Theorem fzopth 10416

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 10385	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ (𝑀...𝑁))
2	1	adantr 276	. . . . . . . 8 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝑀 ∈ (𝑀...𝑁))
3		simpr 110	. . . . . . . 8 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → (𝑀...𝑁) = (𝐽...𝐾))
4	2, 3	eleqtrd 2313	. . . . . . 7 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝑀 ∈ (𝐽...𝐾))
5		elfzuz 10374	. . . . . . 7 ⊢ (𝑀 ∈ (𝐽...𝐾) → 𝑀 ∈ (ℤ≥‘𝐽))
6		uzss 9893	. . . . . . 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 10383	. . . . . . . . 9 ⊢ (𝑀 ∈ (𝐽...𝐾) → 𝐾 ∈ (ℤ≥‘𝐽))
9		eluzfz1 10385	. . . . . . . . 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 2314	. . . . . . 7 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝐽 ∈ (𝑀...𝑁))
12		elfzuz 10374	. . . . . . 7 ⊢ (𝐽 ∈ (𝑀...𝑁) → 𝐽 ∈ (ℤ≥‘𝑀))
13		uzss 9893	. . . . . . 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 3259	. . . . 5 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → (ℤ≥‘𝑀) = (ℤ≥‘𝐽))
16		eluzel2 9876	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ)
17	16	adantr 276	. . . . . 6 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝑀 ∈ ℤ)
18		uz11 9895	. . . . . 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 147	. . . 4 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝑀 = 𝐽)
21		eluzfz2 10386	. . . . . . . . 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 2314	. . . . . . 7 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝐾 ∈ (𝑀...𝑁))
24		elfzuz3 10375	. . . . . . 7 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ≥‘𝐾))
25		uzss 9893	. . . . . . 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 10386	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁))
28	27	adantr 276	. . . . . . . 8 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝑁 ∈ (𝑀...𝑁))
29	28, 3	eleqtrd 2313	. . . . . . 7 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝑁 ∈ (𝐽...𝐾))
30		elfzuz3 10375	. . . . . . 7 ⊢ (𝑁 ∈ (𝐽...𝐾) → 𝐾 ∈ (ℤ≥‘𝑁))
31		uzss 9893	. . . . . . 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 3259	. . . . 5 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → (ℤ≥‘𝑁) = (ℤ≥‘𝐾))
34		eluzelz 9881	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℤ)
35	34	adantr 276	. . . . . 6 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝑁 ∈ ℤ)
36		uz11 9895	. . . . . 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 147	. . . 4 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝑁 = 𝐾)
39	20, 38	jca 306	. . 3 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → (𝑀 = 𝐽 ∧ 𝑁 = 𝐾))
40	39	ex 115	. 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → ((𝑀...𝑁) = (𝐽...𝐾) → (𝑀 = 𝐽 ∧ 𝑁 = 𝐾)))
41		oveq12 6067	. 2 ⊢ ((𝑀 = 𝐽 ∧ 𝑁 = 𝐾) → (𝑀...𝑁) = (𝐽...𝐾))
42	40, 41	impbid1 142	1 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → ((𝑀...𝑁) = (𝐽...𝐾) ↔ (𝑀 = 𝐽 ∧ 𝑁 = 𝐾)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1398   ∈ wcel 2205   ⊆ wss 3214  ‘cfv 5357  (class class class)co 6058  ℤcz 9594  ℤ_≥cuz 9871  ...cfz 10361

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-apti 8258

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3046  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-neg 8463  df-z 9595  df-uz 9872  df-fz 10362

This theorem is referenced by:  fz0to4untppr  10480  2ffzeq  10497  gsumfzval  13654  gsumval2  13660