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Theorem gcdcllem1 16582

Description: Lemma for gcdn0cl 16585, gcddvds 16586 and dvdslegcd 16587. (Contributed by Paul Chapman, 21-Mar-2011.)

**Hypothesis**
Ref	Expression
gcdcllem1.1	⊢ 𝑆 = {𝑧 ∈ ℤ ∣ ∀𝑛 ∈ 𝐴 𝑧 ∥ 𝑛}

**Assertion**
Ref	Expression
gcdcllem1	⊢ ((𝐴 ⊆ ℤ ∧ ∃𝑛 ∈ 𝐴 𝑛 ≠ 0) → (𝑆 ≠ ∅ ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥))

Distinct variable groups: 𝐴,𝑛,𝑥,𝑦,𝑧 𝑥,𝑆 Allowed substitution hints: 𝑆(𝑦, 𝑧, 𝑛)

Proof of Theorem gcdcllem1 Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		1z 12642	. . . . 5 ⊢ 1 ∈ ℤ
2		ssel 3934	. . . . . . 7 ⊢ (𝐴 ⊆ ℤ → (𝑛 ∈ 𝐴 → 𝑛 ∈ ℤ))
3		1dvds 16353	. . . . . . 7 ⊢ (𝑛 ∈ ℤ → 1 ∥ 𝑛)
4	2, 3	syl6 36	. . . . . 6 ⊢ (𝐴 ⊆ ℤ → (𝑛 ∈ 𝐴 → 1 ∥ 𝑛))
5	4	ralrimiv 3159	. . . . 5 ⊢ (𝐴 ⊆ ℤ → ∀𝑛 ∈ 𝐴 1 ∥ 𝑛)
6		breq1 5117	. . . . . . . 8 ⊢ (𝑧 = 1 → (𝑧 ∥ 𝑛 ↔ 1 ∥ 𝑛))
7	6	ralbidv 3191	. . . . . . 7 ⊢ (𝑧 = 1 → (∀𝑛 ∈ 𝐴 𝑧 ∥ 𝑛 ↔ ∀𝑛 ∈ 𝐴 1 ∥ 𝑛))
8		gcdcllem1.1	. . . . . . 7 ⊢ 𝑆 = {𝑧 ∈ ℤ ∣ ∀𝑛 ∈ 𝐴 𝑧 ∥ 𝑛}
9	7, 8	elrab2 3657	. . . . . 6 ⊢ (1 ∈ 𝑆 ↔ (1 ∈ ℤ ∧ ∀𝑛 ∈ 𝐴 1 ∥ 𝑛))
10	9	biimpri 231	. . . . 5 ⊢ ((1 ∈ ℤ ∧ ∀𝑛 ∈ 𝐴 1 ∥ 𝑛) → 1 ∈ 𝑆)
11	1, 5, 10	sylancr 599	. . . 4 ⊢ (𝐴 ⊆ ℤ → 1 ∈ 𝑆)
12	11	ne0d 4298	. . 3 ⊢ (𝐴 ⊆ ℤ → 𝑆 ≠ ∅)
13	12	adantr 486	. 2 ⊢ ((𝐴 ⊆ ℤ ∧ ∃𝑛 ∈ 𝐴 𝑛 ≠ 0) → 𝑆 ≠ ∅)
14		neeq1 3023	. . . 4 ⊢ (𝑛 = 𝑤 → (𝑛 ≠ 0 ↔ 𝑤 ≠ 0))
15	14	cbvrexvw 3247	. . 3 ⊢ (∃𝑛 ∈ 𝐴 𝑛 ≠ 0 ↔ ∃𝑤 ∈ 𝐴 𝑤 ≠ 0)
16		breq1 5117	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑦 → (𝑧 ∥ 𝑛 ↔ 𝑦 ∥ 𝑛))
17	16	ralbidv 3191	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑦 → (∀𝑛 ∈ 𝐴 𝑧 ∥ 𝑛 ↔ ∀𝑛 ∈ 𝐴 𝑦 ∥ 𝑛))
18	17, 8	elrab2 3657	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝑆 ↔ (𝑦 ∈ ℤ ∧ ∀𝑛 ∈ 𝐴 𝑦 ∥ 𝑛))
19	18	simprbi 503	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝑆 → ∀𝑛 ∈ 𝐴 𝑦 ∥ 𝑛)
20	18	simplbi 502	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝑆 → 𝑦 ∈ ℤ)
21		ssel2 3935	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ⊆ ℤ ∧ 𝑛 ∈ 𝐴) → 𝑛 ∈ ℤ)
22		dvdsleabs 16394	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ℤ ∧ 𝑛 ∈ ℤ ∧ 𝑛 ≠ 0) → (𝑦 ∥ 𝑛 → 𝑦 ≤ (abs‘𝑛)))
23	22	3expia 1139	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑛 ≠ 0 → (𝑦 ∥ 𝑛 → 𝑦 ≤ (abs‘𝑛))))
24	21, 23	sylan2 605	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ℤ ∧ (𝐴 ⊆ ℤ ∧ 𝑛 ∈ 𝐴)) → (𝑛 ≠ 0 → (𝑦 ∥ 𝑛 → 𝑦 ≤ (abs‘𝑛))))
