Theorem frgpuplem 18890

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 18835	. . . . . 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 5693	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝑣 → (𝑇 ∘ 𝑢) = (𝑇 ∘ 𝑣))
5	4	oveq2d 7151	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑣 → (𝐻 Σg (𝑇 ∘ 𝑢)) = (𝐻 Σg (𝑇 ∘ 𝑣)))
6		eqid 2798	. . . . . . . . . . . 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 8307	. . . . . . . . . . 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 3939	. . . . . . . . . . 11 ⊢ 𝑊 ⊆ V
10	9	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 𝑊 ⊆ V)
11	8, 10	erinxp 8354	. . . . . . . . 9 ⊢ (𝜑 → ({⟨𝑢, 𝑣⟩ ∣ (𝐻 Σg (𝑇 ∘ 𝑢)) = (𝐻 Σg (𝑇 ∘ 𝑣))} ∩ (𝑊 × 𝑊)) Er 𝑊)
12		df-xp 5525	. . . . . . . . . . . . 13 ⊢ (𝑊 × 𝑊) = {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝑊 ∧ 𝑣 ∈ 𝑊)}
13	12	ineq1i 4135	. . . . . . . . . . . 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 4128	. . . . . . . . . . . 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 5665	. . . . . . . . . . . 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 2829	. . . . . . . . . . 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 3444	. . . . . . . . . . . . . 14 ⊢ 𝑢 ∈ V
18		vex 3444	. . . . . . . . . . . . . 14 ⊢ 𝑣 ∈ V
19	17, 18	prss 4713	. . . . . . . . . . . . 13 ⊢ ((𝑢 ∈ 𝑊 ∧ 𝑣 ∈ 𝑊) ↔ {𝑢, 𝑣} ⊆ 𝑊)
20	19	anbi1i 626	. . . . . . . . . . . 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 5097	. . . . . . . . . . 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 2821	. . . . . . . . . 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 8279	. . . . . . . . . 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 221	. . . . . . . 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 6716	. . . . . . . . . . . . . . 15 ⊢ ( I ‘Word (𝐼 × 2o)) ⊆ Word (𝐼 × 2o)
28	1, 27	eqsstri 3949	. . . . . . . . . . . . . 14 ⊢ 𝑊 ⊆ Word (𝐼 × 2o)
29	28, 26	sseldi 3913	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → 𝑥 ∈ Word (𝐼 × 2o))
30		opelxpi 5556	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o) → ⟨𝑎, 𝑏⟩ ∈ (𝐼 × 2o))
31	30	adantl 485	. . . . . . . . . . . . . 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 8111	. . . . . . . . . . . . . . . 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 5557	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → ⟨𝑎, (1o ∖ 𝑏)⟩ ∈ (𝐼 × 2o))
36	31, 35	s2cld 14224	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩ ∈ Word (𝐼 × 2o))
37		splcl 14105	. . . . . . . . . . . . 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 587	. . . . . . . . . . . 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 18833	. . . . . . . . . . . . . 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 499	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → 𝑊 = Word (𝐼 × 2o))
42	38, 41	eleqtrrd 2893	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩) ∈ 𝑊)
43		pfxcl 14030	. . . . . . . . . . . . . . . . . 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 18888	. . . . . . . . . . . . . . . . . 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 14188	. . . . . . . . . . . . . . . . 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 1368	. . . . . . . . . . . . . . . 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 7151	. . . . . . . . . . . . . . 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		grpmnd 18102	. . . . . . . . . . . . . . . . 17 ⊢ (𝐻 ∈ Grp → 𝐻 ∈ Mnd)
