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Theorem bezoutlem4 16484

Description: Lemma for bezout 16485. (Contributed by Mario Carneiro, 22-Feb-2014.)

**Hypotheses**
Ref	Expression
bezout.1	⊢ 𝑀 = {𝑧 ∈ ℕ ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝑧 = ((𝐴 · 𝑥) + (𝐵 · 𝑦))}
bezout.3	⊢ (𝜑 → 𝐴 ∈ ℤ)
bezout.4	⊢ (𝜑 → 𝐵 ∈ ℤ)
bezout.2	⊢ 𝐺 = inf(𝑀, ℝ, < )
bezout.5	⊢ (𝜑 → ¬ (𝐴 = 0 ∧ 𝐵 = 0))

**Assertion**
Ref	Expression
bezoutlem4	⊢ (𝜑 → (𝐴 gcd 𝐵) ∈ 𝑀)

Distinct variable groups: 𝑥,𝑦,𝑧,𝐴 𝑥,𝐵,𝑦,𝑧 𝑥,𝐺,𝑦,𝑧 𝜑,𝑥,𝑦,𝑧 𝑥,𝑀,𝑦 Allowed substitution hint: 𝑀(𝑧)

Proof of Theorem bezoutlem4 Dummy variables 𝑡 𝑠 𝑢 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		bezout.3	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℤ)
2		bezout.4	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℤ)
3		gcddvds 16444	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴 gcd 𝐵) ∥ 𝐴 ∧ (𝐴 gcd 𝐵) ∥ 𝐵))
4	1, 2, 3	syl2anc 585	. . . . . . 7 ⊢ (𝜑 → ((𝐴 gcd 𝐵) ∥ 𝐴 ∧ (𝐴 gcd 𝐵) ∥ 𝐵))
5	4	simpld 496	. . . . . 6 ⊢ (𝜑 → (𝐴 gcd 𝐵) ∥ 𝐴)
6	1, 2	gcdcld 16449	. . . . . . . 8 ⊢ (𝜑 → (𝐴 gcd 𝐵) ∈ ℕ₀)
7	6	nn0zd 12584	. . . . . . 7 ⊢ (𝜑 → (𝐴 gcd 𝐵) ∈ ℤ)
8		divides 16199	. . . . . . 7 ⊢ (((𝐴 gcd 𝐵) ∈ ℤ ∧ 𝐴 ∈ ℤ) → ((𝐴 gcd 𝐵) ∥ 𝐴 ↔ ∃𝑠 ∈ ℤ (𝑠 · (𝐴 gcd 𝐵)) = 𝐴))
9	7, 1, 8	syl2anc 585	. . . . . 6 ⊢ (𝜑 → ((𝐴 gcd 𝐵) ∥ 𝐴 ↔ ∃𝑠 ∈ ℤ (𝑠 · (𝐴 gcd 𝐵)) = 𝐴))
10	5, 9	mpbid 231	. . . . 5 ⊢ (𝜑 → ∃𝑠 ∈ ℤ (𝑠 · (𝐴 gcd 𝐵)) = 𝐴)
11	4	simprd 497	. . . . . 6 ⊢ (𝜑 → (𝐴 gcd 𝐵) ∥ 𝐵)
12		divides 16199	. . . . . . 7 ⊢ (((𝐴 gcd 𝐵) ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴 gcd 𝐵) ∥ 𝐵 ↔ ∃𝑡 ∈ ℤ (𝑡 · (𝐴 gcd 𝐵)) = 𝐵))
13	7, 2, 12	syl2anc 585	. . . . . 6 ⊢ (𝜑 → ((𝐴 gcd 𝐵) ∥ 𝐵 ↔ ∃𝑡 ∈ ℤ (𝑡 · (𝐴 gcd 𝐵)) = 𝐵))
14	11, 13	mpbid 231	. . . . 5 ⊢ (𝜑 → ∃𝑡 ∈ ℤ (𝑡 · (𝐴 gcd 𝐵)) = 𝐵)
15		reeanv 3227	. . . . . 6 ⊢ (∃𝑠 ∈ ℤ ∃𝑡 ∈ ℤ ((𝑠 · (𝐴 gcd 𝐵)) = 𝐴 ∧ (𝑡 · (𝐴 gcd 𝐵)) = 𝐵) ↔ (∃𝑠 ∈ ℤ (𝑠 · (𝐴 gcd 𝐵)) = 𝐴 ∧ ∃𝑡 ∈ ℤ (𝑡 · (𝐴 gcd 𝐵)) = 𝐵))
