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Theorem fzopth 12416
Description: A finite set of sequential integers has the ordered pair property (compare opth 4974) 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 12386 . . . . . . . . 9 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ (𝑀...𝑁))
21adantr 480 . . . . . . . 8 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝑀 ∈ (𝑀...𝑁))
3 simpr 476 . . . . . . . 8 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → (𝑀...𝑁) = (𝐽...𝐾))
42, 3eleqtrd 2732 . . . . . . 7 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝑀 ∈ (𝐽...𝐾))
5 elfzuz 12376 . . . . . . 7 (𝑀 ∈ (𝐽...𝐾) → 𝑀 ∈ (ℤ𝐽))
6 uzss 11746 . . . . . . 7 (𝑀 ∈ (ℤ𝐽) → (ℤ𝑀) ⊆ (ℤ𝐽))
74, 5, 63syl 18 . . . . . 6 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → (ℤ𝑀) ⊆ (ℤ𝐽))
8 elfzuz2 12384 . . . . . . . . 9 (𝑀 ∈ (𝐽...𝐾) → 𝐾 ∈ (ℤ𝐽))
9 eluzfz1 12386 . . . . . . . . 9 (𝐾 ∈ (ℤ𝐽) → 𝐽 ∈ (𝐽...𝐾))
104, 8, 93syl 18 . . . . . . . 8 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝐽 ∈ (𝐽...𝐾))
1110, 3eleqtrrd 2733 . . . . . . 7 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝐽 ∈ (𝑀...𝑁))
12 elfzuz 12376 . . . . . . 7 (𝐽 ∈ (𝑀...𝑁) → 𝐽 ∈ (ℤ𝑀))
13 uzss 11746 . . . . . . 7 (𝐽 ∈ (ℤ𝑀) → (ℤ𝐽) ⊆ (ℤ𝑀))
1411, 12, 133syl 18 . . . . . 6 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → (ℤ𝐽) ⊆ (ℤ𝑀))
157, 14eqssd 3653 . . . . 5 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → (ℤ𝑀) = (ℤ𝐽))
16 eluzel2 11730 . . . . . . 7 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ ℤ)
1716adantr 480 . . . . . 6 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝑀 ∈ ℤ)
18 uz11 11748 . . . . . 6 (𝑀 ∈ ℤ → ((ℤ𝑀) = (ℤ𝐽) ↔ 𝑀 = 𝐽))
1917, 18syl 17 . . . . 5 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → ((ℤ𝑀) = (ℤ𝐽) ↔ 𝑀 = 𝐽))
2015, 19mpbid 222 . . . 4 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝑀 = 𝐽)
21 eluzfz2 12387 . . . . . . . . 9 (𝐾 ∈ (ℤ𝐽) → 𝐾 ∈ (𝐽...𝐾))
224, 8, 213syl 18 . . . . . . . 8 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝐾 ∈ (𝐽...𝐾))
2322, 3eleqtrrd 2733 . . . . . . 7 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝐾 ∈ (𝑀...𝑁))
24 elfzuz3 12377 . . . . . . 7 (𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ𝐾))
25 uzss 11746 . . . . . . 7 (𝑁 ∈ (ℤ𝐾) → (ℤ𝑁) ⊆ (ℤ𝐾))
2623, 24, 253syl 18 . . . . . 6 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → (ℤ𝑁) ⊆ (ℤ𝐾))
27 eluzfz2 12387 . . . . . . . . 9 (𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ (𝑀...𝑁))
2827adantr 480 . . . . . . . 8 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝑁 ∈ (𝑀...𝑁))
2928, 3eleqtrd 2732 . . . . . . 7 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝑁 ∈ (𝐽...𝐾))
30 elfzuz3 12377 . . . . . . 7 (𝑁 ∈ (𝐽...𝐾) → 𝐾 ∈ (ℤ𝑁))
31 uzss 11746 . . . . . . 7 (𝐾 ∈ (ℤ𝑁) → (ℤ𝐾) ⊆ (ℤ𝑁))
3229, 30, 313syl 18 . . . . . 6 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → (ℤ𝐾) ⊆ (ℤ𝑁))
3326, 32eqssd 3653 . . . . 5 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → (ℤ𝑁) = (ℤ𝐾))
34 eluzelz 11735 . . . . . . 7 (𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ ℤ)
3534adantr 480 . . . . . 6 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝑁 ∈ ℤ)
36 uz11 11748 . . . . . 6 (𝑁 ∈ ℤ → ((ℤ𝑁) = (ℤ𝐾) ↔ 𝑁 = 𝐾))
3735, 36syl 17 . . . . 5 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → ((ℤ𝑁) = (ℤ𝐾) ↔ 𝑁 = 𝐾))
3833, 37mpbid 222 . . . 4 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝑁 = 𝐾)
3920, 38jca 553 . . 3 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → (𝑀 = 𝐽𝑁 = 𝐾))
4039ex 449 . 2 (𝑁 ∈ (ℤ𝑀) → ((𝑀...𝑁) = (𝐽...𝐾) → (𝑀 = 𝐽𝑁 = 𝐾)))
41 oveq12 6699 . 2 ((𝑀 = 𝐽𝑁 = 𝐾) → (𝑀...𝑁) = (𝐽...𝐾))
4240, 41impbid1 215 1 (𝑁 ∈ (ℤ𝑀) → ((𝑀...𝑁) = (𝐽...𝐾) ↔ (𝑀 = 𝐽𝑁 = 𝐾)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wss 3607  cfv 5926  (class class class)co 6690  cz 11415  cuz 11725  ...cfz 12364
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-pre-lttri 10048  ax-pre-lttrn 10049
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-po 5064  df-so 5065  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-neg 10307  df-z 11416  df-uz 11726  df-fz 12365
This theorem is referenced by:  fz0to4untppr  12481  2ffzeq  12499  gsumval2a  17326  eedimeq  25823  sdclem2  33668
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