58	56, 57	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → 𝐻 ∈ Mnd)
59		wrdco 14184	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 prefix 𝑛) ∈ Word (𝐼 × 2o) ∧ 𝑇:(𝐼 × 2o)⟶𝐵) → (𝑇 ∘ (𝑥 prefix 𝑛)) ∈ Word 𝐵)
60	44, 52, 59	syl2anc 587	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (𝑇 ∘ (𝑥 prefix 𝑛)) ∈ Word 𝐵)
61		wrdco 14184	. . . . . . . . . . . . . . . . 17 ⊢ ((⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩ ∈ Word (𝐼 × 2o) ∧ 𝑇:(𝐼 × 2o)⟶𝐵) → (𝑇 ∘ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩) ∈ Word 𝐵)
62	36, 52, 61	syl2anc 587	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (𝑇 ∘ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩) ∈ Word 𝐵)
63		eqid 2798	. . . . . . . . . . . . . . . . 17 ⊢ (+g‘𝐻) = (+g‘𝐻)
64	45, 63	gsumccat 17998	. . . . . . . . . . . . . . . 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)⟩”⟩))))
65	58, 60, 62, 64	syl3anc 1368	. . . . . . . . . . . . . . 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)⟩”⟩))))
66	52, 31, 35	s2co 14273	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (𝑇 ∘ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩) = ⟨“(𝑇‘⟨𝑎, 𝑏⟩)(𝑇‘⟨𝑎, (1o ∖ 𝑏)⟩)”⟩)
67		df-ov 7138	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑎𝑇𝑏) = (𝑇‘⟨𝑎, 𝑏⟩)
68	67	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (𝑎𝑇𝑏) = (𝑇‘⟨𝑎, 𝑏⟩))
69	67	fveq2i 6648	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑁‘(𝑎𝑇𝑏)) = (𝑁‘(𝑇‘⟨𝑎, 𝑏⟩))
70		df-ov 7138	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑎(𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)𝑏) = ((𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)‘⟨𝑎, 𝑏⟩)
71		eqid 2798	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩) = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)
72	71	efgmval 18830	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o) → (𝑎(𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)𝑏) = ⟨𝑎, (1o ∖ 𝑏)⟩)
73	70, 72	syl5eqr 2847	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o) → ((𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)‘⟨𝑎, 𝑏⟩) = ⟨𝑎, (1o ∖ 𝑏)⟩)
74	73	adantl 485	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → ((𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)‘⟨𝑎, 𝑏⟩) = ⟨𝑎, (1o ∖ 𝑏)⟩)
75	74	fveq2d 6649	. . . . . . . . . . . . . . . . . . . . . . 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)⟩))
76	45, 46, 47, 48, 49, 50, 71	frgpuptinv 18889	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ⟨𝑎, 𝑏⟩ ∈ (𝐼 × 2o)) → (𝑇‘((𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)‘⟨𝑎, 𝑏⟩)) = (𝑁‘(𝑇‘⟨𝑎, 𝑏⟩)))
77	30, 76	sylan2 595	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (𝑇‘((𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)‘⟨𝑎, 𝑏⟩)) = (𝑁‘(𝑇‘⟨𝑎, 𝑏⟩)))
78	77	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⟩)))
79	75, 78	eqtr3d 2835	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (𝑇‘⟨𝑎, (1o ∖ 𝑏)⟩) = (𝑁‘(𝑇‘⟨𝑎, 𝑏⟩)))
80	69, 79	eqtr4id 2852	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (𝑁‘(𝑎𝑇𝑏)) = (𝑇‘⟨𝑎, (1o ∖ 𝑏)⟩))
81	68, 80	s2eqd 14216	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → ⟨“(𝑎𝑇𝑏)(𝑁‘(𝑎𝑇𝑏))”⟩ = ⟨“(𝑇‘⟨𝑎, 𝑏⟩)(𝑇‘⟨𝑎, (1o ∖ 𝑏)⟩)”⟩)