16		bezout.1	. . . . . . . . . . 11 ⊢ 𝑀 = {𝑧 ∈ ℕ ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝑧 = ((𝐴 · 𝑥) + (𝐵 · 𝑦))}
17		bezout.2	. . . . . . . . . . 11 ⊢ 𝐺 = inf(𝑀, ℝ, < )
18		bezout.5	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ (𝐴 = 0 ∧ 𝐵 = 0))
19	16, 1, 2, 17, 18	bezoutlem2 16482	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 ∈ 𝑀)
20		oveq2 7417	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑢 → (𝐴 · 𝑥) = (𝐴 · 𝑢))
21	20	oveq1d 7424	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑢 → ((𝐴 · 𝑥) + (𝐵 · 𝑦)) = ((𝐴 · 𝑢) + (𝐵 · 𝑦)))
22	21	eqeq2d 2744	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑢 → (𝑧 = ((𝐴 · 𝑥) + (𝐵 · 𝑦)) ↔ 𝑧 = ((𝐴 · 𝑢) + (𝐵 · 𝑦))))
23		oveq2 7417	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑣 → (𝐵 · 𝑦) = (𝐵 · 𝑣))
24	23	oveq2d 7425	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑣 → ((𝐴 · 𝑢) + (𝐵 · 𝑦)) = ((𝐴 · 𝑢) + (𝐵 · 𝑣)))
25	24	eqeq2d 2744	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑣 → (𝑧 = ((𝐴 · 𝑢) + (𝐵 · 𝑦)) ↔ 𝑧 = ((𝐴 · 𝑢) + (𝐵 · 𝑣))))
26	22, 25	cbvrex2vw 3240	. . . . . . . . . . . 12 ⊢ (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝑧 = ((𝐴 · 𝑥) + (𝐵 · 𝑦)) ↔ ∃𝑢 ∈ ℤ ∃𝑣 ∈ ℤ 𝑧 = ((𝐴 · 𝑢) + (𝐵 · 𝑣)))
27		eqeq1 2737	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝐺 → (𝑧 = ((𝐴 · 𝑢) + (𝐵 · 𝑣)) ↔ 𝐺 = ((𝐴 · 𝑢) + (𝐵 · 𝑣))))
28	27	2rexbidv 3220	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝐺 → (∃𝑢 ∈ ℤ ∃𝑣 ∈ ℤ 𝑧 = ((𝐴 · 𝑢) + (𝐵 · 𝑣)) ↔ ∃𝑢 ∈ ℤ ∃𝑣 ∈ ℤ 𝐺 = ((𝐴 · 𝑢) + (𝐵 · 𝑣))))
29	26, 28	bitrid 283	. . . . . . . . . . 11 ⊢ (𝑧 = 𝐺 → (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝑧 = ((𝐴 · 𝑥) + (𝐵 · 𝑦)) ↔ ∃𝑢 ∈ ℤ ∃𝑣 ∈ ℤ 𝐺 = ((𝐴 · 𝑢) + (𝐵 · 𝑣))))
30	29, 16	elrab2 3687	. . . . . . . . . 10 ⊢ (𝐺 ∈ 𝑀 ↔ (𝐺 ∈ ℕ ∧ ∃𝑢 ∈ ℤ ∃𝑣 ∈ ℤ 𝐺 = ((𝐴 · 𝑢) + (𝐵 · 𝑣))))
