Theorem bezoutlemex 12562

Description: Lemma for Bézout's identity. Existence of a number which we will later show to be the greater common divisor and its decomposition into cofactors. (Contributed by Mario Carneiro and Jim Kingdon, 3-Jan-2022.)

Assertion

Ref	Expression
bezoutlemex	⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → ∃𝑑 ∈ ℕ0 (∀𝑧 ∈ ℕ0 (𝑧 ∥ 𝑑 → (𝑧 ∥ 𝐴 ∧ 𝑧 ∥ 𝐵)) ∧ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝑑 = ((𝐴 · 𝑥) + (𝐵 · 𝑦))))

Distinct variable groups:   𝐴,𝑑,𝑥,𝑦   𝑧,𝐴,𝑑   𝐵,𝑑,𝑥,𝑦   𝑧,𝐵

Proof of Theorem bezoutlemex

Dummy variables 𝑎 𝑏 𝑠 𝑡 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 6021	. . . . . . . 8 ⊢ (𝑦 = 𝑡 → (𝐵 · 𝑦) = (𝐵 · 𝑡))
2	1	oveq2d 6029	. . . . . . 7 ⊢ (𝑦 = 𝑡 → ((𝐴 · 𝑥) + (𝐵 · 𝑦)) = ((𝐴 · 𝑥) + (𝐵 · 𝑡)))
3	2	eqeq2d 2241	. . . . . 6 ⊢ (𝑦 = 𝑡 → (𝑑 = ((𝐴 · 𝑥) + (𝐵 · 𝑦)) ↔ 𝑑 = ((𝐴 · 𝑥) + (𝐵 · 𝑡))))
4	3	cbvrexv 2766	. . . . 5 ⊢ (∃𝑦 ∈ ℤ 𝑑 = ((𝐴 · 𝑥) + (𝐵 · 𝑦)) ↔ ∃𝑡 ∈ ℤ 𝑑 = ((𝐴 · 𝑥) + (𝐵 · 𝑡)))
5	4	rexbii 2537	. . . 4 ⊢ (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝑑 = ((𝐴 · 𝑥) + (𝐵 · 𝑦)) ↔ ∃𝑥 ∈ ℤ ∃𝑡 ∈ ℤ 𝑑 = ((𝐴 · 𝑥) + (𝐵 · 𝑡)))
6		oveq2 6021	. . . . . . . 8 ⊢ (𝑥 = 𝑠 → (𝐴 · 𝑥) = (𝐴 · 𝑠))
7	6	oveq1d 6028	. . . . . . 7 ⊢ (𝑥 = 𝑠 → ((𝐴 · 𝑥) + (𝐵 · 𝑡)) = ((𝐴 · 𝑠) + (𝐵 · 𝑡)))
8	7	eqeq2d 2241	. . . . . 6 ⊢ (𝑥 = 𝑠 → (𝑑 = ((𝐴 · 𝑥) + (𝐵 · 𝑡)) ↔ 𝑑 = ((𝐴 · 𝑠) + (𝐵 · 𝑡))))
9	8	rexbidv 2531	. . . . 5 ⊢ (𝑥 = 𝑠 → (∃𝑡 ∈ ℤ 𝑑 = ((𝐴 · 𝑥) + (𝐵 · 𝑡)) ↔ ∃𝑡 ∈ ℤ 𝑑 = ((𝐴 · 𝑠) + (𝐵 · 𝑡))))
10	9	cbvrexv 2766	. . . 4 ⊢ (∃𝑥 ∈ ℤ ∃𝑡 ∈ ℤ 𝑑 = ((𝐴 · 𝑥) + (𝐵 · 𝑡)) ↔ ∃𝑠 ∈ ℤ ∃𝑡 ∈ ℤ 𝑑 = ((𝐴 · 𝑠) + (𝐵 · 𝑡)))
11	5, 10	bitri 184	. . 3 ⊢ (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝑑 = ((𝐴 · 𝑥) + (𝐵 · 𝑦)) ↔ ∃𝑠 ∈ ℤ ∃𝑡 ∈ ℤ 𝑑 = ((𝐴 · 𝑠) + (𝐵 · 𝑡)))
12		simpl 109	. . 3 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → 𝐴 ∈ ℕ0)
13		simpr 110	. . 3 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → 𝐵 ∈ ℕ0)
14	11, 12, 13	bezoutlemb 12561	. 2 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → [𝐵 / 𝑑]∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝑑 = ((𝐴 · 𝑥) + (𝐵 · 𝑦)))
15		dfsbcq2 3032	. . . 4 ⊢ (𝑏 = 𝐵 → ([𝑏 / 𝑑]∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝑑 = ((𝐴 · 𝑥) + (𝐵 · 𝑦)) ↔ [𝐵 / 𝑑]∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝑑 = ((𝐴 · 𝑥) + (𝐵 · 𝑦))))
