Theorem fzopth 13461

Description: A finite set of sequential integers has the ordered pair property (compare opth 5414) under certain conditions. (Contributed by NM, 31-Oct-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)

Assertion

Ref	Expression
fzopth	⊢ (𝑁 ∈ (ℤ≥‘𝑀) → ((𝑀...𝑁) = (𝐽...𝐾) ↔ (𝑀 = 𝐽 ∧ 𝑁 = 𝐾)))

Proof of Theorem fzopth

Step	Hyp	Ref	Expression
1		eluzfz1 13431	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ (𝑀...𝑁))
2	1	adantr 480	. . . . . . . 8 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝑀 ∈ (𝑀...𝑁))
3		simpr 484	. . . . . . . 8 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → (𝑀...𝑁) = (𝐽...𝐾))
4	2, 3	eleqtrd 2833	. . . . . . 7 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝑀 ∈ (𝐽...𝐾))
5		elfzuz 13420	. . . . . . 7 ⊢ (𝑀 ∈ (𝐽...𝐾) → 𝑀 ∈ (ℤ≥‘𝐽))
6		uzss 12755	. . . . . . 7 ⊢ (𝑀 ∈ (ℤ≥‘𝐽) → (ℤ≥‘𝑀) ⊆ (ℤ≥‘𝐽))
7	4, 5, 6	3syl 18	. . . . . 6 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → (ℤ≥‘𝑀) ⊆ (ℤ≥‘𝐽))
8		elfzuz2 13429	. . . . . . . . 9 ⊢ (𝑀 ∈ (𝐽...𝐾) → 𝐾 ∈ (ℤ≥‘𝐽))
9		eluzfz1 13431	. . . . . . . . 9 ⊢ (𝐾 ∈ (ℤ≥‘𝐽) → 𝐽 ∈ (𝐽...𝐾))
10	4, 8, 9	3syl 18	. . . . . . . 8 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝐽 ∈ (𝐽...𝐾))
11	10, 3	eleqtrrd 2834	. . . . . . 7 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝐽 ∈ (𝑀...𝑁))
12		elfzuz 13420	. . . . . . 7 ⊢ (𝐽 ∈ (𝑀...𝑁) → 𝐽 ∈ (ℤ≥‘𝑀))
13		uzss 12755	. . . . . . 7 ⊢ (𝐽 ∈ (ℤ≥‘𝑀) → (ℤ≥‘𝐽) ⊆ (ℤ≥‘𝑀))
14	11, 12, 13	3syl 18	. . . . . 6 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → (ℤ≥‘𝐽) ⊆ (ℤ≥‘𝑀))
15	7, 14	eqssd 3947	. . . . 5 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → (ℤ≥‘𝑀) = (ℤ≥‘𝐽))
16		eluzel2 12737	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ)
17	16	adantr 480	. . . . . 6 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝑀 ∈ ℤ)
18		uz11 12757	. . . . . 6 ⊢ (𝑀 ∈ ℤ → ((ℤ≥‘𝑀) = (ℤ≥‘𝐽) ↔ 𝑀 = 𝐽))
19	17, 18	syl 17	. . . . 5 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → ((ℤ≥‘𝑀) = (ℤ≥‘𝐽) ↔ 𝑀 = 𝐽))
20	15, 19	mpbid 232	. . . 4 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝑀 = 𝐽)
21		eluzfz2 13432	. . . . . . . . 9 ⊢ (𝐾 ∈ (ℤ≥‘𝐽) → 𝐾 ∈ (𝐽...𝐾))
22	4, 8, 21	3syl 18	. . . . . . . 8 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝐾 ∈ (𝐽...𝐾))
23	22, 3	eleqtrrd 2834	. . . . . . 7 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝐾 ∈ (𝑀...𝑁))
24		elfzuz3 13421	. . . . . . 7 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ≥‘𝐾))
25		uzss 12755	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘𝐾) → (ℤ≥‘𝑁) ⊆ (ℤ≥‘𝐾))
26	23, 24, 25	3syl 18	. . . . . 6 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → (ℤ≥‘𝑁) ⊆ (ℤ≥‘𝐾))
27		eluzfz2 13432	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁))
28	27	adantr 480	. . . . . . . 8 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝑁 ∈ (𝑀...𝑁))
29	28, 3	eleqtrd 2833	. . . . . . 7 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝑁 ∈ (𝐽...𝐾))
30		elfzuz3 13421	. . . . . . 7 ⊢ (𝑁 ∈ (𝐽...𝐾) → 𝐾 ∈ (ℤ≥‘𝑁))
31		uzss 12755	. . . . . . 7 ⊢ (𝐾 ∈ (ℤ≥‘𝑁) → (ℤ≥‘𝐾) ⊆ (ℤ≥‘𝑁))
32	29, 30, 31	3syl 18	. . . . . 6 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → (ℤ≥‘𝐾) ⊆ (ℤ≥‘𝑁))
33	26, 32	eqssd 3947	. . . . 5 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → (ℤ≥‘𝑁) = (ℤ≥‘𝐾))
34		eluzelz 12742	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℤ)
35	34	adantr 480	. . . . . 6 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝑁 ∈ ℤ)
36		uz11 12757	. . . . . 6 ⊢ (𝑁 ∈ ℤ → ((ℤ≥‘𝑁) = (ℤ≥‘𝐾) ↔ 𝑁 = 𝐾))
37	35, 36	syl 17	. . . . 5 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → ((ℤ≥‘𝑁) = (ℤ≥‘𝐾) ↔ 𝑁 = 𝐾))
38	33, 37	mpbid 232	. . . 4 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝑁 = 𝐾)
39	20, 38	jca 511	. . 3 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → (𝑀 = 𝐽 ∧ 𝑁 = 𝐾))
40	39	ex 412	. 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → ((𝑀...𝑁) = (𝐽...𝐾) → (𝑀 = 𝐽 ∧ 𝑁 = 𝐾)))
41		oveq12 7355	. 2 ⊢ ((𝑀 = 𝐽 ∧ 𝑁 = 𝐾) → (𝑀...𝑁) = (𝐽...𝐾))
42	40, 41	impbid1 225	1 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → ((𝑀...𝑁) = (𝐽...𝐾) ↔ (𝑀 = 𝐽 ∧ 𝑁 = 𝐾)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111   ⊆ wss 3897  ‘cfv 6481  (class class class)co 7346  ℤcz 12468  ℤ_≥cuz 12732  ...cfz 13407

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668  ax-cnex 11062  ax-resscn 11063  ax-pre-lttri 11080  ax-pre-lttrn 11081

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-po 5522  df-so 5523  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-1st 7921  df-2nd 7922  df-er 8622  df-en 8870  df-dom 8871  df-sdom 8872  df-pnf 11148  df-mnf 11149  df-xr 11150  df-ltxr 11151  df-le 11152  df-neg 11347  df-z 12469  df-uz 12733  df-fz 13408

This theorem is referenced by:  2ffzeq  13549  gsumval2a  18593  eedimeq  28876  sdclem2  37781