31	19, 30	sylib 217	. . . . . . . . 9 ⊢ (𝜑 → (𝐺 ∈ ℕ ∧ ∃𝑢 ∈ ℤ ∃𝑣 ∈ ℤ 𝐺 = ((𝐴 · 𝑢) + (𝐵 · 𝑣))))
32	31	simprd 497	. . . . . . . 8 ⊢ (𝜑 → ∃𝑢 ∈ ℤ ∃𝑣 ∈ ℤ 𝐺 = ((𝐴 · 𝑢) + (𝐵 · 𝑣)))
33		simprrl 780	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ((𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ) ∧ (𝑠 ∈ ℤ ∧ 𝑡 ∈ ℤ))) → 𝑠 ∈ ℤ)
34		simprll 778	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ((𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ) ∧ (𝑠 ∈ ℤ ∧ 𝑡 ∈ ℤ))) → 𝑢 ∈ ℤ)
35	33, 34	zmulcld 12672	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ) ∧ (𝑠 ∈ ℤ ∧ 𝑡 ∈ ℤ))) → (𝑠 · 𝑢) ∈ ℤ)
36		simprrr 781	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ((𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ) ∧ (𝑠 ∈ ℤ ∧ 𝑡 ∈ ℤ))) → 𝑡 ∈ ℤ)
37		simprlr 779	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ((𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ) ∧ (𝑠 ∈ ℤ ∧ 𝑡 ∈ ℤ))) → 𝑣 ∈ ℤ)
38	36, 37	zmulcld 12672	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ) ∧ (𝑠 ∈ ℤ ∧ 𝑡 ∈ ℤ))) → (𝑡 · 𝑣) ∈ ℤ)
39	35, 38	zaddcld 12670	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ) ∧ (𝑠 ∈ ℤ ∧ 𝑡 ∈ ℤ))) → ((𝑠 · 𝑢) + (𝑡 · 𝑣)) ∈ ℤ)
40	7	adantr 482	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ) ∧ (𝑠 ∈ ℤ ∧ 𝑡 ∈ ℤ))) → (𝐴 gcd 𝐵) ∈ ℤ)
41		dvdsmul2 16222	. . . . . . . . . . . . . . 15 ⊢ ((((𝑠 · 𝑢) + (𝑡 · 𝑣)) ∈ ℤ ∧ (𝐴 gcd 𝐵) ∈ ℤ) → (𝐴 gcd 𝐵) ∥ (((𝑠 · 𝑢) + (𝑡 · 𝑣)) · (𝐴 gcd 𝐵)))
42	39, 40, 41	syl2anc 585	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ) ∧ (𝑠 ∈ ℤ ∧ 𝑡 ∈ ℤ))) → (𝐴 gcd 𝐵) ∥ (((𝑠 · 𝑢) + (𝑡 · 𝑣)) · (𝐴 gcd 𝐵)))
43	35	zcnd 12667	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ) ∧ (𝑠 ∈ ℤ ∧ 𝑡 ∈ ℤ))) → (𝑠 · 𝑢) ∈ ℂ)
44	40	zcnd 12667	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ) ∧ (𝑠 ∈ ℤ ∧ 𝑡 ∈ ℤ))) → (𝐴 gcd 𝐵) ∈ ℂ)
