Theorem ssfzunsnext 13629

Description: A subset of a finite sequence of integers extended by an integer is a subset of a (possibly extended) finite sequence of integers. (Contributed by AV, 13-Nov-2021.)

Assertion

Ref	Expression
ssfzunsnext	⊢ ((𝑆 ⊆ (𝑀...𝑁) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ)) → (𝑆 ∪ {𝐼}) ⊆ (if(𝐼 ≤ 𝑀, 𝐼, 𝑀)...if(𝐼 ≤ 𝑁, 𝑁, 𝐼)))

Proof of Theorem ssfzunsnext

Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 482	. . 3 ⊢ ((𝑆 ⊆ (𝑀...𝑁) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ)) → 𝑆 ⊆ (𝑀...𝑁))
2		simp3 1138	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → 𝐼 ∈ ℤ)
3		simp1 1136	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → 𝑀 ∈ ℤ)
4	2, 3	ifcld 4594	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → if(𝐼 ≤ 𝑀, 𝐼, 𝑀) ∈ ℤ)
5	4	adantr 480	. . . . . . 7 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) ∧ 𝑘 ∈ (𝑀...𝑁)) → if(𝐼 ≤ 𝑀, 𝐼, 𝑀) ∈ ℤ)
6		simp2 1137	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → 𝑁 ∈ ℤ)
7	6, 2	ifcld 4594	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → if(𝐼 ≤ 𝑁, 𝑁, 𝐼) ∈ ℤ)
8	7	adantr 480	. . . . . . 7 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) ∧ 𝑘 ∈ (𝑀...𝑁)) → if(𝐼 ≤ 𝑁, 𝑁, 𝐼) ∈ ℤ)
9		elfzelz 13584	. . . . . . . 8 ⊢ (𝑘 ∈ (𝑀...𝑁) → 𝑘 ∈ ℤ)
10	9	adantl 481	. . . . . . 7 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝑘 ∈ ℤ)
11	4	zred 12747	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → if(𝐼 ≤ 𝑀, 𝐼, 𝑀) ∈ ℝ)
12	11	adantr 480	. . . . . . . 8 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) ∧ 𝑘 ∈ (𝑀...𝑁)) → if(𝐼 ≤ 𝑀, 𝐼, 𝑀) ∈ ℝ)
13		zre 12643	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ)
14	13	3ad2ant1 1133	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → 𝑀 ∈ ℝ)
15	14	adantr 480	. . . . . . . 8 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝑀 ∈ ℝ)
16	9	zred 12747	. . . . . . . . 9 ⊢ (𝑘 ∈ (𝑀...𝑁) → 𝑘 ∈ ℝ)
17	16	adantl 481	. . . . . . . 8 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝑘 ∈ ℝ)
18		zre 12643	. . . . . . . . . . . . 13 ⊢ (𝐼 ∈ ℤ → 𝐼 ∈ ℝ)
19	13, 18	anim12i 612	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℤ ∧ 𝐼 ∈ ℤ) → (𝑀 ∈ ℝ ∧ 𝐼 ∈ ℝ))
20	19	ancomd 461	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℤ ∧ 𝐼 ∈ ℤ) → (𝐼 ∈ ℝ ∧ 𝑀 ∈ ℝ))
21	20	3adant2 1131	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → (𝐼 ∈ ℝ ∧ 𝑀 ∈ ℝ))
22	21	adantr 480	. . . . . . . . 9 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐼 ∈ ℝ ∧ 𝑀 ∈ ℝ))
