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Theorem divalglem9 16437

Description: Lemma for divalg 16439. (Contributed by Paul Chapman, 21-Mar-2011.) (Revised by AV, 2-Oct-2020.)

**Hypotheses**
Ref	Expression
divalglem8.1	⊢ 𝑁 ∈ ℤ
divalglem8.2	⊢ 𝐷 ∈ ℤ
divalglem8.3	⊢ 𝐷 ≠ 0
divalglem8.4	⊢ 𝑆 = {𝑟 ∈ ℕ₀ ∣ 𝐷 ∥ (𝑁 − 𝑟)}
divalglem9.5	⊢ 𝑅 = inf(𝑆, ℝ, < )

**Assertion**
Ref	Expression
divalglem9	⊢ ∃!𝑥 ∈ 𝑆 𝑥 < (abs‘𝐷)

Distinct variable groups: 𝐷,𝑟,𝑥 𝑁,𝑟,𝑥 𝑥,𝑆 𝑥,𝑅 Allowed substitution hints: 𝑅(𝑟) 𝑆(𝑟)

Proof of Theorem divalglem9 Dummy variables 𝑘 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		divalglem9.5	. . . 4 ⊢ 𝑅 = inf(𝑆, ℝ, < )
2		divalglem8.1	. . . . 5 ⊢ 𝑁 ∈ ℤ
3		divalglem8.2	. . . . 5 ⊢ 𝐷 ∈ ℤ
4		divalglem8.3	. . . . 5 ⊢ 𝐷 ≠ 0
5		divalglem8.4	. . . . 5 ⊢ 𝑆 = {𝑟 ∈ ℕ₀ ∣ 𝐷 ∥ (𝑁 − 𝑟)}
6	2, 3, 4, 5	divalglem2 16431	. . . 4 ⊢ inf(𝑆, ℝ, < ) ∈ 𝑆
7	1, 6	eqeltri 2860	. . 3 ⊢ 𝑅 ∈ 𝑆
8	2, 3, 4, 5, 1	divalglem5 16433	. . . 4 ⊢ (0 ≤ 𝑅 ∧ 𝑅 < (abs‘𝐷))
9	8	simpri 489	. . 3 ⊢ 𝑅 < (abs‘𝐷)
10		breq1 5105	. . . 4 ⊢ (𝑥 = 𝑅 → (𝑥 < (abs‘𝐷) ↔ 𝑅 < (abs‘𝐷)))
11	10	rspcev 3583	. . 3 ⊢ ((𝑅 ∈ 𝑆 ∧ 𝑅 < (abs‘𝐷)) → ∃𝑥 ∈ 𝑆 𝑥 < (abs‘𝐷))
12	7, 9, 11	mp2an 702	. 2 ⊢ ∃𝑥 ∈ 𝑆 𝑥 < (abs‘𝐷)
13		oveq2 7406	. . . . . . . . . . . . . . 15 ⊢ (𝑟 = 𝑥 → (𝑁 − 𝑟) = (𝑁 − 𝑥))
14	13	breq2d 5114	. . . . . . . . . . . . . 14 ⊢ (𝑟 = 𝑥 → (𝐷 ∥ (𝑁 − 𝑟) ↔ 𝐷 ∥ (𝑁 − 𝑥)))
15	14, 5	elrab2 3656	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝑆 ↔ (𝑥 ∈ ℕ₀ ∧ 𝐷 ∥ (𝑁 − 𝑥)))
16	15	simplbi 500	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝑆 → 𝑥 ∈ ℕ₀)
17	16	nn0zd 12595	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝑆 → 𝑥 ∈ ℤ)
18		oveq2 7406	. . . . . . . . . . . . . . 15 ⊢ (𝑟 = 𝑦 → (𝑁 − 𝑟) = (𝑁 − 𝑦))
19	18	breq2d 5114	. . . . . . . . . . . . . 14 ⊢ (𝑟 = 𝑦 → (𝐷 ∥ (𝑁 − 𝑟) ↔ 𝐷 ∥ (𝑁 − 𝑦)))
20	19, 5	elrab2 3656	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝑆 ↔ (𝑦 ∈ ℕ₀ ∧ 𝐷 ∥ (𝑁 − 𝑦)))
21	20	simplbi 500	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ 𝑆 → 𝑦 ∈ ℕ₀)
22	21	nn0zd 12595	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝑆 → 𝑦 ∈ ℤ)
23		zsubcl 12615	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℤ ∧ 𝑥 ∈ ℤ) → (𝑁 − 𝑥) ∈ ℤ)
24	2, 23	mpan 700	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℤ → (𝑁 − 𝑥) ∈ ℤ)
25		zsubcl 12615	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (𝑁 − 𝑦) ∈ ℤ)
26	2, 25	mpan 700	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℤ → (𝑁 − 𝑦) ∈ ℤ)
27	24, 26	anim12i 622	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → ((𝑁 − 𝑥) ∈ ℤ ∧ (𝑁 − 𝑦) ∈ ℤ))
28	17, 22, 27	syl2an 605	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → ((𝑁 − 𝑥) ∈ ℤ ∧ (𝑁 − 𝑦) ∈ ℤ))
29	15	simprbi 501	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝑆 → 𝐷 ∥ (𝑁 − 𝑥))
