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Theorem divalglem9 16091

Description: Lemma for divalg 16093. (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 16085	. . . 4 ⊢ inf(𝑆, ℝ, < ) ∈ 𝑆
7	1, 6	eqeltri 2836	. . 3 ⊢ 𝑅 ∈ 𝑆
8	2, 3, 4, 5, 1	divalglem5 16087	. . . 4 ⊢ (0 ≤ 𝑅 ∧ 𝑅 < (abs‘𝐷))
9	8	simpri 485	. . 3 ⊢ 𝑅 < (abs‘𝐷)
10		breq1 5081	. . . 4 ⊢ (𝑥 = 𝑅 → (𝑥 < (abs‘𝐷) ↔ 𝑅 < (abs‘𝐷)))
11	10	rspcev 3560	. . 3 ⊢ ((𝑅 ∈ 𝑆 ∧ 𝑅 < (abs‘𝐷)) → ∃𝑥 ∈ 𝑆 𝑥 < (abs‘𝐷))
12	7, 9, 11	mp2an 688	. 2 ⊢ ∃𝑥 ∈ 𝑆 𝑥 < (abs‘𝐷)
13		oveq2 7276	. . . . . . . . . . . . . . 15 ⊢ (𝑟 = 𝑥 → (𝑁 − 𝑟) = (𝑁 − 𝑥))
14	13	breq2d 5090	. . . . . . . . . . . . . 14 ⊢ (𝑟 = 𝑥 → (𝐷 ∥ (𝑁 − 𝑟) ↔ 𝐷 ∥ (𝑁 − 𝑥)))
15	14, 5	elrab2 3628	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝑆 ↔ (𝑥 ∈ ℕ₀ ∧ 𝐷 ∥ (𝑁 − 𝑥)))
16	15	simplbi 497	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝑆 → 𝑥 ∈ ℕ₀)
17	16	nn0zd 12406	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝑆 → 𝑥 ∈ ℤ)
18		oveq2 7276	. . . . . . . . . . . . . . 15 ⊢ (𝑟 = 𝑦 → (𝑁 − 𝑟) = (𝑁 − 𝑦))
19	18	breq2d 5090	. . . . . . . . . . . . . 14 ⊢ (𝑟 = 𝑦 → (𝐷 ∥ (𝑁 − 𝑟) ↔ 𝐷 ∥ (𝑁 − 𝑦)))
20	19, 5	elrab2 3628	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝑆 ↔ (𝑦 ∈ ℕ₀ ∧ 𝐷 ∥ (𝑁 − 𝑦)))
21	20	simplbi 497	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ 𝑆 → 𝑦 ∈ ℕ₀)
22	21	nn0zd 12406	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝑆 → 𝑦 ∈ ℤ)
23		zsubcl 12345	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℤ ∧ 𝑥 ∈ ℤ) → (𝑁 − 𝑥) ∈ ℤ)
24	2, 23	mpan 686	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℤ → (𝑁 − 𝑥) ∈ ℤ)
25		zsubcl 12345	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (𝑁 − 𝑦) ∈ ℤ)
26	2, 25	mpan 686	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℤ → (𝑁 − 𝑦) ∈ ℤ)
27	24, 26	anim12i 612	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → ((𝑁 − 𝑥) ∈ ℤ ∧ (𝑁 − 𝑦) ∈ ℤ))
28	17, 22, 27	syl2an 595	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → ((𝑁 − 𝑥) ∈ ℤ ∧ (𝑁 − 𝑦) ∈ ℤ))
29	15	simprbi 496	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝑆 → 𝐷 ∥ (𝑁 − 𝑥))
30	20	simprbi 496	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝑆 → 𝐷 ∥ (𝑁 − 𝑦))
31	29, 30	anim12i 612	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝐷 ∥ (𝑁 − 𝑥) ∧ 𝐷 ∥ (𝑁 − 𝑦)))
32		dvds2sub 15981	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ ℤ ∧ (𝑁 − 𝑥) ∈ ℤ ∧ (𝑁 − 𝑦) ∈ ℤ) → ((𝐷 ∥ (𝑁 − 𝑥) ∧ 𝐷 ∥ (𝑁 − 𝑦)) → 𝐷 ∥ ((𝑁 − 𝑥) − (𝑁 − 𝑦))))
33	3, 32	mp3an1 1446	. . . . . . . . . 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 12307	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ ℂ)
36		zcn 12307	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℤ → 𝑦 ∈ ℂ)
37	2	zrei 12308	. . . . . . . . . . . . . . . . 17 ⊢ 𝑁 ∈ ℝ
38	37	recni 10973	. . . . . . . . . . . . . . . 16 ⊢ 𝑁 ∈ ℂ
39	38	subidi 11275	. . . . . . . . . . . . . . 15 ⊢ (𝑁 − 𝑁) = 0
40	39	oveq1i 7278	. . . . . . . . . . . . . 14 ⊢ ((𝑁 − 𝑁) − (𝑥 − 𝑦)) = (0 − (𝑥 − 𝑦))
41		0cn 10951	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ ℂ
42		subsub2 11232	. . . . . . . . . . . . . . 15 ⊢ ((0 ∈ ℂ ∧ 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (0 − (𝑥 − 𝑦)) = (0 + (𝑦 − 𝑥)))
43	41, 42	mp3an1 1446	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (0 − (𝑥 − 𝑦)) = (0 + (𝑦 − 𝑥)))
44	40, 43	eqtrid 2791	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((𝑁 − 𝑁) − (𝑥 − 𝑦)) = (0 + (𝑦 − 𝑥)))
45		sub4 11249	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℂ ∧ 𝑁 ∈ ℂ) ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → ((𝑁 − 𝑁) − (𝑥 − 𝑦)) = ((𝑁 − 𝑥) − (𝑁 − 𝑦)))