23		min2 13252	. . . . . . . . 9 ⊢ ((𝐼 ∈ ℝ ∧ 𝑀 ∈ ℝ) → if(𝐼 ≤ 𝑀, 𝐼, 𝑀) ≤ 𝑀)
24	22, 23	syl 17	. . . . . . . 8 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) ∧ 𝑘 ∈ (𝑀...𝑁)) → if(𝐼 ≤ 𝑀, 𝐼, 𝑀) ≤ 𝑀)
25		elfzle1 13587	. . . . . . . . 9 ⊢ (𝑘 ∈ (𝑀...𝑁) → 𝑀 ≤ 𝑘)
26	25	adantl 481	. . . . . . . 8 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝑀 ≤ 𝑘)
27	12, 15, 17, 24, 26	letrd 11447	. . . . . . 7 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) ∧ 𝑘 ∈ (𝑀...𝑁)) → if(𝐼 ≤ 𝑀, 𝐼, 𝑀) ≤ 𝑘)
28		zre 12643	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ)
29	28	3ad2ant2 1134	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → 𝑁 ∈ ℝ)
30	29	adantr 480	. . . . . . . 8 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝑁 ∈ ℝ)
31	7	zred 12747	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → if(𝐼 ≤ 𝑁, 𝑁, 𝐼) ∈ ℝ)
32	31	adantr 480	. . . . . . . 8 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) ∧ 𝑘 ∈ (𝑀...𝑁)) → if(𝐼 ≤ 𝑁, 𝑁, 𝐼) ∈ ℝ)
33		elfzle2 13588	. . . . . . . . 9 ⊢ (𝑘 ∈ (𝑀...𝑁) → 𝑘 ≤ 𝑁)
34	33	adantl 481	. . . . . . . 8 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝑘 ≤ 𝑁)
35	28, 18	anim12i 612	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → (𝑁 ∈ ℝ ∧ 𝐼 ∈ ℝ))
36	35	3adant1 1130	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → (𝑁 ∈ ℝ ∧ 𝐼 ∈ ℝ))
37	36	ancomd 461	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → (𝐼 ∈ ℝ ∧ 𝑁 ∈ ℝ))
38		max2 13249	. . . . . . . . . 10 ⊢ ((𝐼 ∈ ℝ ∧ 𝑁 ∈ ℝ) → 𝑁 ≤ if(𝐼 ≤ 𝑁, 𝑁, 𝐼))
39	37, 38	syl 17	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → 𝑁 ≤ if(𝐼 ≤ 𝑁, 𝑁, 𝐼))
40	39	adantr 480	. . . . . . . 8 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝑁 ≤ if(𝐼 ≤ 𝑁, 𝑁, 𝐼))
41	17, 30, 32, 34, 40	letrd 11447	. . . . . . 7 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝑘 ≤ if(𝐼 ≤ 𝑁, 𝑁, 𝐼))
42	5, 8, 10, 27, 41	elfzd 13575	. . . . . 6 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝑘 ∈ (if(𝐼 ≤ 𝑀, 𝐼, 𝑀)...if(𝐼 ≤ 𝑁, 𝑁, 𝐼)))
43	42	ex 412	. . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → (𝑘 ∈ (𝑀...𝑁) → 𝑘 ∈ (if(𝐼 ≤ 𝑀, 𝐼, 𝑀)...if(𝐼 ≤ 𝑁, 𝑁, 𝐼))))
44	43	ssrdv 4014	. . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → (𝑀...𝑁) ⊆ (if(𝐼 ≤ 𝑀, 𝐼, 𝑀)...if(𝐼 ≤ 𝑁, 𝑁, 𝐼)))
45	44	adantl 481	. . 3 ⊢ ((𝑆 ⊆ (𝑀...𝑁) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ)) → (𝑀...𝑁) ⊆ (if(𝐼 ≤ 𝑀, 𝐼, 𝑀)...if(𝐼 ≤ 𝑁, 𝑁, 𝐼)))
46	1, 45	sstrd 4019	. 2 ⊢ ((𝑆 ⊆ (𝑀...𝑁) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ)) → 𝑆 ⊆ (if(𝐼 ≤ 𝑀, 𝐼, 𝑀)...if(𝐼 ≤ 𝑁, 𝑁, 𝐼)))
