Theorem frgpuplem 19703

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 19648	. . . . . 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 5806	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝑣 → (𝑇 ∘ 𝑢) = (𝑇 ∘ 𝑣))
5	4	oveq2d 7374	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑣 → (𝐻 Σg (𝑇 ∘ 𝑢)) = (𝐻 Σg (𝑇 ∘ 𝑣)))
6		eqid 2735	. . . . . . . . . . . 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 8672	. . . . . . . . . . 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 3957	. . . . . . . . . . 11 ⊢ 𝑊 ⊆ V
10	9	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 𝑊 ⊆ V)
11	8, 10	erinxp 8730	. . . . . . . . 9 ⊢ (𝜑 → ({⟨𝑢, 𝑣⟩ ∣ (𝐻 Σg (𝑇 ∘ 𝑢)) = (𝐻 Σg (𝑇 ∘ 𝑣))} ∩ (𝑊 × 𝑊)) Er 𝑊)
12		df-xp 5629	. . . . . . . . . . . . 13 ⊢ (𝑊 × 𝑊) = {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝑊 ∧ 𝑣 ∈ 𝑊)}
13	12	ineq1i 4167	. . . . . . . . . . . 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 4160	. . . . . . . . . . . 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 5777	. . . . . . . . . . . 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 2766	. . . . . . . . . . 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 3443	. . . . . . . . . . . . . 14 ⊢ 𝑢 ∈ V
18		vex 3443	. . . . . . . . . . . . . 14 ⊢ 𝑣 ∈ V
19	17, 18	prss 4775	. . . . . . . . . . . . 13 ⊢ ((𝑢 ∈ 𝑊 ∧ 𝑣 ∈ 𝑊) ↔ {𝑢, 𝑣} ⊆ 𝑊)
20	19	anbi1i 625	. . . . . . . . . . . 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 5164	. . . . . . . . . . 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 2758	. . . . . . . . . 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 8643	. . . . . . . . . 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 777	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → 𝑥 ∈ 𝑊)
27		fviss 6910	. . . . . . . . . . . . . . 15 ⊢ ( I ‘Word (𝐼 × 2o)) ⊆ Word (𝐼 × 2o)
28	1, 27	eqsstri 3979	. . . . . . . . . . . . . 14 ⊢ 𝑊 ⊆ Word (𝐼 × 2o)
29	28, 26	sselid 3930	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → 𝑥 ∈ Word (𝐼 × 2o))
30		opelxpi 5660	. . . . . . . . . . . . . . 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 771	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → 𝑎 ∈ 𝐼)
33		2oconcl 8430	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 ∈ 2o → (1o ∖ 𝑏) ∈ 2o)
34	33	ad2antll 730	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (1o ∖ 𝑏) ∈ 2o)
35	32, 34	opelxpd 5662	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → ⟨𝑎, (1o ∖ 𝑏)⟩ ∈ (𝐼 × 2o))
36	31, 35	s2cld 14796	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩ ∈ Word (𝐼 × 2o))
37		splcl 14677	. . . . . . . . . . . . 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 585	. . . . . . . . . . . 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 19646	. . . . . . . . . . . . . 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 2838	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩) ∈ 𝑊)
43		pfxcl 14603	. . . . . . . . . . . . . . . . . 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 19701	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑇:(𝐼 × 2o)⟶𝐵)
52	51	ad2antrr 727	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → 𝑇:(𝐼 × 2o)⟶𝐵)
53		ccatco 14760	. . . . . . . . . . . . . . . . 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 1374	. . . . . . . . . . . . . . . 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 7374	. . . . . . . . . . . . . . 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 727	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → 𝐻 ∈ Grp)
57	56	grpmndd 18878	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → 𝐻 ∈ Mnd)
58		wrdco 14756	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 prefix 𝑛) ∈ Word (𝐼 × 2o) ∧ 𝑇:(𝐼 × 2o)⟶𝐵) → (𝑇 ∘ (𝑥 prefix 𝑛)) ∈ Word 𝐵)
59	44, 52, 58	syl2anc 585	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (𝑇 ∘ (𝑥 prefix 𝑛)) ∈ Word 𝐵)
60		wrdco 14756	. . . . . . . . . . . . . . . . 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 585	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (𝑇 ∘ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩) ∈ Word 𝐵)
62		eqid 2735	. . . . . . . . . . . . . . . . 17 ⊢ (+g‘𝐻) = (+g‘𝐻)
