MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dvdsabseq Structured version   Visualization version   GIF version

Theorem dvdsabseq 16258
Description: If two integers divide each other, they must be equal, up to a difference in sign. Theorem 1.1(j) in [ApostolNT] p. 14. (Contributed by Mario Carneiro, 30-May-2014.) (Revised by AV, 7-Aug-2021.)
Assertion
Ref Expression
dvdsabseq ((𝑀𝑁𝑁𝑀) → (abs‘𝑀) = (abs‘𝑁))

Proof of Theorem dvdsabseq
StepHypRef Expression
1 dvdszrcl 16204 . . 3 (𝑀𝑁 → (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ))
2 simpr 485 . . . . . 6 ((𝑀𝑁𝑁𝑀) → 𝑁𝑀)
3 breq1 5151 . . . . . . . 8 (𝑁 = 0 → (𝑁𝑀 ↔ 0 ∥ 𝑀))
4 0dvds 16222 . . . . . . . . . 10 (𝑀 ∈ ℤ → (0 ∥ 𝑀𝑀 = 0))
54adantr 481 . . . . . . . . 9 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (0 ∥ 𝑀𝑀 = 0))
6 zcn 12565 . . . . . . . . . . . 12 (𝑀 ∈ ℤ → 𝑀 ∈ ℂ)
76abs00ad 15239 . . . . . . . . . . 11 (𝑀 ∈ ℤ → ((abs‘𝑀) = 0 ↔ 𝑀 = 0))
87bicomd 222 . . . . . . . . . 10 (𝑀 ∈ ℤ → (𝑀 = 0 ↔ (abs‘𝑀) = 0))
98adantr 481 . . . . . . . . 9 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 = 0 ↔ (abs‘𝑀) = 0))
105, 9bitrd 278 . . . . . . . 8 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (0 ∥ 𝑀 ↔ (abs‘𝑀) = 0))
113, 10sylan9bb 510 . . . . . . 7 ((𝑁 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (𝑁𝑀 ↔ (abs‘𝑀) = 0))
12 fveq2 6891 . . . . . . . . . 10 (𝑁 = 0 → (abs‘𝑁) = (abs‘0))
13 abs0 15234 . . . . . . . . . 10 (abs‘0) = 0
1412, 13eqtrdi 2788 . . . . . . . . 9 (𝑁 = 0 → (abs‘𝑁) = 0)
1514adantr 481 . . . . . . . 8 ((𝑁 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (abs‘𝑁) = 0)
1615eqeq2d 2743 . . . . . . 7 ((𝑁 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → ((abs‘𝑀) = (abs‘𝑁) ↔ (abs‘𝑀) = 0))
1711, 16bitr4d 281 . . . . . 6 ((𝑁 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (𝑁𝑀 ↔ (abs‘𝑀) = (abs‘𝑁)))
182, 17imbitrid 243 . . . . 5 ((𝑁 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → ((𝑀𝑁𝑁𝑀) → (abs‘𝑀) = (abs‘𝑁)))
1918expd 416 . . . 4 ((𝑁 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (𝑀𝑁 → (𝑁𝑀 → (abs‘𝑀) = (abs‘𝑁))))
20 simprl 769 . . . . . 6 ((¬ 𝑁 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → 𝑀 ∈ ℤ)
21 simpr 485 . . . . . . 7 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑁 ∈ ℤ)
2221adantl 482 . . . . . 6 ((¬ 𝑁 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → 𝑁 ∈ ℤ)
23 neqne 2948 . . . . . . 7 𝑁 = 0 → 𝑁 ≠ 0)
2423adantr 481 . . . . . 6 ((¬ 𝑁 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → 𝑁 ≠ 0)
25 dvdsleabs2 16257 . . . . . 6 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → (𝑀𝑁 → (abs‘𝑀) ≤ (abs‘𝑁)))
2620, 22, 24, 25syl3anc 1371 . . . . 5 ((¬ 𝑁 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (𝑀𝑁 → (abs‘𝑀) ≤ (abs‘𝑁)))
