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Theorem dvdsabseq 14959
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 14912 . . 3 (𝑀𝑁 → (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ))
2 simpr 477 . . . . . 6 ((𝑀𝑁𝑁𝑀) → 𝑁𝑀)
3 breq1 4616 . . . . . . . 8 (𝑁 = 0 → (𝑁𝑀 ↔ 0 ∥ 𝑀))
4 0dvds 14926 . . . . . . . . . 10 (𝑀 ∈ ℤ → (0 ∥ 𝑀𝑀 = 0))
54adantr 481 . . . . . . . . 9 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (0 ∥ 𝑀𝑀 = 0))
6 zcn 11326 . . . . . . . . . . . 12 (𝑀 ∈ ℤ → 𝑀 ∈ ℂ)
76abs00ad 13964 . . . . . . . . . . 11 (𝑀 ∈ ℤ → ((abs‘𝑀) = 0 ↔ 𝑀 = 0))
87bicomd 213 . . . . . . . . . 10 (𝑀 ∈ ℤ → (𝑀 = 0 ↔ (abs‘𝑀) = 0))
98adantr 481 . . . . . . . . 9 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 = 0 ↔ (abs‘𝑀) = 0))
105, 9bitrd 268 . . . . . . . 8 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (0 ∥ 𝑀 ↔ (abs‘𝑀) = 0))
113, 10sylan9bb 735 . . . . . . 7 ((𝑁 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (𝑁𝑀 ↔ (abs‘𝑀) = 0))
12 fveq2 6148 . . . . . . . . . 10 (𝑁 = 0 → (abs‘𝑁) = (abs‘0))
13 abs0 13959 . . . . . . . . . 10 (abs‘0) = 0
1412, 13syl6eq 2671 . . . . . . . . 9 (𝑁 = 0 → (abs‘𝑁) = 0)
1514adantr 481 . . . . . . . 8 ((𝑁 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (abs‘𝑁) = 0)
1615eqeq2d 2631 . . . . . . 7 ((𝑁 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → ((abs‘𝑀) = (abs‘𝑁) ↔ (abs‘𝑀) = 0))
1711, 16bitr4d 271 . . . . . 6 ((𝑁 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (𝑁𝑀 ↔ (abs‘𝑀) = (abs‘𝑁)))
182, 17syl5ib 234 . . . . 5 ((𝑁 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → ((𝑀𝑁𝑁𝑀) → (abs‘𝑀) = (abs‘𝑁)))
1918expd 452 . . . 4 ((𝑁 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (𝑀𝑁 → (𝑁𝑀 → (abs‘𝑀) = (abs‘𝑁))))
20 simprl 793 . . . . . 6 ((¬ 𝑁 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → 𝑀 ∈ ℤ)
21 simpr 477 . . . . . . 7 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑁 ∈ ℤ)
2221adantl 482 . . . . . 6 ((¬ 𝑁 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → 𝑁 ∈ ℤ)
23 neqne 2798 . . . . . . 7 𝑁 = 0 → 𝑁 ≠ 0)
2423adantr 481 . . . . . 6 ((¬ 𝑁 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → 𝑁 ≠ 0)
25 dvdsleabs2 14958 . . . . . 6 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → (𝑀𝑁 → (abs‘𝑀) ≤ (abs‘𝑁)))
2620, 22, 24, 25syl3anc 1323 . . . . 5 ((¬ 𝑁 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (𝑀𝑁 → (abs‘𝑀) ≤ (abs‘𝑁)))
27 simpr 477 . . . . . . . . . . 11 ((𝑁𝑀𝑀𝑁) → 𝑀𝑁)
28 breq1 4616 . . . . . . . . . . . . 13 (𝑀 = 0 → (𝑀𝑁 ↔ 0 ∥ 𝑁))
29 0dvds 14926 . . . . . . . . . . . . . . 15 (𝑁 ∈ ℤ → (0 ∥ 𝑁𝑁 = 0))
30 eqcom 2628 . . . . . . . . . . . . . . . 16 ((abs‘𝑁) = 0 ↔ 0 = (abs‘𝑁))
31 zcn 11326 . . . . . . . . . . . . . . . . 17 (𝑁 ∈ ℤ → 𝑁 ∈ ℂ)
3231abs00ad 13964 . . . . . . . . . . . . . . . 16 (𝑁 ∈ ℤ → ((abs‘𝑁) = 0 ↔ 𝑁 = 0))
3330, 32syl5rbbr 275 . . . . . . . . . . . . . . 15 (𝑁 ∈ ℤ → (𝑁 = 0 ↔ 0 = (abs‘𝑁)))
3429, 33bitrd 268 . . . . . . . . . . . . . 14 (𝑁 ∈ ℤ → (0 ∥ 𝑁 ↔ 0 = (abs‘𝑁)))
3534adantl 482 . . . . . . . . . . . . 13 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (0 ∥ 𝑁 ↔ 0 = (abs‘𝑁)))
3628, 35sylan9bb 735 . . . . . . . . . . . 12 ((𝑀 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (𝑀𝑁 ↔ 0 = (abs‘𝑁)))