25	24	anassrs 473	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∈ ℤ ∧ 𝐴 ⊆ ℤ) ∧ 𝑛 ∈ 𝐴) → (𝑛 ≠ 0 → (𝑦 ∥ 𝑛 → 𝑦 ≤ (abs‘𝑛))))
26	25	com23 87	. . . . . . . . . . . 12 ⊢ (((𝑦 ∈ ℤ ∧ 𝐴 ⊆ ℤ) ∧ 𝑛 ∈ 𝐴) → (𝑦 ∥ 𝑛 → (𝑛 ≠ 0 → 𝑦 ≤ (abs‘𝑛))))
27	26	ralrimiva 3160	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℤ ∧ 𝐴 ⊆ ℤ) → ∀𝑛 ∈ 𝐴 (𝑦 ∥ 𝑛 → (𝑛 ≠ 0 → 𝑦 ≤ (abs‘𝑛))))
28	27	ancoms 464	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ ℤ ∧ 𝑦 ∈ ℤ) → ∀𝑛 ∈ 𝐴 (𝑦 ∥ 𝑛 → (𝑛 ≠ 0 → 𝑦 ≤ (abs‘𝑛))))
29	20, 28	sylan2 605	. . . . . . . . 9 ⊢ ((𝐴 ⊆ ℤ ∧ 𝑦 ∈ 𝑆) → ∀𝑛 ∈ 𝐴 (𝑦 ∥ 𝑛 → (𝑛 ≠ 0 → 𝑦 ≤ (abs‘𝑛))))
30		r19.26 3128	. . . . . . . . . 10 ⊢ (∀𝑛 ∈ 𝐴 (𝑦 ∥ 𝑛 ∧ (𝑦 ∥ 𝑛 → (𝑛 ≠ 0 → 𝑦 ≤ (abs‘𝑛)))) ↔ (∀𝑛 ∈ 𝐴 𝑦 ∥ 𝑛 ∧ ∀𝑛 ∈ 𝐴 (𝑦 ∥ 𝑛 → (𝑛 ≠ 0 → 𝑦 ≤ (abs‘𝑛)))))
31		pm3.35 815	. . . . . . . . . . 11 ⊢ ((𝑦 ∥ 𝑛 ∧ (𝑦 ∥ 𝑛 → (𝑛 ≠ 0 → 𝑦 ≤ (abs‘𝑛)))) → (𝑛 ≠ 0 → 𝑦 ≤ (abs‘𝑛)))
32	31	ralimi 3105	. . . . . . . . . 10 ⊢ (∀𝑛 ∈ 𝐴 (𝑦 ∥ 𝑛 ∧ (𝑦 ∥ 𝑛 → (𝑛 ≠ 0 → 𝑦 ≤ (abs‘𝑛)))) → ∀𝑛 ∈ 𝐴 (𝑛 ≠ 0 → 𝑦 ≤ (abs‘𝑛)))
33	30, 32	sylbir 238	. . . . . . . . 9 ⊢ ((∀𝑛 ∈ 𝐴 𝑦 ∥ 𝑛 ∧ ∀𝑛 ∈ 𝐴 (𝑦 ∥ 𝑛 → (𝑛 ≠ 0 → 𝑦 ≤ (abs‘𝑛)))) → ∀𝑛 ∈ 𝐴 (𝑛 ≠ 0 → 𝑦 ≤ (abs‘𝑛)))
34	19, 29, 33	syl2an2 699	. . . . . . . 8 ⊢ ((𝐴 ⊆ ℤ ∧ 𝑦 ∈ 𝑆) → ∀𝑛 ∈ 𝐴 (𝑛 ≠ 0 → 𝑦 ≤ (abs‘𝑛)))
35	34	ralrimiva 3160	. . . . . . 7 ⊢ (𝐴 ⊆ ℤ → ∀𝑦 ∈ 𝑆 ∀𝑛 ∈ 𝐴 (𝑛 ≠ 0 → 𝑦 ≤ (abs‘𝑛)))
36		fveq2 6888	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑤 → (abs‘𝑛) = (abs‘𝑤))
37	36	breq2d 5126	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑤 → (𝑦 ≤ (abs‘𝑛) ↔ 𝑦 ≤ (abs‘𝑤)))
38	14, 37	imbi12d 347	. . . . . . . . . 10 ⊢ (𝑛 = 𝑤 → ((𝑛 ≠ 0 → 𝑦 ≤ (abs‘𝑛)) ↔ (𝑤 ≠ 0 → 𝑦 ≤ (abs‘𝑤))))
39	38	cbvralvw 3246	. . . . . . . . 9 ⊢ (∀𝑛 ∈ 𝐴 (𝑛 ≠ 0 → 𝑦 ≤ (abs‘𝑛)) ↔ ∀𝑤 ∈ 𝐴 (𝑤 ≠ 0 → 𝑦 ≤ (abs‘𝑤)))
40	39	ralbii 3114	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝑆 ∀𝑛 ∈ 𝐴 (𝑛 ≠ 0 → 𝑦 ≤ (abs‘𝑛)) ↔ ∀𝑦 ∈ 𝑆 ∀𝑤 ∈ 𝐴 (𝑤 ≠ 0 → 𝑦 ≤ (abs‘𝑤)))
41		ralcom 3296	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝑆 ∀𝑤 ∈ 𝐴 (𝑤 ≠ 0 → 𝑦 ≤ (abs‘𝑤)) ↔ ∀𝑤 ∈ 𝐴 ∀𝑦 ∈ 𝑆 (𝑤 ≠ 0 → 𝑦 ≤ (abs‘𝑤)))
42		r19.21v 3193	. . . . . . . . 9 ⊢ (∀𝑦 ∈ 𝑆 (𝑤 ≠ 0 → 𝑦 ≤ (abs‘𝑤)) ↔ (𝑤 ≠ 0 → ∀𝑦 ∈ 𝑆 𝑦 ≤ (abs‘𝑤)))
43	42	ralbii 3114	. . . . . . . 8 ⊢ (∀𝑤 ∈ 𝐴 ∀𝑦 ∈ 𝑆 (𝑤 ≠ 0 → 𝑦 ≤ (abs‘𝑤)) ↔ ∀𝑤 ∈ 𝐴 (𝑤 ≠ 0 → ∀𝑦 ∈ 𝑆 𝑦 ≤ (abs‘𝑤)))
