Theorem efgi 18132

Description: Value of the free group construction. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)

Hypotheses

Ref	Expression
efgval.w	⊢ 𝑊 = ( I ‘Word (𝐼 × 2𝑜))
efgval.r	⊢ ∼ = ( ~FG ‘𝐼)

Assertion

Ref	Expression
efgi	⊢ (((𝐴 ∈ 𝑊 ∧ 𝑁 ∈ (0...(#‘𝐴))) ∧ (𝐽 ∈ 𝐼 ∧ 𝐾 ∈ 2𝑜)) → 𝐴 ∼ (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝐾⟩⟨𝐽, (1𝑜 ∖ 𝐾)⟩”⟩⟩))

Proof of Theorem efgi

Dummy variables 𝑎 𝑏 𝑖 𝑟 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6191	. . . . . . . . . . 11 ⊢ (𝑢 = 𝐴 → (#‘𝑢) = (#‘𝐴))
2	1	oveq2d 6666	. . . . . . . . . 10 ⊢ (𝑢 = 𝐴 → (0...(#‘𝑢)) = (0...(#‘𝐴)))
3		id 22	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝐴 → 𝑢 = 𝐴)
4		oveq1 6657	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝐴 → (𝑢 splice ⟨𝑖, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜 ∖ 𝑏)⟩”⟩⟩) = (𝐴 splice ⟨𝑖, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜 ∖ 𝑏)⟩”⟩⟩))
5	3, 4	breq12d 4666	. . . . . . . . . . 11 ⊢ (𝑢 = 𝐴 → (𝑢𝑟(𝑢 splice ⟨𝑖, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜 ∖ 𝑏)⟩”⟩⟩) ↔ 𝐴𝑟(𝐴 splice ⟨𝑖, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜 ∖ 𝑏)⟩”⟩⟩)))
6	5	2ralbidv 2989	. . . . . . . . . 10 ⊢ (𝑢 = 𝐴 → (∀𝑎 ∈ 𝐼 ∀𝑏 ∈ 2𝑜 𝑢𝑟(𝑢 splice ⟨𝑖, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜 ∖ 𝑏)⟩”⟩⟩) ↔ ∀𝑎 ∈ 𝐼 ∀𝑏 ∈ 2𝑜 𝐴𝑟(𝐴 splice ⟨𝑖, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜 ∖ 𝑏)⟩”⟩⟩)))
7	2, 6	raleqbidv 3152	. . . . . . . . 9 ⊢ (𝑢 = 𝐴 → (∀𝑖 ∈ (0...(#‘𝑢))∀𝑎 ∈ 𝐼 ∀𝑏 ∈ 2𝑜 𝑢𝑟(𝑢 splice ⟨𝑖, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜 ∖ 𝑏)⟩”⟩⟩) ↔ ∀𝑖 ∈ (0...(#‘𝐴))∀𝑎 ∈ 𝐼 ∀𝑏 ∈ 2𝑜 𝐴𝑟(𝐴 splice ⟨𝑖, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜 ∖ 𝑏)⟩”⟩⟩)))
8	7	rspcv 3305	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑊 → (∀𝑢 ∈ 𝑊 ∀𝑖 ∈ (0...(#‘𝑢))∀𝑎 ∈ 𝐼 ∀𝑏 ∈ 2𝑜 𝑢𝑟(𝑢 splice ⟨𝑖, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜 ∖ 𝑏)⟩”⟩⟩) → ∀𝑖 ∈ (0...(#‘𝐴))∀𝑎 ∈ 𝐼 ∀𝑏 ∈ 2𝑜 𝐴𝑟(𝐴 splice ⟨𝑖, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜 ∖ 𝑏)⟩”⟩⟩)))
9		oteq1 4411	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑁 → ⟨𝑖, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜 ∖ 𝑏)⟩”⟩⟩ = ⟨𝑁, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜 ∖ 𝑏)⟩”⟩⟩)
10		oteq2 4412	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑁 → ⟨𝑁, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜 ∖ 𝑏)⟩”⟩⟩ = ⟨𝑁, 𝑁, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜 ∖ 𝑏)⟩”⟩⟩)