45	38	zcnd 12667	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ) ∧ (𝑠 ∈ ℤ ∧ 𝑡 ∈ ℤ))) → (𝑡 · 𝑣) ∈ ℂ)
46	33	zcnd 12667	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ((𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ) ∧ (𝑠 ∈ ℤ ∧ 𝑡 ∈ ℤ))) → 𝑠 ∈ ℂ)
47	34	zcnd 12667	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ((𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ) ∧ (𝑠 ∈ ℤ ∧ 𝑡 ∈ ℤ))) → 𝑢 ∈ ℂ)
48	46, 47, 44	mul32d 11424	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ) ∧ (𝑠 ∈ ℤ ∧ 𝑡 ∈ ℤ))) → ((𝑠 · 𝑢) · (𝐴 gcd 𝐵)) = ((𝑠 · (𝐴 gcd 𝐵)) · 𝑢))
49	36	zcnd 12667	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ((𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ) ∧ (𝑠 ∈ ℤ ∧ 𝑡 ∈ ℤ))) → 𝑡 ∈ ℂ)
50	37	zcnd 12667	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ((𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ) ∧ (𝑠 ∈ ℤ ∧ 𝑡 ∈ ℤ))) → 𝑣 ∈ ℂ)
51	49, 50, 44	mul32d 11424	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ) ∧ (𝑠 ∈ ℤ ∧ 𝑡 ∈ ℤ))) → ((𝑡 · 𝑣) · (𝐴 gcd 𝐵)) = ((𝑡 · (𝐴 gcd 𝐵)) · 𝑣))
52	48, 51	oveq12d 7427	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ) ∧ (𝑠 ∈ ℤ ∧ 𝑡 ∈ ℤ))) → (((𝑠 · 𝑢) · (𝐴 gcd 𝐵)) + ((𝑡 · 𝑣) · (𝐴 gcd 𝐵))) = (((𝑠 · (𝐴 gcd 𝐵)) · 𝑢) + ((𝑡 · (𝐴 gcd 𝐵)) · 𝑣)))
53	43, 44, 45, 52	joinlmuladdmuld 11241	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ) ∧ (𝑠 ∈ ℤ ∧ 𝑡 ∈ ℤ))) → (((𝑠 · 𝑢) + (𝑡 · 𝑣)) · (𝐴 gcd 𝐵)) = (((𝑠 · (𝐴 gcd 𝐵)) · 𝑢) + ((𝑡 · (𝐴 gcd 𝐵)) · 𝑣)))
54	42, 53	breqtrd 5175	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ) ∧ (𝑠 ∈ ℤ ∧ 𝑡 ∈ ℤ))) → (𝐴 gcd 𝐵) ∥ (((𝑠 · (𝐴 gcd 𝐵)) · 𝑢) + ((𝑡 · (𝐴 gcd 𝐵)) · 𝑣)))
55		oveq1 7416	. . . . . . . . . . . . . . 15 ⊢ ((𝑠 · (𝐴 gcd 𝐵)) = 𝐴 → ((𝑠 · (𝐴 gcd 𝐵)) · 𝑢) = (𝐴 · 𝑢))
56		oveq1 7416	. . . . . . . . . . . . . . 15 ⊢ ((𝑡 · (𝐴 gcd 𝐵)) = 𝐵 → ((𝑡 · (𝐴 gcd 𝐵)) · 𝑣) = (𝐵 · 𝑣))
