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Theorem dvdsabseq 16350
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 16295 . . 3 (𝑀𝑁 → (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ))
2 simpr 484 . . . . . 6 ((𝑀𝑁𝑁𝑀) → 𝑁𝑀)
3 breq1 5146 . . . . . . . 8 (𝑁 = 0 → (𝑁𝑀 ↔ 0 ∥ 𝑀))
4 0dvds 16314 . . . . . . . . . 10 (𝑀 ∈ ℤ → (0 ∥ 𝑀𝑀 = 0))
54adantr 480 . . . . . . . . 9 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (0 ∥ 𝑀𝑀 = 0))
6 zcn 12618 . . . . . . . . . . . 12 (𝑀 ∈ ℤ → 𝑀 ∈ ℂ)
76abs00ad 15329 . . . . . . . . . . 11 (𝑀 ∈ ℤ → ((abs‘𝑀) = 0 ↔ 𝑀 = 0))
87bicomd 223 . . . . . . . . . 10 (𝑀 ∈ ℤ → (𝑀 = 0 ↔ (abs‘𝑀) = 0))
98adantr 480 . . . . . . . . 9 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 = 0 ↔ (abs‘𝑀) = 0))
105, 9bitrd 279 . . . . . . . 8 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (0 ∥ 𝑀 ↔ (abs‘𝑀) = 0))
113, 10sylan9bb 509 . . . . . . 7 ((𝑁 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (𝑁𝑀 ↔ (abs‘𝑀) = 0))
12 fveq2 6906 . . . . . . . . . 10 (𝑁 = 0 → (abs‘𝑁) = (abs‘0))
13 abs0 15324 . . . . . . . . . 10 (abs‘0) = 0
1412, 13eqtrdi 2793 . . . . . . . . 9 (𝑁 = 0 → (abs‘𝑁) = 0)
1514adantr 480 . . . . . . . 8 ((𝑁 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (abs‘𝑁) = 0)
1615eqeq2d 2748 . . . . . . 7 ((𝑁 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → ((abs‘𝑀) = (abs‘𝑁) ↔ (abs‘𝑀) = 0))
1711, 16bitr4d 282 . . . . . 6 ((𝑁 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (𝑁𝑀 ↔ (abs‘𝑀) = (abs‘𝑁)))
182, 17imbitrid 244 . . . . 5 ((𝑁 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → ((𝑀𝑁𝑁𝑀) → (abs‘𝑀) = (abs‘𝑁)))
1918expd 415 . . . 4 ((𝑁 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (𝑀𝑁 → (𝑁𝑀 → (abs‘𝑀) = (abs‘𝑁))))
20 simprl 771 . . . . . 6 ((¬ 𝑁 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → 𝑀 ∈ ℤ)
21 simpr 484 . . . . . . 7 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑁 ∈ ℤ)
2221adantl 481 . . . . . 6 ((¬ 𝑁 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → 𝑁 ∈ ℤ)
23 neqne 2948 . . . . . . 7 𝑁 = 0 → 𝑁 ≠ 0)
2423adantr 480 . . . . . 6 ((¬ 𝑁 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → 𝑁 ≠ 0)
25 dvdsleabs2 16349 . . . . . 6 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → (𝑀𝑁 → (abs‘𝑀) ≤ (abs‘𝑁)))
2620, 22, 24, 25syl3anc 1373 . . . . 5 ((¬ 𝑁 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (𝑀𝑁 → (abs‘𝑀) ≤ (abs‘𝑁)))
27 simpr 484 . . . . . . . . . . 11 ((𝑁𝑀𝑀𝑁) → 𝑀𝑁)
28 breq1 5146 . . . . . . . . . . . . 13 (𝑀 = 0 → (𝑀𝑁 ↔ 0 ∥ 𝑁))
29 0dvds 16314 . . . . . . . . . . . . . . 15 (𝑁 ∈ ℤ → (0 ∥ 𝑁𝑁 = 0))
30 zcn 12618 . . . . . . . . . . . . . . . . 17 (𝑁 ∈ ℤ → 𝑁 ∈ ℂ)
3130abs00ad 15329 . . . . . . . . . . . . . . . 16 (𝑁 ∈ ℤ → ((abs‘𝑁) = 0 ↔ 𝑁 = 0))
32 eqcom 2744 . . . . . . . . . . . . . . . 16 ((abs‘𝑁) = 0 ↔ 0 = (abs‘𝑁))
3331, 32bitr3di 286 . . . . . . . . . . . . . . 15 (𝑁 ∈ ℤ → (𝑁 = 0 ↔ 0 = (abs‘𝑁)))
3429, 33bitrd 279 . . . . . . . . . . . . . 14 (𝑁 ∈ ℤ → (0 ∥ 𝑁 ↔ 0 = (abs‘𝑁)))
3534adantl 481 . . . . . . . . . . . . 13 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (0 ∥ 𝑁 ↔ 0 = (abs‘𝑁)))
3628, 35sylan9bb 509 . . . . . . . . . . . 12 ((𝑀 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (𝑀𝑁 ↔ 0 = (abs‘𝑁)))
