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Theorem divalglem9 16524

Description: Lemma for divalg 16526. (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 16518	. . . 4 ⊢ inf(𝑆, ℝ, < ) ∈ 𝑆
7	1, 6	eqeltri 2856	. . 3 ⊢ 𝑅 ∈ 𝑆
8	2, 3, 4, 5, 1	divalglem5 16520	. . . 4 ⊢ (0 ≤ 𝑅 ∧ 𝑅 < (abs‘𝐷))
9	8	simpri 491	. . 3 ⊢ 𝑅 < (abs‘𝐷)
10		breq1 5106	. . . 4 ⊢ (𝑥 = 𝑅 → (𝑥 < (abs‘𝐷) ↔ 𝑅 < (abs‘𝐷)))
11	10	rspcev 3576	. . 3 ⊢ ((𝑅 ∈ 𝑆 ∧ 𝑅 < (abs‘𝐷)) → ∃𝑥 ∈ 𝑆 𝑥 < (abs‘𝐷))
12	7, 9, 11	mp2an 705	. 2 ⊢ ∃𝑥 ∈ 𝑆 𝑥 < (abs‘𝐷)
13		oveq2 7417	. . . . . . . . . . . . . . 15 ⊢ (𝑟 = 𝑥 → (𝑁 − 𝑟) = (𝑁 − 𝑥))
14	13	breq2d 5115	. . . . . . . . . . . . . 14 ⊢ (𝑟 = 𝑥 → (𝐷 ∥ (𝑁 − 𝑟) ↔ 𝐷 ∥ (𝑁 − 𝑥)))
15	14, 5	elrab2 3649	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝑆 ↔ (𝑥 ∈ ℕ₀ ∧ 𝐷 ∥ (𝑁 − 𝑥)))
16	15	simplbi 502	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝑆 → 𝑥 ∈ ℕ₀)
17	16	nn0zd 12673	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝑆 → 𝑥 ∈ ℤ)
18		oveq2 7417	. . . . . . . . . . . . . . 15 ⊢ (𝑟 = 𝑦 → (𝑁 − 𝑟) = (𝑁 − 𝑦))
19	18	breq2d 5115	. . . . . . . . . . . . . 14 ⊢ (𝑟 = 𝑦 → (𝐷 ∥ (𝑁 − 𝑟) ↔ 𝐷 ∥ (𝑁 − 𝑦)))
20	19, 5	elrab2 3649	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝑆 ↔ (𝑦 ∈ ℕ₀ ∧ 𝐷 ∥ (𝑁 − 𝑦)))
21	20	simplbi 502	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ 𝑆 → 𝑦 ∈ ℕ₀)
22	21	nn0zd 12673	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝑆 → 𝑦 ∈ ℤ)
23		zsubcl 12693	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℤ ∧ 𝑥 ∈ ℤ) → (𝑁 − 𝑥) ∈ ℤ)
24	2, 23	mpan 703	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℤ → (𝑁 − 𝑥) ∈ ℤ)
25		zsubcl 12693	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (𝑁 − 𝑦) ∈ ℤ)
26	2, 25	mpan 703	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℤ → (𝑁 − 𝑦) ∈ ℤ)
27	24, 26	anim12i 625	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → ((𝑁 − 𝑥) ∈ ℤ ∧ (𝑁 − 𝑦) ∈ ℤ))
28	17, 22, 27	syl2an 608	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → ((𝑁 − 𝑥) ∈ ℤ ∧ (𝑁 − 𝑦) ∈ ℤ))
29	15	simprbi 503	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝑆 → 𝐷 ∥ (𝑁 − 𝑥))
30	20	simprbi 503	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝑆 → 𝐷 ∥ (𝑁 − 𝑦))
31	29, 30	anim12i 625	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝐷 ∥ (𝑁 − 𝑥) ∧ 𝐷 ∥ (𝑁 − 𝑦)))
32		dvds2sub 16414	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ ℤ ∧ (𝑁 − 𝑥) ∈ ℤ ∧ (𝑁 − 𝑦) ∈ ℤ) → ((𝐷 ∥ (𝑁 − 𝑥) ∧ 𝐷 ∥ (𝑁 − 𝑦)) → 𝐷 ∥ ((𝑁 − 𝑥) − (𝑁 − 𝑦))))
33	3, 32	mp3an1 1477	. . . . . . . . . 10 ⊢ (((𝑁 − 𝑥) ∈ ℤ ∧ (𝑁 − 𝑦) ∈ ℤ) → ((𝐷 ∥ (𝑁 − 𝑥) ∧ 𝐷 ∥ (𝑁 − 𝑦)) → 𝐷 ∥ ((𝑁 − 𝑥) − (𝑁 − 𝑦))))
34	28, 31, 33	sylc 66	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → 𝐷 ∥ ((𝑁 − 𝑥) − (𝑁 − 𝑦)))
35		zcn 12653	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ ℂ)
36		zcn 12653	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℤ → 𝑦 ∈ ℂ)
37	2	zrei 12654	. . . . . . . . . . . . . . . . 17 ⊢ 𝑁 ∈ ℝ
38	37	recni 11280	. . . . . . . . . . . . . . . 16 ⊢ 𝑁 ∈ ℂ
39	38	subidi 11586	. . . . . . . . . . . . . . 15 ⊢ (𝑁 − 𝑁) = 0
40	39	oveq1i 7419	. . . . . . . . . . . . . 14 ⊢ ((𝑁 − 𝑁) − (𝑥 − 𝑦)) = (0 − (𝑥 − 𝑦))
41		0cn 11255	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ ℂ
42		subsub2 11543	. . . . . . . . . . . . . . 15 ⊢ ((0 ∈ ℂ ∧ 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (0 − (𝑥 − 𝑦)) = (0 + (𝑦 − 𝑥)))
43	41, 42	mp3an1 1477	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (0 − (𝑥 − 𝑦)) = (0 + (𝑦 − 𝑥)))
44	40, 43	eqtrid 2807	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((𝑁 − 𝑁) − (𝑥 − 𝑦)) = (0 + (𝑦 − 𝑥)))
45		sub4 11560	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℂ ∧ 𝑁 ∈ ℂ) ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → ((𝑁 − 𝑁) − (𝑥 − 𝑦)) = ((𝑁 − 𝑥) − (𝑁 − 𝑦)))