82	66, 81	eqtr4d 2836	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (𝑇 ∘ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩) = ⟨“(𝑎𝑇𝑏)(𝑁‘(𝑎𝑇𝑏))”⟩)
83	82	oveq2d 7151	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (𝐻 Σg (𝑇 ∘ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩)) = (𝐻 Σg ⟨“(𝑎𝑇𝑏)(𝑁‘(𝑎𝑇𝑏))”⟩))
84		simprr 772	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → 𝑏 ∈ 2o)
85	52, 32, 84	fovrnd 7300	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (𝑎𝑇𝑏) ∈ 𝐵)
86	45, 46	grpinvcl 18143	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐻 ∈ Grp ∧ (𝑎𝑇𝑏) ∈ 𝐵) → (𝑁‘(𝑎𝑇𝑏)) ∈ 𝐵)
87	56, 85, 86	syl2anc 587	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (𝑁‘(𝑎𝑇𝑏)) ∈ 𝐵)
88	45, 63	gsumws2 17999	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐻 ∈ Mnd ∧ (𝑎𝑇𝑏) ∈ 𝐵 ∧ (𝑁‘(𝑎𝑇𝑏)) ∈ 𝐵) → (𝐻 Σg ⟨“(𝑎𝑇𝑏)(𝑁‘(𝑎𝑇𝑏))”⟩) = ((𝑎𝑇𝑏)(+g‘𝐻)(𝑁‘(𝑎𝑇𝑏))))
89	58, 85, 87, 88	syl3anc 1368	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (𝐻 Σg ⟨“(𝑎𝑇𝑏)(𝑁‘(𝑎𝑇𝑏))”⟩) = ((𝑎𝑇𝑏)(+g‘𝐻)(𝑁‘(𝑎𝑇𝑏))))
90		eqid 2798	. . . . . . . . . . . . . . . . . . . 20 ⊢ (0g‘𝐻) = (0g‘𝐻)
91	45, 63, 90, 46	grprinv 18145	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐻 ∈ Grp ∧ (𝑎𝑇𝑏) ∈ 𝐵) → ((𝑎𝑇𝑏)(+g‘𝐻)(𝑁‘(𝑎𝑇𝑏))) = (0g‘𝐻))
92	56, 85, 91	syl2anc 587	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → ((𝑎𝑇𝑏)(+g‘𝐻)(𝑁‘(𝑎𝑇𝑏))) = (0g‘𝐻))
93	83, 89, 92	3eqtrd 2837	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (𝐻 Σg (𝑇 ∘ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩)) = (0g‘𝐻))
94	93	oveq2d 7151	. . . . . . . . . . . . . . . 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)))
95	45	gsumwcl 17995	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐻 ∈ Mnd ∧ (𝑇 ∘ (𝑥 prefix 𝑛)) ∈ Word 𝐵) → (𝐻 Σg (𝑇 ∘ (𝑥 prefix 𝑛))) ∈ 𝐵)
96	58, 60, 95	syl2anc 587	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (𝐻 Σg (𝑇 ∘ (𝑥 prefix 𝑛))) ∈ 𝐵)
97	45, 63, 90	grprid 18126	. . . . . . . . . . . . . . . . 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))))
98	56, 96, 97	syl2anc 587	. . . . . . . . . . . . . . . 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))))
99	94, 98	eqtrd 2833	. . . . . . . . . . . . . . 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))))
100	55, 65, 99	3eqtrrd 2838	. . . . . . . . . . . . . 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)⟩”⟩))))
101	100	oveq1d 7150	. . . . . . . . . . . . 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)⟩)))))
102		swrdcl 13998	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ Word (𝐼 × 2o) → (𝑥 substr ⟨𝑛, (♯‘𝑥)⟩) ∈ Word (𝐼 × 2o))
103	29, 102	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (𝑥 substr ⟨𝑛, (♯‘𝑥)⟩) ∈ Word (𝐼 × 2o))
104		wrdco 14184	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 substr ⟨𝑛, (♯‘𝑥)⟩) ∈ Word (𝐼 × 2o) ∧ 𝑇:(𝐼 × 2o)⟶𝐵) → (𝑇 ∘ (𝑥 substr ⟨𝑛, (♯‘𝑥)⟩)) ∈ Word 𝐵)
105	103, 52, 104	syl2anc 587	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (𝑇 ∘ (𝑥 substr ⟨𝑛, (♯‘𝑥)⟩)) ∈ Word 𝐵)
106	45, 63	gsumccat 17998	. . . . . . . . . . . . . 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)⟩)))))
107	58, 60, 105, 106	syl3anc 1368	. . . . . . . . . . . . 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)⟩)))))
108		ccatcl 13917	. . . . . . . . . . . . . . . 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))
