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Theorem fzopth 13537
Description: A finite set of sequential integers has the ordered pair property (compare opth 5476) under certain conditions. (Contributed by NM, 31-Oct-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
Assertion
Ref Expression
fzopth (𝑁 ∈ (ℤ𝑀) → ((𝑀...𝑁) = (𝐽...𝐾) ↔ (𝑀 = 𝐽𝑁 = 𝐾)))

Proof of Theorem fzopth
StepHypRef Expression
1 eluzfz1 13507 . . . . . . . . 9 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ (𝑀...𝑁))
21adantr 481 . . . . . . . 8 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝑀 ∈ (𝑀...𝑁))
3 simpr 485 . . . . . . . 8 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → (𝑀...𝑁) = (𝐽...𝐾))
42, 3eleqtrd 2835 . . . . . . 7 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝑀 ∈ (𝐽...𝐾))
5 elfzuz 13496 . . . . . . 7 (𝑀 ∈ (𝐽...𝐾) → 𝑀 ∈ (ℤ𝐽))
6 uzss 12844 . . . . . . 7 (𝑀 ∈ (ℤ𝐽) → (ℤ𝑀) ⊆ (ℤ𝐽))
74, 5, 63syl 18 . . . . . 6 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → (ℤ𝑀) ⊆ (ℤ𝐽))
8 elfzuz2 13505 . . . . . . . . 9 (𝑀 ∈ (𝐽...𝐾) → 𝐾 ∈ (ℤ𝐽))
9 eluzfz1 13507 . . . . . . . . 9 (𝐾 ∈ (ℤ𝐽) → 𝐽 ∈ (𝐽...𝐾))
104, 8, 93syl 18 . . . . . . . 8 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝐽 ∈ (𝐽...𝐾))
1110, 3eleqtrrd 2836 . . . . . . 7 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝐽 ∈ (𝑀...𝑁))
12 elfzuz 13496 . . . . . . 7 (𝐽 ∈ (𝑀...𝑁) → 𝐽 ∈ (ℤ𝑀))
13 uzss 12844 . . . . . . 7 (𝐽 ∈ (ℤ𝑀) → (ℤ𝐽) ⊆ (ℤ𝑀))
1411, 12, 133syl 18 . . . . . 6 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → (ℤ𝐽) ⊆ (ℤ𝑀))
157, 14eqssd 3999 . . . . 5 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → (ℤ𝑀) = (ℤ𝐽))
16 eluzel2 12826 . . . . . . 7 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ ℤ)
1716adantr 481 . . . . . 6 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝑀 ∈ ℤ)
18 uz11 12846 . . . . . 6 (𝑀 ∈ ℤ → ((ℤ𝑀) = (ℤ𝐽) ↔ 𝑀 = 𝐽))
1917, 18syl 17 . . . . 5 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → ((ℤ𝑀) = (ℤ𝐽) ↔ 𝑀 = 𝐽))
2015, 19mpbid 231 . . . 4 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝑀 = 𝐽)
21 eluzfz2 13508 . . . . . . . . 9 (𝐾 ∈ (ℤ𝐽) → 𝐾 ∈ (𝐽...𝐾))
224, 8, 213syl 18 . . . . . . . 8 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝐾 ∈ (𝐽...𝐾))
2322, 3eleqtrrd 2836 . . . . . . 7 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝐾 ∈ (𝑀...𝑁))
24 elfzuz3 13497 . . . . . . 7 (𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ𝐾))
25 uzss 12844 . . . . . . 7 (𝑁 ∈ (ℤ𝐾) → (ℤ𝑁) ⊆ (ℤ𝐾))
2623, 24, 253syl 18 . . . . . 6 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → (ℤ𝑁) ⊆ (ℤ𝐾))
27 eluzfz2 13508 . . . . . . . . 9 (𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ (𝑀...𝑁))
2827adantr 481 . . . . . . . 8 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝑁 ∈ (𝑀...𝑁))
2928, 3eleqtrd 2835 . . . . . . 7 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝑁 ∈ (𝐽...𝐾))
30 elfzuz3 13497 . . . . . . 7 (𝑁 ∈ (𝐽...𝐾) → 𝐾 ∈ (ℤ𝑁))
31 uzss 12844 . . . . . . 7 (𝐾 ∈ (ℤ𝑁) → (ℤ𝐾) ⊆ (ℤ𝑁))
3229, 30, 313syl 18 . . . . . 6 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → (ℤ𝐾) ⊆ (ℤ𝑁))
3326, 32eqssd 3999 . . . . 5 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → (ℤ𝑁) = (ℤ𝐾))
34 eluzelz 12831 . . . . . . 7 (𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ ℤ)
3534adantr 481 . . . . . 6 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝑁 ∈ ℤ)
36 uz11 12846 . . . . . 6 (𝑁 ∈ ℤ → ((ℤ𝑁) = (ℤ𝐾) ↔ 𝑁 = 𝐾))
3735, 36syl 17 . . . . 5 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → ((ℤ𝑁) = (ℤ𝐾) ↔ 𝑁 = 𝐾))
3833, 37mpbid 231 . . . 4 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝑁 = 𝐾)
3920, 38jca 512 . . 3 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → (𝑀 = 𝐽𝑁 = 𝐾))
4039ex 413 . 2 (𝑁 ∈ (ℤ𝑀) → ((𝑀...𝑁) = (𝐽...𝐾) → (𝑀 = 𝐽𝑁 = 𝐾)))
41 oveq12 7417 . 2 ((𝑀 = 𝐽𝑁 = 𝐾) → (𝑀...𝑁) = (𝐽...𝐾))
4240, 41impbid1 224 1 (𝑁 ∈ (ℤ𝑀) → ((𝑀...𝑁) = (𝐽...𝐾) ↔ (𝑀 = 𝐽𝑁 = 𝐾)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wss 3948  cfv 6543  (class class class)co 7408  cz 12557  cuz 12821  ...cfz 13483
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 7724  ax-cnex 11165  ax-resscn 11166  ax-pre-lttri 11183  ax-pre-lttrn 11184
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-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  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-id 5574  df-po 5588  df-so 5589  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-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413  df-1st 7974  df-2nd 7975  df-er 8702  df-en 8939  df-dom 8940  df-sdom 8941  df-pnf 11249  df-mnf 11250  df-xr 11251  df-ltxr 11252  df-le 11253  df-neg 11446  df-z 12558  df-uz 12822  df-fz 13484
This theorem is referenced by:  fz0to4untppr  13603  2ffzeq  13621  gsumval2a  18603  eedimeq  28153  sdclem2  36605
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