46	38, 38, 45	mpanl12 698	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((𝑁 − 𝑁) − (𝑥 − 𝑦)) = ((𝑁 − 𝑥) − (𝑁 − 𝑦)))
47		subcl 11203	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝑦 − 𝑥) ∈ ℂ)
48	47	ancoms 458	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑦 − 𝑥) ∈ ℂ)
49	48	addid2d 11159	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (0 + (𝑦 − 𝑥)) = (𝑦 − 𝑥))
50	44, 46, 49	3eqtr3d 2787	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((𝑁 − 𝑥) − (𝑁 − 𝑦)) = (𝑦 − 𝑥))
51	35, 36, 50	syl2an 595	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → ((𝑁 − 𝑥) − (𝑁 − 𝑦)) = (𝑦 − 𝑥))
52	17, 22, 51	syl2an 595	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → ((𝑁 − 𝑥) − (𝑁 − 𝑦)) = (𝑦 − 𝑥))
53	52	breq2d 5090	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝐷 ∥ ((𝑁 − 𝑥) − (𝑁 − 𝑦)) ↔ 𝐷 ∥ (𝑦 − 𝑥)))
54	34, 53	mpbid 231	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → 𝐷 ∥ (𝑦 − 𝑥))
55		zsubcl 12345	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℤ ∧ 𝑥 ∈ ℤ) → (𝑦 − 𝑥) ∈ ℤ)
56	55	ancoms 458	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (𝑦 − 𝑥) ∈ ℤ)
57		absdvdsb 15965	. . . . . . . . . 10 ⊢ ((𝐷 ∈ ℤ ∧ (𝑦 − 𝑥) ∈ ℤ) → (𝐷 ∥ (𝑦 − 𝑥) ↔ (abs‘𝐷) ∥ (𝑦 − 𝑥)))
58	3, 56, 57	sylancr 586	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (𝐷 ∥ (𝑦 − 𝑥) ↔ (abs‘𝐷) ∥ (𝑦 − 𝑥)))
59	17, 22, 58	syl2an 595	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝐷 ∥ (𝑦 − 𝑥) ↔ (abs‘𝐷) ∥ (𝑦 − 𝑥)))
60	54, 59	mpbid 231	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (abs‘𝐷) ∥ (𝑦 − 𝑥))
61		nnabscl 15018	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ ℤ ∧ 𝐷 ≠ 0) → (abs‘𝐷) ∈ ℕ)
62	3, 4, 61	mp2an 688	. . . . . . . . . 10 ⊢ (abs‘𝐷) ∈ ℕ
63	62	nnzi 12327	. . . . . . . . 9 ⊢ (abs‘𝐷) ∈ ℤ
64		divides 15946	. . . . . . . . 9 ⊢ (((abs‘𝐷) ∈ ℤ ∧ (𝑦 − 𝑥) ∈ ℤ) → ((abs‘𝐷) ∥ (𝑦 − 𝑥) ↔ ∃𝑘 ∈ ℤ (𝑘 · (abs‘𝐷)) = (𝑦 − 𝑥)))
65	63, 56, 64	sylancr 586	. . . . . . . 8 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → ((abs‘𝐷) ∥ (𝑦 − 𝑥) ↔ ∃𝑘 ∈ ℤ (𝑘 · (abs‘𝐷)) = (𝑦 − 𝑥)))
66	17, 22, 65	syl2an 595	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → ((abs‘𝐷) ∥ (𝑦 − 𝑥) ↔ ∃𝑘 ∈ ℤ (𝑘 · (abs‘𝐷)) = (𝑦 − 𝑥)))
67	60, 66	mpbid 231	. . . . . 6 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → ∃𝑘 ∈ ℤ (𝑘 · (abs‘𝐷)) = (𝑦 − 𝑥))
68	67	adantr 480	. . . . 5 ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑥 < (abs‘𝐷) ∧ 𝑦 < (abs‘𝐷))) → ∃𝑘 ∈ ℤ (𝑘 · (abs‘𝐷)) = (𝑦 − 𝑥))
69	2, 3, 4, 5	divalglem8 16090	. . . . . 6 ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑥 < (abs‘𝐷) ∧ 𝑦 < (abs‘𝐷))) → (𝑘 ∈ ℤ → ((𝑘 · (abs‘𝐷)) = (𝑦 − 𝑥) → 𝑥 = 𝑦)))
70	69	rexlimdv 3213	. . . . 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 412	. . 3 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → ((𝑥 < (abs‘𝐷) ∧ 𝑦 < (abs‘𝐷)) → 𝑥 = 𝑦))
73	72	rgen2 3128	. 2 ⊢ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ((𝑥 < (abs‘𝐷) ∧ 𝑦 < (abs‘𝐷)) → 𝑥 = 𝑦)
74		breq1 5081	. . 3 ⊢ (𝑥 = 𝑦 → (𝑥 < (abs‘𝐷) ↔ 𝑦 < (abs‘𝐷)))
75	74	reu4 3669	. 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 707	1 ⊢ ∃!𝑥 ∈ 𝑆 𝑥 < (abs‘𝐷)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1541 ∈ wcel 2109 ≠ wne 2944 ∀wral 3065 ∃wrex 3066 ∃!wreu 3067 {crab 3069 class class class wbr 5078 ‘cfv 6430 (class class class)co 7268 infcinf 9161 ℂcc 10853 ℝcr 10854 0cc0 10855 + caddc 10858 · cmul 10860 < clt 10993 ≤ cle 10994 − cmin 11188 ℕcn 11956 ℕ₀cn0 12216 ℤcz 12302 abscabs 14926 ∥ cdvds 15944