37 fveq2 6148 . . . . . . . . . . . . . . 15 (𝑀 = 0 → (abs‘𝑀) = (abs‘0))
3837, 13syl6eq 2671 . . . . . . . . . . . . . 14 (𝑀 = 0 → (abs‘𝑀) = 0)
3938adantr 481 . . . . . . . . . . . . 13 ((𝑀 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (abs‘𝑀) = 0)
4039eqeq1d 2623 . . . . . . . . . . . 12 ((𝑀 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → ((abs‘𝑀) = (abs‘𝑁) ↔ 0 = (abs‘𝑁)))
4136, 40bitr4d 271 . . . . . . . . . . 11 ((𝑀 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (𝑀𝑁 ↔ (abs‘𝑀) = (abs‘𝑁)))
4227, 41syl5ib 234 . . . . . . . . . 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 454 . . . . . . . 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 793 . . . . . . . . . . 11 ((¬ 𝑀 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → 𝑀 ∈ ℤ)
47 neqne 2798 . . . . . . . . . . . 12 𝑀 = 0 → 𝑀 ≠ 0)
4847adantr 481 . . . . . . . . . . 11 ((¬ 𝑀 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → 𝑀 ≠ 0)
49 dvdsleabs2 14958 . . . . . . . . . . 11 ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑀 ≠ 0) → (𝑁𝑀 → (abs‘𝑁) ≤ (abs‘𝑀)))
5045, 46, 48, 49syl3anc 1323 . . . . . . . . . 10 ((¬ 𝑀 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (𝑁𝑀 → (abs‘𝑁) ≤ (abs‘𝑀)))
51 eqcom 2628 . . . . . . . . . . . . . 14 ((abs‘𝑀) = (abs‘𝑁) ↔ (abs‘𝑁) = (abs‘𝑀))
5231abscld 14109 . . . . . . . . . . . . . . 15 (𝑁 ∈ ℤ → (abs‘𝑁) ∈ ℝ)
536abscld 14109 . . . . . . . . . . . . . . 15 (𝑀 ∈ ℤ → (abs‘𝑀) ∈ ℝ)
54 letri3 10067 . . . . . . . . . . . . . . 15 (((abs‘𝑁) ∈ ℝ ∧ (abs‘𝑀) ∈ ℝ) → ((abs‘𝑁) = (abs‘𝑀) ↔ ((abs‘𝑁) ≤ (abs‘𝑀) ∧ (abs‘𝑀) ≤ (abs‘𝑁))))
5552, 53, 54syl2anr 495 . . . . . . . . . . . . . 14 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((abs‘𝑁) = (abs‘𝑀) ↔ ((abs‘𝑁) ≤ (abs‘𝑀) ∧ (abs‘𝑀) ≤ (abs‘𝑁))))
5651, 55syl5bb 272 . . . . . . . . . . . . 13 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((abs‘𝑀) = (abs‘𝑁) ↔ ((abs‘𝑁) ≤ (abs‘𝑀) ∧ (abs‘𝑀) ≤ (abs‘𝑁))))
5756biimprd 238 . . . . . . . . . . . 12 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (((abs‘𝑁) ≤ (abs‘𝑀) ∧ (abs‘𝑀) ≤ (abs‘𝑁)) → (abs‘𝑀) = (abs‘𝑁)))
5857expd 452 . . . . . . . . . . 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 830 . . . . . . 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 830 . . 3 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀𝑁 → (𝑁𝑀 → (abs‘𝑀) = (abs‘𝑁))))
671, 66mpcom 38 . 2 (𝑀𝑁 → (𝑁𝑀 → (abs‘𝑀) = (abs‘𝑁)))
6867imp 445 1 ((𝑀𝑁𝑁𝑀) → (abs‘𝑀) = (abs‘𝑁))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wne 2790   class class class wbr 4613  cfv 5847  cr 9879  0cc0 9880  cle 10019  cz 11321  abscabs 13908  cdvds 14907
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957  ax-pre-sup 9958
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-er 7687  df-en 7900  df-dom 7901  df-sdom 7902  df-sup 8292  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-div 10629  df-nn 10965  df-2 11023  df-3 11024  df-n0 11237  df-z 11322  df-uz 11632  df-rp 11777  df-seq 12742  df-exp 12801  df-cj 13773  df-re 13774  df-im 13775  df-sqrt 13909  df-abs 13910  df-dvds 14908
This theorem is referenced by:  dvdseq  14960
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