Theorem fzopth 9872

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 ∈ (ℤ≥‘𝑀) → ((𝑀...𝑁) = (𝐽...𝐾) ↔ (𝑀 = 𝐽 ∧ 𝑁 = 𝐾)))

Proof of Theorem fzopth

Step	Hyp	Ref	Expression
1		eluzfz1 9842	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ (𝑀...𝑁))
2	1	adantr 274	. . . . . . . 8 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝑀 ∈ (𝑀...𝑁))
3		simpr 109	. . . . . . . 8 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → (𝑀...𝑁) = (𝐽...𝐾))
4	2, 3	eleqtrd 2219	. . . . . . 7 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝑀 ∈ (𝐽...𝐾))
5		elfzuz 9833	. . . . . . 7 ⊢ (𝑀 ∈ (𝐽...𝐾) → 𝑀 ∈ (ℤ≥‘𝐽))
6		uzss 9370	. . . . . . 7 ⊢ (𝑀 ∈ (ℤ≥‘𝐽) → (ℤ≥‘𝑀) ⊆ (ℤ≥‘𝐽))
7	4, 5, 6	3syl 17	. . . . . 6 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → (ℤ≥‘𝑀) ⊆ (ℤ≥‘𝐽))
8		elfzuz2 9840	. . . . . . . . 9 ⊢ (𝑀 ∈ (𝐽...𝐾) → 𝐾 ∈ (ℤ≥‘𝐽))
9		eluzfz1 9842	. . . . . . . . 9 ⊢ (𝐾 ∈ (ℤ≥‘𝐽) → 𝐽 ∈ (𝐽...𝐾))
10	4, 8, 9	3syl 17	. . . . . . . 8 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝐽 ∈ (𝐽...𝐾))
11	10, 3	eleqtrrd 2220	. . . . . . 7 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝐽 ∈ (𝑀...𝑁))
12		elfzuz 9833	. . . . . . 7 ⊢ (𝐽 ∈ (𝑀...𝑁) → 𝐽 ∈ (ℤ≥‘𝑀))
13		uzss 9370	. . . . . . 7 ⊢ (𝐽 ∈ (ℤ≥‘𝑀) → (ℤ≥‘𝐽) ⊆ (ℤ≥‘𝑀))
14	11, 12, 13	3syl 17	. . . . . 6 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → (ℤ≥‘𝐽) ⊆ (ℤ≥‘𝑀))
15	7, 14	eqssd 3119	. . . . 5 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → (ℤ≥‘𝑀) = (ℤ≥‘𝐽))
16		eluzel2 9355	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ)
17	16	adantr 274	. . . . . 6 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝑀 ∈ ℤ)
18		uz11 9372	. . . . . 6 ⊢ (𝑀 ∈ ℤ → ((ℤ≥‘𝑀) = (ℤ≥‘𝐽) ↔ 𝑀 = 𝐽))
19	17, 18	syl 14	. . . . 5 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → ((ℤ≥‘𝑀) = (ℤ≥‘𝐽) ↔ 𝑀 = 𝐽))
20	15, 19	mpbid 146	. . . 4 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝑀 = 𝐽)
21		eluzfz2 9843	. . . . . . . . 9 ⊢ (𝐾 ∈ (ℤ≥‘𝐽) → 𝐾 ∈ (𝐽...𝐾))
22	4, 8, 21	3syl 17	. . . . . . . 8 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝐾 ∈ (𝐽...𝐾))
23	22, 3	eleqtrrd 2220	. . . . . . 7 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝐾 ∈ (𝑀...𝑁))
24		elfzuz3 9834	. . . . . . 7 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ≥‘𝐾))
25		uzss 9370	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘𝐾) → (ℤ≥‘𝑁) ⊆ (ℤ≥‘𝐾))
26	23, 24, 25	3syl 17	. . . . . 6 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → (ℤ≥‘𝑁) ⊆ (ℤ≥‘𝐾))
27		eluzfz2 9843	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁))
28	27	adantr 274	. . . . . . . 8 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝑁 ∈ (𝑀...𝑁))
29	28, 3	eleqtrd 2219	. . . . . . 7 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝑁 ∈ (𝐽...𝐾))
30		elfzuz3 9834	. . . . . . 7 ⊢ (𝑁 ∈ (𝐽...𝐾) → 𝐾 ∈ (ℤ≥‘𝑁))
31		uzss 9370	. . . . . . 7 ⊢ (𝐾 ∈ (ℤ≥‘𝑁) → (ℤ≥‘𝐾) ⊆ (ℤ≥‘𝑁))
32	29, 30, 31	3syl 17	. . . . . 6 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → (ℤ≥‘𝐾) ⊆ (ℤ≥‘𝑁))
33	26, 32	eqssd 3119	. . . . 5 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → (ℤ≥‘𝑁) = (ℤ≥‘𝐾))
34		eluzelz 9359	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℤ)
35	34	adantr 274	. . . . . 6 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝑁 ∈ ℤ)
36		uz11 9372	. . . . . 6 ⊢ (𝑁 ∈ ℤ → ((ℤ≥‘𝑁) = (ℤ≥‘𝐾) ↔ 𝑁 = 𝐾))
37	35, 36	syl 14	. . . . 5 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → ((ℤ≥‘𝑁) = (ℤ≥‘𝐾) ↔ 𝑁 = 𝐾))
38	33, 37	mpbid 146	. . . 4 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝑁 = 𝐾)
39	20, 38	jca 304	. . 3 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → (𝑀 = 𝐽 ∧ 𝑁 = 𝐾))
40	39	ex 114	. 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → ((𝑀...𝑁) = (𝐽...𝐾) → (𝑀 = 𝐽 ∧ 𝑁 = 𝐾)))
41		oveq12 5791	. 2 ⊢ ((𝑀 = 𝐽 ∧ 𝑁 = 𝐾) → (𝑀...𝑁) = (𝐽...𝐾))
42	40, 41	impbid1 141	1 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → ((𝑀...𝑁) = (𝐽...𝐾) ↔ (𝑀 = 𝐽 ∧ 𝑁 = 𝐾)))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1332   ∈ wcel 1481   ⊆ wss 3076  ‘cfv 5131  (class class class)co 5782  ℤcz 9078  ℤ_≥cuz 9350  ...cfz 9821

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-cnex 7735  ax-resscn 7736  ax-pre-ltirr 7756  ax-pre-ltwlin 7757  ax-pre-apti 7759

This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-sbc 2914  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-mpt 3999  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-fv 5139  df-ov 5785  df-oprab 5786  df-mpo 5787  df-pnf 7826  df-mnf 7827  df-xr 7828  df-ltxr 7829  df-le 7830  df-neg 7960  df-z 9079  df-uz 9351  df-fz 9822

This theorem is referenced by:  2ffzeq  9949