Theorem efgi 19926

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 6883	. . . . . . . . . . 11 ⊢ (𝑢 = 𝐴 → (♯‘𝑢) = (♯‘𝐴))
2	1	oveq2d 7434	. . . . . . . . . 10 ⊢ (𝑢 = 𝐴 → (0...(♯‘𝑢)) = (0...(♯‘𝐴)))
3		id 23	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝐴 → 𝑢 = 𝐴)
4		oveq1 7425	. . . . . . . . . . . 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 5116	. . . . . . . . . . 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 3227	. . . . . . . . . 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 3335	. . . . . . . . 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 3573	. . . . . . . 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 4842	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑁 → ⟨𝑖, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩ = ⟨𝑁, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩)
10		oteq2 4843	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑁 → ⟨𝑁, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩ = ⟨𝑁, 𝑁, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩)
11	9, 10	eqtrd 2796	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑁 → ⟨𝑖, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩ = ⟨𝑁, 𝑁, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩)
12	11	oveq2d 7434	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑁 → (𝐴 splice ⟨𝑖, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩) = (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩))
13	12	breq2d 5115	. . . . . . . . . 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 3227	. . . . . . . . 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 3573	. . . . . . . 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 517	. . . . . . 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 4833	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝐽 → ⟨𝑎, 𝑏⟩ = ⟨𝐽, 𝑏⟩)
18		opeq1 4833	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝐽 → ⟨𝑎, (1o ∖ 𝑏)⟩ = ⟨𝐽, (1o ∖ 𝑏)⟩)
19	17, 18	s2eqd 15007	. . . . . . . . . . 11 ⊢ (𝑎 = 𝐽 → ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩ = ⟨“⟨𝐽, 𝑏⟩⟨𝐽, (1o ∖ 𝑏)⟩”⟩)
20	19	oteq3d 4847	. . . . . . . . . 10 ⊢ (𝑎 = 𝐽 → ⟨𝑁, 𝑁, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩ = ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝑏⟩⟨𝐽, (1o ∖ 𝑏)⟩”⟩⟩)
21	20	oveq2d 7434	. . . . . . . . 9 ⊢ (𝑎 = 𝐽 → (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩) = (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝑏⟩⟨𝐽, (1o ∖ 𝑏)⟩”⟩⟩))
22	21	breq2d 5115	. . . . . . . 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 4834	. . . . . . . . . . . . 13 ⊢ (𝑏 = 𝐾 → ⟨𝐽, 𝑏⟩ = ⟨𝐽, 𝐾⟩)
24		difeq2 4068	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 𝐾 → (1o ∖ 𝑏) = (1o ∖ 𝐾))
25	24	opeq2d 4840	. . . . . . . . . . . . 13 ⊢ (𝑏 = 𝐾 → ⟨𝐽, (1o ∖ 𝑏)⟩ = ⟨𝐽, (1o ∖ 𝐾)⟩)
26	23, 25	s2eqd 15007	. . . . . . . . . . . 12 ⊢ (𝑏 = 𝐾 → ⟨“⟨𝐽, 𝑏⟩⟨𝐽, (1o ∖ 𝑏)⟩”⟩ = ⟨“⟨𝐽, 𝐾⟩⟨𝐽, (1o ∖ 𝐾)⟩”⟩)
27	26	oteq3d 4847	. . . . . . . . . . 11 ⊢ (𝑏 = 𝐾 → ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝑏⟩⟨𝐽, (1o ∖ 𝑏)⟩”⟩⟩ = ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝐾⟩⟨𝐽, (1o ∖ 𝐾)⟩”⟩⟩)
28	27	oveq2d 7434	. . . . . . . . . 10 ⊢ (𝑏 = 𝐾 → (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝑏⟩⟨𝐽, (1o ∖ 𝑏)⟩”⟩⟩) = (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝐾⟩⟨𝐽, (1o ∖ 𝐾)⟩”⟩⟩))
29	28	breq2d 5115	. . . . . . . . 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 5104	. . . . . . . . 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	bitrdi 290	. . . . . . . 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 3587	. . . . . . 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 517	. . . . . 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 496	. . . . 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 1960	. . . 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 5432	. . . . 5 ⊢ ⟨𝐴, (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝐾⟩⟨𝐽, (1o ∖ 𝐾)⟩”⟩⟩)⟩ ∈ V
37	36	elintab 4919	. . . 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 237	. . 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 19924	. . 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	eleqtrrdi 2872	. 2 ⊢ (((𝐴 ∈ 𝑊 ∧ 𝑁 ∈ (0...(♯‘𝐴))) ∧ (𝐽 ∈ 𝐼 ∧ 𝐾 ∈ 2o)) → ⟨𝐴, (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝐾⟩⟨𝐽, (1o ∖ 𝐾)⟩”⟩⟩)⟩ ∈ ∼ )
43		df-br 5104	. 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 237	1 ⊢ (((𝐴 ∈ 𝑊 ∧ 𝑁 ∈ (0...(♯‘𝐴))) ∧ (𝐽 ∈ 𝐼 ∧ 𝐾 ∈ 2o)) → 𝐴 ∼ (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝐾⟩⟨𝐽, (1o ∖ 𝐾)⟩”⟩⟩))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∀wal 1568   = wceq 1570   ∈ wcel 2145  {cab 2739  ∀wral 3077   ∖ cdif 3896  ⟨cop 4590  ⟨cotp 4592  ∩ cint 4907   class class class wbr 5103   I cid 5545   × cxp 5649  ‘cfv 6537  (class class class)co 7418  1_oc1o 8462  2_oc2o 8463   Er wer 8707  0cc0 11193  ...cfz 13632  ♯chash 14467  Word cword 14651   splice csplice 14891  ⟨“cs2 14985   ~_FG cefg 19913

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749  ax-cnex 11249  ax-resscn 11250  ax-1cn 11251  ax-icn 11252  ax-addcl 11253  ax-addrcl 11254  ax-mulcl 11255  ax-mulrcl 11256  ax-mulcom 11257  ax-addass 11258  ax-mulass 11259  ax-distr 11260  ax-i2m1 11261  ax-1ne0 11262  ax-1rid 11263  ax-rnegex 11264  ax-rrecex 11265  ax-cnre 11266  ax-pre-lttri 11267  ax-pre-lttrn 11268  ax-pre-ltadd 11269  ax-pre-mulgt0 11270

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-ot 4593  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7375  df-ov 7421  df-oprab 7422  df-mpo 7423  df-om 7876  df-1st 7999  df-2nd 8000  df-frecs 8292  df-wrecs 8323  df-recs 8372  df-rdg 8411  df-1o 8469  df-2o 8470  df-er 8710  df-map 8842  df-en 8967  df-dom 8968  df-sdom 8969  df-fin 8970  df-card 10013  df-pnf 11338  df-mnf 11339  df-xr 11340  df-ltxr 11341  df-le 11342  df-sub 11536  df-neg 11537  df-nn 12329  df-n0 12600  df-z 12687  df-uz 12959  df-fz 13633  df-fzo 13782  df-hash 14468  df-word 14652  df-concat 14709  df-s1 14736  df-substr 14782  df-pfx 14814  df-splice 14892  df-s2 14992  df-efg 19916

This theorem is used by:  efgi0  19927  efgi1  19928