   substr csubstr 14688   prefix cpfx 14718   splice csplice 14797  ⟨“cs2 14890  Basecbs 17258  +_gcplusg 17311  0_gc0g 17499   Σ_g cgsu 17500  Mndcmnd 18772  Grpcgrp 18973  inv_gcminusg 18974   ~_FG cefg 19748

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-ot 4657  df-uni 4932  df-int 4971  df-iun 5017  df-iin 5018  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-2o 8523  df-er 8763  df-map 8886  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-card 10008  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-nn 12294  df-2 12356  df-n0 12554  df-z 12640  df-uz 12904  df-fz 13568  df-fzo 13712  df-seq 14053  df-hash 14380  df-word 14563  df-concat 14619  df-s1 14644  df-substr 14689  df-pfx 14719  df-splice 14798  df-s2 14897  df-sets 17211  df-slot 17229  df-ndx 17241  df-base 17259  df-ress 17288  df-plusg 17324  df-0g 17501  df-gsum 17502  df-mgm 18678  df-sgrp 18757  df-mnd 18773  df-submnd 18819  df-grp 18976  df-minusg 18977  df-efg 19751

This theorem is referenced by:  frgpupf  19815