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Mirrors > Home > MPE Home > Th. List > divalglem0 | Structured version Visualization version GIF version |
Description: Lemma for divalg 15742. (Contributed by Paul Chapman, 21-Mar-2011.) |
Ref | Expression |
---|---|
divalglem0.1 | ⊢ 𝑁 ∈ ℤ |
divalglem0.2 | ⊢ 𝐷 ∈ ℤ |
Ref | Expression |
---|---|
divalglem0 | ⊢ ((𝑅 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝐷 ∥ (𝑁 − 𝑅) → 𝐷 ∥ (𝑁 − (𝑅 − (𝐾 · (abs‘𝐷)))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | divalglem0.2 | . . . . . 6 ⊢ 𝐷 ∈ ℤ | |
2 | iddvds 15611 | . . . . . . 7 ⊢ (𝐷 ∈ ℤ → 𝐷 ∥ 𝐷) | |
3 | dvdsabsb 15617 | . . . . . . . 8 ⊢ ((𝐷 ∈ ℤ ∧ 𝐷 ∈ ℤ) → (𝐷 ∥ 𝐷 ↔ 𝐷 ∥ (abs‘𝐷))) | |
4 | 3 | anidms 567 | . . . . . . 7 ⊢ (𝐷 ∈ ℤ → (𝐷 ∥ 𝐷 ↔ 𝐷 ∥ (abs‘𝐷))) |
5 | 2, 4 | mpbid 233 | . . . . . 6 ⊢ (𝐷 ∈ ℤ → 𝐷 ∥ (abs‘𝐷)) |
6 | 1, 5 | ax-mp 5 | . . . . 5 ⊢ 𝐷 ∥ (abs‘𝐷) |
7 | nn0abscl 14660 | . . . . . . . 8 ⊢ (𝐷 ∈ ℤ → (abs‘𝐷) ∈ ℕ0) | |
8 | 1, 7 | ax-mp 5 | . . . . . . 7 ⊢ (abs‘𝐷) ∈ ℕ0 |
9 | 8 | nn0zi 11995 | . . . . . 6 ⊢ (abs‘𝐷) ∈ ℤ |
10 | dvdsmultr2 15637 | . . . . . 6 ⊢ ((𝐷 ∈ ℤ ∧ 𝐾 ∈ ℤ ∧ (abs‘𝐷) ∈ ℤ) → (𝐷 ∥ (abs‘𝐷) → 𝐷 ∥ (𝐾 · (abs‘𝐷)))) | |
11 | 1, 9, 10 | mp3an13 1443 | . . . . 5 ⊢ (𝐾 ∈ ℤ → (𝐷 ∥ (abs‘𝐷) → 𝐷 ∥ (𝐾 · (abs‘𝐷)))) |
12 | 6, 11 | mpi 20 | . . . 4 ⊢ (𝐾 ∈ ℤ → 𝐷 ∥ (𝐾 · (abs‘𝐷))) |
13 | 12 | adantl 482 | . . 3 ⊢ ((𝑅 ∈ ℤ ∧ 𝐾 ∈ ℤ) → 𝐷 ∥ (𝐾 · (abs‘𝐷))) |
14 | divalglem0.1 | . . . . 5 ⊢ 𝑁 ∈ ℤ | |
15 | zsubcl 12012 | . . . . 5 ⊢ ((𝑁 ∈ ℤ ∧ 𝑅 ∈ ℤ) → (𝑁 − 𝑅) ∈ ℤ) | |
16 | 14, 15 | mpan 686 | . . . 4 ⊢ (𝑅 ∈ ℤ → (𝑁 − 𝑅) ∈ ℤ) |
17 | zmulcl 12019 | . . . . 5 ⊢ ((𝐾 ∈ ℤ ∧ (abs‘𝐷) ∈ ℤ) → (𝐾 · (abs‘𝐷)) ∈ ℤ) | |
18 | 9, 17 | mpan2 687 | . . . 4 ⊢ (𝐾 ∈ ℤ → (𝐾 · (abs‘𝐷)) ∈ ℤ) |
19 | dvds2add 15631 | . . . 4 ⊢ ((𝐷 ∈ ℤ ∧ (𝑁 − 𝑅) ∈ ℤ ∧ (𝐾 · (abs‘𝐷)) ∈ ℤ) → ((𝐷 ∥ (𝑁 − 𝑅) ∧ 𝐷 ∥ (𝐾 · (abs‘𝐷))) → 𝐷 ∥ ((𝑁 − 𝑅) + (𝐾 · (abs‘𝐷))))) | |
20 | 1, 16, 18, 19 | mp3an3an 1458 | . . 3 ⊢ ((𝑅 ∈ ℤ ∧ 𝐾 ∈ ℤ) → ((𝐷 ∥ (𝑁 − 𝑅) ∧ 𝐷 ∥ (𝐾 · (abs‘𝐷))) → 𝐷 ∥ ((𝑁 − 𝑅) + (𝐾 · (abs‘𝐷))))) |
21 | 13, 20 | mpan2d 690 | . 2 ⊢ ((𝑅 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝐷 ∥ (𝑁 − 𝑅) → 𝐷 ∥ ((𝑁 − 𝑅) + (𝐾 · (abs‘𝐷))))) |
22 | zcn 11974 | . . . . 5 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ) | |
23 | 14, 22 | ax-mp 5 | . . . 4 ⊢ 𝑁 ∈ ℂ |
24 | zcn 11974 | . . . 4 ⊢ (𝑅 ∈ ℤ → 𝑅 ∈ ℂ) | |
25 | 18 | zcnd 12076 | . . . 4 ⊢ (𝐾 ∈ ℤ → (𝐾 · (abs‘𝐷)) ∈ ℂ) |
26 | subsub 10904 | . . . 4 ⊢ ((𝑁 ∈ ℂ ∧ 𝑅 ∈ ℂ ∧ (𝐾 · (abs‘𝐷)) ∈ ℂ) → (𝑁 − (𝑅 − (𝐾 · (abs‘𝐷)))) = ((𝑁 − 𝑅) + (𝐾 · (abs‘𝐷)))) | |
27 | 23, 24, 25, 26 | mp3an3an 1458 | . . 3 ⊢ ((𝑅 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑁 − (𝑅 − (𝐾 · (abs‘𝐷)))) = ((𝑁 − 𝑅) + (𝐾 · (abs‘𝐷)))) |
28 | 27 | breq2d 5069 | . 2 ⊢ ((𝑅 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝐷 ∥ (𝑁 − (𝑅 − (𝐾 · (abs‘𝐷)))) ↔ 𝐷 ∥ ((𝑁 − 𝑅) + (𝐾 · (abs‘𝐷))))) |
29 | 21, 28 | sylibrd 260 | 1 ⊢ ((𝑅 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝐷 ∥ (𝑁 − 𝑅) → 𝐷 ∥ (𝑁 − (𝑅 − (𝐾 · (abs‘𝐷)))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 class class class wbr 5057 ‘cfv 6348 (class class class)co 7145 ℂcc 10523 + caddc 10528 · cmul 10530 − cmin 10858 ℕ0cn0 11885 ℤcz 11969 abscabs 14581 ∥ cdvds 15595 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-sup 8894 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-seq 13358 df-exp 13418 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-dvds 15596 |
This theorem is referenced by: divalglem5 15736 |
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