ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  fzopth GIF version

Theorem fzopth 10060
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 10030 . . . . . . . . 9 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ (𝑀...𝑁))
21adantr 276 . . . . . . . 8 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝑀 ∈ (𝑀...𝑁))
3 simpr 110 . . . . . . . 8 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → (𝑀...𝑁) = (𝐽...𝐾))
42, 3eleqtrd 2256 . . . . . . 7 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝑀 ∈ (𝐽...𝐾))
5 elfzuz 10020 . . . . . . 7 (𝑀 ∈ (𝐽...𝐾) → 𝑀 ∈ (ℤ𝐽))
6 uzss 9547 . . . . . . 7 (𝑀 ∈ (ℤ𝐽) → (ℤ𝑀) ⊆ (ℤ𝐽))
74, 5, 63syl 17 . . . . . 6 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → (ℤ𝑀) ⊆ (ℤ𝐽))
8 elfzuz2 10028 . . . . . . . . 9 (𝑀 ∈ (𝐽...𝐾) → 𝐾 ∈ (ℤ𝐽))
9 eluzfz1 10030 . . . . . . . . 9 (𝐾 ∈ (ℤ𝐽) → 𝐽 ∈ (𝐽...𝐾))
104, 8, 93syl 17 . . . . . . . 8 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝐽 ∈ (𝐽...𝐾))
1110, 3eleqtrrd 2257 . . . . . . 7 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝐽 ∈ (𝑀...𝑁))
12 elfzuz 10020 . . . . . . 7 (𝐽 ∈ (𝑀...𝑁) → 𝐽 ∈ (ℤ𝑀))
13 uzss 9547 . . . . . . 7 (𝐽 ∈ (ℤ𝑀) → (ℤ𝐽) ⊆ (ℤ𝑀))
1411, 12, 133syl 17 . . . . . 6 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → (ℤ𝐽) ⊆ (ℤ𝑀))
157, 14eqssd 3172 . . . . 5 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → (ℤ𝑀) = (ℤ𝐽))
16 eluzel2 9532 . . . . . . 7 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ ℤ)
1716adantr 276 . . . . . 6 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝑀 ∈ ℤ)
18 uz11 9549 . . . . . 6 (𝑀 ∈ ℤ → ((ℤ𝑀) = (ℤ𝐽) ↔ 𝑀 = 𝐽))
1917, 18syl 14 . . . . 5 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → ((ℤ𝑀) = (ℤ𝐽) ↔ 𝑀 = 𝐽))
2015, 19mpbid 147 . . . 4 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝑀 = 𝐽)
21 eluzfz2 10031 . . . . . . . . 9 (𝐾 ∈ (ℤ𝐽) → 𝐾 ∈ (𝐽...𝐾))
224, 8, 213syl 17 . . . . . . . 8 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝐾 ∈ (𝐽...𝐾))
2322, 3eleqtrrd 2257 . . . . . . 7 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝐾 ∈ (𝑀...𝑁))
24 elfzuz3 10021 . . . . . . 7 (𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ𝐾))
25 uzss 9547 . . . . . . 7 (𝑁 ∈ (ℤ𝐾) → (ℤ𝑁) ⊆ (ℤ𝐾))
2623, 24, 253syl 17 . . . . . 6 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → (ℤ𝑁) ⊆ (ℤ𝐾))
27 eluzfz2 10031 . . . . . . . . 9 (𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ (𝑀...𝑁))
2827adantr 276 . . . . . . . 8 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝑁 ∈ (𝑀...𝑁))
2928, 3eleqtrd 2256 . . . . . . 7 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝑁 ∈ (𝐽...𝐾))
30 elfzuz3 10021 . . . . . . 7 (𝑁 ∈ (𝐽...𝐾) → 𝐾 ∈ (ℤ𝑁))
31 uzss 9547 . . . . . . 7 (𝐾 ∈ (ℤ𝑁) → (ℤ𝐾) ⊆ (ℤ𝑁))
3229, 30, 313syl 17 . . . . . 6 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → (ℤ𝐾) ⊆ (ℤ𝑁))
3326, 32eqssd 3172 . . . . 5 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → (ℤ𝑁) = (ℤ𝐾))
34 eluzelz 9536 . . . . . . 7 (𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ ℤ)
3534adantr 276 . . . . . 6 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝑁 ∈ ℤ)
36 uz11 9549 . . . . . 6 (𝑁 ∈ ℤ → ((ℤ𝑁) = (ℤ𝐾) ↔ 𝑁 = 𝐾))
3735, 36syl 14 . . . . 5 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → ((ℤ𝑁) = (ℤ𝐾) ↔ 𝑁 = 𝐾))
3833, 37mpbid 147 . . . 4 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝑁 = 𝐾)
3920, 38jca 306 . . 3 ((𝑁 ∈ (ℤ𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → (𝑀 = 𝐽𝑁 = 𝐾))
4039ex 115 . 2 (𝑁 ∈ (ℤ𝑀) → ((𝑀...𝑁) = (𝐽...𝐾) → (𝑀 = 𝐽𝑁 = 𝐾)))
41 oveq12 5883 . 2 ((𝑀 = 𝐽𝑁 = 𝐾) → (𝑀...𝑁) = (𝐽...𝐾))
4240, 41impbid1 142 1 (𝑁 ∈ (ℤ𝑀) → ((𝑀...𝑁) = (𝐽...𝐾) ↔ (𝑀 = 𝐽𝑁 = 𝐾)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1353  wcel 2148  wss 3129  cfv 5216  (class class class)co 5874  cz 9252  cuz 9527  ...cfz 10007
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-cnex 7901  ax-resscn 7902  ax-pre-ltirr 7922  ax-pre-ltwlin 7923  ax-pre-apti 7925
This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-fv 5224  df-ov 5877  df-oprab 5878  df-mpo 5879  df-pnf 7993  df-mnf 7994  df-xr 7995  df-ltxr 7996  df-le 7997  df-neg 8130  df-z 9253  df-uz 9528  df-fz 10008
This theorem is referenced by:  fz0to4untppr  10123  2ffzeq  10140
  Copyright terms: Public domain W3C validator