37 fveq2 6906 . . . . . . . . . . . . . . 15 (𝑀 = 0 → (abs‘𝑀) = (abs‘0))
3837, 13eqtrdi 2793 . . . . . . . . . . . . . 14 (𝑀 = 0 → (abs‘𝑀) = 0)
3938adantr 480 . . . . . . . . . . . . 13 ((𝑀 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (abs‘𝑀) = 0)
4039eqeq1d 2739 . . . . . . . . . . . 12 ((𝑀 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → ((abs‘𝑀) = (abs‘𝑁) ↔ 0 = (abs‘𝑁)))
4136, 40bitr4d 282 . . . . . . . . . . 11 ((𝑀 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (𝑀𝑁 ↔ (abs‘𝑀) = (abs‘𝑁)))
4227, 41imbitrid 244 . . . . . . . . . 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 416 . . . . . . . 8 ((𝑀 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (𝑀𝑁 → (𝑁𝑀 → ((abs‘𝑀) ≤ (abs‘𝑁) → (abs‘𝑀) = (abs‘𝑁)))))
4521adantl 481 . . . . . . . . . . 11 ((¬ 𝑀 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → 𝑁 ∈ ℤ)
46 simprl 771 . . . . . . . . . . 11 ((¬ 𝑀 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → 𝑀 ∈ ℤ)
47 neqne 2948 . . . . . . . . . . . 12 𝑀 = 0 → 𝑀 ≠ 0)
4847adantr 480 . . . . . . . . . . 11 ((¬ 𝑀 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → 𝑀 ≠ 0)
49 dvdsleabs2 16349 . . . . . . . . . . 11 ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑀 ≠ 0) → (𝑁𝑀 → (abs‘𝑁) ≤ (abs‘𝑀)))
5045, 46, 48, 49syl3anc 1373 . . . . . . . . . 10 ((¬ 𝑀 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (𝑁𝑀 → (abs‘𝑁) ≤ (abs‘𝑀)))
51 eqcom 2744 . . . . . . . . . . . . . 14 ((abs‘𝑀) = (abs‘𝑁) ↔ (abs‘𝑁) = (abs‘𝑀))
5230abscld 15475 . . . . . . . . . . . . . . 15 (𝑁 ∈ ℤ → (abs‘𝑁) ∈ ℝ)
536abscld 15475 . . . . . . . . . . . . . . 15 (𝑀 ∈ ℤ → (abs‘𝑀) ∈ ℝ)
54 letri3 11346 . . . . . . . . . . . . . . 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 283 . . . . . . . . . . . . 13 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((abs‘𝑀) = (abs‘𝑁) ↔ ((abs‘𝑁) ≤ (abs‘𝑀) ∧ (abs‘𝑀) ≤ (abs‘𝑁))))
5756biimprd 248 . . . . . . . . . . . 12 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (((abs‘𝑁) ≤ (abs‘𝑀) ∧ (abs‘𝑀) ≤ (abs‘𝑁)) → (abs‘𝑀) = (abs‘𝑁)))
5857expd 415 . . . . . . . . . . 11 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((abs‘𝑁) ≤ (abs‘𝑀) → ((abs‘𝑀) ≤ (abs‘𝑁) → (abs‘𝑀) = (abs‘𝑁))))
5958adantl 481 . . . . . . . . . 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 812 . . . . . . 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 481 . . . . 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 812 . . 3 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀𝑁 → (𝑁𝑀 → (abs‘𝑀) = (abs‘𝑁))))
671, 66mpcom 38 . 2 (𝑀𝑁 → (𝑁𝑀 → (abs‘𝑀) = (abs‘𝑁)))
6867imp 406 1 ((𝑀𝑁𝑁𝑀) → (abs‘𝑀) = (abs‘𝑁))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wne 2940   class class class wbr 5143  cfv 6561  cr 11154  0cc0 11155  cle 11296  cz 12613  abscabs 15273  cdvds 16290
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-sup 9482  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-n0 12527  df-z 12614  df-uz 12879  df-rp 13035  df-seq 14043  df-exp 14103  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-dvds 16291
This theorem is referenced by:  dvdseq  16351
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