16		breq2 4090	. . . . . . . . 9 ⊢ (𝑏 = 𝐵 → (𝑧 ∥ 𝑏 ↔ 𝑧 ∥ 𝐵))
17	16	anbi2d 464	. . . . . . . 8 ⊢ (𝑏 = 𝐵 → ((𝑧 ∥ 𝐴 ∧ 𝑧 ∥ 𝑏) ↔ (𝑧 ∥ 𝐴 ∧ 𝑧 ∥ 𝐵)))
18	17	imbi2d 230	. . . . . . 7 ⊢ (𝑏 = 𝐵 → ((𝑧 ∥ 𝑑 → (𝑧 ∥ 𝐴 ∧ 𝑧 ∥ 𝑏)) ↔ (𝑧 ∥ 𝑑 → (𝑧 ∥ 𝐴 ∧ 𝑧 ∥ 𝐵))))
19	18	ralbidv 2530	. . . . . 6 ⊢ (𝑏 = 𝐵 → (∀𝑧 ∈ ℕ0 (𝑧 ∥ 𝑑 → (𝑧 ∥ 𝐴 ∧ 𝑧 ∥ 𝑏)) ↔ ∀𝑧 ∈ ℕ0 (𝑧 ∥ 𝑑 → (𝑧 ∥ 𝐴 ∧ 𝑧 ∥ 𝐵))))
20	19	anbi1d 465	. . . . 5 ⊢ (𝑏 = 𝐵 → ((∀𝑧 ∈ ℕ0 (𝑧 ∥ 𝑑 → (𝑧 ∥ 𝐴 ∧ 𝑧 ∥ 𝑏)) ∧ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝑑 = ((𝐴 · 𝑥) + (𝐵 · 𝑦))) ↔ (∀𝑧 ∈ ℕ0 (𝑧 ∥ 𝑑 → (𝑧 ∥ 𝐴 ∧ 𝑧 ∥ 𝐵)) ∧ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝑑 = ((𝐴 · 𝑥) + (𝐵 · 𝑦)))))
21	20	rexbidv 2531	. . . 4 ⊢ (𝑏 = 𝐵 → (∃𝑑 ∈ ℕ0 (∀𝑧 ∈ ℕ0 (𝑧 ∥ 𝑑 → (𝑧 ∥ 𝐴 ∧ 𝑧 ∥ 𝑏)) ∧ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝑑 = ((𝐴 · 𝑥) + (𝐵 · 𝑦))) ↔ ∃𝑑 ∈ ℕ0 (∀𝑧 ∈ ℕ0 (𝑧 ∥ 𝑑 → (𝑧 ∥ 𝐴 ∧ 𝑧 ∥ 𝐵)) ∧ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝑑 = ((𝐴 · 𝑥) + (𝐵 · 𝑦)))))
22	15, 21	imbi12d 234	. . 3 ⊢ (𝑏 = 𝐵 → (([𝑏 / 𝑑]∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝑑 = ((𝐴 · 𝑥) + (𝐵 · 𝑦)) → ∃𝑑 ∈ ℕ0 (∀𝑧 ∈ ℕ0 (𝑧 ∥ 𝑑 → (𝑧 ∥ 𝐴 ∧ 𝑧 ∥ 𝑏)) ∧ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝑑 = ((𝐴 · 𝑥) + (𝐵 · 𝑦)))) ↔ ([𝐵 / 𝑑]∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝑑 = ((𝐴 · 𝑥) + (𝐵 · 𝑦)) → ∃𝑑 ∈ ℕ0 (∀𝑧 ∈ ℕ0 (𝑧 ∥ 𝑑 → (𝑧 ∥ 𝐴 ∧ 𝑧 ∥ 𝐵)) ∧ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝑑 = ((𝐴 · 𝑥) + (𝐵 · 𝑦))))))
23	11, 12, 13	bezoutlema 12560	. . . 4 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → [𝐴 / 𝑑]∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝑑 = ((𝐴 · 𝑥) + (𝐵 · 𝑦)))
24		dfsbcq2 3032	. . . . . 6 ⊢ (𝑎 = 𝐴 → ([𝑎 / 𝑑]∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝑑 = ((𝐴 · 𝑥) + (𝐵 · 𝑦)) ↔ [𝐴 / 𝑑]∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝑑 = ((𝐴 · 𝑥) + (𝐵 · 𝑦))))
25		breq2 4090	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝐴 → (𝑧 ∥ 𝑎 ↔ 𝑧 ∥ 𝐴))
26	25	anbi1d 465	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝐴 → ((𝑧 ∥ 𝑎 ∧ 𝑧 ∥ 𝑏) ↔ (𝑧 ∥ 𝐴 ∧ 𝑧 ∥ 𝑏)))
27	26	imbi2d 230	. . . . . . . . . . 11 ⊢ (𝑎 = 𝐴 → ((𝑧 ∥ 𝑑 → (𝑧 ∥ 𝑎 ∧ 𝑧 ∥ 𝑏)) ↔ (𝑧 ∥ 𝑑 → (𝑧 ∥ 𝐴 ∧ 𝑧 ∥ 𝑏))))
28	27	ralbidv 2530	. . . . . . . . . 10 ⊢ (𝑎 = 𝐴 → (∀𝑧 ∈ ℕ0 (𝑧 ∥ 𝑑 → (𝑧 ∥ 𝑎 ∧ 𝑧 ∥ 𝑏)) ↔ ∀𝑧 ∈ ℕ0 (𝑧 ∥ 𝑑 → (𝑧 ∥ 𝐴 ∧ 𝑧 ∥ 𝑏))))
