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Theorem fzopth 8922
Description: A finite set of sequential integers can represent an ordered pair. (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 8893 . . . . . . . . 9 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ (𝑀...𝑁))
21adantr 261 . . . . . . . 8 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝑀 ∈ (𝑀...𝑁))
3 simpr 103 . . . . . . . 8 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → (𝑀...𝑁) = (𝐽...𝐾))
42, 3eleqtrd 2116 . . . . . . 7 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝑀 ∈ (𝐽...𝐾))
5 elfzuz 8884 . . . . . . 7 (𝑀 ∈ (𝐽...𝐾) → 𝑀 ∈ (ℤ𝐽))
6 uzss 8491 . . . . . . 7 (𝑀 ∈ (ℤ𝐽) → (ℤ𝑀) ⊆ (ℤ𝐽))
74, 5, 63syl 17 . . . . . 6 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → (ℤ𝑀) ⊆ (ℤ𝐽))
8 elfzuz2 8891 . . . . . . . . 9 (𝑀 ∈ (𝐽...𝐾) → 𝐾 ∈ (ℤ𝐽))
9 eluzfz1 8893 . . . . . . . . 9 (𝐾 ∈ (ℤ𝐽) → 𝐽 ∈ (𝐽...𝐾))
104, 8, 93syl 17 . . . . . . . 8 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝐽 ∈ (𝐽...𝐾))
1110, 3eleqtrrd 2117 . . . . . . 7 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝐽 ∈ (𝑀...𝑁))
12 elfzuz 8884 . . . . . . 7 (𝐽 ∈ (𝑀...𝑁) → 𝐽 ∈ (ℤ𝑀))
13 uzss 8491 . . . . . . 7 (𝐽 ∈ (ℤ𝑀) → (ℤ𝐽) ⊆ (ℤ𝑀))
1411, 12, 133syl 17 . . . . . 6 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → (ℤ𝐽) ⊆ (ℤ𝑀))
157, 14eqssd 2962 . . . . 5 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → (ℤ𝑀) = (ℤ𝐽))
16 eluzel2 8476 . . . . . . 7 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ ℤ)
1716adantr 261 . . . . . 6 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝑀 ∈ ℤ)
18 uz11 8493 . . . . . 6 (𝑀 ∈ ℤ → ((ℤ𝑀) = (ℤ𝐽) ↔ 𝑀 = 𝐽))
1917, 18syl 14 . . . . 5 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → ((ℤ𝑀) = (ℤ𝐽) ↔ 𝑀 = 𝐽))
2015, 19mpbid 135 . . . 4 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝑀 = 𝐽)
21 eluzfz2 8894 . . . . . . . . 9 (𝐾 ∈ (ℤ𝐽) → 𝐾 ∈ (𝐽...𝐾))
224, 8, 213syl 17 . . . . . . . 8 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝐾 ∈ (𝐽...𝐾))
2322, 3eleqtrrd 2117 . . . . . . 7 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝐾 ∈ (𝑀...𝑁))
24 elfzuz3 8885 . . . . . . 7 (𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ𝐾))
25 uzss 8491 . . . . . . 7 (𝑁 ∈ (ℤ𝐾) → (ℤ𝑁) ⊆ (ℤ𝐾))
2623, 24, 253syl 17 . . . . . 6 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → (ℤ𝑁) ⊆ (ℤ𝐾))
27 eluzfz2 8894 . . . . . . . . 9 (𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ (𝑀...𝑁))
2827adantr 261 . . . . . . . 8 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝑁 ∈ (𝑀...𝑁))
2928, 3eleqtrd 2116 . . . . . . 7 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝑁 ∈ (𝐽...𝐾))
30 elfzuz3 8885 . . . . . . 7 (𝑁 ∈ (𝐽...𝐾) → 𝐾 ∈ (ℤ𝑁))
31 uzss 8491 . . . . . . 7 (𝐾 ∈ (ℤ𝑁) → (ℤ𝐾) ⊆ (ℤ𝑁))
3229, 30, 313syl 17 . . . . . 6 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → (ℤ𝐾) ⊆ (ℤ𝑁))
3326, 32eqssd 2962 . . . . 5 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → (ℤ𝑁) = (ℤ𝐾))
34 eluzelz 8480 . . . . . . 7 (𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ ℤ)
3534adantr 261 . . . . . 6 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝑁 ∈ ℤ)
36 uz11 8493 . . . . . 6 (𝑁 ∈ ℤ → ((ℤ𝑁) = (ℤ𝐾) ↔ 𝑁 = 𝐾))
3735, 36syl 14 . . . . 5 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → ((ℤ𝑁) = (ℤ𝐾) ↔ 𝑁 = 𝐾))
3833, 37mpbid 135 . . . 4 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝑁 = 𝐾)
3920, 38jca 290 . . 3 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → (𝑀 = 𝐽𝑁 = 𝐾))
4039ex 108 . 2 (𝑁 ∈ (ℤ𝑀) → ((𝑀...𝑁) = (𝐽...𝐾) → (𝑀 = 𝐽𝑁 = 𝐾)))
41 oveq12 5521 . 2 ((𝑀 = 𝐽𝑁 = 𝐾) → (𝑀...𝑁) = (𝐽...𝐾))
4240, 41impbid1 130 1 (𝑁 ∈ (ℤ𝑀) → ((𝑀...𝑁) = (𝐽...𝐾) ↔ (𝑀 = 𝐽𝑁 = 𝐾)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wb 98   = wceq 1243  wcel 1393  wss 2917  cfv 4902  (class class class)co 5512  cz 8243  cuz 8471  ...cfz 8872
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-cnex 6973  ax-resscn 6974  ax-pre-ltirr 6994  ax-pre-ltwlin 6995  ax-pre-apti 6997
This theorem depends on definitions:  df-bi 110  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-nel 2207  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-sbc 2765  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-pnf 7060  df-mnf 7061  df-xr 7062  df-ltxr 7063  df-le 7064  df-neg 7183  df-z 8244  df-uz 8472  df-fz 8873
This theorem is referenced by:  2ffzeq  8996
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