63	45, 62	gsumccat 18768	. . . . . . . . . . . . . . . 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 1374	. . . . . . . . . . . . . . 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 14845	. . . . . . . . . . . . . . . . . . . 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 7361	. . . . . . . . . . . . . . . . . . . . . 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 6836	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑁‘(𝑎𝑇𝑏)) = (𝑁‘(𝑇‘⟨𝑎, 𝑏⟩))
69		df-ov 7361	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑎(𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)𝑏) = ((𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)‘⟨𝑎, 𝑏⟩)
70		eqid 2735	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩) = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)
71	70	efgmval 19643	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o) → (𝑎(𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)𝑏) = ⟨𝑎, (1o ∖ 𝑏)⟩)
72	69, 71	eqtr3id 2784	. . . . . . . . . . . . . . . . . . . . . . . . 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 6837	. . . . . . . . . . . . . . . . . . . . . . 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 19702	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ⟨𝑎, 𝑏⟩ ∈ (𝐼 × 2o)) → (𝑇‘((𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)‘⟨𝑎, 𝑏⟩)) = (𝑁‘(𝑇‘⟨𝑎, 𝑏⟩)))
76	30, 75	sylan2 594	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (𝑇‘((𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)‘⟨𝑎, 𝑏⟩)) = (𝑁‘(𝑇‘⟨𝑎, 𝑏⟩)))
77	76	adantlr 716	. . . . . . . . . . . . . . . . . . . . . . 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 2772	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (𝑇‘⟨𝑎, (1o ∖ 𝑏)⟩) = (𝑁‘(𝑇‘⟨𝑎, 𝑏⟩)))
79	68, 78	eqtr4id 2789	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (𝑁‘(𝑎𝑇𝑏)) = (𝑇‘⟨𝑎, (1o ∖ 𝑏)⟩))
80	67, 79	s2eqd 14788	. . . . . . . . . . . . . . . . . . . 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 2773	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (𝑇 ∘ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩) = ⟨“(𝑎𝑇𝑏)(𝑁‘(𝑎𝑇𝑏))”⟩)
82	81	oveq2d 7374	. . . . . . . . . . . . . . . . . 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 773	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → 𝑏 ∈ 2o)
84	52, 32, 83	fovcdmd 7530	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (𝑎𝑇𝑏) ∈ 𝐵)
85	45, 46	grpinvcl 18919	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐻 ∈ Grp ∧ (𝑎𝑇𝑏) ∈ 𝐵) → (𝑁‘(𝑎𝑇𝑏)) ∈ 𝐵)
86	56, 84, 85	syl2anc 585	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (𝑁‘(𝑎𝑇𝑏)) ∈ 𝐵)
87	45, 62	gsumws2 18769	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐻 ∈ Mnd ∧ (𝑎𝑇𝑏) ∈ 𝐵 ∧ (𝑁‘(𝑎𝑇𝑏)) ∈ 𝐵) → (𝐻 Σg ⟨“(𝑎𝑇𝑏)(𝑁‘(𝑎𝑇𝑏))”⟩) = ((𝑎𝑇𝑏)(+g‘𝐻)(𝑁‘(𝑎𝑇𝑏))))
88	57, 84, 86, 87	syl3anc 1374	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (𝐻 Σg ⟨“(𝑎𝑇𝑏)(𝑁‘(𝑎𝑇𝑏))”⟩) = ((𝑎𝑇𝑏)(+g‘𝐻)(𝑁‘(𝑎𝑇𝑏))))
89		eqid 2735	. . . . . . . . . . . . . . . . . . . 20 ⊢ (0g‘𝐻) = (0g‘𝐻)
90	45, 62, 89, 46	grprinv 18922	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐻 ∈ Grp ∧ (𝑎𝑇𝑏) ∈ 𝐵) → ((𝑎𝑇𝑏)(+g‘𝐻)(𝑁‘(𝑎𝑇𝑏))) = (0g‘𝐻))
91	56, 84, 90	syl2anc 585	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → ((𝑎𝑇𝑏)(+g‘𝐻)(𝑁‘(𝑎𝑇𝑏))) = (0g‘𝐻))
92	82, 88, 91	3eqtrd 2774	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (𝐻 Σg (𝑇 ∘ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩)) = (0g‘𝐻))
93	92	oveq2d 7374	. . . . . . . . . . . . . . . 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 18766	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐻 ∈ Mnd ∧ (𝑇 ∘ (𝑥 prefix 𝑛)) ∈ Word 𝐵) → (𝐻 Σg (𝑇 ∘ (𝑥 prefix 𝑛))) ∈ 𝐵)
95	57, 59, 94	syl2anc 585	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (𝐻 Σg (𝑇 ∘ (𝑥 prefix 𝑛))) ∈ 𝐵)
96	45, 62, 89	grprid 18900	. . . . . . . . . . . . . . . . 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 585	. . . . . . . . . . . . . . . 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 2770	. . . . . . . . . . . . . . 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 2775	. . . . . . . . . . . . . 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 7373	. . . . . . . . . . . . 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 14571	. . . . . . . . . . . . . . . 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 14756	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 substr ⟨𝑛, (♯‘𝑥)⟩) ∈ Word (𝐼 × 2o) ∧ 𝑇:(𝐼 × 2o)⟶𝐵) → (𝑇 ∘ (𝑥 substr ⟨𝑛, (♯‘𝑥)⟩)) ∈ Word 𝐵)