109	44, 36, 108	syl2anc 587	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → ((𝑥 prefix 𝑛) ++ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩) ∈ Word (𝐼 × 2o))
110		wrdco 14184	. . . . . . . . . . . . . . 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)
111	109, 52, 110	syl2anc 587	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (𝑇 ∘ ((𝑥 prefix 𝑛) ++ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩)) ∈ Word 𝐵)
112	45, 63	gsumccat 17998	. . . . . . . . . . . . . 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)⟩)))))
113	58, 111, 105, 112	syl3anc 1368	. . . . . . . . . . . . 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)⟩)))))
114	101, 107, 113	3eqtr4d 2843	. . . . . . . . . . . 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)⟩)))))
115		simplrr 777	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → 𝑛 ∈ (0...(♯‘𝑥)))
116		lencl 13876	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ Word (𝐼 × 2o) → (♯‘𝑥) ∈ ℕ0)
117	29, 116	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (♯‘𝑥) ∈ ℕ0)
118		nn0uz 12268	. . . . . . . . . . . . . . . . . . 19 ⊢ ℕ0 = (ℤ≥‘0)
119	117, 118	eleqtrdi 2900	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (♯‘𝑥) ∈ (ℤ≥‘0))
120		eluzfz2 12910	. . . . . . . . . . . . . . . . . 18 ⊢ ((♯‘𝑥) ∈ (ℤ≥‘0) → (♯‘𝑥) ∈ (0...(♯‘𝑥)))
121	119, 120	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (♯‘𝑥) ∈ (0...(♯‘𝑥)))
122		ccatpfx 14054	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ Word (𝐼 × 2o) ∧ 𝑛 ∈ (0...(♯‘𝑥)) ∧ (♯‘𝑥) ∈ (0...(♯‘𝑥))) → ((𝑥 prefix 𝑛) ++ (𝑥 substr ⟨𝑛, (♯‘𝑥)⟩)) = (𝑥 prefix (♯‘𝑥)))
123	29, 115, 121, 122	syl3anc 1368	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → ((𝑥 prefix 𝑛) ++ (𝑥 substr ⟨𝑛, (♯‘𝑥)⟩)) = (𝑥 prefix (♯‘𝑥)))
124		pfxid 14037	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ Word (𝐼 × 2o) → (𝑥 prefix (♯‘𝑥)) = 𝑥)
125	29, 124	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (𝑥 prefix (♯‘𝑥)) = 𝑥)
126	123, 125	eqtrd 2833	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → ((𝑥 prefix 𝑛) ++ (𝑥 substr ⟨𝑛, (♯‘𝑥)⟩)) = 𝑥)
127	126	coeq2d 5697	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (𝑇 ∘ ((𝑥 prefix 𝑛) ++ (𝑥 substr ⟨𝑛, (♯‘𝑥)⟩))) = (𝑇 ∘ 𝑥))
128		ccatco 14188	. . . . . . . . . . . . . . 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)⟩))))
129	44, 103, 52, 128	syl3anc 1368	. . . . . . . . . . . . . 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)⟩))))
130	127, 129	eqtr3d 2835	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (𝑇 ∘ 𝑥) = ((𝑇 ∘ (𝑥 prefix 𝑛)) ++ (𝑇 ∘ (𝑥 substr ⟨𝑛, (♯‘𝑥)⟩))))
131	130	oveq2d 7151	. . . . . . . . . . . 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)⟩)))))
132		splval 14104	. . . . . . . . . . . . . . . 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)⟩)))
133	26, 115, 115, 36, 132	syl13anc 1369	. . . . . . . . . . . . . . 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)⟩)))
134	133	coeq2d 5697	. . . . . . . . . . . . . 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)⟩))))
135		ccatco 14188	. . . . . . . . . . . . . . 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)⟩))))
136	109, 103, 52, 135	syl3anc 1368	. . . . . . . . . . . . . 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)⟩))))
137	134, 136	eqtrd 2833	. . . . . . . . . . . . 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)⟩))))