29	28	anbi1d 465	. . . . . . . . 9 ⊢ (𝑎 = 𝐴 → ((∀𝑧 ∈ ℕ0 (𝑧 ∥ 𝑑 → (𝑧 ∥ 𝑎 ∧ 𝑧 ∥ 𝑏)) ∧ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝑑 = ((𝐴 · 𝑥) + (𝐵 · 𝑦))) ↔ (∀𝑧 ∈ ℕ0 (𝑧 ∥ 𝑑 → (𝑧 ∥ 𝐴 ∧ 𝑧 ∥ 𝑏)) ∧ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝑑 = ((𝐴 · 𝑥) + (𝐵 · 𝑦)))))
30	29	rexbidv 2531	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → (∃𝑑 ∈ ℕ0 (∀𝑧 ∈ ℕ0 (𝑧 ∥ 𝑑 → (𝑧 ∥ 𝑎 ∧ 𝑧 ∥ 𝑏)) ∧ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝑑 = ((𝐴 · 𝑥) + (𝐵 · 𝑦))) ↔ ∃𝑑 ∈ ℕ0 (∀𝑧 ∈ ℕ0 (𝑧 ∥ 𝑑 → (𝑧 ∥ 𝐴 ∧ 𝑧 ∥ 𝑏)) ∧ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝑑 = ((𝐴 · 𝑥) + (𝐵 · 𝑦)))))
31	30	imbi2d 230	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (([𝑏 / 𝑑]∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝑑 = ((𝐴 · 𝑥) + (𝐵 · 𝑦)) → ∃𝑑 ∈ ℕ0 (∀𝑧 ∈ ℕ0 (𝑧 ∥ 𝑑 → (𝑧 ∥ 𝑎 ∧ 𝑧 ∥ 𝑏)) ∧ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝑑 = ((𝐴 · 𝑥) + (𝐵 · 𝑦)))) ↔ ([𝑏 / 𝑑]∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝑑 = ((𝐴 · 𝑥) + (𝐵 · 𝑦)) → ∃𝑑 ∈ ℕ0 (∀𝑧 ∈ ℕ0 (𝑧 ∥ 𝑑 → (𝑧 ∥ 𝐴 ∧ 𝑧 ∥ 𝑏)) ∧ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝑑 = ((𝐴 · 𝑥) + (𝐵 · 𝑦))))))
32	31	ralbidv 2530	. . . . . 6 ⊢ (𝑎 = 𝐴 → (∀𝑏 ∈ ℕ0 ([𝑏 / 𝑑]∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝑑 = ((𝐴 · 𝑥) + (𝐵 · 𝑦)) → ∃𝑑 ∈ ℕ0 (∀𝑧 ∈ ℕ0 (𝑧 ∥ 𝑑 → (𝑧 ∥ 𝑎 ∧ 𝑧 ∥ 𝑏)) ∧ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝑑 = ((𝐴 · 𝑥) + (𝐵 · 𝑦)))) ↔ ∀𝑏 ∈ ℕ0 ([𝑏 / 𝑑]∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝑑 = ((𝐴 · 𝑥) + (𝐵 · 𝑦)) → ∃𝑑 ∈ ℕ0 (∀𝑧 ∈ ℕ0 (𝑧 ∥ 𝑑 → (𝑧 ∥ 𝐴 ∧ 𝑧 ∥ 𝑏)) ∧ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝑑 = ((𝐴 · 𝑥) + (𝐵 · 𝑦))))))
33	24, 32	imbi12d 234	. . . . 5 ⊢ (𝑎 = 𝐴 → (([𝑎 / 𝑑]∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝑑 = ((𝐴 · 𝑥) + (𝐵 · 𝑦)) → ∀𝑏 ∈ ℕ0 ([𝑏 / 𝑑]∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝑑 = ((𝐴 · 𝑥) + (𝐵 · 𝑦)) → ∃𝑑 ∈ ℕ0 (∀𝑧 ∈ ℕ0 (𝑧 ∥ 𝑑 → (𝑧 ∥ 𝑎 ∧ 𝑧 ∥ 𝑏)) ∧ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝑑 = ((𝐴 · 𝑥) + (𝐵 · 𝑦))))) ↔ ([𝐴 / 𝑑]∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝑑 = ((𝐴 · 𝑥) + (𝐵 · 𝑦)) → ∀𝑏 ∈ ℕ0 ([𝑏 / 𝑑]∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝑑 = ((𝐴 · 𝑥) + (𝐵 · 𝑦)) → ∃𝑑 ∈ ℕ0 (∀𝑧 ∈ ℕ0 (𝑧 ∥ 𝑑 → (𝑧 ∥ 𝐴 ∧ 𝑧 ∥ 𝑏)) ∧ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝑑 = ((𝐴 · 𝑥) + (𝐵 · 𝑦)))))))
34		breq1 4089	. . . . . . . 8 ⊢ (𝑧 = 𝑤 → (𝑧 ∥ 𝑑 ↔ 𝑤 ∥ 𝑑))