11	9, 10	eqtrd 2656	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑁 → ⟨𝑖, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜 ∖ 𝑏)⟩”⟩⟩ = ⟨𝑁, 𝑁, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜 ∖ 𝑏)⟩”⟩⟩)
12	11	oveq2d 6666	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑁 → (𝐴 splice ⟨𝑖, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜 ∖ 𝑏)⟩”⟩⟩) = (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜 ∖ 𝑏)⟩”⟩⟩))
13	12	breq2d 4665	. . . . . . . . . 10 ⊢ (𝑖 = 𝑁 → (𝐴𝑟(𝐴 splice ⟨𝑖, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜 ∖ 𝑏)⟩”⟩⟩) ↔ 𝐴𝑟(𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜 ∖ 𝑏)⟩”⟩⟩)))
14	13	2ralbidv 2989	. . . . . . . . 9 ⊢ (𝑖 = 𝑁 → (∀𝑎 ∈ 𝐼 ∀𝑏 ∈ 2𝑜 𝐴𝑟(𝐴 splice ⟨𝑖, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜 ∖ 𝑏)⟩”⟩⟩) ↔ ∀𝑎 ∈ 𝐼 ∀𝑏 ∈ 2𝑜 𝐴𝑟(𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜 ∖ 𝑏)⟩”⟩⟩)))
15	14	rspcv 3305	. . . . . . . 8 ⊢ (𝑁 ∈ (0...(#‘𝐴)) → (∀𝑖 ∈ (0...(#‘𝐴))∀𝑎 ∈ 𝐼 ∀𝑏 ∈ 2𝑜 𝐴𝑟(𝐴 splice ⟨𝑖, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜 ∖ 𝑏)⟩”⟩⟩) → ∀𝑎 ∈ 𝐼 ∀𝑏 ∈ 2𝑜 𝐴𝑟(𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜 ∖ 𝑏)⟩”⟩⟩)))
16	8, 15	sylan9 689	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑊 ∧ 𝑁 ∈ (0...(#‘𝐴))) → (∀𝑢 ∈ 𝑊 ∀𝑖 ∈ (0...(#‘𝑢))∀𝑎 ∈ 𝐼 ∀𝑏 ∈ 2𝑜 𝑢𝑟(𝑢 splice ⟨𝑖, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜 ∖ 𝑏)⟩”⟩⟩) → ∀𝑎 ∈ 𝐼 ∀𝑏 ∈ 2𝑜 𝐴𝑟(𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜 ∖ 𝑏)⟩”⟩⟩)))
17		opeq1 4402	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝐽 → ⟨𝑎, 𝑏⟩ = ⟨𝐽, 𝑏⟩)
18		opeq1 4402	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝐽 → ⟨𝑎, (1𝑜 ∖ 𝑏)⟩ = ⟨𝐽, (1𝑜 ∖ 𝑏)⟩)
19	17, 18	s2eqd 13608	. . . . . . . . . . 11 ⊢ (𝑎 = 𝐽 → ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜 ∖ 𝑏)⟩”⟩ = ⟨“⟨𝐽, 𝑏⟩⟨𝐽, (1𝑜 ∖ 𝑏)⟩”⟩)
20	19	oteq3d 4416	. . . . . . . . . 10 ⊢ (𝑎 = 𝐽 → ⟨𝑁, 𝑁, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜 ∖ 𝑏)⟩”⟩⟩ = ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝑏⟩⟨𝐽, (1𝑜 ∖ 𝑏)⟩”⟩⟩)
21	20	oveq2d 6666	. . . . . . . . 9 ⊢ (𝑎 = 𝐽 → (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜 ∖ 𝑏)⟩”⟩⟩) = (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝑏⟩⟨𝐽, (1𝑜 ∖ 𝑏)⟩”⟩⟩))
22	21	breq2d 4665	. . . . . . . 8 ⊢ (𝑎 = 𝐽 → (𝐴𝑟(𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜 ∖ 𝑏)⟩”⟩⟩) ↔ 𝐴𝑟(𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝑏⟩⟨𝐽, (1𝑜 ∖ 𝑏)⟩”⟩⟩)))
23		opeq2 4403	. . . . . . . . . . . . 13 ⊢ (𝑏 = 𝐾 → ⟨𝐽, 𝑏⟩ = ⟨𝐽, 𝐾⟩)
