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Theorem bezoutlem4 16178

Description: Lemma for bezout 16179. (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 16138	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴 gcd 𝐵) ∥ 𝐴 ∧ (𝐴 gcd 𝐵) ∥ 𝐵))
4	1, 2, 3	syl2anc 583	. . . . . . 7 ⊢ (𝜑 → ((𝐴 gcd 𝐵) ∥ 𝐴 ∧ (𝐴 gcd 𝐵) ∥ 𝐵))
5	4	simpld 494	. . . . . 6 ⊢ (𝜑 → (𝐴 gcd 𝐵) ∥ 𝐴)
6	1, 2	gcdcld 16143	. . . . . . . 8 ⊢ (𝜑 → (𝐴 gcd 𝐵) ∈ ℕ₀)
7	6	nn0zd 12353	. . . . . . 7 ⊢ (𝜑 → (𝐴 gcd 𝐵) ∈ ℤ)
8		divides 15893	. . . . . . 7 ⊢ (((𝐴 gcd 𝐵) ∈ ℤ ∧ 𝐴 ∈ ℤ) → ((𝐴 gcd 𝐵) ∥ 𝐴 ↔ ∃𝑠 ∈ ℤ (𝑠 · (𝐴 gcd 𝐵)) = 𝐴))
9	7, 1, 8	syl2anc 583	. . . . . 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 495	. . . . . 6 ⊢ (𝜑 → (𝐴 gcd 𝐵) ∥ 𝐵)
12		divides 15893	. . . . . . 7 ⊢ (((𝐴 gcd 𝐵) ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴 gcd 𝐵) ∥ 𝐵 ↔ ∃𝑡 ∈ ℤ (𝑡 · (𝐴 gcd 𝐵)) = 𝐵))
13	7, 2, 12	syl2anc 583	. . . . . 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 3292	. . . . . 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 16176	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 ∈ 𝑀)
20		oveq2 7263	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑢 → (𝐴 · 𝑥) = (𝐴 · 𝑢))
21	20	oveq1d 7270	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑢 → ((𝐴 · 𝑥) + (𝐵 · 𝑦)) = ((𝐴 · 𝑢) + (𝐵 · 𝑦)))
22	21	eqeq2d 2749	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑢 → (𝑧 = ((𝐴 · 𝑥) + (𝐵 · 𝑦)) ↔ 𝑧 = ((𝐴 · 𝑢) + (𝐵 · 𝑦))))
23		oveq2 7263	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑣 → (𝐵 · 𝑦) = (𝐵 · 𝑣))
24	23	oveq2d 7271	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑣 → ((𝐴 · 𝑢) + (𝐵 · 𝑦)) = ((𝐴 · 𝑢) + (𝐵 · 𝑣)))
25	24	eqeq2d 2749	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑣 → (𝑧 = ((𝐴 · 𝑢) + (𝐵 · 𝑦)) ↔ 𝑧 = ((𝐴 · 𝑢) + (𝐵 · 𝑣))))
26	22, 25	cbvrex2vw 3386	. . . . . . . . . . . 12 ⊢ (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝑧 = ((𝐴 · 𝑥) + (𝐵 · 𝑦)) ↔ ∃𝑢 ∈ ℤ ∃𝑣 ∈ ℤ 𝑧 = ((𝐴 · 𝑢) + (𝐵 · 𝑣)))
27		eqeq1 2742	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝐺 → (𝑧 = ((𝐴 · 𝑢) + (𝐵 · 𝑣)) ↔ 𝐺 = ((𝐴 · 𝑢) + (𝐵 · 𝑣))))
28	27	2rexbidv 3228	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝐺 → (∃𝑢 ∈ ℤ ∃𝑣 ∈ ℤ 𝑧 = ((𝐴 · 𝑢) + (𝐵 · 𝑣)) ↔ ∃𝑢 ∈ ℤ ∃𝑣 ∈ ℤ 𝐺 = ((𝐴 · 𝑢) + (𝐵 · 𝑣))))
29	26, 28	syl5bb 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 3620	. . . . . . . . . 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 495	. . . . . . . 8 ⊢ (𝜑 → ∃𝑢 ∈ ℤ ∃𝑣 ∈ ℤ 𝐺 = ((𝐴 · 𝑢) + (𝐵 · 𝑣)))
33		simprrl 777	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ((𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ) ∧ (𝑠 ∈ ℤ ∧ 𝑡 ∈ ℤ))) → 𝑠 ∈ ℤ)
34		simprll 775	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ((𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ) ∧ (𝑠 ∈ ℤ ∧ 𝑡 ∈ ℤ))) → 𝑢 ∈ ℤ)