27 simpr 485 . . . . . . . . . . 11 ((𝑁𝑀𝑀𝑁) → 𝑀𝑁)
28 breq1 5151 . . . . . . . . . . . . 13 (𝑀 = 0 → (𝑀𝑁 ↔ 0 ∥ 𝑁))
29 0dvds 16222 . . . . . . . . . . . . . . 15 (𝑁 ∈ ℤ → (0 ∥ 𝑁𝑁 = 0))
30 zcn 12565 . . . . . . . . . . . . . . . . 17 (𝑁 ∈ ℤ → 𝑁 ∈ ℂ)
3130abs00ad 15239 . . . . . . . . . . . . . . . 16 (𝑁 ∈ ℤ → ((abs‘𝑁) = 0 ↔ 𝑁 = 0))
32 eqcom 2739 . . . . . . . . . . . . . . . 16 ((abs‘𝑁) = 0 ↔ 0 = (abs‘𝑁))
3331, 32bitr3di 285 . . . . . . . . . . . . . . 15 (𝑁 ∈ ℤ → (𝑁 = 0 ↔ 0 = (abs‘𝑁)))
3429, 33bitrd 278 . . . . . . . . . . . . . 14 (𝑁 ∈ ℤ → (0 ∥ 𝑁 ↔ 0 = (abs‘𝑁)))
3534adantl 482 . . . . . . . . . . . . 13 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (0 ∥ 𝑁 ↔ 0 = (abs‘𝑁)))
3628, 35sylan9bb 510 . . . . . . . . . . . 12 ((𝑀 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (𝑀𝑁 ↔ 0 = (abs‘𝑁)))
37 fveq2 6891 . . . . . . . . . . . . . . 15 (𝑀 = 0 → (abs‘𝑀) = (abs‘0))
3837, 13eqtrdi 2788 . . . . . . . . . . . . . 14 (𝑀 = 0 → (abs‘𝑀) = 0)
3938adantr 481 . . . . . . . . . . . . 13 ((𝑀 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (abs‘𝑀) = 0)
4039eqeq1d 2734 . . . . . . . . . . . 12 ((𝑀 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → ((abs‘𝑀) = (abs‘𝑁) ↔ 0 = (abs‘𝑁)))
4136, 40bitr4d 281 . . . . . . . . . . 11 ((𝑀 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (𝑀𝑁 ↔ (abs‘𝑀) = (abs‘𝑁)))
4227, 41imbitrid 243 . . . . . . . . . 10 ((𝑀 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → ((𝑁𝑀𝑀𝑁) → (abs‘𝑀) = (abs‘𝑁)))
4342a1dd 50 . . . . . . . . 9 ((𝑀 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → ((𝑁𝑀𝑀𝑁) → ((abs‘𝑀) ≤ (abs‘𝑁) → (abs‘𝑀) = (abs‘𝑁))))
4443expcomd 417 . . . . . . . 8 ((𝑀 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (𝑀𝑁 → (𝑁𝑀 → ((abs‘𝑀) ≤ (abs‘𝑁) → (abs‘𝑀) = (abs‘𝑁)))))
4521adantl 482 . . . . . . . . . . 11 ((¬ 𝑀 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → 𝑁 ∈ ℤ)
46 simprl 769 . . . . . . . . . . 11 ((¬ 𝑀 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → 𝑀 ∈ ℤ)
47 neqne 2948 . . . . . . . . . . . 12 𝑀 = 0 → 𝑀 ≠ 0)
4847adantr 481 . . . . . . . . . . 11 ((¬ 𝑀 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → 𝑀 ≠ 0)
49 dvdsleabs2 16257 . . . . . . . . . . 11 ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑀 ≠ 0) → (𝑁𝑀 → (abs‘𝑁) ≤ (abs‘𝑀)))
5045, 46, 48, 49syl3anc 1371 . . . . . . . . . 10 ((¬ 𝑀 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (𝑁𝑀 → (abs‘𝑁) ≤ (abs‘𝑀)))
51 eqcom 2739 . . . . . . . . . . . . . 14 ((abs‘𝑀) = (abs‘𝑁) ↔ (abs‘𝑁) = (abs‘𝑀))
5230abscld 15385 . . . . . . . . . . . . . . 15 (𝑁 ∈ ℤ → (abs‘𝑁) ∈ ℝ)
536abscld 15385 . . . . . . . . . . . . . . 15 (𝑀 ∈ ℤ → (abs‘𝑀) ∈ ℝ)