46	38, 38, 45	mpanl12 715	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((𝑁 − 𝑁) − (𝑥 − 𝑦)) = ((𝑁 − 𝑥) − (𝑁 − 𝑦)))
47		subcl 11513	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝑦 − 𝑥) ∈ ℂ)
48	47	ancoms 464	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑦 − 𝑥) ∈ ℂ)
49	48	addlidd 11468	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (0 + (𝑦 − 𝑥)) = (𝑦 − 𝑥))
50	44, 46, 49	3eqtr3d 2803	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((𝑁 − 𝑥) − (𝑁 − 𝑦)) = (𝑦 − 𝑥))
51	35, 36, 50	syl2an 608	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → ((𝑁 − 𝑥) − (𝑁 − 𝑦)) = (𝑦 − 𝑥))
52	17, 22, 51	syl2an 608	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → ((𝑁 − 𝑥) − (𝑁 − 𝑦)) = (𝑦 − 𝑥))
53	52	breq2d 5115	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝐷 ∥ ((𝑁 − 𝑥) − (𝑁 − 𝑦)) ↔ 𝐷 ∥ (𝑦 − 𝑥)))
54	34, 53	mpbid 235	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → 𝐷 ∥ (𝑦 − 𝑥))
55		zsubcl 12693	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℤ ∧ 𝑥 ∈ ℤ) → (𝑦 − 𝑥) ∈ ℤ)
56	55	ancoms 464	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (𝑦 − 𝑥) ∈ ℤ)
57		absdvdsb 16397	. . . . . . . . . 10 ⊢ ((𝐷 ∈ ℤ ∧ (𝑦 − 𝑥) ∈ ℤ) → (𝐷 ∥ (𝑦 − 𝑥) ↔ (abs‘𝐷) ∥ (𝑦 − 𝑥)))
58	3, 56, 57	sylancr 599	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (𝐷 ∥ (𝑦 − 𝑥) ↔ (abs‘𝐷) ∥ (𝑦 − 𝑥)))
59	17, 22, 58	syl2an 608	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝐷 ∥ (𝑦 − 𝑥) ↔ (abs‘𝐷) ∥ (𝑦 − 𝑥)))
60	54, 59	mpbid 235	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (abs‘𝐷) ∥ (𝑦 − 𝑥))
61		nnabscl 15446	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ ℤ ∧ 𝐷 ≠ 0) → (abs‘𝐷) ∈ ℕ)
62	3, 4, 61	mp2an 705	. . . . . . . . . 10 ⊢ (abs‘𝐷) ∈ ℕ
63	62	nnzi 12675	. . . . . . . . 9 ⊢ (abs‘𝐷) ∈ ℤ
64		divides 16377	. . . . . . . . 9 ⊢ (((abs‘𝐷) ∈ ℤ ∧ (𝑦 − 𝑥) ∈ ℤ) → ((abs‘𝐷) ∥ (𝑦 − 𝑥) ↔ ∃𝑘 ∈ ℤ (𝑘 · (abs‘𝐷)) = (𝑦 − 𝑥)))
65	63, 56, 64	sylancr 599	. . . . . . . 8 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → ((abs‘𝐷) ∥ (𝑦 − 𝑥) ↔ ∃𝑘 ∈ ℤ (𝑘 · (abs‘𝐷)) = (𝑦 − 𝑥)))
66	17, 22, 65	syl2an 608	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → ((abs‘𝐷) ∥ (𝑦 − 𝑥) ↔ ∃𝑘 ∈ ℤ (𝑘 · (abs‘𝐷)) = (𝑦 − 𝑥)))
67	60, 66	mpbid 235	. . . . . 6 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → ∃𝑘 ∈ ℤ (𝑘 · (abs‘𝐷)) = (𝑦 − 𝑥))
68	67	adantr 486	. . . . 5 ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑥 < (abs‘𝐷) ∧ 𝑦 < (abs‘𝐷))) → ∃𝑘 ∈ ℤ (𝑘 · (abs‘𝐷)) = (𝑦 − 𝑥))
69	2, 3, 4, 5	divalglem8 16523	. . . . . 6 ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑥 < (abs‘𝐷) ∧ 𝑦 < (abs‘𝐷))) → (𝑘 ∈ ℤ → ((𝑘 · (abs‘𝐷)) = (𝑦 − 𝑥) → 𝑥 = 𝑦)))
70	69	rexlimdv 3161	. . . . 5 ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑥 < (abs‘𝐷) ∧ 𝑦 < (abs‘𝐷))) → (∃𝑘 ∈ ℤ (𝑘 · (abs‘𝐷)) = (𝑦 − 𝑥) → 𝑥 = 𝑦))
71	68, 70	mpd 16	. . . 4 ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑥 < (abs‘𝐷) ∧ 𝑦 < (abs‘𝐷))) → 𝑥 = 𝑦)
72	71	ex 418	. . 3 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → ((𝑥 < (abs‘𝐷) ∧ 𝑦 < (abs‘𝐷)) → 𝑥 = 𝑦))
73	72	rgen2 3202	. 2 ⊢ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ((𝑥 < (abs‘𝐷) ∧ 𝑦 < (abs‘𝐷)) → 𝑥 = 𝑦)
74		breq1 5106	. . 3 ⊢ (𝑥 = 𝑦 → (𝑥 < (abs‘𝐷) ↔ 𝑦 < (abs‘𝐷)))
75	74	reu4 3689	. 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 724	1 ⊢ ∃!𝑥 ∈ 𝑆 𝑥 < (abs‘𝐷)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ≠ wne 2955 ∀wral 3076 ∃wrex 3086 ∃!wreu 3363 {crab 3412 class class class wbr 5103 ‘cfv 6528 (class class class)co 7409 infcinf 9411 ℂcc 11155 ℝcr 11156 0cc0 11157 + caddc 11160 · cmul 11162 < clt 11300 ≤ cle 11301 − cmin 11498 ℕcn 12290 ℕ₀cn0 12561 ℤcz 12648 abscabs 15354 ∥ cdvds 16375