30	20	simprbi 501	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝑆 → 𝐷 ∥ (𝑁 − 𝑦))
31	29, 30	anim12i 622	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝐷 ∥ (𝑁 − 𝑥) ∧ 𝐷 ∥ (𝑁 − 𝑦)))
32		dvds2sub 16327	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ ℤ ∧ (𝑁 − 𝑥) ∈ ℤ ∧ (𝑁 − 𝑦) ∈ ℤ) → ((𝐷 ∥ (𝑁 − 𝑥) ∧ 𝐷 ∥ (𝑁 − 𝑦)) → 𝐷 ∥ ((𝑁 − 𝑥) − (𝑁 − 𝑦))))
33	3, 32	mp3an1 1471	. . . . . . . . . 10 ⊢ (((𝑁 − 𝑥) ∈ ℤ ∧ (𝑁 − 𝑦) ∈ ℤ) → ((𝐷 ∥ (𝑁 − 𝑥) ∧ 𝐷 ∥ (𝑁 − 𝑦)) → 𝐷 ∥ ((𝑁 − 𝑥) − (𝑁 − 𝑦))))
34	28, 31, 33	sylc 65	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → 𝐷 ∥ ((𝑁 − 𝑥) − (𝑁 − 𝑦)))
35		zcn 12575	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ ℂ)
36		zcn 12575	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℤ → 𝑦 ∈ ℂ)
37	2	zrei 12576	. . . . . . . . . . . . . . . . 17 ⊢ 𝑁 ∈ ℝ
38	37	recni 11198	. . . . . . . . . . . . . . . 16 ⊢ 𝑁 ∈ ℂ
39	38	subidi 11504	. . . . . . . . . . . . . . 15 ⊢ (𝑁 − 𝑁) = 0
40	39	oveq1i 7408	. . . . . . . . . . . . . 14 ⊢ ((𝑁 − 𝑁) − (𝑥 − 𝑦)) = (0 − (𝑥 − 𝑦))
41		0cn 11173	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ ℂ
42		subsub2 11461	. . . . . . . . . . . . . . 15 ⊢ ((0 ∈ ℂ ∧ 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (0 − (𝑥 − 𝑦)) = (0 + (𝑦 − 𝑥)))
43	41, 42	mp3an1 1471	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (0 − (𝑥 − 𝑦)) = (0 + (𝑦 − 𝑥)))
44	40, 43	eqtrid 2811	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((𝑁 − 𝑁) − (𝑥 − 𝑦)) = (0 + (𝑦 − 𝑥)))
45		sub4 11478	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℂ ∧ 𝑁 ∈ ℂ) ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → ((𝑁 − 𝑁) − (𝑥 − 𝑦)) = ((𝑁 − 𝑥) − (𝑁 − 𝑦)))
46	38, 38, 45	mpanl12 712	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((𝑁 − 𝑁) − (𝑥 − 𝑦)) = ((𝑁 − 𝑥) − (𝑁 − 𝑦)))
47		subcl 11431	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝑦 − 𝑥) ∈ ℂ)
48	47	ancoms 462	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑦 − 𝑥) ∈ ℂ)
49	48	addlidd 11386	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (0 + (𝑦 − 𝑥)) = (𝑦 − 𝑥))
50	44, 46, 49	3eqtr3d 2807	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((𝑁 − 𝑥) − (𝑁 − 𝑦)) = (𝑦 − 𝑥))
51	35, 36, 50	syl2an 605	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → ((𝑁 − 𝑥) − (𝑁 − 𝑦)) = (𝑦 − 𝑥))
52	17, 22, 51	syl2an 605	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → ((𝑁 − 𝑥) − (𝑁 − 𝑦)) = (𝑦 − 𝑥))
53	52	breq2d 5114	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝐷 ∥ ((𝑁 − 𝑥) − (𝑁 − 𝑦)) ↔ 𝐷 ∥ (𝑦 − 𝑥)))
54	34, 53	mpbid 234	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → 𝐷 ∥ (𝑦 − 𝑥))
55		zsubcl 12615	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℤ ∧ 𝑥 ∈ ℤ) → (𝑦 − 𝑥) ∈ ℤ)