57	55, 56	oveqan12d 7428	. . . . . . . . . . . . . 14 ⊢ (((𝑠 · (𝐴 gcd 𝐵)) = 𝐴 ∧ (𝑡 · (𝐴 gcd 𝐵)) = 𝐵) → (((𝑠 · (𝐴 gcd 𝐵)) · 𝑢) + ((𝑡 · (𝐴 gcd 𝐵)) · 𝑣)) = ((𝐴 · 𝑢) + (𝐵 · 𝑣)))
58	57	breq2d 5161	. . . . . . . . . . . . 13 ⊢ (((𝑠 · (𝐴 gcd 𝐵)) = 𝐴 ∧ (𝑡 · (𝐴 gcd 𝐵)) = 𝐵) → ((𝐴 gcd 𝐵) ∥ (((𝑠 · (𝐴 gcd 𝐵)) · 𝑢) + ((𝑡 · (𝐴 gcd 𝐵)) · 𝑣)) ↔ (𝐴 gcd 𝐵) ∥ ((𝐴 · 𝑢) + (𝐵 · 𝑣))))
59	54, 58	syl5ibcom 244	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ) ∧ (𝑠 ∈ ℤ ∧ 𝑡 ∈ ℤ))) → (((𝑠 · (𝐴 gcd 𝐵)) = 𝐴 ∧ (𝑡 · (𝐴 gcd 𝐵)) = 𝐵) → (𝐴 gcd 𝐵) ∥ ((𝐴 · 𝑢) + (𝐵 · 𝑣))))
60		breq2 5153	. . . . . . . . . . . . 13 ⊢ (𝐺 = ((𝐴 · 𝑢) + (𝐵 · 𝑣)) → ((𝐴 gcd 𝐵) ∥ 𝐺 ↔ (𝐴 gcd 𝐵) ∥ ((𝐴 · 𝑢) + (𝐵 · 𝑣))))
61	60	imbi2d 341	. . . . . . . . . . . 12 ⊢ (𝐺 = ((𝐴 · 𝑢) + (𝐵 · 𝑣)) → ((((𝑠 · (𝐴 gcd 𝐵)) = 𝐴 ∧ (𝑡 · (𝐴 gcd 𝐵)) = 𝐵) → (𝐴 gcd 𝐵) ∥ 𝐺) ↔ (((𝑠 · (𝐴 gcd 𝐵)) = 𝐴 ∧ (𝑡 · (𝐴 gcd 𝐵)) = 𝐵) → (𝐴 gcd 𝐵) ∥ ((𝐴 · 𝑢) + (𝐵 · 𝑣)))))
62	59, 61	syl5ibrcom 246	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ) ∧ (𝑠 ∈ ℤ ∧ 𝑡 ∈ ℤ))) → (𝐺 = ((𝐴 · 𝑢) + (𝐵 · 𝑣)) → (((𝑠 · (𝐴 gcd 𝐵)) = 𝐴 ∧ (𝑡 · (𝐴 gcd 𝐵)) = 𝐵) → (𝐴 gcd 𝐵) ∥ 𝐺)))
63	62	expr 458	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ)) → ((𝑠 ∈ ℤ ∧ 𝑡 ∈ ℤ) → (𝐺 = ((𝐴 · 𝑢) + (𝐵 · 𝑣)) → (((𝑠 · (𝐴 gcd 𝐵)) = 𝐴 ∧ (𝑡 · (𝐴 gcd 𝐵)) = 𝐵) → (𝐴 gcd 𝐵) ∥ 𝐺))))
64	63	com23 86	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ)) → (𝐺 = ((𝐴 · 𝑢) + (𝐵 · 𝑣)) → ((𝑠 ∈ ℤ ∧ 𝑡 ∈ ℤ) → (((𝑠 · (𝐴 gcd 𝐵)) = 𝐴 ∧ (𝑡 · (𝐴 gcd 𝐵)) = 𝐵) → (𝐴 gcd 𝐵) ∥ 𝐺))))
65	64	rexlimdvva 3212	. . . . . . . 8 ⊢ (𝜑 → (∃𝑢 ∈ ℤ ∃𝑣 ∈ ℤ 𝐺 = ((𝐴 · 𝑢) + (𝐵 · 𝑣)) → ((𝑠 ∈ ℤ ∧ 𝑡 ∈ ℤ) → (((𝑠 · (𝐴 gcd 𝐵)) = 𝐴 ∧ (𝑡 · (𝐴 gcd 𝐵)) = 𝐵) → (𝐴 gcd 𝐵) ∥ 𝐺))))
66	32, 65	mpd 15	. . . . . . 7 ⊢ (𝜑 → ((𝑠 ∈ ℤ ∧ 𝑡 ∈ ℤ) → (((𝑠 · (𝐴 gcd 𝐵)) = 𝐴 ∧ (𝑡 · (𝐴 gcd 𝐵)) = 𝐵) → (𝐴 gcd 𝐵) ∥ 𝐺)))
67	66	rexlimdvv 3211	. . . . . 6 ⊢ (𝜑 → (∃𝑠 ∈ ℤ ∃𝑡 ∈ ℤ ((𝑠 · (𝐴 gcd 𝐵)) = 𝐴 ∧ (𝑡 · (𝐴 gcd 𝐵)) = 𝐵) → (𝐴 gcd 𝐵) ∥ 𝐺))
