Theorem fzopth 13649

Description: A finite set of sequential integers has the ordered pair property (compare opth 5445) 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 13618	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ (𝑀...𝑁))
2	1	adantr 486	. . . . . . . 8 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝑀 ∈ (𝑀...𝑁))
3		simpr 490	. . . . . . . 8 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → (𝑀...𝑁) = (𝐽...𝐾))
4	2, 3	eleqtrd 2862	. . . . . . 7 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝑀 ∈ (𝐽...𝐾))
5		elfzuz 13607	. . . . . . 7 ⊢ (𝑀 ∈ (𝐽...𝐾) → 𝑀 ∈ (ℤ≥‘𝐽))
6		uzss 12943	. . . . . . 7 ⊢ (𝑀 ∈ (ℤ≥‘𝐽) → (ℤ≥‘𝑀) ⊆ (ℤ≥‘𝐽))
7	4, 5, 6	3syl 19	. . . . . 6 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → (ℤ≥‘𝑀) ⊆ (ℤ≥‘𝐽))
8		elfzuz2 13616	. . . . . . . . 9 ⊢ (𝑀 ∈ (𝐽...𝐾) → 𝐾 ∈ (ℤ≥‘𝐽))
9		eluzfz1 13618	. . . . . . . . 9 ⊢ (𝐾 ∈ (ℤ≥‘𝐽) → 𝐽 ∈ (𝐽...𝐾))
10	4, 8, 9	3syl 19	. . . . . . . 8 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝐽 ∈ (𝐽...𝐾))
11	10, 3	eleqtrrd 2863	. . . . . . 7 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝐽 ∈ (𝑀...𝑁))
12		elfzuz 13607	. . . . . . 7 ⊢ (𝐽 ∈ (𝑀...𝑁) → 𝐽 ∈ (ℤ≥‘𝑀))
13		uzss 12943	. . . . . . 7 ⊢ (𝐽 ∈ (ℤ≥‘𝑀) → (ℤ≥‘𝐽) ⊆ (ℤ≥‘𝑀))
14	11, 12, 13	3syl 19	. . . . . 6 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → (ℤ≥‘𝐽) ⊆ (ℤ≥‘𝑀))
15	7, 14	eqssd 3948	. . . . 5 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → (ℤ≥‘𝑀) = (ℤ≥‘𝐽))
16		eluzel2 12925	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ)
17	16	adantr 486	. . . . . 6 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝑀 ∈ ℤ)
18		uz11 12945	. . . . . 6 ⊢ (𝑀 ∈ ℤ → ((ℤ≥‘𝑀) = (ℤ≥‘𝐽) ↔ 𝑀 = 𝐽))
19	17, 18	syl 18	. . . . 5 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → ((ℤ≥‘𝑀) = (ℤ≥‘𝐽) ↔ 𝑀 = 𝐽))
20	15, 19	mpbid 235	. . . 4 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝑀 = 𝐽)
21		eluzfz2 13619	. . . . . . . . 9 ⊢ (𝐾 ∈ (ℤ≥‘𝐽) → 𝐾 ∈ (𝐽...𝐾))
22	4, 8, 21	3syl 19	. . . . . . . 8 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝐾 ∈ (𝐽...𝐾))
23	22, 3	eleqtrrd 2863	. . . . . . 7 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝐾 ∈ (𝑀...𝑁))
24		elfzuz3 13608	. . . . . . 7 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ≥‘𝐾))
25		uzss 12943	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘𝐾) → (ℤ≥‘𝑁) ⊆ (ℤ≥‘𝐾))
26	23, 24, 25	3syl 19	. . . . . 6 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → (ℤ≥‘𝑁) ⊆ (ℤ≥‘𝐾))
27		eluzfz2 13619	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁))
28	27	adantr 486	. . . . . . . 8 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝑁 ∈ (𝑀...𝑁))
29	28, 3	eleqtrd 2862	. . . . . . 7 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝑁 ∈ (𝐽...𝐾))
30		elfzuz3 13608	. . . . . . 7 ⊢ (𝑁 ∈ (𝐽...𝐾) → 𝐾 ∈ (ℤ≥‘𝑁))
31		uzss 12943	. . . . . . 7 ⊢ (𝐾 ∈ (ℤ≥‘𝑁) → (ℤ≥‘𝐾) ⊆ (ℤ≥‘𝑁))
32	29, 30, 31	3syl 19	. . . . . 6 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → (ℤ≥‘𝐾) ⊆ (ℤ≥‘𝑁))
33	26, 32	eqssd 3948	. . . . 5 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → (ℤ≥‘𝑁) = (ℤ≥‘𝐾))
34		eluzelz 12930	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℤ)
35	34	adantr 486	. . . . . 6 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝑁 ∈ ℤ)
36		uz11 12945	. . . . . 6 ⊢ (𝑁 ∈ ℤ → ((ℤ≥‘𝑁) = (ℤ≥‘𝐾) ↔ 𝑁 = 𝐾))
37	35, 36	syl 18	. . . . 5 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → ((ℤ≥‘𝑁) = (ℤ≥‘𝐾) ↔ 𝑁 = 𝐾))
38	33, 37	mpbid 235	. . . 4 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝑁 = 𝐾)
39	20, 38	jca 521	. . 3 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → (𝑀 = 𝐽 ∧ 𝑁 = 𝐾))
40	39	ex 418	. 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → ((𝑀...𝑁) = (𝐽...𝐾) → (𝑀 = 𝐽 ∧ 𝑁 = 𝐾)))
41		oveq12 7418	. 2 ⊢ ((𝑀 = 𝐽 ∧ 𝑁 = 𝐾) → (𝑀...𝑁) = (𝐽...𝐾))
42	40, 41	impbid1 228	1 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → ((𝑀...𝑁) = (𝐽...𝐾) ↔ (𝑀 = 𝐽 ∧ 𝑁 = 𝐾)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ⊆ wss 3899  ‘cfv 6528  (class class class)co 7409  ℤcz 12648  ℤ_≥cuz 12920  ...cfz 13594

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7735  ax-cnex 11213  ax-resscn 11214  ax-pre-lttri 11231  ax-pre-lttrn 11232

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5543  df-po 5556  df-so 5557  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-iota 6484  df-fun 6530  df-fn 6531  df-f 6532  df-f1 6533  df-fo 6534  df-f1o 6535  df-fv 6536  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1st 7985  df-2nd 7986  df-er 8696  df-en 8953  df-dom 8954  df-sdom 8955  df-pnf 11302  df-mnf 11303  df-xr 11304  df-ltxr 11305  df-le 11306  df-neg 11501  df-z 12649  df-uz 12921  df-fz 13595

This theorem is used by:  2ffzeq  13737  gsumval2a  18821  eedimeq  29395  sdclem2  38590