104	102, 52, 103	syl2anc 585	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (𝑇 ∘ (𝑥 substr ⟨𝑛, (♯‘𝑥)⟩)) ∈ Word 𝐵)
105	45, 62	gsumccat 18768	. . . . . . . . . . . . . 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 1374	. . . . . . . . . . . . 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 14499	. . . . . . . . . . . . . . . 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 585	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → ((𝑥 prefix 𝑛) ++ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩) ∈ Word (𝐼 × 2o))
109		wrdco 14756	. . . . . . . . . . . . . . 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 585	. . . . . . . . . . . . . 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 18768	. . . . . . . . . . . . . 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 1374	. . . . . . . . . . . . 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 2780	. . . . . . . . . . . 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 778	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → 𝑛 ∈ (0...(♯‘𝑥)))
115		lencl 14458	. . . . . . . . . . . . . . . . . . . 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 12791	. . . . . . . . . . . . . . . . . . 19 ⊢ ℕ0 = (ℤ≥‘0)
118	116, 117	eleqtrdi 2845	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (♯‘𝑥) ∈ (ℤ≥‘0))
119		eluzfz2 13450	. . . . . . . . . . . . . . . . . 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 14626	. . . . . . . . . . . . . . . . 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 1374	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → ((𝑥 prefix 𝑛) ++ (𝑥 substr ⟨𝑛, (♯‘𝑥)⟩)) = (𝑥 prefix (♯‘𝑥)))
123		pfxid 14610	. . . . . . . . . . . . . . . . 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 2770	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → ((𝑥 prefix 𝑛) ++ (𝑥 substr ⟨𝑛, (♯‘𝑥)⟩)) = 𝑥)
126	125	coeq2d 5810	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o)) → (𝑇 ∘ ((𝑥 prefix 𝑛) ++ (𝑥 substr ⟨𝑛, (♯‘𝑥)⟩))) = (𝑇 ∘ 𝑥))
127		ccatco 14760	. . . . . . . . . . . . . . 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 1374	. . . . . . . . . . . . . 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 2772	. . . . . . . . . . . . 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 7374	. . . . . . . . . . . 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 14676	. . . . . . . . . . . . . . . 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 1375	. . . . . . . . . . . . . . 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 5810	. . . . . . . . . . . . . 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 14760	. . . . . . . . . . . . . . 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 1374	. . . . . . . . . . . . . 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 2770	. . . . . . . . . . . . 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 7374	. . . . . . . . . . . 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 2780	. . . . . . . . . . 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 3443	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
140		ovex 7391	. . . . . . . . . . . 12 ⊢ (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩) ∈ V
141		eleq1 2823	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = 𝑥 → (𝑢 ∈ 𝑊 ↔ 𝑥 ∈ 𝑊))
142		eleq1 2823	. . . . . . . . . . . . . . 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 639	. . . . . . . . . . . . . 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 5806	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = 𝑥 → (𝑇 ∘ 𝑢) = (𝑇 ∘ 𝑥))
146	145	oveq2d 7374	. . . . . . . . . . . . . 14 ⊢ (𝑢 = 𝑥 → (𝐻 Σg (𝑇 ∘ 𝑢)) = (𝐻 Σg (𝑇 ∘ 𝑥)))
147		coeq2 5806	. . . . . . . . . . . . . . 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 7374	. . . . . . . . . . . . . 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 2749	. . . . . . . . . . . . 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 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)⟩”⟩⟩))))))
151		eqid 2735	. . . . . . . . . . . 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 5484	. . . . . . . . . . 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 1346	. . . . . . . . . 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 3178	. . . . . . . . 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 3178	. . . . . . . 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 6847	. . . . . . . . . 10 ⊢ 𝑊 ∈ V
