Theorem efgi 19632

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 6822	. . . . . . . . . . 11 ⊢ (𝑢 = 𝐴 → (♯‘𝑢) = (♯‘𝐴))
2	1	oveq2d 7362	. . . . . . . . . 10 ⊢ (𝑢 = 𝐴 → (0...(♯‘𝑢)) = (0...(♯‘𝐴)))
3		id 22	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝐴 → 𝑢 = 𝐴)
4		oveq1 7353	. . . . . . . . . . . 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 5104	. . . . . . . . . . 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 3196	. . . . . . . . . 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 3312	. . . . . . . . 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 4834	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑁 → ⟨𝑖, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩ = ⟨𝑁, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩)
10		oteq2 4835	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑁 → ⟨𝑁, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩ = ⟨𝑁, 𝑁, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩)
11	9, 10	eqtrd 2766	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑁 → ⟨𝑖, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩ = ⟨𝑁, 𝑁, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩)
12	11	oveq2d 7362	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑁 → (𝐴 splice ⟨𝑖, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩) = (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩))
13	12	breq2d 5103	. . . . . . . . . 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 3196	. . . . . . . . 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 507	. . . . . . 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 4825	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝐽 → ⟨𝑎, 𝑏⟩ = ⟨𝐽, 𝑏⟩)
18		opeq1 4825	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝐽 → ⟨𝑎, (1o ∖ 𝑏)⟩ = ⟨𝐽, (1o ∖ 𝑏)⟩)
19	17, 18	s2eqd 14770	. . . . . . . . . . 11 ⊢ (𝑎 = 𝐽 → ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩ = ⟨“⟨𝐽, 𝑏⟩⟨𝐽, (1o ∖ 𝑏)⟩”⟩)
20	19	oteq3d 4839	. . . . . . . . . 10 ⊢ (𝑎 = 𝐽 → ⟨𝑁, 𝑁, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩ = ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝑏⟩⟨𝐽, (1o ∖ 𝑏)⟩”⟩⟩)
21	20	oveq2d 7362	. . . . . . . . 9 ⊢ (𝑎 = 𝐽 → (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩) = (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝑏⟩⟨𝐽, (1o ∖ 𝑏)⟩”⟩⟩))
22	21	breq2d 5103	. . . . . . . 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 4826	. . . . . . . . . . . . 13 ⊢ (𝑏 = 𝐾 → ⟨𝐽, 𝑏⟩ = ⟨𝐽, 𝐾⟩)
24		difeq2 4070	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 𝐾 → (1o ∖ 𝑏) = (1o ∖ 𝐾))
25	24	opeq2d 4832	. . . . . . . . . . . . 13 ⊢ (𝑏 = 𝐾 → ⟨𝐽, (1o ∖ 𝑏)⟩ = ⟨𝐽, (1o ∖ 𝐾)⟩)
26	23, 25	s2eqd 14770	. . . . . . . . . . . 12 ⊢ (𝑏 = 𝐾 → ⟨“⟨𝐽, 𝑏⟩⟨𝐽, (1o ∖ 𝑏)⟩”⟩ = ⟨“⟨𝐽, 𝐾⟩⟨𝐽, (1o ∖ 𝐾)⟩”⟩)
27	26	oteq3d 4839	. . . . . . . . . . 11 ⊢ (𝑏 = 𝐾 → ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝑏⟩⟨𝐽, (1o ∖ 𝑏)⟩”⟩⟩ = ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝐾⟩⟨𝐽, (1o ∖ 𝐾)⟩”⟩⟩)
28	27	oveq2d 7362	. . . . . . . . . 10 ⊢ (𝑏 = 𝐾 → (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝑏⟩⟨𝐽, (1o ∖ 𝑏)⟩”⟩⟩) = (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝐾⟩⟨𝐽, (1o ∖ 𝐾)⟩”⟩⟩))
29	28	breq2d 5103	. . . . . . . . 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 5092	. . . . . . . . 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 287	. . . . . . . 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 3588	. . . . . . 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 507	. . . . . 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 490	. . . . 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 1928	. . . 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 5404	. . . . 5 ⊢ ⟨𝐴, (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝐾⟩⟨𝐽, (1o ∖ 𝐾)⟩”⟩⟩)⟩ ∈ V
37	36	elintab 4909	. . . 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 234	. . 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 19630	. . 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 2842	. 2 ⊢ (((𝐴 ∈ 𝑊 ∧ 𝑁 ∈ (0...(♯‘𝐴))) ∧ (𝐽 ∈ 𝐼 ∧ 𝐾 ∈ 2o)) → ⟨𝐴, (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝐾⟩⟨𝐽, (1o ∖ 𝐾)⟩”⟩⟩)⟩ ∈ ∼ )
43		df-br 5092	. 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 234	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 395  ∀wal 1539   = wceq 1541   ∈ wcel 2111  {cab 2709  ∀wral 3047   ∖ cdif 3899  ⟨cop 4582  ⟨cotp 4584  ∩ cint 4897   class class class wbr 5091   I cid 5510   × cxp 5614  ‘cfv 6481  (class class class)co 7346  1_oc1o 8378  2_oc2o 8379   Er wer 8619  0cc0 11006  ...cfz 13407  ♯chash 14237  Word cword 14420   splice csplice 14656  ⟨“cs2 14748   ~_FG cefg 19619

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668  ax-cnex 11062  ax-resscn 11063  ax-1cn 11064  ax-icn 11065  ax-addcl 11066  ax-addrcl 11067  ax-mulcl 11068  ax-mulrcl 11069  ax-mulcom 11070  ax-addass 11071  ax-mulass 11072  ax-distr 11073  ax-i2m1 11074  ax-1ne0 11075  ax-1rid 11076  ax-rnegex 11077  ax-rrecex 11078  ax-cnre 11079  ax-pre-lttri 11080  ax-pre-lttrn 11081  ax-pre-ltadd 11082  ax-pre-mulgt0 11083

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-ot 4585  df-uni 4860  df-int 4898  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-tr 5199  df-id 5511  df-eprel 5516  df-po 5524  df-so 5525  df-fr 5569  df-we 5571  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-1st 7921  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-1o 8385  df-2o 8386  df-er 8622  df-map 8752  df-en 8870  df-dom 8871  df-sdom 8872  df-fin 8873  df-card 9832  df-pnf 11148  df-mnf 11149  df-xr 11150  df-ltxr 11151  df-le 11152  df-sub 11346  df-neg 11347  df-nn 12126  df-n0 12382  df-z 12469  df-uz 12733  df-fz 13408  df-fzo 13555  df-hash 14238  df-word 14421  df-concat 14478  df-s1 14504  df-substr 14549  df-pfx 14579  df-splice 14657  df-s2 14755  df-efg 19622

This theorem is referenced by:  efgi0  19633  efgi1  19634