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Mirrors > Home > ILE Home > Th. List > absdvdsb | GIF version |
Description: An integer divides another iff its absolute value does. (Contributed by Paul Chapman, 21-Mar-2011.) |
Ref | Expression |
---|---|
absdvdsb | ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∥ 𝑁 ↔ (abs‘𝑀) ∥ 𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq1 4024 | . . . 4 ⊢ ((abs‘𝑀) = 𝑀 → ((abs‘𝑀) ∥ 𝑁 ↔ 𝑀 ∥ 𝑁)) | |
2 | 1 | bicomd 141 | . . 3 ⊢ ((abs‘𝑀) = 𝑀 → (𝑀 ∥ 𝑁 ↔ (abs‘𝑀) ∥ 𝑁)) |
3 | 2 | a1i 9 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((abs‘𝑀) = 𝑀 → (𝑀 ∥ 𝑁 ↔ (abs‘𝑀) ∥ 𝑁))) |
4 | negdvdsb 11856 | . . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∥ 𝑁 ↔ -𝑀 ∥ 𝑁)) | |
5 | breq1 4024 | . . . . 5 ⊢ ((abs‘𝑀) = -𝑀 → ((abs‘𝑀) ∥ 𝑁 ↔ -𝑀 ∥ 𝑁)) | |
6 | 5 | bicomd 141 | . . . 4 ⊢ ((abs‘𝑀) = -𝑀 → (-𝑀 ∥ 𝑁 ↔ (abs‘𝑀) ∥ 𝑁)) |
7 | 4, 6 | sylan9bb 462 | . . 3 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (abs‘𝑀) = -𝑀) → (𝑀 ∥ 𝑁 ↔ (abs‘𝑀) ∥ 𝑁)) |
8 | 7 | ex 115 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((abs‘𝑀) = -𝑀 → (𝑀 ∥ 𝑁 ↔ (abs‘𝑀) ∥ 𝑁))) |
9 | zq 9663 | . . . 4 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℚ) | |
10 | 9 | qabsord 11127 | . . 3 ⊢ (𝑀 ∈ ℤ → ((abs‘𝑀) = 𝑀 ∨ (abs‘𝑀) = -𝑀)) |
11 | 10 | adantr 276 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((abs‘𝑀) = 𝑀 ∨ (abs‘𝑀) = -𝑀)) |
12 | 3, 8, 11 | mpjaod 719 | 1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∥ 𝑁 ↔ (abs‘𝑀) ∥ 𝑁)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 709 = wceq 1364 ∈ wcel 2160 class class class wbr 4021 ‘cfv 5238 -cneg 8165 ℤcz 9289 abscabs 11048 ∥ cdvds 11836 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4136 ax-sep 4139 ax-nul 4147 ax-pow 4195 ax-pr 4230 ax-un 4454 ax-setind 4557 ax-iinf 4608 ax-cnex 7937 ax-resscn 7938 ax-1cn 7939 ax-1re 7940 ax-icn 7941 ax-addcl 7942 ax-addrcl 7943 ax-mulcl 7944 ax-mulrcl 7945 ax-addcom 7946 ax-mulcom 7947 ax-addass 7948 ax-mulass 7949 ax-distr 7950 ax-i2m1 7951 ax-0lt1 7952 ax-1rid 7953 ax-0id 7954 ax-rnegex 7955 ax-precex 7956 ax-cnre 7957 ax-pre-ltirr 7958 ax-pre-ltwlin 7959 ax-pre-lttrn 7960 ax-pre-apti 7961 ax-pre-ltadd 7962 ax-pre-mulgt0 7963 ax-pre-mulext 7964 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-if 3550 df-pw 3595 df-sn 3616 df-pr 3617 df-op 3619 df-uni 3828 df-int 3863 df-iun 3906 df-br 4022 df-opab 4083 df-mpt 4084 df-tr 4120 df-id 4314 df-po 4317 df-iso 4318 df-iord 4387 df-on 4389 df-ilim 4390 df-suc 4392 df-iom 4611 df-xp 4653 df-rel 4654 df-cnv 4655 df-co 4656 df-dm 4657 df-rn 4658 df-res 4659 df-ima 4660 df-iota 5199 df-fun 5240 df-fn 5241 df-f 5242 df-f1 5243 df-fo 5244 df-f1o 5245 df-fv 5246 df-riota 5855 df-ov 5903 df-oprab 5904 df-mpo 5905 df-1st 6169 df-2nd 6170 df-recs 6334 df-frec 6420 df-pnf 8030 df-mnf 8031 df-xr 8032 df-ltxr 8033 df-le 8034 df-sub 8166 df-neg 8167 df-reap 8568 df-ap 8575 df-div 8666 df-inn 8956 df-2 9014 df-n0 9213 df-z 9290 df-uz 9565 df-q 9657 df-rp 9691 df-seqfrec 10486 df-exp 10561 df-cj 10893 df-re 10894 df-im 10895 df-rsqrt 11049 df-abs 11050 df-dvds 11837 |
This theorem is referenced by: dvdsleabs2 11894 gcd0id 12022 dvdssq 12074 lcmdvds 12123 lcmgcdeq 12127 pc2dvds 12374 |
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