138	137	oveq2d 7151	. . . . . . . . . . . 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)⟩)))))
139	114, 131, 138	3eqtr4d 2843	. . . . . . . . . . 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)⟩”⟩⟩))))
140		vex 3444	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
141		ovex 7168	. . . . . . . . . . . 12 ⊢ (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩) ∈ V
142		eleq1 2877	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = 𝑥 → (𝑢 ∈ 𝑊 ↔ 𝑥 ∈ 𝑊))
143		eleq1 2877	. . . . . . . . . . . . . . 15 ⊢ (𝑣 = (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩) → (𝑣 ∈ 𝑊 ↔ (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩) ∈ 𝑊))
144	142, 143	bi2anan9 638	. . . . . . . . . . . . . 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)))
145	19, 144	bitr3id 288	. . . . . . . . . . . . 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)))
146		coeq2 5693	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = 𝑥 → (𝑇 ∘ 𝑢) = (𝑇 ∘ 𝑥))
147	146	oveq2d 7151	. . . . . . . . . . . . . 14 ⊢ (𝑢 = 𝑥 → (𝐻 Σg (𝑇 ∘ 𝑢)) = (𝐻 Σg (𝑇 ∘ 𝑥)))
148		coeq2 5693	. . . . . . . . . . . . . . 15 ⊢ (𝑣 = (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩) → (𝑇 ∘ 𝑣) = (𝑇 ∘ (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩)))
149	148	oveq2d 7151	. . . . . . . . . . . . . 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)⟩”⟩⟩))))
150	147, 149	eqeqan12d 2815	. . . . . . . . . . . . 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)⟩”⟩⟩)))))
151	145, 150	anbi12d 633	. . . . . . . . . . . 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)⟩”⟩⟩))))))
152		eqid 2798	. . . . . . . . . . . 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)))}
153	140, 141, 151, 152	braba 5389	. . . . . . . . . . 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)⟩”⟩⟩)))))
154	26, 42, 139, 153	syl21anbrc 1341	. . . . . . . . . 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)⟩”⟩⟩))
155	154	ralrimivva 3156	. . . . . . . . 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)⟩”⟩⟩))
156	155	ralrimivva 3156	. . . . . . . 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)⟩”⟩⟩))
157	1	fvexi 6659	. . . . . . . . . 10 ⊢ 𝑊 ∈ V
158		erex 8296	. . . . . . . . . 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))
159	25, 157, 158	mpisyl 21	. . . . . . . . 9 ⊢ (𝜑 → {⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇 ∘ 𝑢)) = (𝐻 Σg (𝑇 ∘ 𝑣)))} ∈ V)
160		ereq1 8279	. . . . . . . . . . 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))
161		breq 5032	. . . . . . . . . . . . 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)⟩”⟩⟩)))
162	161	2ralbidv 3164	. . . . . . . . . . . 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)⟩”⟩⟩)))
163	162	2ralbidv 3164	. . . . . . . . . . 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)⟩”⟩⟩)))
164	160, 163	anbi12d 633	. . . . . . . . . 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)⟩”⟩⟩))))
165	164	elabg 3614	. . . . . . . . 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)⟩”⟩⟩))))
166	159, 165	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)⟩”⟩⟩))))
167	25, 156, 166	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)⟩”⟩⟩))})
168		intss1 4853	. . . . . . 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)))})
169	167, 168	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)))})
