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Theorem bezoutlem4 16486

Description: Lemma for bezout 16487. (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 16446	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴 gcd 𝐵) ∥ 𝐴 ∧ (𝐴 gcd 𝐵) ∥ 𝐵))
4	1, 2, 3	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → ((𝐴 gcd 𝐵) ∥ 𝐴 ∧ (𝐴 gcd 𝐵) ∥ 𝐵))
5	4	simpld 495	. . . . . 6 ⊢ (𝜑 → (𝐴 gcd 𝐵) ∥ 𝐴)
6	1, 2	gcdcld 16451	. . . . . . . 8 ⊢ (𝜑 → (𝐴 gcd 𝐵) ∈ ℕ₀)
7	6	nn0zd 12586	. . . . . . 7 ⊢ (𝜑 → (𝐴 gcd 𝐵) ∈ ℤ)
8		divides 16201	. . . . . . 7 ⊢ (((𝐴 gcd 𝐵) ∈ ℤ ∧ 𝐴 ∈ ℤ) → ((𝐴 gcd 𝐵) ∥ 𝐴 ↔ ∃𝑠 ∈ ℤ (𝑠 · (𝐴 gcd 𝐵)) = 𝐴))
9	7, 1, 8	syl2anc 584	. . . . . 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 496	. . . . . 6 ⊢ (𝜑 → (𝐴 gcd 𝐵) ∥ 𝐵)
12		divides 16201	. . . . . . 7 ⊢ (((𝐴 gcd 𝐵) ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴 gcd 𝐵) ∥ 𝐵 ↔ ∃𝑡 ∈ ℤ (𝑡 · (𝐴 gcd 𝐵)) = 𝐵))
13	7, 2, 12	syl2anc 584	. . . . . 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 3226	. . . . . 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 16484	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 ∈ 𝑀)
20		oveq2 7419	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑢 → (𝐴 · 𝑥) = (𝐴 · 𝑢))
21	20	oveq1d 7426	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑢 → ((𝐴 · 𝑥) + (𝐵 · 𝑦)) = ((𝐴 · 𝑢) + (𝐵 · 𝑦)))
22	21	eqeq2d 2743	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑢 → (𝑧 = ((𝐴 · 𝑥) + (𝐵 · 𝑦)) ↔ 𝑧 = ((𝐴 · 𝑢) + (𝐵 · 𝑦))))
23		oveq2 7419	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑣 → (𝐵 · 𝑦) = (𝐵 · 𝑣))
24	23	oveq2d 7427	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑣 → ((𝐴 · 𝑢) + (𝐵 · 𝑦)) = ((𝐴 · 𝑢) + (𝐵 · 𝑣)))
25	24	eqeq2d 2743	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑣 → (𝑧 = ((𝐴 · 𝑢) + (𝐵 · 𝑦)) ↔ 𝑧 = ((𝐴 · 𝑢) + (𝐵 · 𝑣))))
26	22, 25	cbvrex2vw 3239	. . . . . . . . . . . 12 ⊢ (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝑧 = ((𝐴 · 𝑥) + (𝐵 · 𝑦)) ↔ ∃𝑢 ∈ ℤ ∃𝑣 ∈ ℤ 𝑧 = ((𝐴 · 𝑢) + (𝐵 · 𝑣)))
27		eqeq1 2736	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝐺 → (𝑧 = ((𝐴 · 𝑢) + (𝐵 · 𝑣)) ↔ 𝐺 = ((𝐴 · 𝑢) + (𝐵 · 𝑣))))
28	27	2rexbidv 3219	. . . . . . . . . . . 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 282	. . . . . . . . . . 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 3686	. . . . . . . . . 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 496	. . . . . . . 8 ⊢ (𝜑 → ∃𝑢 ∈ ℤ ∃𝑣 ∈ ℤ 𝐺 = ((𝐴 · 𝑢) + (𝐵 · 𝑣)))
33		simprrl 779	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ((𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ) ∧ (𝑠 ∈ ℤ ∧ 𝑡 ∈ ℤ))) → 𝑠 ∈ ℤ)
34		simprll 777	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ((𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ) ∧ (𝑠 ∈ ℤ ∧ 𝑡 ∈ ℤ))) → 𝑢 ∈ ℤ)
35	33, 34	zmulcld 12674	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ) ∧ (𝑠 ∈ ℤ ∧ 𝑡 ∈ ℤ))) → (𝑠 · 𝑢) ∈ ℤ)