24		difeq2 3722	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 𝐾 → (1𝑜 ∖ 𝑏) = (1𝑜 ∖ 𝐾))
25	24	opeq2d 4409	. . . . . . . . . . . . 13 ⊢ (𝑏 = 𝐾 → ⟨𝐽, (1𝑜 ∖ 𝑏)⟩ = ⟨𝐽, (1𝑜 ∖ 𝐾)⟩)
26	23, 25	s2eqd 13608	. . . . . . . . . . . 12 ⊢ (𝑏 = 𝐾 → ⟨“⟨𝐽, 𝑏⟩⟨𝐽, (1𝑜 ∖ 𝑏)⟩”⟩ = ⟨“⟨𝐽, 𝐾⟩⟨𝐽, (1𝑜 ∖ 𝐾)⟩”⟩)
27	26	oteq3d 4416	. . . . . . . . . . 11 ⊢ (𝑏 = 𝐾 → ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝑏⟩⟨𝐽, (1𝑜 ∖ 𝑏)⟩”⟩⟩ = ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝐾⟩⟨𝐽, (1𝑜 ∖ 𝐾)⟩”⟩⟩)
28	27	oveq2d 6666	. . . . . . . . . 10 ⊢ (𝑏 = 𝐾 → (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝑏⟩⟨𝐽, (1𝑜 ∖ 𝑏)⟩”⟩⟩) = (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝐾⟩⟨𝐽, (1𝑜 ∖ 𝐾)⟩”⟩⟩))
29	28	breq2d 4665	. . . . . . . . 9 ⊢ (𝑏 = 𝐾 → (𝐴𝑟(𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝑏⟩⟨𝐽, (1𝑜 ∖ 𝑏)⟩”⟩⟩) ↔ 𝐴𝑟(𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝐾⟩⟨𝐽, (1𝑜 ∖ 𝐾)⟩”⟩⟩)))
30		df-br 4654	. . . . . . . . 9 ⊢ (𝐴𝑟(𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝐾⟩⟨𝐽, (1𝑜 ∖ 𝐾)⟩”⟩⟩) ↔ ⟨𝐴, (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝐾⟩⟨𝐽, (1𝑜 ∖ 𝐾)⟩”⟩⟩)⟩ ∈ 𝑟)
31	29, 30	syl6bb 276	. . . . . . . 8 ⊢ (𝑏 = 𝐾 → (𝐴𝑟(𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝑏⟩⟨𝐽, (1𝑜 ∖ 𝑏)⟩”⟩⟩) ↔ ⟨𝐴, (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝐾⟩⟨𝐽, (1𝑜 ∖ 𝐾)⟩”⟩⟩)⟩ ∈ 𝑟))
32	22, 31	rspc2v 3322	. . . . . . 7 ⊢ ((𝐽 ∈ 𝐼 ∧ 𝐾 ∈ 2𝑜) → (∀𝑎 ∈ 𝐼 ∀𝑏 ∈ 2𝑜 𝐴𝑟(𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜 ∖ 𝑏)⟩”⟩⟩) → ⟨𝐴, (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝐾⟩⟨𝐽, (1𝑜 ∖ 𝐾)⟩”⟩⟩)⟩ ∈ 𝑟))
33	16, 32	sylan9 689	. . . . . 6 ⊢ (((𝐴 ∈ 𝑊 ∧ 𝑁 ∈ (0...(#‘𝐴))) ∧ (𝐽 ∈ 𝐼 ∧ 𝐾 ∈ 2𝑜)) → (∀𝑢 ∈ 𝑊 ∀𝑖 ∈ (0...(#‘𝑢))∀𝑎 ∈ 𝐼 ∀𝑏 ∈ 2𝑜 𝑢𝑟(𝑢 splice ⟨𝑖, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜 ∖ 𝑏)⟩”⟩⟩) → ⟨𝐴, (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝐾⟩⟨𝐽, (1𝑜 ∖ 𝐾)⟩”⟩⟩)⟩ ∈ 𝑟))
34	33	adantld 483	. . . . 5 ⊢ (((𝐴 ∈ 𝑊 ∧ 𝑁 ∈ (0...(#‘𝐴))) ∧ (𝐽 ∈ 𝐼 ∧ 𝐾 ∈ 2𝑜)) → ((𝑟 Er 𝑊 ∧ ∀𝑢 ∈ 𝑊 ∀𝑖 ∈ (0...(#‘𝑢))∀𝑎 ∈ 𝐼 ∀𝑏 ∈ 2𝑜 𝑢𝑟(𝑢 splice ⟨𝑖, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜 ∖ 𝑏)⟩”⟩⟩)) → ⟨𝐴, (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝐾⟩⟨𝐽, (1𝑜 ∖ 𝐾)⟩”⟩⟩)⟩ ∈ 𝑟))
35	34	alrimiv 1855	. . . 4 ⊢ (((𝐴 ∈ 𝑊 ∧ 𝑁 ∈ (0...(#‘𝐴))) ∧ (𝐽 ∈ 𝐼 ∧ 𝐾 ∈ 2𝑜)) → ∀𝑟((𝑟 Er 𝑊 ∧ ∀𝑢 ∈ 𝑊 ∀𝑖 ∈ (0...(#‘𝑢))∀𝑎 ∈ 𝐼 ∀𝑏 ∈ 2𝑜 𝑢𝑟(𝑢 splice ⟨𝑖, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜 ∖ 𝑏)⟩”⟩⟩)) → ⟨𝐴, (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝐾⟩⟨𝐽, (1𝑜 ∖ 𝐾)⟩”⟩⟩)⟩ ∈ 𝑟))
