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Mirrors > Home > MPE Home > Th. List > dvdssub2 | Structured version Visualization version GIF version |
Description: If an integer divides a difference, then it divides one term iff it divides the other. (Contributed by Mario Carneiro, 13-Jul-2014.) |
Ref | Expression |
---|---|
dvdssub2 | ⊢ (((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐾 ∥ (𝑀 − 𝑁)) → (𝐾 ∥ 𝑀 ↔ 𝐾 ∥ 𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zsubcl 12475 | . . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 − 𝑁) ∈ ℤ) | |
2 | 1 | 3adant1 1130 | . . . . . . 7 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 − 𝑁) ∈ ℤ) |
3 | dvds2sub 16107 | . . . . . . 7 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ (𝑀 − 𝑁) ∈ ℤ) → ((𝐾 ∥ 𝑀 ∧ 𝐾 ∥ (𝑀 − 𝑁)) → 𝐾 ∥ (𝑀 − (𝑀 − 𝑁)))) | |
4 | 2, 3 | syld3an3 1409 | . . . . . 6 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝐾 ∥ 𝑀 ∧ 𝐾 ∥ (𝑀 − 𝑁)) → 𝐾 ∥ (𝑀 − (𝑀 − 𝑁)))) |
5 | 4 | ancomsd 466 | . . . . 5 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝐾 ∥ (𝑀 − 𝑁) ∧ 𝐾 ∥ 𝑀) → 𝐾 ∥ (𝑀 − (𝑀 − 𝑁)))) |
6 | 5 | imp 407 | . . . 4 ⊢ (((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝐾 ∥ (𝑀 − 𝑁) ∧ 𝐾 ∥ 𝑀)) → 𝐾 ∥ (𝑀 − (𝑀 − 𝑁))) |
7 | zcn 12437 | . . . . . . 7 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℂ) | |
8 | zcn 12437 | . . . . . . 7 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ) | |
9 | nncan 11363 | . . . . . . 7 ⊢ ((𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ) → (𝑀 − (𝑀 − 𝑁)) = 𝑁) | |
10 | 7, 8, 9 | syl2an 596 | . . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 − (𝑀 − 𝑁)) = 𝑁) |
11 | 10 | 3adant1 1130 | . . . . 5 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 − (𝑀 − 𝑁)) = 𝑁) |
12 | 11 | adantr 481 | . . . 4 ⊢ (((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝐾 ∥ (𝑀 − 𝑁) ∧ 𝐾 ∥ 𝑀)) → (𝑀 − (𝑀 − 𝑁)) = 𝑁) |
13 | 6, 12 | breqtrd 5129 | . . 3 ⊢ (((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝐾 ∥ (𝑀 − 𝑁) ∧ 𝐾 ∥ 𝑀)) → 𝐾 ∥ 𝑁) |
14 | 13 | expr 457 | . 2 ⊢ (((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐾 ∥ (𝑀 − 𝑁)) → (𝐾 ∥ 𝑀 → 𝐾 ∥ 𝑁)) |
15 | dvds2add 16106 | . . . . . 6 ⊢ ((𝐾 ∈ ℤ ∧ (𝑀 − 𝑁) ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝐾 ∥ (𝑀 − 𝑁) ∧ 𝐾 ∥ 𝑁) → 𝐾 ∥ ((𝑀 − 𝑁) + 𝑁))) | |
16 | 2, 15 | syld3an2 1411 | . . . . 5 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝐾 ∥ (𝑀 − 𝑁) ∧ 𝐾 ∥ 𝑁) → 𝐾 ∥ ((𝑀 − 𝑁) + 𝑁))) |
17 | 16 | imp 407 | . . . 4 ⊢ (((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝐾 ∥ (𝑀 − 𝑁) ∧ 𝐾 ∥ 𝑁)) → 𝐾 ∥ ((𝑀 − 𝑁) + 𝑁)) |
18 | npcan 11343 | . . . . . . 7 ⊢ ((𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ) → ((𝑀 − 𝑁) + 𝑁) = 𝑀) | |
19 | 7, 8, 18 | syl2an 596 | . . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 − 𝑁) + 𝑁) = 𝑀) |
20 | 19 | 3adant1 1130 | . . . . 5 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 − 𝑁) + 𝑁) = 𝑀) |
21 | 20 | adantr 481 | . . . 4 ⊢ (((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝐾 ∥ (𝑀 − 𝑁) ∧ 𝐾 ∥ 𝑁)) → ((𝑀 − 𝑁) + 𝑁) = 𝑀) |
22 | 17, 21 | breqtrd 5129 | . . 3 ⊢ (((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝐾 ∥ (𝑀 − 𝑁) ∧ 𝐾 ∥ 𝑁)) → 𝐾 ∥ 𝑀) |
23 | 22 | expr 457 | . 2 ⊢ (((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐾 ∥ (𝑀 − 𝑁)) → (𝐾 ∥ 𝑁 → 𝐾 ∥ 𝑀)) |
24 | 14, 23 | impbid 211 | 1 ⊢ (((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐾 ∥ (𝑀 − 𝑁)) → (𝐾 ∥ 𝑀 ↔ 𝐾 ∥ 𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 class class class wbr 5103 (class class class)co 7349 ℂcc 10982 + caddc 10987 − cmin 11318 ℤcz 12432 ∥ cdvds 16070 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7662 ax-resscn 11041 ax-1cn 11042 ax-icn 11043 ax-addcl 11044 ax-addrcl 11045 ax-mulcl 11046 ax-mulrcl 11047 ax-mulcom 11048 ax-addass 11049 ax-mulass 11050 ax-distr 11051 ax-i2m1 11052 ax-1ne0 11053 ax-1rid 11054 ax-rnegex 11055 ax-rrecex 11056 ax-cnre 11057 ax-pre-lttri 11058 ax-pre-lttrn 11059 ax-pre-ltadd 11060 ax-pre-mulgt0 11061 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5528 df-eprel 5534 df-po 5542 df-so 5543 df-fr 5585 df-we 5587 df-xp 5636 df-rel 5637 df-cnv 5638 df-co 5639 df-dm 5640 df-rn 5641 df-res 5642 df-ima 5643 df-pred 6249 df-ord 6316 df-on 6317 df-lim 6318 df-suc 6319 df-iota 6443 df-fun 6493 df-fn 6494 df-f 6495 df-f1 6496 df-fo 6497 df-f1o 6498 df-fv 6499 df-riota 7305 df-ov 7352 df-oprab 7353 df-mpo 7354 df-om 7793 df-2nd 7912 df-frecs 8179 df-wrecs 8210 df-recs 8284 df-rdg 8323 df-er 8581 df-en 8817 df-dom 8818 df-sdom 8819 df-pnf 11124 df-mnf 11125 df-xr 11126 df-ltxr 11127 df-le 11128 df-sub 11320 df-neg 11321 df-nn 12087 df-n0 12347 df-z 12433 df-dvds 16071 |
This theorem is referenced by: dvdsadd 16118 3dvds 16147 bitsmod 16250 bitsinv1lem 16255 sylow2blem3 19333 znunit 20893 perfectlem1 26499 lgsqr 26621 lgsqrmodndvds 26623 2sqlem8 26696 poimirlem28 36001 jm2.20nn 41186 |
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