35	33, 34	zmulcld 12361	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ) ∧ (𝑠 ∈ ℤ ∧ 𝑡 ∈ ℤ))) → (𝑠 · 𝑢) ∈ ℤ)
36		simprrr 778	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ((𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ) ∧ (𝑠 ∈ ℤ ∧ 𝑡 ∈ ℤ))) → 𝑡 ∈ ℤ)
37		simprlr 776	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ((𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ) ∧ (𝑠 ∈ ℤ ∧ 𝑡 ∈ ℤ))) → 𝑣 ∈ ℤ)
38	36, 37	zmulcld 12361	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ) ∧ (𝑠 ∈ ℤ ∧ 𝑡 ∈ ℤ))) → (𝑡 · 𝑣) ∈ ℤ)
39	35, 38	zaddcld 12359	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ) ∧ (𝑠 ∈ ℤ ∧ 𝑡 ∈ ℤ))) → ((𝑠 · 𝑢) + (𝑡 · 𝑣)) ∈ ℤ)
40	7	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ) ∧ (𝑠 ∈ ℤ ∧ 𝑡 ∈ ℤ))) → (𝐴 gcd 𝐵) ∈ ℤ)
41		dvdsmul2 15916	. . . . . . . . . . . . . . 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 583	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ) ∧ (𝑠 ∈ ℤ ∧ 𝑡 ∈ ℤ))) → (𝐴 gcd 𝐵) ∥ (((𝑠 · 𝑢) + (𝑡 · 𝑣)) · (𝐴 gcd 𝐵)))
43	35	zcnd 12356	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ) ∧ (𝑠 ∈ ℤ ∧ 𝑡 ∈ ℤ))) → (𝑠 · 𝑢) ∈ ℂ)
44	40	zcnd 12356	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ) ∧ (𝑠 ∈ ℤ ∧ 𝑡 ∈ ℤ))) → (𝐴 gcd 𝐵) ∈ ℂ)
45	38	zcnd 12356	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ) ∧ (𝑠 ∈ ℤ ∧ 𝑡 ∈ ℤ))) → (𝑡 · 𝑣) ∈ ℂ)
46	33	zcnd 12356	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ((𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ) ∧ (𝑠 ∈ ℤ ∧ 𝑡 ∈ ℤ))) → 𝑠 ∈ ℂ)
47	34	zcnd 12356	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ((𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ) ∧ (𝑠 ∈ ℤ ∧ 𝑡 ∈ ℤ))) → 𝑢 ∈ ℂ)
48	46, 47, 44	mul32d 11115	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ) ∧ (𝑠 ∈ ℤ ∧ 𝑡 ∈ ℤ))) → ((𝑠 · 𝑢) · (𝐴 gcd 𝐵)) = ((𝑠 · (𝐴 gcd 𝐵)) · 𝑢))
49	36	zcnd 12356	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ((𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ) ∧ (𝑠 ∈ ℤ ∧ 𝑡 ∈ ℤ))) → 𝑡 ∈ ℂ)
50	37	zcnd 12356	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ((𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ) ∧ (𝑠 ∈ ℤ ∧ 𝑡 ∈ ℤ))) → 𝑣 ∈ ℂ)
51	49, 50, 44	mul32d 11115	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ) ∧ (𝑠 ∈ ℤ ∧ 𝑡 ∈ ℤ))) → ((𝑡 · 𝑣) · (𝐴 gcd 𝐵)) = ((𝑡 · (𝐴 gcd 𝐵)) · 𝑣))
52	48, 51	oveq12d 7273	. . . . . . . . . . . . . . 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 10933	. . . . . . . . . . . . . 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 5096	. . . . . . . . . . . . 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 7262	. . . . . . . . . . . . . . 15 ⊢ ((𝑠 · (𝐴 gcd 𝐵)) = 𝐴 → ((𝑠 · (𝐴 gcd 𝐵)) · 𝑢) = (𝐴 · 𝑢))
56		oveq1 7262	. . . . . . . . . . . . . . 15 ⊢ ((𝑡 · (𝐴 gcd 𝐵)) = 𝐵 → ((𝑡 · (𝐴 gcd 𝐵)) · 𝑣) = (𝐵 · 𝑣))