56	55	ancoms 462	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (𝑦 − 𝑥) ∈ ℤ)
57		absdvdsb 16310	. . . . . . . . . 10 ⊢ ((𝐷 ∈ ℤ ∧ (𝑦 − 𝑥) ∈ ℤ) → (𝐷 ∥ (𝑦 − 𝑥) ↔ (abs‘𝐷) ∥ (𝑦 − 𝑥)))
58	3, 56, 57	sylancr 596	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (𝐷 ∥ (𝑦 − 𝑥) ↔ (abs‘𝐷) ∥ (𝑦 − 𝑥)))
59	17, 22, 58	syl2an 605	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝐷 ∥ (𝑦 − 𝑥) ↔ (abs‘𝐷) ∥ (𝑦 − 𝑥)))
60	54, 59	mpbid 234	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (abs‘𝐷) ∥ (𝑦 − 𝑥))
61		nnabscl 15355	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ ℤ ∧ 𝐷 ≠ 0) → (abs‘𝐷) ∈ ℕ)
62	3, 4, 61	mp2an 702	. . . . . . . . . 10 ⊢ (abs‘𝐷) ∈ ℕ
63	62	nnzi 12597	. . . . . . . . 9 ⊢ (abs‘𝐷) ∈ ℤ
64		divides 16290	. . . . . . . . 9 ⊢ (((abs‘𝐷) ∈ ℤ ∧ (𝑦 − 𝑥) ∈ ℤ) → ((abs‘𝐷) ∥ (𝑦 − 𝑥) ↔ ∃𝑘 ∈ ℤ (𝑘 · (abs‘𝐷)) = (𝑦 − 𝑥)))
65	63, 56, 64	sylancr 596	. . . . . . . 8 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → ((abs‘𝐷) ∥ (𝑦 − 𝑥) ↔ ∃𝑘 ∈ ℤ (𝑘 · (abs‘𝐷)) = (𝑦 − 𝑥)))
66	17, 22, 65	syl2an 605	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → ((abs‘𝐷) ∥ (𝑦 − 𝑥) ↔ ∃𝑘 ∈ ℤ (𝑘 · (abs‘𝐷)) = (𝑦 − 𝑥)))
67	60, 66	mpbid 234	. . . . . 6 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → ∃𝑘 ∈ ℤ (𝑘 · (abs‘𝐷)) = (𝑦 − 𝑥))
68	67	adantr 484	. . . . 5 ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑥 < (abs‘𝐷) ∧ 𝑦 < (abs‘𝐷))) → ∃𝑘 ∈ ℤ (𝑘 · (abs‘𝐷)) = (𝑦 − 𝑥))
69	2, 3, 4, 5	divalglem8 16436	. . . . . 6 ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑥 < (abs‘𝐷) ∧ 𝑦 < (abs‘𝐷))) → (𝑘 ∈ ℤ → ((𝑘 · (abs‘𝐷)) = (𝑦 − 𝑥) → 𝑥 = 𝑦)))
70	69	rexlimdv 3163	. . . . 5 ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑥 < (abs‘𝐷) ∧ 𝑦 < (abs‘𝐷))) → (∃𝑘 ∈ ℤ (𝑘 · (abs‘𝐷)) = (𝑦 − 𝑥) → 𝑥 = 𝑦))
71	68, 70	mpd 15	. . . 4 ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑥 < (abs‘𝐷) ∧ 𝑦 < (abs‘𝐷))) → 𝑥 = 𝑦)
72	71	ex 416	. . 3 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → ((𝑥 < (abs‘𝐷) ∧ 𝑦 < (abs‘𝐷)) → 𝑥 = 𝑦))
73	72	rgen2 3204	. 2 ⊢ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ((𝑥 < (abs‘𝐷) ∧ 𝑦 < (abs‘𝐷)) → 𝑥 = 𝑦)
74		breq1 5105	. . 3 ⊢ (𝑥 = 𝑦 → (𝑥 < (abs‘𝐷) ↔ 𝑦 < (abs‘𝐷)))
75	74	reu4 3696	. 2 ⊢ (∃!𝑥 ∈ 𝑆 𝑥 < (abs‘𝐷) ↔ (∃𝑥 ∈ 𝑆 𝑥 < (abs‘𝐷) ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ((𝑥 < (abs‘𝐷) ∧ 𝑦 < (abs‘𝐷)) → 𝑥 = 𝑦)))
76	12, 73, 75	mpbir2an 721	1 ⊢ ∃!𝑥 ∈ 𝑆 𝑥 < (abs‘𝐷)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 399 = wceq 1562 ∈ wcel 2144 ≠ wne 2959 ∀wral 3078 ∃wrex 3088 ∃!wreu 3367 {crab 3416 class class class wbr 5102 ‘cfv 6523 (class class class)co 7398 infcinf 9389 ℂcc 11073 ℝcr 11074 0cc0 11075 + caddc 11078 · cmul 11080 < clt 11218 ≤ cle 11219 − cmin 11416 ℕcn 12212 ℕ₀cn0 12483 ℤcz 12570 abscabs 15263 ∥ cdvds 16288