68	15, 67	biimtrrid 242	. . . . 5 ⊢ (𝜑 → ((∃𝑠 ∈ ℤ (𝑠 · (𝐴 gcd 𝐵)) = 𝐴 ∧ ∃𝑡 ∈ ℤ (𝑡 · (𝐴 gcd 𝐵)) = 𝐵) → (𝐴 gcd 𝐵) ∥ 𝐺))
69	10, 14, 68	mp2and 698	. . . 4 ⊢ (𝜑 → (𝐴 gcd 𝐵) ∥ 𝐺)
70	31	simpld 496	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ ℕ)
71		dvdsle 16253	. . . . 5 ⊢ (((𝐴 gcd 𝐵) ∈ ℤ ∧ 𝐺 ∈ ℕ) → ((𝐴 gcd 𝐵) ∥ 𝐺 → (𝐴 gcd 𝐵) ≤ 𝐺))
72	7, 70, 71	syl2anc 585	. . . 4 ⊢ (𝜑 → ((𝐴 gcd 𝐵) ∥ 𝐺 → (𝐴 gcd 𝐵) ≤ 𝐺))
73	69, 72	mpd 15	. . 3 ⊢ (𝜑 → (𝐴 gcd 𝐵) ≤ 𝐺)
74		breq2 5153	. . . . 5 ⊢ (𝐴 = 0 → (𝐺 ∥ 𝐴 ↔ 𝐺 ∥ 0))
75	16, 1, 2	bezoutlem1 16481	. . . . . . . 8 ⊢ (𝜑 → (𝐴 ≠ 0 → (abs‘𝐴) ∈ 𝑀))
76	16, 1, 2, 17, 18	bezoutlem3 16483	. . . . . . . 8 ⊢ (𝜑 → ((abs‘𝐴) ∈ 𝑀 → 𝐺 ∥ (abs‘𝐴)))
77	75, 76	syld 47	. . . . . . 7 ⊢ (𝜑 → (𝐴 ≠ 0 → 𝐺 ∥ (abs‘𝐴)))
78	70	nnzd 12585	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ ℤ)
79		dvdsabsb 16219	. . . . . . . 8 ⊢ ((𝐺 ∈ ℤ ∧ 𝐴 ∈ ℤ) → (𝐺 ∥ 𝐴 ↔ 𝐺 ∥ (abs‘𝐴)))
80	78, 1, 79	syl2anc 585	. . . . . . 7 ⊢ (𝜑 → (𝐺 ∥ 𝐴 ↔ 𝐺 ∥ (abs‘𝐴)))
81	77, 80	sylibrd 259	. . . . . 6 ⊢ (𝜑 → (𝐴 ≠ 0 → 𝐺 ∥ 𝐴))
82	81	imp 408	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → 𝐺 ∥ 𝐴)
83		dvds0 16215	. . . . . 6 ⊢ (𝐺 ∈ ℤ → 𝐺 ∥ 0)
84	78, 83	syl 17	. . . . 5 ⊢ (𝜑 → 𝐺 ∥ 0)
85	74, 82, 84	pm2.61ne 3028	. . . 4 ⊢ (𝜑 → 𝐺 ∥ 𝐴)
86		breq2 5153	. . . . 5 ⊢ (𝐵 = 0 → (𝐺 ∥ 𝐵 ↔ 𝐺 ∥ 0))
87		eqid 2733	. . . . . . . . . 10 ⊢ {𝑧 ∈ ℕ ∣ ∃𝑦 ∈ ℤ ∃𝑥 ∈ ℤ 𝑧 = ((𝐵 · 𝑦) + (𝐴 · 𝑥))} = {𝑧 ∈ ℕ ∣ ∃𝑦 ∈ ℤ ∃𝑥 ∈ ℤ 𝑧 = ((𝐵 · 𝑦) + (𝐴 · 𝑥))}
88	87, 2, 1	bezoutlem1 16481	. . . . . . . . 9 ⊢ (𝜑 → (𝐵 ≠ 0 → (abs‘𝐵) ∈ {𝑧 ∈ ℕ ∣ ∃𝑦 ∈ ℤ ∃𝑥 ∈ ℤ 𝑧 = ((𝐵 · 𝑦) + (𝐴 · 𝑥))}))
89		rexcom 3288	. . . . . . . . . . . . 13 ⊢ (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝑧 = ((𝐴 · 𝑥) + (𝐵 · 𝑦)) ↔ ∃𝑦 ∈ ℤ ∃𝑥 ∈ ℤ 𝑧 = ((𝐴 · 𝑥) + (𝐵 · 𝑦)))
90	1	zcnd 12667	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐴 ∈ ℂ)
