Theorem bezoutlemex 12571

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 6025	. . . . . . . 8 ⊢ (𝑦 = 𝑡 → (𝐵 · 𝑦) = (𝐵 · 𝑡))
2	1	oveq2d 6033	. . . . . . 7 ⊢ (𝑦 = 𝑡 → ((𝐴 · 𝑥) + (𝐵 · 𝑦)) = ((𝐴 · 𝑥) + (𝐵 · 𝑡)))
3	2	eqeq2d 2243	. . . . . 6 ⊢ (𝑦 = 𝑡 → (𝑑 = ((𝐴 · 𝑥) + (𝐵 · 𝑦)) ↔ 𝑑 = ((𝐴 · 𝑥) + (𝐵 · 𝑡))))
4	3	cbvrexv 2768	. . . . 5 ⊢ (∃𝑦 ∈ ℤ 𝑑 = ((𝐴 · 𝑥) + (𝐵 · 𝑦)) ↔ ∃𝑡 ∈ ℤ 𝑑 = ((𝐴 · 𝑥) + (𝐵 · 𝑡)))
5	4	rexbii 2539	. . . 4 ⊢ (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝑑 = ((𝐴 · 𝑥) + (𝐵 · 𝑦)) ↔ ∃𝑥 ∈ ℤ ∃𝑡 ∈ ℤ 𝑑 = ((𝐴 · 𝑥) + (𝐵 · 𝑡)))
6		oveq2 6025	. . . . . . . 8 ⊢ (𝑥 = 𝑠 → (𝐴 · 𝑥) = (𝐴 · 𝑠))
7	6	oveq1d 6032	. . . . . . 7 ⊢ (𝑥 = 𝑠 → ((𝐴 · 𝑥) + (𝐵 · 𝑡)) = ((𝐴 · 𝑠) + (𝐵 · 𝑡)))
8	7	eqeq2d 2243	. . . . . 6 ⊢ (𝑥 = 𝑠 → (𝑑 = ((𝐴 · 𝑥) + (𝐵 · 𝑡)) ↔ 𝑑 = ((𝐴 · 𝑠) + (𝐵 · 𝑡))))
9	8	rexbidv 2533	. . . . 5 ⊢ (𝑥 = 𝑠 → (∃𝑡 ∈ ℤ 𝑑 = ((𝐴 · 𝑥) + (𝐵 · 𝑡)) ↔ ∃𝑡 ∈ ℤ 𝑑 = ((𝐴 · 𝑠) + (𝐵 · 𝑡))))
10	9	cbvrexv 2768	. . . 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 12570	. 2 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → [𝐵 / 𝑑]∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝑑 = ((𝐴 · 𝑥) + (𝐵 · 𝑦)))
15		dfsbcq2 3034	. . . 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 4092	. . . . . . . . 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 2532	. . . . . 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 2533	. . . 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 12569	. . . 4 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → [𝐴 / 𝑑]∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝑑 = ((𝐴 · 𝑥) + (𝐵 · 𝑦)))
24		dfsbcq2 3034	. . . . . 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 4092	. . . . . . . . . . . . 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 2532	. . . . . . . . . 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 2533	. . . . . . . 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 2532	. . . . . 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 4091	. . . . . . . 8 ⊢ (𝑧 = 𝑤 → (𝑧 ∥ 𝑑 ↔ 𝑤 ∥ 𝑑))
35		breq1 4091	. . . . . . . . 9 ⊢ (𝑧 = 𝑤 → (𝑧 ∥ 𝑎 ↔ 𝑤 ∥ 𝑎))
36		breq1 4091	. . . . . . . . 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 2767	. . . . . 6 ⊢ (∀𝑧 ∈ ℕ0 (𝑧 ∥ 𝑑 → (𝑧 ∥ 𝑎 ∧ 𝑧 ∥ 𝑏)) ↔ ∀𝑤 ∈ ℕ0 (𝑤 ∥ 𝑑 → (𝑤 ∥ 𝑎 ∧ 𝑤 ∥ 𝑏)))
40	11, 39, 12, 13	bezoutlemmain 12568	. . . . 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 2915	. . . 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 2915	. 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 1397  [wsb 1810   ∈ wcel 2202  ∀wral 2510  ∃wrex 2511  [wsbc 3031   class class class wbr 4088  (class class class)co 6017   + caddc 8034   · cmul 8036  ℕ₀cn0 9401  ℤcz 9478   ∥ cdvds 12347

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-mulrcl 8130  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-precex 8141  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147  ax-pre-mulgt0 8148  ax-pre-mulext 8149  ax-arch 8150

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-recs 6470  df-frec 6556  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-reap 8754  df-ap 8761  df-div 8852  df-inn 9143  df-2 9201  df-n0 9402  df-z 9479  df-uz 9755  df-q 9853  df-rp 9888  df-fz 10243  df-fl 10529  df-mod 10584  df-seqfrec 10709  df-exp 10800  df-cj 11402  df-re 11403  df-im 11404  df-rsqrt 11558  df-abs 11559  df-dvds 12348

This theorem is referenced by:  bezoutlemzz  12572