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Mirrors > Home > MPE Home > Th. List > divalglem0 | Structured version Visualization version GIF version |
Description: Lemma for divalg 15126. (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 14995 | . . . . . . 7 ⊢ (𝐷 ∈ ℤ → 𝐷 ∥ 𝐷) | |
3 | dvdsabsb 15001 | . . . . . . . 8 ⊢ ((𝐷 ∈ ℤ ∧ 𝐷 ∈ ℤ) → (𝐷 ∥ 𝐷 ↔ 𝐷 ∥ (abs‘𝐷))) | |
4 | 3 | anidms 677 | . . . . . . 7 ⊢ (𝐷 ∈ ℤ → (𝐷 ∥ 𝐷 ↔ 𝐷 ∥ (abs‘𝐷))) |
5 | 2, 4 | mpbid 222 | . . . . . 6 ⊢ (𝐷 ∈ ℤ → 𝐷 ∥ (abs‘𝐷)) |
6 | 1, 5 | ax-mp 5 | . . . . 5 ⊢ 𝐷 ∥ (abs‘𝐷) |
7 | nn0abscl 14052 | . . . . . . . 8 ⊢ (𝐷 ∈ ℤ → (abs‘𝐷) ∈ ℕ0) | |
8 | 1, 7 | ax-mp 5 | . . . . . . 7 ⊢ (abs‘𝐷) ∈ ℕ0 |
9 | 8 | nn0zi 11402 | . . . . . 6 ⊢ (abs‘𝐷) ∈ ℤ |
10 | dvdsmultr2 15021 | . . . . . 6 ⊢ ((𝐷 ∈ ℤ ∧ 𝐾 ∈ ℤ ∧ (abs‘𝐷) ∈ ℤ) → (𝐷 ∥ (abs‘𝐷) → 𝐷 ∥ (𝐾 · (abs‘𝐷)))) | |
11 | 1, 9, 10 | mp3an13 1415 | . . . . 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 11419 | . . . . 5 ⊢ ((𝑁 ∈ ℤ ∧ 𝑅 ∈ ℤ) → (𝑁 − 𝑅) ∈ ℤ) | |
16 | 14, 15 | mpan 706 | . . . 4 ⊢ (𝑅 ∈ ℤ → (𝑁 − 𝑅) ∈ ℤ) |
17 | zmulcl 11426 | . . . . 5 ⊢ ((𝐾 ∈ ℤ ∧ (abs‘𝐷) ∈ ℤ) → (𝐾 · (abs‘𝐷)) ∈ ℤ) | |
18 | 9, 17 | mpan2 707 | . . . 4 ⊢ (𝐾 ∈ ℤ → (𝐾 · (abs‘𝐷)) ∈ ℤ) |
19 | dvds2add 15015 | . . . . 5 ⊢ ((𝐷 ∈ ℤ ∧ (𝑁 − 𝑅) ∈ ℤ ∧ (𝐾 · (abs‘𝐷)) ∈ ℤ) → ((𝐷 ∥ (𝑁 − 𝑅) ∧ 𝐷 ∥ (𝐾 · (abs‘𝐷))) → 𝐷 ∥ ((𝑁 − 𝑅) + (𝐾 · (abs‘𝐷))))) | |
20 | 1, 19 | mp3an1 1411 | . . . 4 ⊢ (((𝑁 − 𝑅) ∈ ℤ ∧ (𝐾 · (abs‘𝐷)) ∈ ℤ) → ((𝐷 ∥ (𝑁 − 𝑅) ∧ 𝐷 ∥ (𝐾 · (abs‘𝐷))) → 𝐷 ∥ ((𝑁 − 𝑅) + (𝐾 · (abs‘𝐷))))) |
21 | 16, 18, 20 | syl2an 494 | . . 3 ⊢ ((𝑅 ∈ ℤ ∧ 𝐾 ∈ ℤ) → ((𝐷 ∥ (𝑁 − 𝑅) ∧ 𝐷 ∥ (𝐾 · (abs‘𝐷))) → 𝐷 ∥ ((𝑁 − 𝑅) + (𝐾 · (abs‘𝐷))))) |
22 | 13, 21 | mpan2d 710 | . 2 ⊢ ((𝑅 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝐷 ∥ (𝑁 − 𝑅) → 𝐷 ∥ ((𝑁 − 𝑅) + (𝐾 · (abs‘𝐷))))) |
23 | zcn 11382 | . . . 4 ⊢ (𝑅 ∈ ℤ → 𝑅 ∈ ℂ) | |
24 | 18 | zcnd 11483 | . . . 4 ⊢ (𝐾 ∈ ℤ → (𝐾 · (abs‘𝐷)) ∈ ℂ) |
25 | zcn 11382 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ) | |
26 | 14, 25 | ax-mp 5 | . . . . 5 ⊢ 𝑁 ∈ ℂ |
27 | subsub 10311 | . . . . 5 ⊢ ((𝑁 ∈ ℂ ∧ 𝑅 ∈ ℂ ∧ (𝐾 · (abs‘𝐷)) ∈ ℂ) → (𝑁 − (𝑅 − (𝐾 · (abs‘𝐷)))) = ((𝑁 − 𝑅) + (𝐾 · (abs‘𝐷)))) | |
28 | 26, 27 | mp3an1 1411 | . . . 4 ⊢ ((𝑅 ∈ ℂ ∧ (𝐾 · (abs‘𝐷)) ∈ ℂ) → (𝑁 − (𝑅 − (𝐾 · (abs‘𝐷)))) = ((𝑁 − 𝑅) + (𝐾 · (abs‘𝐷)))) |
29 | 23, 24, 28 | syl2an 494 | . . 3 ⊢ ((𝑅 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑁 − (𝑅 − (𝐾 · (abs‘𝐷)))) = ((𝑁 − 𝑅) + (𝐾 · (abs‘𝐷)))) |
30 | 29 | breq2d 4665 | . 2 ⊢ ((𝑅 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝐷 ∥ (𝑁 − (𝑅 − (𝐾 · (abs‘𝐷)))) ↔ 𝐷 ∥ ((𝑁 − 𝑅) + (𝐾 · (abs‘𝐷))))) |
31 | 22, 30 | sylibrd 249 | 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 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 + caddc 9939 · cmul 9941 − cmin 10266 ℕ0cn0 11292 ℤcz 11377 abscabs 13974 ∥ cdvds 14983 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-seq 12802 df-exp 12861 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-dvds 14984 |
This theorem is referenced by: divalglem5 15120 |
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