91	90	adantr 482	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑥 ∈ ℤ)) → 𝐴 ∈ ℂ)
92		zcn 12563	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ ℂ)
93	92	ad2antll 728	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑥 ∈ ℤ)) → 𝑥 ∈ ℂ)
94	91, 93	mulcld 11234	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑥 ∈ ℤ)) → (𝐴 · 𝑥) ∈ ℂ)
95	2	zcnd 12667	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐵 ∈ ℂ)
96	95	adantr 482	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑥 ∈ ℤ)) → 𝐵 ∈ ℂ)
97		zcn 12563	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ ℤ → 𝑦 ∈ ℂ)
98	97	ad2antrl 727	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑥 ∈ ℤ)) → 𝑦 ∈ ℂ)
99	96, 98	mulcld 11234	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑥 ∈ ℤ)) → (𝐵 · 𝑦) ∈ ℂ)
100	94, 99	addcomd 11416	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑥 ∈ ℤ)) → ((𝐴 · 𝑥) + (𝐵 · 𝑦)) = ((𝐵 · 𝑦) + (𝐴 · 𝑥)))
101	100	eqeq2d 2744	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑥 ∈ ℤ)) → (𝑧 = ((𝐴 · 𝑥) + (𝐵 · 𝑦)) ↔ 𝑧 = ((𝐵 · 𝑦) + (𝐴 · 𝑥))))
102	101	2rexbidva 3218	. . . . . . . . . . . . 13 ⊢ (𝜑 → (∃𝑦 ∈ ℤ ∃𝑥 ∈ ℤ 𝑧 = ((𝐴 · 𝑥) + (𝐵 · 𝑦)) ↔ ∃𝑦 ∈ ℤ ∃𝑥 ∈ ℤ 𝑧 = ((𝐵 · 𝑦) + (𝐴 · 𝑥))))
103	89, 102	bitrid 283	. . . . . . . . . . . 12 ⊢ (𝜑 → (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝑧 = ((𝐴 · 𝑥) + (𝐵 · 𝑦)) ↔ ∃𝑦 ∈ ℤ ∃𝑥 ∈ ℤ 𝑧 = ((𝐵 · 𝑦) + (𝐴 · 𝑥))))
104	103	rabbidv 3441	. . . . . . . . . . 11 ⊢ (𝜑 → {𝑧 ∈ ℕ ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝑧 = ((𝐴 · 𝑥) + (𝐵 · 𝑦))} = {𝑧 ∈ ℕ ∣ ∃𝑦 ∈ ℤ ∃𝑥 ∈ ℤ 𝑧 = ((𝐵 · 𝑦) + (𝐴 · 𝑥))})
105	16, 104	eqtrid 2785	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 = {𝑧 ∈ ℕ ∣ ∃𝑦 ∈ ℤ ∃𝑥 ∈ ℤ 𝑧 = ((𝐵 · 𝑦) + (𝐴 · 𝑥))})
106	105	eleq2d 2820	. . . . . . . . 9 ⊢ (𝜑 → ((abs‘𝐵) ∈ 𝑀 ↔ (abs‘𝐵) ∈ {𝑧 ∈ ℕ ∣ ∃𝑦 ∈ ℤ ∃𝑥 ∈ ℤ 𝑧 = ((𝐵 · 𝑦) + (𝐴 · 𝑥))}))
107	88, 106	sylibrd 259	. . . . . . . 8 ⊢ (𝜑 → (𝐵 ≠ 0 → (abs‘𝐵) ∈ 𝑀))