36		simprrr 780	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ((𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ) ∧ (𝑠 ∈ ℤ ∧ 𝑡 ∈ ℤ))) → 𝑡 ∈ ℤ)
37		simprlr 778	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ((𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ) ∧ (𝑠 ∈ ℤ ∧ 𝑡 ∈ ℤ))) → 𝑣 ∈ ℤ)
38	36, 37	zmulcld 12674	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ) ∧ (𝑠 ∈ ℤ ∧ 𝑡 ∈ ℤ))) → (𝑡 · 𝑣) ∈ ℤ)
39	35, 38	zaddcld 12672	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ) ∧ (𝑠 ∈ ℤ ∧ 𝑡 ∈ ℤ))) → ((𝑠 · 𝑢) + (𝑡 · 𝑣)) ∈ ℤ)
40	7	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ) ∧ (𝑠 ∈ ℤ ∧ 𝑡 ∈ ℤ))) → (𝐴 gcd 𝐵) ∈ ℤ)
41		dvdsmul2 16224	. . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ) ∧ (𝑠 ∈ ℤ ∧ 𝑡 ∈ ℤ))) → (𝐴 gcd 𝐵) ∥ (((𝑠 · 𝑢) + (𝑡 · 𝑣)) · (𝐴 gcd 𝐵)))
43	35	zcnd 12669	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ) ∧ (𝑠 ∈ ℤ ∧ 𝑡 ∈ ℤ))) → (𝑠 · 𝑢) ∈ ℂ)
44	40	zcnd 12669	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ) ∧ (𝑠 ∈ ℤ ∧ 𝑡 ∈ ℤ))) → (𝐴 gcd 𝐵) ∈ ℂ)
45	38	zcnd 12669	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ) ∧ (𝑠 ∈ ℤ ∧ 𝑡 ∈ ℤ))) → (𝑡 · 𝑣) ∈ ℂ)
46	33	zcnd 12669	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ((𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ) ∧ (𝑠 ∈ ℤ ∧ 𝑡 ∈ ℤ))) → 𝑠 ∈ ℂ)
47	34	zcnd 12669	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ((𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ) ∧ (𝑠 ∈ ℤ ∧ 𝑡 ∈ ℤ))) → 𝑢 ∈ ℂ)
48	46, 47, 44	mul32d 11426	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ) ∧ (𝑠 ∈ ℤ ∧ 𝑡 ∈ ℤ))) → ((𝑠 · 𝑢) · (𝐴 gcd 𝐵)) = ((𝑠 · (𝐴 gcd 𝐵)) · 𝑢))
49	36	zcnd 12669	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ((𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ) ∧ (𝑠 ∈ ℤ ∧ 𝑡 ∈ ℤ))) → 𝑡 ∈ ℂ)
50	37	zcnd 12669	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ((𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ) ∧ (𝑠 ∈ ℤ ∧ 𝑡 ∈ ℤ))) → 𝑣 ∈ ℂ)
51	49, 50, 44	mul32d 11426	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ) ∧ (𝑠 ∈ ℤ ∧ 𝑡 ∈ ℤ))) → ((𝑡 · 𝑣) · (𝐴 gcd 𝐵)) = ((𝑡 · (𝐴 gcd 𝐵)) · 𝑣))
52	48, 51	oveq12d 7429	. . . . . . . . . . . . . . 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 11243	. . . . . . . . . . . . . 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 5174	. . . . . . . . . . . . 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 7418	. . . . . . . . . . . . . . 15 ⊢ ((𝑠 · (𝐴 gcd 𝐵)) = 𝐴 → ((𝑠 · (𝐴 gcd 𝐵)) · 𝑢) = (𝐴 · 𝑢))
56		oveq1 7418	. . . . . . . . . . . . . . 15 ⊢ ((𝑡 · (𝐴 gcd 𝐵)) = 𝐵 → ((𝑡 · (𝐴 gcd 𝐵)) · 𝑣) = (𝐵 · 𝑣))
57	55, 56	oveqan12d 7430	. . . . . . . . . . . . . 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 5160	. . . . . . . . . . . . 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 5152	. . . . . . . . . . . . 13 ⊢ (𝐺 = ((𝐴 · 𝑢) + (𝐵 · 𝑣)) → ((𝐴 gcd 𝐵) ∥ 𝐺 ↔ (𝐴 gcd 𝐵) ∥ ((𝐴 · 𝑢) + (𝐵 · 𝑣))))