47	4	adantl 481	. . . 4 ⊢ ((𝑆 ⊆ (𝑀...𝑁) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ)) → if(𝐼 ≤ 𝑀, 𝐼, 𝑀) ∈ ℤ)
48	7	adantl 481	. . . 4 ⊢ ((𝑆 ⊆ (𝑀...𝑁) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ)) → if(𝐼 ≤ 𝑁, 𝑁, 𝐼) ∈ ℤ)
49	2	adantl 481	. . . 4 ⊢ ((𝑆 ⊆ (𝑀...𝑁) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ)) → 𝐼 ∈ ℤ)
50	19	3adant2 1131	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → (𝑀 ∈ ℝ ∧ 𝐼 ∈ ℝ))
51	50	adantl 481	. . . . . 6 ⊢ ((𝑆 ⊆ (𝑀...𝑁) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ)) → (𝑀 ∈ ℝ ∧ 𝐼 ∈ ℝ))
52	51	ancomd 461	. . . . 5 ⊢ ((𝑆 ⊆ (𝑀...𝑁) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ)) → (𝐼 ∈ ℝ ∧ 𝑀 ∈ ℝ))
53		min1 13251	. . . . 5 ⊢ ((𝐼 ∈ ℝ ∧ 𝑀 ∈ ℝ) → if(𝐼 ≤ 𝑀, 𝐼, 𝑀) ≤ 𝐼)
54	52, 53	syl 17	. . . 4 ⊢ ((𝑆 ⊆ (𝑀...𝑁) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ)) → if(𝐼 ≤ 𝑀, 𝐼, 𝑀) ≤ 𝐼)
55	36	adantl 481	. . . . . 6 ⊢ ((𝑆 ⊆ (𝑀...𝑁) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ)) → (𝑁 ∈ ℝ ∧ 𝐼 ∈ ℝ))
56	55	ancomd 461	. . . . 5 ⊢ ((𝑆 ⊆ (𝑀...𝑁) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ)) → (𝐼 ∈ ℝ ∧ 𝑁 ∈ ℝ))
57		max1 13247	. . . . 5 ⊢ ((𝐼 ∈ ℝ ∧ 𝑁 ∈ ℝ) → 𝐼 ≤ if(𝐼 ≤ 𝑁, 𝑁, 𝐼))
58	56, 57	syl 17	. . . 4 ⊢ ((𝑆 ⊆ (𝑀...𝑁) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ)) → 𝐼 ≤ if(𝐼 ≤ 𝑁, 𝑁, 𝐼))
59	47, 48, 49, 54, 58	elfzd 13575	. . 3 ⊢ ((𝑆 ⊆ (𝑀...𝑁) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ)) → 𝐼 ∈ (if(𝐼 ≤ 𝑀, 𝐼, 𝑀)...if(𝐼 ≤ 𝑁, 𝑁, 𝐼)))
60	59	snssd 4834	. 2 ⊢ ((𝑆 ⊆ (𝑀...𝑁) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ)) → {𝐼} ⊆ (if(𝐼 ≤ 𝑀, 𝐼, 𝑀)...if(𝐼 ≤ 𝑁, 𝑁, 𝐼)))
61	46, 60	unssd 4215	1 ⊢ ((𝑆 ⊆ (𝑀...𝑁) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ)) → (𝑆 ∪ {𝐼}) ⊆ (if(𝐼 ≤ 𝑀, 𝐼, 𝑀)...if(𝐼 ≤ 𝑁, 𝑁, 𝐼)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   ∈ wcel 2108   ∪ cun 3974   ⊆ wss 3976  ifcif 4548  {csn 4648   class class class wbr 5166  (class class class)co 7448  ℝcr 11183   ≤ cle 11325  ℤcz 12639  ...cfz 13567

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-pre-lttri 11258  ax-pre-lttrn 11259

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-po 5607  df-so 5608  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-1st 8030  df-2nd 8031  df-er 8763  df-en 9004  df-dom 9005  df-sdom 9006  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-neg 11523  df-z 12640  df-uz 12904  df-fz 13568

This theorem is referenced by:  ssfzunsn  13630  setsstruct2  17221