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Theorem fzopth 13538
Description: A finite set of sequential integers has the ordered pair property (compare opth 5477) 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 13508 . . . . . . . . 9 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ (𝑀...𝑁))
21adantr 482 . . . . . . . 8 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝑀 ∈ (𝑀...𝑁))
3 simpr 486 . . . . . . . 8 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → (𝑀...𝑁) = (𝐽...𝐾))
42, 3eleqtrd 2836 . . . . . . 7 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝑀 ∈ (𝐽...𝐾))
5 elfzuz 13497 . . . . . . 7 (𝑀 ∈ (𝐽...𝐾) → 𝑀 ∈ (ℤ𝐽))
6 uzss 12845 . . . . . . 7 (𝑀 ∈ (ℤ𝐽) → (ℤ𝑀) ⊆ (ℤ𝐽))
74, 5, 63syl 18 . . . . . 6 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → (ℤ𝑀) ⊆ (ℤ𝐽))
8 elfzuz2 13506 . . . . . . . . 9 (𝑀 ∈ (𝐽...𝐾) → 𝐾 ∈ (ℤ𝐽))
9 eluzfz1 13508 . . . . . . . . 9 (𝐾 ∈ (ℤ𝐽) → 𝐽 ∈ (𝐽...𝐾))
104, 8, 93syl 18 . . . . . . . 8 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝐽 ∈ (𝐽...𝐾))
1110, 3eleqtrrd 2837 . . . . . . 7 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝐽 ∈ (𝑀...𝑁))
12 elfzuz 13497 . . . . . . 7 (𝐽 ∈ (𝑀...𝑁) → 𝐽 ∈ (ℤ𝑀))
13 uzss 12845 . . . . . . 7 (𝐽 ∈ (ℤ𝑀) → (ℤ𝐽) ⊆ (ℤ𝑀))
1411, 12, 133syl 18 . . . . . 6 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → (ℤ𝐽) ⊆ (ℤ𝑀))
157, 14eqssd 4000 . . . . 5 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → (ℤ𝑀) = (ℤ𝐽))
16 eluzel2 12827 . . . . . . 7 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ ℤ)
1716adantr 482 . . . . . 6 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝑀 ∈ ℤ)
18 uz11 12847 . . . . . 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 13509 . . . . . . . . 9 (𝐾 ∈ (ℤ𝐽) → 𝐾 ∈ (𝐽...𝐾))
224, 8, 213syl 18 . . . . . . . 8 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝐾 ∈ (𝐽...𝐾))
2322, 3eleqtrrd 2837 . . . . . . 7 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝐾 ∈ (𝑀...𝑁))
24 elfzuz3 13498 . . . . . . 7 (𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ𝐾))
25 uzss 12845 . . . . . . 7 (𝑁 ∈ (ℤ𝐾) → (ℤ𝑁) ⊆ (ℤ𝐾))
2623, 24, 253syl 18 . . . . . 6 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → (ℤ𝑁) ⊆ (ℤ𝐾))
27 eluzfz2 13509 . . . . . . . . 9 (𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ (𝑀...𝑁))
2827adantr 482 . . . . . . . 8 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝑁 ∈ (𝑀...𝑁))
2928, 3eleqtrd 2836 . . . . . . 7 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝑁 ∈ (𝐽...𝐾))
30 elfzuz3 13498 . . . . . . 7 (𝑁 ∈ (𝐽...𝐾) → 𝐾 ∈ (ℤ𝑁))
31 uzss 12845 . . . . . . 7 (𝐾 ∈ (ℤ𝑁) → (ℤ𝐾) ⊆ (ℤ𝑁))
3229, 30, 313syl 18 . . . . . 6 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → (ℤ𝐾) ⊆ (ℤ𝑁))
3326, 32eqssd 4000 . . . . 5 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → (ℤ𝑁) = (ℤ𝐾))
34 eluzelz 12832 . . . . . . 7 (𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ ℤ)
3534adantr 482 . . . . . 6 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝑁 ∈ ℤ)
36 uz11 12847 . . . . . 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 513 . . 3 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → (𝑀 = 𝐽𝑁 = 𝐾))
4039ex 414 . 2 (𝑁 ∈ (ℤ𝑀) → ((𝑀...𝑁) = (𝐽...𝐾) → (𝑀 = 𝐽𝑁 = 𝐾)))
41 oveq12 7418 . 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 397   = wceq 1542  wcel 2107  wss 3949  cfv 6544  (class class class)co 7409  cz 12558  cuz 12822  ...cfz 13484
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-pre-lttri 11184  ax-pre-lttrn 11185
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-po 5589  df-so 5590  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976  df-er 8703  df-en 8940  df-dom 8941  df-sdom 8942  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-neg 11447  df-z 12559  df-uz 12823  df-fz 13485
This theorem is referenced by:  fz0to4untppr  13604  2ffzeq  13622  gsumval2a  18604  eedimeq  28156  sdclem2  36610
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