170	3, 169	eqsstrid 3963	. . . . 5 ⊢ (𝜑 → ∼ ⊆ {⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇 ∘ 𝑢)) = (𝐻 Σg (𝑇 ∘ 𝑣)))})
171	170	ssbrd 5073	. . . 4 ⊢ (𝜑 → (𝐴 ∼ 𝐶 → 𝐴{⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇 ∘ 𝑢)) = (𝐻 Σg (𝑇 ∘ 𝑣)))}𝐶))
172	171	imp 410	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∼ 𝐶) → 𝐴{⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇 ∘ 𝑢)) = (𝐻 Σg (𝑇 ∘ 𝑣)))}𝐶)
173	1, 2	efger 18836	. . . . . 6 ⊢ ∼ Er 𝑊
174		errel 8281	. . . . . 6 ⊢ ( ∼ Er 𝑊 → Rel ∼ )
175	173, 174	mp1i 13	. . . . 5 ⊢ (𝜑 → Rel ∼ )
176		brrelex12 5568	. . . . 5 ⊢ ((Rel ∼ ∧ 𝐴 ∼ 𝐶) → (𝐴 ∈ V ∧ 𝐶 ∈ V))
177	175, 176	sylan 583	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∼ 𝐶) → (𝐴 ∈ V ∧ 𝐶 ∈ V))
178		preq12 4631	. . . . . . 7 ⊢ ((𝑢 = 𝐴 ∧ 𝑣 = 𝐶) → {𝑢, 𝑣} = {𝐴, 𝐶})
179	178	sseq1d 3946	. . . . . 6 ⊢ ((𝑢 = 𝐴 ∧ 𝑣 = 𝐶) → ({𝑢, 𝑣} ⊆ 𝑊 ↔ {𝐴, 𝐶} ⊆ 𝑊))
180		coeq2 5693	. . . . . . . 8 ⊢ (𝑢 = 𝐴 → (𝑇 ∘ 𝑢) = (𝑇 ∘ 𝐴))
181	180	oveq2d 7151	. . . . . . 7 ⊢ (𝑢 = 𝐴 → (𝐻 Σg (𝑇 ∘ 𝑢)) = (𝐻 Σg (𝑇 ∘ 𝐴)))
182		coeq2 5693	. . . . . . . 8 ⊢ (𝑣 = 𝐶 → (𝑇 ∘ 𝑣) = (𝑇 ∘ 𝐶))
183	182	oveq2d 7151	. . . . . . 7 ⊢ (𝑣 = 𝐶 → (𝐻 Σg (𝑇 ∘ 𝑣)) = (𝐻 Σg (𝑇 ∘ 𝐶)))
184	181, 183	eqeqan12d 2815	. . . . . 6 ⊢ ((𝑢 = 𝐴 ∧ 𝑣 = 𝐶) → ((𝐻 Σg (𝑇 ∘ 𝑢)) = (𝐻 Σg (𝑇 ∘ 𝑣)) ↔ (𝐻 Σg (𝑇 ∘ 𝐴)) = (𝐻 Σg (𝑇 ∘ 𝐶))))
185	179, 184	anbi12d 633	. . . . 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)))))
186	185, 152	brabga 5386	. . . 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)))))
187	177, 186	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)))))
188	172, 187	mpbid 235	. 2 ⊢ ((𝜑 ∧ 𝐴 ∼ 𝐶) → ({𝐴, 𝐶} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇 ∘ 𝐴)) = (𝐻 Σg (𝑇 ∘ 𝐶))))
189	188	simprd 499	1 ⊢ ((𝜑 ∧ 𝐴 ∼ 𝐶) → (𝐻 Σg (𝑇 ∘ 𝐴)) = (𝐻 Σg (𝑇 ∘ 𝐶)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  {cab 2776  ∀wral 3106  Vcvv 3441   ∖ cdif 3878   ∩ cin 3880   ⊆ wss 3881  ∅c0 4243  ifcif 4425  {cpr 4527  ⟨cop 4531  ⟨cotp 4533  ∩ cint 4838   class class class wbr 5030  {copab 5092   I cid 5424   × cxp 5517   ∘ ccom 5523  Rel wrel 5524  ⟶wf 6320  ‘cfv 6324  (class class class)co 7135   ∈ cmpo 7137  1_oc1o 8078  2_oc2o 8079   Er wer 8269  0cc0 10526  ℕ₀cn0 11885  ℤ_≥cuz 12231  ...cfz 12885  ♯chash 13686  Word cword 13857   ++ cconcat 13913   substr csubstr 13993   prefix cpfx 14023   splice csplice 14102  ⟨“cs2 14194  Basecbs 16475  +_gcplusg 16557  0_gc0g 16705   Σ_g cgsu 16706  Mndcmnd 17903  Grpcgrp 18095  inv_gcminusg 18096   ~_FG cefg 18824

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-ot 4534  df-uni 4801  df-int 4839  df-iun 4883  df-iin 4884  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-2o 8086  df-oadd 8089  df-er 8272  df-map 8391  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-card 9352  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11626  df-2 11688  df-n0 11886  df-z 11970  df-uz 12232  df-fz 12886  df-fzo 13029  df-seq 13365  df-hash 13687  df-word 13858  df-concat 13914  df-s1 13941  df-substr 13994  df-pfx 14024  df-splice 14103  df-s2 14201  df-ndx 16478  df-slot 16479  df-base 16481  df-sets 16482  df-ress 16483  df-plusg 16570  df-0g 16707  df-gsum 16708  df-mgm 17844  df-sgrp 17893  df-mnd 17904  df-submnd 17949  df-grp 18098  df-minusg 18099  df-efg 18827

This theorem is referenced by:  frgpupf  18891