54 letri3 11301 . . . . . . . . . . . . . . 15 (((abs‘𝑁) ∈ ℝ ∧ (abs‘𝑀) ∈ ℝ) → ((abs‘𝑁) = (abs‘𝑀) ↔ ((abs‘𝑁) ≤ (abs‘𝑀) ∧ (abs‘𝑀) ≤ (abs‘𝑁))))
5552, 53, 54syl2anr 597 . . . . . . . . . . . . . 14 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((abs‘𝑁) = (abs‘𝑀) ↔ ((abs‘𝑁) ≤ (abs‘𝑀) ∧ (abs‘𝑀) ≤ (abs‘𝑁))))
5651, 55bitrid 282 . . . . . . . . . . . . 13 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((abs‘𝑀) = (abs‘𝑁) ↔ ((abs‘𝑁) ≤ (abs‘𝑀) ∧ (abs‘𝑀) ≤ (abs‘𝑁))))
5756biimprd 247 . . . . . . . . . . . 12 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (((abs‘𝑁) ≤ (abs‘𝑀) ∧ (abs‘𝑀) ≤ (abs‘𝑁)) → (abs‘𝑀) = (abs‘𝑁)))
5857expd 416 . . . . . . . . . . 11 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((abs‘𝑁) ≤ (abs‘𝑀) → ((abs‘𝑀) ≤ (abs‘𝑁) → (abs‘𝑀) = (abs‘𝑁))))
5958adantl 482 . . . . . . . . . 10 ((¬ 𝑀 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → ((abs‘𝑁) ≤ (abs‘𝑀) → ((abs‘𝑀) ≤ (abs‘𝑁) → (abs‘𝑀) = (abs‘𝑁))))
6050, 59syld 47 . . . . . . . . 9 ((¬ 𝑀 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (𝑁𝑀 → ((abs‘𝑀) ≤ (abs‘𝑁) → (abs‘𝑀) = (abs‘𝑁))))
6160a1d 25 . . . . . . . 8 ((¬ 𝑀 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (𝑀𝑁 → (𝑁𝑀 → ((abs‘𝑀) ≤ (abs‘𝑁) → (abs‘𝑀) = (abs‘𝑁)))))
6244, 61pm2.61ian 810 . . . . . . 7 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀𝑁 → (𝑁𝑀 → ((abs‘𝑀) ≤ (abs‘𝑁) → (abs‘𝑀) = (abs‘𝑁)))))
6362com34 91 . . . . . 6 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀𝑁 → ((abs‘𝑀) ≤ (abs‘𝑁) → (𝑁𝑀 → (abs‘𝑀) = (abs‘𝑁)))))
6463adantl 482 . . . . 5 ((¬ 𝑁 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (𝑀𝑁 → ((abs‘𝑀) ≤ (abs‘𝑁) → (𝑁𝑀 → (abs‘𝑀) = (abs‘𝑁)))))
6526, 64mpdd 43 . . . 4 ((¬ 𝑁 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (𝑀𝑁 → (𝑁𝑀 → (abs‘𝑀) = (abs‘𝑁))))
6619, 65pm2.61ian 810 . . 3 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀𝑁 → (𝑁𝑀 → (abs‘𝑀) = (abs‘𝑁))))
671, 66mpcom 38 . 2 (𝑀𝑁 → (𝑁𝑀 → (abs‘𝑀) = (abs‘𝑁)))
6867imp 407 1 ((𝑀𝑁𝑁𝑀) → (abs‘𝑀) = (abs‘𝑁))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wne 2940   class class class wbr 5148  cfv 6543  cr 11111  0cc0 11112  cle 11251  cz 12560  abscabs 15183  cdvds 16199
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 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189  ax-pre-sup 11190
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-sup 9439  df-pnf 11252  df-mnf 11253  df-xr 11254  df-ltxr 11255  df-le 11256  df-sub 11448  df-neg 11449  df-div 11874  df-nn 12215  df-2 12277  df-3 12278  df-n0 12475  df-z 12561  df-uz 12825  df-rp 12977  df-seq 13969  df-exp 14030  df-cj 15048  df-re 15049  df-im 15050  df-sqrt 15184  df-abs 15185  df-dvds 16200
This theorem is referenced by:  dvdseq  16259
  Copyright terms: Public domain W3C validator