61	60	imbi2d 340	. . . . . . . . . . . 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 457	. . . . . . . . . 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 3211	. . . . . . . 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 3210	. . . . . 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 697	. . . 4 ⊢ (𝜑 → (𝐴 gcd 𝐵) ∥ 𝐺)
70	31	simpld 495	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ ℕ)
71		dvdsle 16255	. . . . 5 ⊢ (((𝐴 gcd 𝐵) ∈ ℤ ∧ 𝐺 ∈ ℕ) → ((𝐴 gcd 𝐵) ∥ 𝐺 → (𝐴 gcd 𝐵) ≤ 𝐺))
72	7, 70, 71	syl2anc 584	. . . 4 ⊢ (𝜑 → ((𝐴 gcd 𝐵) ∥ 𝐺 → (𝐴 gcd 𝐵) ≤ 𝐺))
73	69, 72	mpd 15	. . 3 ⊢ (𝜑 → (𝐴 gcd 𝐵) ≤ 𝐺)
74		breq2 5152	. . . . 5 ⊢ (𝐴 = 0 → (𝐺 ∥ 𝐴 ↔ 𝐺 ∥ 0))
75	16, 1, 2	bezoutlem1 16483	. . . . . . . 8 ⊢ (𝜑 → (𝐴 ≠ 0 → (abs‘𝐴) ∈ 𝑀))
76	16, 1, 2, 17, 18	bezoutlem3 16485	. . . . . . . 8 ⊢ (𝜑 → ((abs‘𝐴) ∈ 𝑀 → 𝐺 ∥ (abs‘𝐴)))
77	75, 76	syld 47	. . . . . . 7 ⊢ (𝜑 → (𝐴 ≠ 0 → 𝐺 ∥ (abs‘𝐴)))
78	70	nnzd 12587	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ ℤ)
79		dvdsabsb 16221	. . . . . . . 8 ⊢ ((𝐺 ∈ ℤ ∧ 𝐴 ∈ ℤ) → (𝐺 ∥ 𝐴 ↔ 𝐺 ∥ (abs‘𝐴)))
80	78, 1, 79	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → (𝐺 ∥ 𝐴 ↔ 𝐺 ∥ (abs‘𝐴)))
81	77, 80	sylibrd 258	. . . . . 6 ⊢ (𝜑 → (𝐴 ≠ 0 → 𝐺 ∥ 𝐴))
82	81	imp 407	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → 𝐺 ∥ 𝐴)
83		dvds0 16217	. . . . . 6 ⊢ (𝐺 ∈ ℤ → 𝐺 ∥ 0)
84	78, 83	syl 17	. . . . 5 ⊢ (𝜑 → 𝐺 ∥ 0)
85	74, 82, 84	pm2.61ne 3027	. . . 4 ⊢ (𝜑 → 𝐺 ∥ 𝐴)
86		breq2 5152	. . . . 5 ⊢ (𝐵 = 0 → (𝐺 ∥ 𝐵 ↔ 𝐺 ∥ 0))
87		eqid 2732	. . . . . . . . . 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 16483	. . . . . . . . 9 ⊢ (𝜑 → (𝐵 ≠ 0 → (abs‘𝐵) ∈ {𝑧 ∈ ℕ ∣ ∃𝑦 ∈ ℤ ∃𝑥 ∈ ℤ 𝑧 = ((𝐵 · 𝑦) + (𝐴 · 𝑥))}))
89		rexcom 3287	. . . . . . . . . . . . 13 ⊢ (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝑧 = ((𝐴 · 𝑥) + (𝐵 · 𝑦)) ↔ ∃𝑦 ∈ ℤ ∃𝑥 ∈ ℤ 𝑧 = ((𝐴 · 𝑥) + (𝐵 · 𝑦)))
90	1	zcnd 12669	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐴 ∈ ℂ)
91	90	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑥 ∈ ℤ)) → 𝐴 ∈ ℂ)
92		zcn 12565	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ ℂ)
93	92	ad2antll 727	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑥 ∈ ℤ)) → 𝑥 ∈ ℂ)
94	91, 93	mulcld 11236	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑥 ∈ ℤ)) → (𝐴 · 𝑥) ∈ ℂ)
95	2	zcnd 12669	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐵 ∈ ℂ)
96	95	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑥 ∈ ℤ)) → 𝐵 ∈ ℂ)
97		zcn 12565	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ ℤ → 𝑦 ∈ ℂ)
98	97	ad2antrl 726	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑥 ∈ ℤ)) → 𝑦 ∈ ℂ)
99	96, 98	mulcld 11236	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑥 ∈ ℤ)) → (𝐵 · 𝑦) ∈ ℂ)
100	94, 99	addcomd 11418	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑥 ∈ ℤ)) → ((𝐴 · 𝑥) + (𝐵 · 𝑦)) = ((𝐵 · 𝑦) + (𝐴 · 𝑥)))