35		breq1 4089	. . . . . . . . 9 ⊢ (𝑧 = 𝑤 → (𝑧 ∥ 𝑎 ↔ 𝑤 ∥ 𝑎))
36		breq1 4089	. . . . . . . . 9 ⊢ (𝑧 = 𝑤 → (𝑧 ∥ 𝑏 ↔ 𝑤 ∥ 𝑏))
37	35, 36	anbi12d 473	. . . . . . . 8 ⊢ (𝑧 = 𝑤 → ((𝑧 ∥ 𝑎 ∧ 𝑧 ∥ 𝑏) ↔ (𝑤 ∥ 𝑎 ∧ 𝑤 ∥ 𝑏)))
38	34, 37	imbi12d 234	. . . . . . 7 ⊢ (𝑧 = 𝑤 → ((𝑧 ∥ 𝑑 → (𝑧 ∥ 𝑎 ∧ 𝑧 ∥ 𝑏)) ↔ (𝑤 ∥ 𝑑 → (𝑤 ∥ 𝑎 ∧ 𝑤 ∥ 𝑏))))
39	38	cbvralv 2765	. . . . . 6 ⊢ (∀𝑧 ∈ ℕ0 (𝑧 ∥ 𝑑 → (𝑧 ∥ 𝑎 ∧ 𝑧 ∥ 𝑏)) ↔ ∀𝑤 ∈ ℕ0 (𝑤 ∥ 𝑑 → (𝑤 ∥ 𝑎 ∧ 𝑤 ∥ 𝑏)))
40	11, 39, 12, 13	bezoutlemmain 12559	. . . . 5 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → ∀𝑎 ∈ ℕ0 ([𝑎 / 𝑑]∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝑑 = ((𝐴 · 𝑥) + (𝐵 · 𝑦)) → ∀𝑏 ∈ ℕ0 ([𝑏 / 𝑑]∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝑑 = ((𝐴 · 𝑥) + (𝐵 · 𝑦)) → ∃𝑑 ∈ ℕ0 (∀𝑧 ∈ ℕ0 (𝑧 ∥ 𝑑 → (𝑧 ∥ 𝑎 ∧ 𝑧 ∥ 𝑏)) ∧ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝑑 = ((𝐴 · 𝑥) + (𝐵 · 𝑦))))))
41	33, 40, 12	rspcdva 2913	. . . 4 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → ([𝐴 / 𝑑]∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝑑 = ((𝐴 · 𝑥) + (𝐵 · 𝑦)) → ∀𝑏 ∈ ℕ0 ([𝑏 / 𝑑]∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝑑 = ((𝐴 · 𝑥) + (𝐵 · 𝑦)) → ∃𝑑 ∈ ℕ0 (∀𝑧 ∈ ℕ0 (𝑧 ∥ 𝑑 → (𝑧 ∥ 𝐴 ∧ 𝑧 ∥ 𝑏)) ∧ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝑑 = ((𝐴 · 𝑥) + (𝐵 · 𝑦))))))
42	23, 41	mpd 13	. . 3 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → ∀𝑏 ∈ ℕ0 ([𝑏 / 𝑑]∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝑑 = ((𝐴 · 𝑥) + (𝐵 · 𝑦)) → ∃𝑑 ∈ ℕ0 (∀𝑧 ∈ ℕ0 (𝑧 ∥ 𝑑 → (𝑧 ∥ 𝐴 ∧ 𝑧 ∥ 𝑏)) ∧ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝑑 = ((𝐴 · 𝑥) + (𝐵 · 𝑦)))))
43	22, 42, 13	rspcdva 2913	. 2 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → ([𝐵 / 𝑑]∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝑑 = ((𝐴 · 𝑥) + (𝐵 · 𝑦)) → ∃𝑑 ∈ ℕ0 (∀𝑧 ∈ ℕ0 (𝑧 ∥ 𝑑 → (𝑧 ∥ 𝐴 ∧ 𝑧 ∥ 𝐵)) ∧ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝑑 = ((𝐴 · 𝑥) + (𝐵 · 𝑦)))))
44	14, 43	mpd 13	1 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → ∃𝑑 ∈ ℕ0 (∀𝑧 ∈ ℕ0 (𝑧 ∥ 𝑑 → (𝑧 ∥ 𝐴 ∧ 𝑧 ∥ 𝐵)) ∧ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝑑 = ((𝐴 · 𝑥) + (𝐵 · 𝑦))))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1395  [wsb 1808   ∈ wcel 2200  ∀wral 2508  ∃wrex 2509  [wsbc 3029   class class class wbr 4086  (class class class)co 6013   + caddc 8025   · cmul 8027  ℕ₀cn0 9392  ℤcz 9469   ∥ cdvds 12338