44	40, 41, 43	3bitri 300	. . . . . . 7 ⊢ (∀𝑦 ∈ 𝑆 ∀𝑛 ∈ 𝐴 (𝑛 ≠ 0 → 𝑦 ≤ (abs‘𝑛)) ↔ ∀𝑤 ∈ 𝐴 (𝑤 ≠ 0 → ∀𝑦 ∈ 𝑆 𝑦 ≤ (abs‘𝑤)))
45	35, 44	sylib 221	. . . . . 6 ⊢ (𝐴 ⊆ ℤ → ∀𝑤 ∈ 𝐴 (𝑤 ≠ 0 → ∀𝑦 ∈ 𝑆 𝑦 ≤ (abs‘𝑤)))
46		ssel2 3935	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ ℤ ∧ 𝑤 ∈ 𝐴) → 𝑤 ∈ ℤ)
47		nn0abscl 15389	. . . . . . . . . . 11 ⊢ (𝑤 ∈ ℤ → (abs‘𝑤) ∈ ℕ₀)
48	46, 47	syl 18	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ ℤ ∧ 𝑤 ∈ 𝐴) → (abs‘𝑤) ∈ ℕ₀)
49	48	nn0zd 12634	. . . . . . . . 9 ⊢ ((𝐴 ⊆ ℤ ∧ 𝑤 ∈ 𝐴) → (abs‘𝑤) ∈ ℤ)
50		breq2 5118	. . . . . . . . . . 11 ⊢ (𝑥 = (abs‘𝑤) → (𝑦 ≤ 𝑥 ↔ 𝑦 ≤ (abs‘𝑤)))
51	50	ralbidv 3191	. . . . . . . . . 10 ⊢ (𝑥 = (abs‘𝑤) → (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ↔ ∀𝑦 ∈ 𝑆 𝑦 ≤ (abs‘𝑤)))
52	51	adantl 487	. . . . . . . . 9 ⊢ (((𝐴 ⊆ ℤ ∧ 𝑤 ∈ 𝐴) ∧ 𝑥 = (abs‘𝑤)) → (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ↔ ∀𝑦 ∈ 𝑆 𝑦 ≤ (abs‘𝑤)))
53	49, 52	rspcedv 3577	. . . . . . . 8 ⊢ ((𝐴 ⊆ ℤ ∧ 𝑤 ∈ 𝐴) → (∀𝑦 ∈ 𝑆 𝑦 ≤ (abs‘𝑤) → ∃𝑥 ∈ ℤ ∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥))
54	53	imim2d 58	. . . . . . 7 ⊢ ((𝐴 ⊆ ℤ ∧ 𝑤 ∈ 𝐴) → ((𝑤 ≠ 0 → ∀𝑦 ∈ 𝑆 𝑦 ≤ (abs‘𝑤)) → (𝑤 ≠ 0 → ∃𝑥 ∈ ℤ ∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥)))
55	54	ralimdva 3180	. . . . . 6 ⊢ (𝐴 ⊆ ℤ → (∀𝑤 ∈ 𝐴 (𝑤 ≠ 0 → ∀𝑦 ∈ 𝑆 𝑦 ≤ (abs‘𝑤)) → ∀𝑤 ∈ 𝐴 (𝑤 ≠ 0 → ∃𝑥 ∈ ℤ ∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥)))
56	45, 55	mpd 16	. . . . 5 ⊢ (𝐴 ⊆ ℤ → ∀𝑤 ∈ 𝐴 (𝑤 ≠ 0 → ∃𝑥 ∈ ℤ ∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥))
57		r19.23v 3195	. . . . 5 ⊢ (∀𝑤 ∈ 𝐴 (𝑤 ≠ 0 → ∃𝑥 ∈ ℤ ∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥) ↔ (∃𝑤 ∈ 𝐴 𝑤 ≠ 0 → ∃𝑥 ∈ ℤ ∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥))
58	56, 57	sylib 221	. . . 4 ⊢ (𝐴 ⊆ ℤ → (∃𝑤 ∈ 𝐴 𝑤 ≠ 0 → ∃𝑥 ∈ ℤ ∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥))
59	58	imp 412	. . 3 ⊢ ((𝐴 ⊆ ℤ ∧ ∃𝑤 ∈ 𝐴 𝑤 ≠ 0) → ∃𝑥 ∈ ℤ ∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥)
60	15, 59	sylan2b 606	. 2 ⊢ ((𝐴 ⊆ ℤ ∧ ∃𝑛 ∈ 𝐴 𝑛 ≠ 0) → ∃𝑥 ∈ ℤ ∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥)
61	13, 60	jca 521	1 ⊢ ((𝐴 ⊆ ℤ ∧ ∃𝑛 ∈ 𝐴 𝑛 ≠ 0) → (𝑆 ≠ ∅ ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2146 ≠ wne 2961 ∀wral 3082 ∃wrex 3092 {crab 3419 ⊆ wss 3908 ∅c0 4289 class class class wbr 5114 ‘cfv 6543 0cc0 11118 1c1 11119 ≤ cle 11262 ℕ₀cn0 12522 ℤcz 12609 abscabs 15311 ∥ cdvds 16335