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1817 ax-4 1831 ax-5 1932 ax-6 1989 ax-7 2030 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2214 ax-ext 2736 ax-sep 5248 ax-nul 5258 ax-pow 5324 ax-pr 5392 ax-un 7720 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-pre-sup 11153

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1100 df-3an 1101 df-tru 1565 df-fal 1575 df-ex 1802 df-nf 1806 df-sb 2093 df-mo 2568 df-eu 2598 df-clab 2743 df-cleq 2756 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3458 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4288 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-iun 4953 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5544 df-eprel 5549 df-po 5557 df-so 5558 df-fr 5602 df-we 5604 df-xp 5655 df-rel 5656 df-cnv 5657 df-co 5658 df-dm 5659 df-rn 5660 df-res 5661 df-ima 5662 df-pred 6290 df-ord 6351 df-on 6352 df-lim 6353 df-suc 6354 df-iota 6479 df-fun 6525 df-fn 6526 df-f 6527 df-f1 6528 df-fo 6529 df-f1o 6530 df-fv 6531 df-riota 7355 df-ov 7401 df-oprab 7402 df-mpo 7403 df-om 7849 df-1st 7972 df-2nd 7973 df-frecs 8264 df-wrecs 8295 df-recs 8344 df-rdg 8383 df-er 8680 df-en 8930 df-dom 8931 df-sdom 8932 df-sup 9390 df-inf 9391 df-pnf 11220 df-mnf 11221 df-xr 11222 df-ltxr 11223 df-le 11224 df-sub 11418 df-neg 11419 df-div 11847 df-nn 12213 df-2 12282 df-3 12283 df-n0 12484 df-z 12571 df-uz 12842 df-rp 12996 df-fz 13515 df-seq 14017 df-exp 14077 df-cj 15128 df-re 15129 df-im 15130 df-sqrt 15264 df-abs 15265 df-dvds 16289

This theorem is referenced by: divalglem10 16438

W3C validator