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-mulrcl 8121  ax-addcom 8122  ax-mulcom 8123  ax-addass 8124  ax-mulass 8125  ax-distr 8126  ax-i2m1 8127  ax-0lt1 8128  ax-1rid 8129  ax-0id 8130  ax-rnegex 8131  ax-precex 8132  ax-cnre 8133  ax-pre-ltirr 8134  ax-pre-ltwlin 8135  ax-pre-lttrn 8136  ax-pre-apti 8137  ax-pre-ltadd 8138  ax-pre-mulgt0 8139  ax-pre-mulext 8140  ax-arch 8141

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-po 4391  df-iso 4392  df-iord 4461  df-on 4463  df-ilim 4464  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-frec 6552  df-pnf 8206  df-mnf 8207  df-xr 8208  df-ltxr 8209  df-le 8210  df-sub 8342  df-neg 8343  df-reap 8745  df-ap 8752  df-div 8843  df-inn 9134  df-2 9192  df-n0 9393  df-z 9470  df-uz 9746  df-q 9844  df-rp 9879  df-fz 10234  df-fl 10520  df-mod 10575  df-seqfrec 10700  df-exp 10791  df-cj 11393  df-re 11394  df-im 11395  df-rsqrt 11549  df-abs 11550  df-dvds 12339

This theorem is referenced by:  bezoutlemzz  12563