101	100	eqeq2d 2743	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑥 ∈ ℤ)) → (𝑧 = ((𝐴 · 𝑥) + (𝐵 · 𝑦)) ↔ 𝑧 = ((𝐵 · 𝑦) + (𝐴 · 𝑥))))
102	101	2rexbidva 3217	. . . . . . . . . . . . 13 ⊢ (𝜑 → (∃𝑦 ∈ ℤ ∃𝑥 ∈ ℤ 𝑧 = ((𝐴 · 𝑥) + (𝐵 · 𝑦)) ↔ ∃𝑦 ∈ ℤ ∃𝑥 ∈ ℤ 𝑧 = ((𝐵 · 𝑦) + (𝐴 · 𝑥))))
103	89, 102	bitrid 282	. . . . . . . . . . . 12 ⊢ (𝜑 → (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝑧 = ((𝐴 · 𝑥) + (𝐵 · 𝑦)) ↔ ∃𝑦 ∈ ℤ ∃𝑥 ∈ ℤ 𝑧 = ((𝐵 · 𝑦) + (𝐴 · 𝑥))))
104	103	rabbidv 3440	. . . . . . . . . . 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 2784	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 = {𝑧 ∈ ℕ ∣ ∃𝑦 ∈ ℤ ∃𝑥 ∈ ℤ 𝑧 = ((𝐵 · 𝑦) + (𝐴 · 𝑥))})
106	105	eleq2d 2819	. . . . . . . . 9 ⊢ (𝜑 → ((abs‘𝐵) ∈ 𝑀 ↔ (abs‘𝐵) ∈ {𝑧 ∈ ℕ ∣ ∃𝑦 ∈ ℤ ∃𝑥 ∈ ℤ 𝑧 = ((𝐵 · 𝑦) + (𝐴 · 𝑥))}))
107	88, 106	sylibrd 258	. . . . . . . 8 ⊢ (𝜑 → (𝐵 ≠ 0 → (abs‘𝐵) ∈ 𝑀))
108	16, 1, 2, 17, 18	bezoutlem3 16485	. . . . . . . 8 ⊢ (𝜑 → ((abs‘𝐵) ∈ 𝑀 → 𝐺 ∥ (abs‘𝐵)))
109	107, 108	syld 47	. . . . . . 7 ⊢ (𝜑 → (𝐵 ≠ 0 → 𝐺 ∥ (abs‘𝐵)))
110		dvdsabsb 16221	. . . . . . . 8 ⊢ ((𝐺 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐺 ∥ 𝐵 ↔ 𝐺 ∥ (abs‘𝐵)))
111	78, 2, 110	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → (𝐺 ∥ 𝐵 ↔ 𝐺 ∥ (abs‘𝐵)))
112	109, 111	sylibrd 258	. . . . . 6 ⊢ (𝜑 → (𝐵 ≠ 0 → 𝐺 ∥ 𝐵))
113	112	imp 407	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≠ 0) → 𝐺 ∥ 𝐵)
114	86, 113, 84	pm2.61ne 3027	. . . 4 ⊢ (𝜑 → 𝐺 ∥ 𝐵)
115		dvdslegcd 16447	. . . . 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 1373	. . . 4 ⊢ (𝜑 → ((𝐺 ∥ 𝐴 ∧ 𝐺 ∥ 𝐵) → 𝐺 ≤ (𝐴 gcd 𝐵)))
117	85, 114, 116	mp2and 697	. . 3 ⊢ (𝜑 → 𝐺 ≤ (𝐴 gcd 𝐵))
118	6	nn0red 12535	. . . 4 ⊢ (𝜑 → (𝐴 gcd 𝐵) ∈ ℝ)
119	70	nnred 12229	. . . 4 ⊢ (𝜑 → 𝐺 ∈ ℝ)
120	118, 119	letri3d 11358	. . 3 ⊢ (𝜑 → ((𝐴 gcd 𝐵) = 𝐺 ↔ ((𝐴 gcd 𝐵) ≤ 𝐺 ∧ 𝐺 ≤ (𝐴 gcd 𝐵))))
121	73, 117, 120	mpbir2and 711	. 2 ⊢ (𝜑 → (𝐴 gcd 𝐵) = 𝐺)
122	121, 19	eqeltrd 2833	1 ⊢ (𝜑 → (𝐴 gcd 𝐵) ∈ 𝑀)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ∃wrex 3070 {crab 3432 class class class wbr 5148 ‘cfv 6543 (class class class)co 7411 infcinf 9438 ℂcc 11110 ℝcr 11111 0cc0 11112 + caddc 11115 · cmul 11117 < clt 11250 ≤ cle 11251 ℕcn 12214 ℤcz 12560 abscabs 15183 ∥ cdvds 16199 gcd cgcd 16437

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 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 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-sup 9439 df-inf 9440 df-pnf 11252 df-mnf 11253 df-xr 11254 df-ltxr 11255 df-le 11256 df-sub 11448 df-neg 11449 df-div 11874 df-nn 12215 df-2 12277 df-3 12278 df-n0 12475 df-z 12561 df-uz 12825 df-rp 12977 df-fl 13759 df-mod 13837 df-seq 13969 df-exp 14030 df-cj 15048 df-re 15049 df-im 15050 df-sqrt 15184 df-abs 15185 df-dvds 16200 df-gcd 16438

This theorem is referenced by: bezout 16487

W3C validator