57	55, 56	oveqan12d 7274	. . . . . . . . . . . . . 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 5082	. . . . . . . . . . . . 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 5074	. . . . . . . . . . . . 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 456	. . . . . . . . . 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 3222	. . . . . . . 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 3221	. . . . . 6 ⊢ (𝜑 → (∃𝑠 ∈ ℤ ∃𝑡 ∈ ℤ ((𝑠 · (𝐴 gcd 𝐵)) = 𝐴 ∧ (𝑡 · (𝐴 gcd 𝐵)) = 𝐵) → (𝐴 gcd 𝐵) ∥ 𝐺))
68	15, 67	syl5bir 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 695	. . . 4 ⊢ (𝜑 → (𝐴 gcd 𝐵) ∥ 𝐺)
70	31	simpld 494	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ ℕ)
71		dvdsle 15947	. . . . 5 ⊢ (((𝐴 gcd 𝐵) ∈ ℤ ∧ 𝐺 ∈ ℕ) → ((𝐴 gcd 𝐵) ∥ 𝐺 → (𝐴 gcd 𝐵) ≤ 𝐺))
72	7, 70, 71	syl2anc 583	. . . 4 ⊢ (𝜑 → ((𝐴 gcd 𝐵) ∥ 𝐺 → (𝐴 gcd 𝐵) ≤ 𝐺))
73	69, 72	mpd 15	. . 3 ⊢ (𝜑 → (𝐴 gcd 𝐵) ≤ 𝐺)
74		breq2 5074	. . . . 5 ⊢ (𝐴 = 0 → (𝐺 ∥ 𝐴 ↔ 𝐺 ∥ 0))
75	16, 1, 2	bezoutlem1 16175	. . . . . . . 8 ⊢ (𝜑 → (𝐴 ≠ 0 → (abs‘𝐴) ∈ 𝑀))
76	16, 1, 2, 17, 18	bezoutlem3 16177	. . . . . . . 8 ⊢ (𝜑 → ((abs‘𝐴) ∈ 𝑀 → 𝐺 ∥ (abs‘𝐴)))
77	75, 76	syld 47	. . . . . . 7 ⊢ (𝜑 → (𝐴 ≠ 0 → 𝐺 ∥ (abs‘𝐴)))
78	70	nnzd 12354	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ ℤ)
79		dvdsabsb 15913	. . . . . . . 8 ⊢ ((𝐺 ∈ ℤ ∧ 𝐴 ∈ ℤ) → (𝐺 ∥ 𝐴 ↔ 𝐺 ∥ (abs‘𝐴)))
80	78, 1, 79	syl2anc 583	. . . . . . 7 ⊢ (𝜑 → (𝐺 ∥ 𝐴 ↔ 𝐺 ∥ (abs‘𝐴)))
81	77, 80	sylibrd 258	. . . . . 6 ⊢ (𝜑 → (𝐴 ≠ 0 → 𝐺 ∥ 𝐴))
82	81	imp 406	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → 𝐺 ∥ 𝐴)
83		dvds0 15909	. . . . . 6 ⊢ (𝐺 ∈ ℤ → 𝐺 ∥ 0)
84	78, 83	syl 17	. . . . 5 ⊢ (𝜑 → 𝐺 ∥ 0)
85	74, 82, 84	pm2.61ne 3029	. . . 4 ⊢ (𝜑 → 𝐺 ∥ 𝐴)
86		breq2 5074	. . . . 5 ⊢ (𝐵 = 0 → (𝐺 ∥ 𝐵 ↔ 𝐺 ∥ 0))
87		eqid 2738	. . . . . . . . . 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 16175	. . . . . . . . 9 ⊢ (𝜑 → (𝐵 ≠ 0 → (abs‘𝐵) ∈ {𝑧 ∈ ℕ ∣ ∃𝑦 ∈ ℤ ∃𝑥 ∈ ℤ 𝑧 = ((𝐵 · 𝑦) + (𝐴 · 𝑥))}))
89		rexcom 3281	. . . . . . . . . . . . 13 ⊢ (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝑧 = ((𝐴 · 𝑥) + (𝐵 · 𝑦)) ↔ ∃𝑦 ∈ ℤ ∃𝑥 ∈ ℤ 𝑧 = ((𝐴 · 𝑥) + (𝐵 · 𝑦)))
90	1	zcnd 12356	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐴 ∈ ℂ)
91	90	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑥 ∈ ℤ)) → 𝐴 ∈ ℂ)
92		zcn 12254	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ ℂ)
93	92	ad2antll 725	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑥 ∈ ℤ)) → 𝑥 ∈ ℂ)
94	91, 93	mulcld 10926	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑥 ∈ ℤ)) → (𝐴 · 𝑥) ∈ ℂ)
95	2	zcnd 12356	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐵 ∈ ℂ)
96	95	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑥 ∈ ℤ)) → 𝐵 ∈ ℂ)
97		zcn 12254	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ ℤ → 𝑦 ∈ ℂ)
