Theorem efgi 19325

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 6774	. . . . . . . . . . 11 ⊢ (𝑢 = 𝐴 → (♯‘𝑢) = (♯‘𝐴))
2	1	oveq2d 7291	. . . . . . . . . 10 ⊢ (𝑢 = 𝐴 → (0...(♯‘𝑢)) = (0...(♯‘𝐴)))
3		id 22	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝐴 → 𝑢 = 𝐴)
4		oveq1 7282	. . . . . . . . . . . 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 5087	. . . . . . . . . . 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 3129	. . . . . . . . . 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 3336	. . . . . . . . 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 3557	. . . . . . . 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 4813	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑁 → ⟨𝑖, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩ = ⟨𝑁, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩)
10		oteq2 4814	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑁 → ⟨𝑁, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩ = ⟨𝑁, 𝑁, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩)
11	9, 10	eqtrd 2778	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑁 → ⟨𝑖, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩ = ⟨𝑁, 𝑁, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩)
12	11	oveq2d 7291	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑁 → (𝐴 splice ⟨𝑖, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩) = (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩))
13	12	breq2d 5086	. . . . . . . . . 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 3129	. . . . . . . . 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 3557	. . . . . . . 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 508	. . . . . . 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 4804	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝐽 → ⟨𝑎, 𝑏⟩ = ⟨𝐽, 𝑏⟩)
18		opeq1 4804	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝐽 → ⟨𝑎, (1o ∖ 𝑏)⟩ = ⟨𝐽, (1o ∖ 𝑏)⟩)
19	17, 18	s2eqd 14576	. . . . . . . . . . 11 ⊢ (𝑎 = 𝐽 → ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩ = ⟨“⟨𝐽, 𝑏⟩⟨𝐽, (1o ∖ 𝑏)⟩”⟩)
20	19	oteq3d 4818	. . . . . . . . . 10 ⊢ (𝑎 = 𝐽 → ⟨𝑁, 𝑁, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩ = ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝑏⟩⟨𝐽, (1o ∖ 𝑏)⟩”⟩⟩)
21	20	oveq2d 7291	. . . . . . . . 9 ⊢ (𝑎 = 𝐽 → (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩) = (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝑏⟩⟨𝐽, (1o ∖ 𝑏)⟩”⟩⟩))
22	21	breq2d 5086	. . . . . . . 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 4805	. . . . . . . . . . . . 13 ⊢ (𝑏 = 𝐾 → ⟨𝐽, 𝑏⟩ = ⟨𝐽, 𝐾⟩)
24		difeq2 4051	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 𝐾 → (1o ∖ 𝑏) = (1o ∖ 𝐾))
25	24	opeq2d 4811	. . . . . . . . . . . . 13 ⊢ (𝑏 = 𝐾 → ⟨𝐽, (1o ∖ 𝑏)⟩ = ⟨𝐽, (1o ∖ 𝐾)⟩)
26	23, 25	s2eqd 14576	. . . . . . . . . . . 12 ⊢ (𝑏 = 𝐾 → ⟨“⟨𝐽, 𝑏⟩⟨𝐽, (1o ∖ 𝑏)⟩”⟩ = ⟨“⟨𝐽, 𝐾⟩⟨𝐽, (1o ∖ 𝐾)⟩”⟩)
27	26	oteq3d 4818	. . . . . . . . . . 11 ⊢ (𝑏 = 𝐾 → ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝑏⟩⟨𝐽, (1o ∖ 𝑏)⟩”⟩⟩ = ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝐾⟩⟨𝐽, (1o ∖ 𝐾)⟩”⟩⟩)
28	27	oveq2d 7291	. . . . . . . . . 10 ⊢ (𝑏 = 𝐾 → (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝑏⟩⟨𝐽, (1o ∖ 𝑏)⟩”⟩⟩) = (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝐾⟩⟨𝐽, (1o ∖ 𝐾)⟩”⟩⟩))
29	28	breq2d 5086	. . . . . . . . 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 5075	. . . . . . . . 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 3570	. . . . . . 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 508	. . . . . 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 491	. . . . 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 1930	. . . 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 5379	. . . . 5 ⊢ ⟨𝐴, (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝐾⟩⟨𝐽, (1o ∖ 𝐾)⟩”⟩⟩)⟩ ∈ V
37	36	elintab 4890	. . . 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 233	. . 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 19323	. . 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 2850	. 2 ⊢ (((𝐴 ∈ 𝑊 ∧ 𝑁 ∈ (0...(♯‘𝐴))) ∧ (𝐽 ∈ 𝐼 ∧ 𝐾 ∈ 2o)) → ⟨𝐴, (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝐾⟩⟨𝐽, (1o ∖ 𝐾)⟩”⟩⟩)⟩ ∈ ∼ )
43		df-br 5075	. 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 233	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 396  ∀wal 1537   = wceq 1539   ∈ wcel 2106  {cab 2715  ∀wral 3064   ∖ cdif 3884  ⟨cop 4567  ⟨cotp 4569  ∩ cint 4879   class class class wbr 5074   I cid 5488   × cxp 5587  ‘cfv 6433  (class class class)co 7275  1_oc1o 8290  2_oc2o 8291   Er wer 8495  0cc0 10871  ...cfz 13239  ♯chash 14044  Word cword 14217   splice csplice 14462  ⟨“cs2 14554   ~_FG cefg 19312

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-ot 4570  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-2o 8298  df-er 8498  df-map 8617  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-card 9697  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-n0 12234  df-z 12320  df-uz 12583  df-fz 13240  df-fzo 13383  df-hash 14045  df-word 14218  df-concat 14274  df-s1 14301  df-substr 14354  df-pfx 14384  df-splice 14463  df-s2 14561  df-efg 19315

This theorem is referenced by:  efgi0  19326  efgi1  19327