108	16, 1, 2, 17, 18	bezoutlem3 16483	. . . . . . . 8 ⊢ (𝜑 → ((abs‘𝐵) ∈ 𝑀 → 𝐺 ∥ (abs‘𝐵)))
109	107, 108	syld 47	. . . . . . 7 ⊢ (𝜑 → (𝐵 ≠ 0 → 𝐺 ∥ (abs‘𝐵)))
110		dvdsabsb 16219	. . . . . . . 8 ⊢ ((𝐺 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐺 ∥ 𝐵 ↔ 𝐺 ∥ (abs‘𝐵)))
111	78, 2, 110	syl2anc 585	. . . . . . 7 ⊢ (𝜑 → (𝐺 ∥ 𝐵 ↔ 𝐺 ∥ (abs‘𝐵)))
112	109, 111	sylibrd 259	. . . . . 6 ⊢ (𝜑 → (𝐵 ≠ 0 → 𝐺 ∥ 𝐵))
113	112	imp 408	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≠ 0) → 𝐺 ∥ 𝐵)
114	86, 113, 84	pm2.61ne 3028	. . . 4 ⊢ (𝜑 → 𝐺 ∥ 𝐵)
115		dvdslegcd 16445	. . . . 5 ⊢ (((𝐺 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → ((𝐺 ∥ 𝐴 ∧ 𝐺 ∥ 𝐵) → 𝐺 ≤ (𝐴 gcd 𝐵)))
116	78, 1, 2, 18, 115	syl31anc 1374	. . . 4 ⊢ (𝜑 → ((𝐺 ∥ 𝐴 ∧ 𝐺 ∥ 𝐵) → 𝐺 ≤ (𝐴 gcd 𝐵)))
117	85, 114, 116	mp2and 698	. . 3 ⊢ (𝜑 → 𝐺 ≤ (𝐴 gcd 𝐵))
118	6	nn0red 12533	. . . 4 ⊢ (𝜑 → (𝐴 gcd 𝐵) ∈ ℝ)
119	70	nnred 12227	. . . 4 ⊢ (𝜑 → 𝐺 ∈ ℝ)
120	118, 119	letri3d 11356	. . 3 ⊢ (𝜑 → ((𝐴 gcd 𝐵) = 𝐺 ↔ ((𝐴 gcd 𝐵) ≤ 𝐺 ∧ 𝐺 ≤ (𝐴 gcd 𝐵))))
121	73, 117, 120	mpbir2and 712	. 2 ⊢ (𝜑 → (𝐴 gcd 𝐵) = 𝐺)
122	121, 19	eqeltrd 2834	1 ⊢ (𝜑 → (𝐴 gcd 𝐵) ∈ 𝑀)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∃wrex 3071 {crab 3433 class class class wbr 5149 ‘cfv 6544 (class class class)co 7409 infcinf 9436 ℂcc 11108 ℝcr 11109 0cc0 11110 + caddc 11113 · cmul 11115 < clt 11248 ≤ cle 11249 ℕcn 12212 ℤcz 12558 abscabs 15181 ∥ cdvds 16197 gcd cgcd 16435

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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-sup 9437 df-inf 9438 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-n0 12473 df-z 12559 df-uz 12823 df-rp 12975 df-fl 13757 df-mod 13835 df-seq 13967 df-exp 14028 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-dvds 16198 df-gcd 16436

This theorem is referenced by: bezout 16485

W3C validator