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 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7735 ax-cnex 11213 ax-resscn 11214 ax-1cn 11215 ax-icn 11216 ax-addcl 11217 ax-addrcl 11218 ax-mulcl 11219 ax-mulrcl 11220 ax-mulcom 11221 ax-addass 11222 ax-mulass 11223 ax-distr 11224 ax-i2m1 11225 ax-1ne0 11226 ax-1rid 11227 ax-rnegex 11228 ax-rrecex 11229 ax-cnre 11230 ax-pre-lttri 11231 ax-pre-lttrn 11232 ax-pre-ltadd 11233 ax-pre-mulgt0 11234 ax-pre-sup 11235

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 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5543 df-eprel 5548 df-po 5556 df-so 5557 df-fr 5601 df-we 5603 df-xp 5654 df-rel 5655 df-cnv 5656 df-co 5657 df-dm 5658 df-rn 5659 df-res 5660 df-ima 5661 df-pred 6294 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6484 df-fun 6530 df-fn 6531 df-f 6532 df-f1 6533 df-fo 6534 df-f1o 6535 df-fv 6536 df-riota 7366 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7862 df-1st 7985 df-2nd 7986 df-frecs 8278 df-wrecs 8309 df-recs 8358 df-rdg 8397 df-er 8696 df-en 8953 df-dom 8954 df-sdom 8955 df-sup 9412 df-inf 9413 df-pnf 11302 df-mnf 11303 df-xr 11304 df-ltxr 11305 df-le 11306 df-sub 11500 df-neg 11501 df-div 11929 df-nn 12291 df-2 12360 df-3 12361 df-n0 12562 df-z 12649 df-uz 12921 df-rp 13076 df-fz 13595 df-seq 14099 df-exp 14159 df-cj 15219 df-re 15220 df-im 15221 df-sqrt 15355 df-abs 15356 df-dvds 16376

This theorem is used by: divalglem10 16525

W3C validator