157		erex 8660	. . . . . . . . . 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 8643	. . . . . . . . . . 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 5099	. . . . . . . . . . . . 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 3199	. . . . . . . . . . . 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 3199	. . . . . . . . . . 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 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)⟩”⟩⟩))))
164	163	elabg 3630	. . . . . . . . 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 714	. . . . . . 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 4917	. . . . . . 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 3971	. . . . 5 ⊢ (𝜑 → ∼ ⊆ {⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇 ∘ 𝑢)) = (𝐻 Σg (𝑇 ∘ 𝑣)))})
170	169	ssbrd 5140	. . . 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 19649	. . . . . 6 ⊢ ∼ Er 𝑊
173		errel 8645	. . . . . 6 ⊢ ( ∼ Er 𝑊 → Rel ∼ )
174	172, 173	mp1i 13	. . . . 5 ⊢ (𝜑 → Rel ∼ )
175		brrelex12 5675	. . . . 5 ⊢ ((Rel ∼ ∧ 𝐴 ∼ 𝐶) → (𝐴 ∈ V ∧ 𝐶 ∈ V))
176	174, 175	sylan 581	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∼ 𝐶) → (𝐴 ∈ V ∧ 𝐶 ∈ V))
177		preq12 4691	. . . . . . 7 ⊢ ((𝑢 = 𝐴 ∧ 𝑣 = 𝐶) → {𝑢, 𝑣} = {𝐴, 𝐶})
178	177	sseq1d 3964	. . . . . 6 ⊢ ((𝑢 = 𝐴 ∧ 𝑣 = 𝐶) → ({𝑢, 𝑣} ⊆ 𝑊 ↔ {𝐴, 𝐶} ⊆ 𝑊))
179		coeq2 5806	. . . . . . . 8 ⊢ (𝑢 = 𝐴 → (𝑇 ∘ 𝑢) = (𝑇 ∘ 𝐴))
180	179	oveq2d 7374	. . . . . . 7 ⊢ (𝑢 = 𝐴 → (𝐻 Σg (𝑇 ∘ 𝑢)) = (𝐻 Σg (𝑇 ∘ 𝐴)))
181		coeq2 5806	. . . . . . . 8 ⊢ (𝑣 = 𝐶 → (𝑇 ∘ 𝑣) = (𝑇 ∘ 𝐶))
182	181	oveq2d 7374	. . . . . . 7 ⊢ (𝑣 = 𝐶 → (𝐻 Σg (𝑇 ∘ 𝑣)) = (𝐻 Σg (𝑇 ∘ 𝐶)))
183	180, 182	eqeqan12d 2749	. . . . . 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 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)))))
185	184, 151	brabga 5481	. . . 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 1542   ∈ wcel 2114  {cab 2713  ∀wral 3050  Vcvv 3439   ∖ cdif 3897   ∩ cin 3899   ⊆ wss 3900  ∅c0 4284  ifcif 4478  {cpr 4581  ⟨cop 4585  ⟨cotp 4587  ∩ cint 4901   class class class wbr 5097  {copab 5159   I cid 5517   × cxp 5621   ∘ ccom 5627  Rel wrel 5628  ⟶wf 6487  ‘cfv 6491  (class class class)co 7358   ∈ cmpo 7360  1_oc1o 8390  2_oc2o 8391   Er wer 8632  0cc0 11028  ℕ₀cn0 12403  ℤ_≥cuz 12753  ...cfz 13425  ♯chash 14255  Word cword 14438   ++ cconcat 14495   substr csubstr 14566   prefix cpfx 14596   splice csplice 14674  ⟨“cs2 14766  Basecbs 17138  +_gcplusg 17179  0_gc0g 17361   Σ_g cgsu 17362  Mndcmnd 18661  Grpcgrp 18865  inv_gcminusg 18866   ~_FG cefg 19637

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2183  ax-ext 2707  ax-rep 5223  ax-sep 5240  ax-nul 5250  ax-pow 5309  ax-pr 5376  ax-un 7680  ax-cnex 11084  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104  ax-pre-mulgt0 11105

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2932  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3349  df-reu 3350  df-rab 3399  df-v 3441  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4285  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-ot 4588  df-uni 4863  df-int 4902  df-iun 4947  df-iin 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6258  df-ord 6319  df-on 6320  df-lim 6321  df-suc 6322  df-iota 6447  df-fun 6493  df-fn 6494  df-f 6495  df-f1 6496  df-fo 6497  df-f1o 6498  df-fv 6499  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-2o 8398  df-er 8635  df-map 8767  df-en 8886  df-dom 8887  df-sdom 8888  df-fin 8889  df-card 9853  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11368  df-neg 11369  df-nn 12148  df-2 12210  df-n0 12404  df-z 12491  df-uz 12754  df-fz 13426  df-fzo 13573  df-seq 13927  df-hash 14256  df-word 14439  df-concat 14496  df-s1 14522  df-substr 14567  df-pfx 14597  df-splice 14675  df-s2 14773  df-sets 17093  df-slot 17111  df-ndx 17123  df-base 17139  df-ress 17160  df-plusg 17192  df-0g 17363  df-gsum 17364  df-mgm 18567  df-sgrp 18646  df-mnd 18662  df-submnd 18711  df-grp 18868  df-minusg 18869  df-efg 19640

This theorem is referenced by:  frgpupf  19704