36		opex 4932	. . . . 5 ⊢ ⟨𝐴, (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝐾⟩⟨𝐽, (1𝑜 ∖ 𝐾)⟩”⟩⟩)⟩ ∈ V
37	36	elintab 4487	. . . 4 ⊢ (⟨𝐴, (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝐾⟩⟨𝐽, (1𝑜 ∖ 𝐾)⟩”⟩⟩)⟩ ∈ ∩ {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑢 ∈ 𝑊 ∀𝑖 ∈ (0...(#‘𝑢))∀𝑎 ∈ 𝐼 ∀𝑏 ∈ 2𝑜 𝑢𝑟(𝑢 splice ⟨𝑖, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜 ∖ 𝑏)⟩”⟩⟩))} ↔ ∀𝑟((𝑟 Er 𝑊 ∧ ∀𝑢 ∈ 𝑊 ∀𝑖 ∈ (0...(#‘𝑢))∀𝑎 ∈ 𝐼 ∀𝑏 ∈ 2𝑜 𝑢𝑟(𝑢 splice ⟨𝑖, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜 ∖ 𝑏)⟩”⟩⟩)) → ⟨𝐴, (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝐾⟩⟨𝐽, (1𝑜 ∖ 𝐾)⟩”⟩⟩)⟩ ∈ 𝑟))
38	35, 37	sylibr 224	. . 3 ⊢ (((𝐴 ∈ 𝑊 ∧ 𝑁 ∈ (0...(#‘𝐴))) ∧ (𝐽 ∈ 𝐼 ∧ 𝐾 ∈ 2𝑜)) → ⟨𝐴, (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝐾⟩⟨𝐽, (1𝑜 ∖ 𝐾)⟩”⟩⟩)⟩ ∈ ∩ {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑢 ∈ 𝑊 ∀𝑖 ∈ (0...(#‘𝑢))∀𝑎 ∈ 𝐼 ∀𝑏 ∈ 2𝑜 𝑢𝑟(𝑢 splice ⟨𝑖, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜 ∖ 𝑏)⟩”⟩⟩))})
39		efgval.w	. . . 4 ⊢ 𝑊 = ( I ‘Word (𝐼 × 2𝑜))
40		efgval.r	. . . 4 ⊢ ∼ = ( ~FG ‘𝐼)
41	39, 40	efgval 18130	. . 3 ⊢ ∼ = ∩ {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑢 ∈ 𝑊 ∀𝑖 ∈ (0...(#‘𝑢))∀𝑎 ∈ 𝐼 ∀𝑏 ∈ 2𝑜 𝑢𝑟(𝑢 splice ⟨𝑖, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜 ∖ 𝑏)⟩”⟩⟩))}
42	38, 41	syl6eleqr 2712	. 2 ⊢ (((𝐴 ∈ 𝑊 ∧ 𝑁 ∈ (0...(#‘𝐴))) ∧ (𝐽 ∈ 𝐼 ∧ 𝐾 ∈ 2𝑜)) → ⟨𝐴, (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝐾⟩⟨𝐽, (1𝑜 ∖ 𝐾)⟩”⟩⟩)⟩ ∈ ∼ )
43		df-br 4654	. 2 ⊢ (𝐴 ∼ (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝐾⟩⟨𝐽, (1𝑜 ∖ 𝐾)⟩”⟩⟩) ↔ ⟨𝐴, (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝐾⟩⟨𝐽, (1𝑜 ∖ 𝐾)⟩”⟩⟩)⟩ ∈ ∼ )
44	42, 43	sylibr 224	1 ⊢ (((𝐴 ∈ 𝑊 ∧ 𝑁 ∈ (0...(#‘𝐴))) ∧ (𝐽 ∈ 𝐼 ∧ 𝐾 ∈ 2𝑜)) → 𝐴 ∼ (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝐾⟩⟨𝐽, (1𝑜 ∖ 𝐾)⟩”⟩⟩))

Syntax hints:   → wi 4   ∧ wa 384  ∀wal 1481   = wceq 1483   ∈ wcel 1990  {cab 2608  ∀wral 2912   ∖ cdif 3571  ⟨cop 4183  ⟨cotp 4185  ∩ cint 4475   class class class wbr 4653   I cid 5023   × cxp 5112  ‘cfv 5888  (class class class)co 6650  1_𝑜c1o 7553  2_𝑜c2o 7554   Er wer 7739  0cc0 9936  ...cfz 12326  #chash 13117  Word cword 13291   splice csplice 13296  ⟨“cs2 13586   ~_FG cefg 18119

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-ot 4186  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-concat 13301  df-s1 13302  df-substr 13303  df-splice 13304  df-s2 13593  df-efg 18122

This theorem is referenced by:  efgi0  18133  efgi1  18134