Theorem efgi 19616

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 6826	. . . . . . . . . . 11 ⊢ (𝑢 = 𝐴 → (♯‘𝑢) = (♯‘𝐴))
2	1	oveq2d 7369	. . . . . . . . . 10 ⊢ (𝑢 = 𝐴 → (0...(♯‘𝑢)) = (0...(♯‘𝐴)))
3		id 22	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝐴 → 𝑢 = 𝐴)
4		oveq1 7360	. . . . . . . . . . . 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 5108	. . . . . . . . . . 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 3193	. . . . . . . . . 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 3310	. . . . . . . . 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 3575	. . . . . . . 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 4836	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑁 → ⟨𝑖, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩ = ⟨𝑁, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩)
10		oteq2 4837	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑁 → ⟨𝑁, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩ = ⟨𝑁, 𝑁, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩)
11	9, 10	eqtrd 2764	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑁 → ⟨𝑖, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩ = ⟨𝑁, 𝑁, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩)
12	11	oveq2d 7369	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑁 → (𝐴 splice ⟨𝑖, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩) = (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩))
13	12	breq2d 5107	. . . . . . . . . 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 3193	. . . . . . . . 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 3575	. . . . . . . 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 4827	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝐽 → ⟨𝑎, 𝑏⟩ = ⟨𝐽, 𝑏⟩)
18		opeq1 4827	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝐽 → ⟨𝑎, (1o ∖ 𝑏)⟩ = ⟨𝐽, (1o ∖ 𝑏)⟩)
19	17, 18	s2eqd 14788	. . . . . . . . . . 11 ⊢ (𝑎 = 𝐽 → ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩ = ⟨“⟨𝐽, 𝑏⟩⟨𝐽, (1o ∖ 𝑏)⟩”⟩)
20	19	oteq3d 4841	. . . . . . . . . 10 ⊢ (𝑎 = 𝐽 → ⟨𝑁, 𝑁, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩ = ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝑏⟩⟨𝐽, (1o ∖ 𝑏)⟩”⟩⟩)
21	20	oveq2d 7369	. . . . . . . . 9 ⊢ (𝑎 = 𝐽 → (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩) = (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝑏⟩⟨𝐽, (1o ∖ 𝑏)⟩”⟩⟩))
22	21	breq2d 5107	. . . . . . . 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 4828	. . . . . . . . . . . . 13 ⊢ (𝑏 = 𝐾 → ⟨𝐽, 𝑏⟩ = ⟨𝐽, 𝐾⟩)
24		difeq2 4073	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 𝐾 → (1o ∖ 𝑏) = (1o ∖ 𝐾))
25	24	opeq2d 4834	. . . . . . . . . . . . 13 ⊢ (𝑏 = 𝐾 → ⟨𝐽, (1o ∖ 𝑏)⟩ = ⟨𝐽, (1o ∖ 𝐾)⟩)
26	23, 25	s2eqd 14788	. . . . . . . . . . . 12 ⊢ (𝑏 = 𝐾 → ⟨“⟨𝐽, 𝑏⟩⟨𝐽, (1o ∖ 𝑏)⟩”⟩ = ⟨“⟨𝐽, 𝐾⟩⟨𝐽, (1o ∖ 𝐾)⟩”⟩)
27	26	oteq3d 4841	. . . . . . . . . . 11 ⊢ (𝑏 = 𝐾 → ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝑏⟩⟨𝐽, (1o ∖ 𝑏)⟩”⟩⟩ = ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝐾⟩⟨𝐽, (1o ∖ 𝐾)⟩”⟩⟩)
28	27	oveq2d 7369	. . . . . . . . . 10 ⊢ (𝑏 = 𝐾 → (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝑏⟩⟨𝐽, (1o ∖ 𝑏)⟩”⟩⟩) = (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝐾⟩⟨𝐽, (1o ∖ 𝐾)⟩”⟩⟩))
29	28	breq2d 5107	. . . . . . . . 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 5096	. . . . . . . . 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 3590	. . . . . . 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 1927	. . . 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 5411	. . . . 5 ⊢ ⟨𝐴, (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝐾⟩⟨𝐽, (1o ∖ 𝐾)⟩”⟩⟩)⟩ ∈ V
37	36	elintab 4911	. . . 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 19614	. . 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 2839	. 2 ⊢ (((𝐴 ∈ 𝑊 ∧ 𝑁 ∈ (0...(♯‘𝐴))) ∧ (𝐽 ∈ 𝐼 ∧ 𝐾 ∈ 2o)) → ⟨𝐴, (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝐾⟩⟨𝐽, (1o ∖ 𝐾)⟩”⟩⟩)⟩ ∈ ∼ )
43		df-br 5096	. 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 1538   = wceq 1540   ∈ wcel 2109  {cab 2707  ∀wral 3044   ∖ cdif 3902  ⟨cop 4585  ⟨cotp 4587  ∩ cint 4899   class class class wbr 5095   I cid 5517   × cxp 5621  ‘cfv 6486  (class class class)co 7353  1_oc1o 8388  2_oc2o 8389   Er wer 8629  0cc0 11028  ...cfz 13428  ♯chash 14255  Word cword 14438   splice csplice 14673  ⟨“cs2 14766   ~_FG cefg 19603

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675  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 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-ot 4588  df-uni 4862  df-int 4900  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  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 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7310  df-ov 7356  df-oprab 7357  df-mpo 7358  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-2o 8396  df-er 8632  df-map 8762  df-en 8880  df-dom 8881  df-sdom 8882  df-fin 8883  df-card 9854  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11367  df-neg 11368  df-nn 12147  df-n0 12403  df-z 12490  df-uz 12754  df-fz 13429  df-fzo 13576  df-hash 14256  df-word 14439  df-concat 14496  df-s1 14521  df-substr 14566  df-pfx 14596  df-splice 14674  df-s2 14773  df-efg 19606

This theorem is referenced by:  efgi0  19617  efgi1  19618