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7579 ax-cnex 10911 ax-resscn 10912 ax-1cn 10913 ax-icn 10914 ax-addcl 10915 ax-addrcl 10916 ax-mulcl 10917 ax-mulrcl 10918 ax-mulcom 10919 ax-addass 10920 ax-mulass 10921 ax-distr 10922 ax-i2m1 10923 ax-1ne0 10924 ax-1rid 10925 ax-rnegex 10926 ax-rrecex 10927 ax-cnre 10928 ax-pre-lttri 10929 ax-pre-lttrn 10930 ax-pre-ltadd 10931 ax-pre-mulgt0 10932 ax-pre-sup 10933

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3070 df-rex 3071 df-reu 3072 df-rmo 3073 df-rab 3074 df-v 3432 df-sbc 3720 df-csb 3837 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-pss 3910 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4845 df-iun 4931 df-br 5079 df-opab 5141 df-mpt 5162 df-tr 5196 df-id 5488 df-eprel 5494 df-po 5502 df-so 5503 df-fr 5543 df-we 5545 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-pred 6199 df-ord 6266 df-on 6267 df-lim 6268 df-suc 6269 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-f1 6435 df-fo 6436 df-f1o 6437 df-fv 6438 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-om 7701 df-1st 7817 df-2nd 7818 df-frecs 8081 df-wrecs 8112 df-recs 8186 df-rdg 8225 df-er 8472 df-en 8708 df-dom 8709 df-sdom 8710 df-sup 9162 df-inf 9163 df-pnf 10995 df-mnf 10996 df-xr 10997 df-ltxr 10998 df-le 10999 df-sub 11190 df-neg 11191 df-div 11616 df-nn 11957 df-2 12019 df-3 12020 df-n0 12217 df-z 12303 df-uz 12565 df-rp 12713 df-fz 13222 df-seq 13703 df-exp 13764 df-cj 14791 df-re 14792 df-im 14793 df-sqrt 14927 df-abs 14928 df-dvds 15945

This theorem is referenced by: divalglem10 16092

W3C validator