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2738 ax-sep 5262 ax-nul 5274 ax-pow 5341 ax-pr 5409 ax-un 7745 ax-cnex 11174 ax-resscn 11175 ax-1cn 11176 ax-icn 11177 ax-addcl 11178 ax-addrcl 11179 ax-mulcl 11180 ax-mulrcl 11181 ax-mulcom 11182 ax-addass 11183 ax-mulass 11184 ax-distr 11185 ax-i2m1 11186 ax-1ne0 11187 ax-1rid 11188 ax-rnegex 11189 ax-rrecex 11190 ax-cnre 11191 ax-pre-lttri 11192 ax-pre-lttrn 11193 ax-pre-ltadd 11194 ax-pre-mulgt0 11195 ax-pre-sup 11196

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2570 df-eu 2600 df-clab 2745 df-cleq 2758 df-clel 2841 df-nfc 2915 df-ne 2962 df-nel 3068 df-ral 3083 df-rex 3093 df-rmo 3372 df-reu 3373 df-rab 3420 df-v 3460 df-sbc 3748 df-csb 3857 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3928 df-nul 4290 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4878 df-iun 4963 df-br 5115 df-opab 5179 df-mpt 5198 df-tr 5224 df-id 5561 df-eprel 5566 df-po 5574 df-so 5575 df-fr 5619 df-we 5621 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6309 df-ord 6370 df-on 6371 df-lim 6372 df-suc 6373 df-iota 6499 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7380 df-ov 7426 df-oprab 7427 df-mpo 7428 df-om 7872 df-2nd 7996 df-frecs 8287 df-wrecs 8318 df-recs 8367 df-rdg 8406 df-er 8703 df-en 8953 df-dom 8954 df-sdom 8955 df-sup 9412 df-pnf 11263 df-mnf 11264 df-xr 11265 df-ltxr 11266 df-le 11267 df-sub 11461 df-neg 11462 df-div 11890 df-nn 12252 df-2 12321 df-3 12322 df-n0 12523 df-z 12610 df-uz 12881 df-rp 13035 df-seq 14058 df-exp 14118 df-cj 15176 df-re 15177 df-im 15178 df-sqrt 15312 df-abs 15313 df-dvds 16336

This theorem is used by: gcdcllem3 16584

W3C validator