98	97	ad2antrl 724	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑥 ∈ ℤ)) → 𝑦 ∈ ℂ)
99	96, 98	mulcld 10926	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑥 ∈ ℤ)) → (𝐵 · 𝑦) ∈ ℂ)
100	94, 99	addcomd 11107	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑥 ∈ ℤ)) → ((𝐴 · 𝑥) + (𝐵 · 𝑦)) = ((𝐵 · 𝑦) + (𝐴 · 𝑥)))
101	100	eqeq2d 2749	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑥 ∈ ℤ)) → (𝑧 = ((𝐴 · 𝑥) + (𝐵 · 𝑦)) ↔ 𝑧 = ((𝐵 · 𝑦) + (𝐴 · 𝑥))))
102	101	2rexbidva 3227	. . . . . . . . . . . . 13 ⊢ (𝜑 → (∃𝑦 ∈ ℤ ∃𝑥 ∈ ℤ 𝑧 = ((𝐴 · 𝑥) + (𝐵 · 𝑦)) ↔ ∃𝑦 ∈ ℤ ∃𝑥 ∈ ℤ 𝑧 = ((𝐵 · 𝑦) + (𝐴 · 𝑥))))
103	89, 102	syl5bb 282	. . . . . . . . . . . 12 ⊢ (𝜑 → (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝑧 = ((𝐴 · 𝑥) + (𝐵 · 𝑦)) ↔ ∃𝑦 ∈ ℤ ∃𝑥 ∈ ℤ 𝑧 = ((𝐵 · 𝑦) + (𝐴 · 𝑥))))
104	103	rabbidv 3404	. . . . . . . . . . 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 2790	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 = {𝑧 ∈ ℕ ∣ ∃𝑦 ∈ ℤ ∃𝑥 ∈ ℤ 𝑧 = ((𝐵 · 𝑦) + (𝐴 · 𝑥))})
106	105	eleq2d 2824	. . . . . . . . 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 16177	. . . . . . . 8 ⊢ (𝜑 → ((abs‘𝐵) ∈ 𝑀 → 𝐺 ∥ (abs‘𝐵)))
109	107, 108	syld 47	. . . . . . 7 ⊢ (𝜑 → (𝐵 ≠ 0 → 𝐺 ∥ (abs‘𝐵)))
110		dvdsabsb 15913	. . . . . . . 8 ⊢ ((𝐺 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐺 ∥ 𝐵 ↔ 𝐺 ∥ (abs‘𝐵)))
111	78, 2, 110	syl2anc 583	. . . . . . 7 ⊢ (𝜑 → (𝐺 ∥ 𝐵 ↔ 𝐺 ∥ (abs‘𝐵)))
112	109, 111	sylibrd 258	. . . . . 6 ⊢ (𝜑 → (𝐵 ≠ 0 → 𝐺 ∥ 𝐵))
113	112	imp 406	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≠ 0) → 𝐺 ∥ 𝐵)
114	86, 113, 84	pm2.61ne 3029	. . . 4 ⊢ (𝜑 → 𝐺 ∥ 𝐵)
115		dvdslegcd 16139	. . . . 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 1371	. . . 4 ⊢ (𝜑 → ((𝐺 ∥ 𝐴 ∧ 𝐺 ∥ 𝐵) → 𝐺 ≤ (𝐴 gcd 𝐵)))
117	85, 114, 116	mp2and 695	. . 3 ⊢ (𝜑 → 𝐺 ≤ (𝐴 gcd 𝐵))
118	6	nn0red 12224	. . . 4 ⊢ (𝜑 → (𝐴 gcd 𝐵) ∈ ℝ)
119	70	nnred 11918	. . . 4 ⊢ (𝜑 → 𝐺 ∈ ℝ)
120	118, 119	letri3d 11047	. . 3 ⊢ (𝜑 → ((𝐴 gcd 𝐵) = 𝐺 ↔ ((𝐴 gcd 𝐵) ≤ 𝐺 ∧ 𝐺 ≤ (𝐴 gcd 𝐵))))
121	73, 117, 120	mpbir2and 709	. 2 ⊢ (𝜑 → (𝐴 gcd 𝐵) = 𝐺)
122	121, 19	eqeltrd 2839	1 ⊢ (𝜑 → (𝐴 gcd 𝐵) ∈ 𝑀)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1539 ∈ wcel 2108 ≠ wne 2942 ∃wrex 3064 {crab 3067 class class class wbr 5070 ‘cfv 6418 (class class class)co 7255 infcinf 9130 ℂcc 10800 ℝcr 10801 0cc0 10802 + caddc 10805 · cmul 10807 < clt 10940 ≤ cle 10941 ℕcn 11903 ℤcz 12249 abscabs 14873 ∥ cdvds 15891 gcd cgcd 16129

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-er 8456 df-en 8692 df-dom 8693 df-sdom 8694 df-sup 9131 df-inf 9132 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-n0 12164 df-z 12250 df-uz 12512 df-rp 12660 df-fl 13440 df-mod 13518 df-seq 13650 df-exp 13711 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-dvds 15892 df-gcd 16130

This theorem is referenced by: bezout 16179

W3C validator