Theorem ssfzunsnext 13514

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 483	. . 3 ⊢ ((𝑆 ⊆ (𝑀...𝑁) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ)) → 𝑆 ⊆ (𝑀...𝑁))
2		simp3 1144	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → 𝐼 ∈ ℤ)
3		simp1 1142	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → 𝑀 ∈ ℤ)
4	2, 3	ifcld 4501	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → if(𝐼 ≤ 𝑀, 𝐼, 𝑀) ∈ ℤ)
5	4	adantr 481	. . . . . . 7 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) ∧ 𝑘 ∈ (𝑀...𝑁)) → if(𝐼 ≤ 𝑀, 𝐼, 𝑀) ∈ ℤ)
6		simp2 1143	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → 𝑁 ∈ ℤ)
7	6, 2	ifcld 4501	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → if(𝐼 ≤ 𝑁, 𝑁, 𝐼) ∈ ℤ)
8	7	adantr 481	. . . . . . 7 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) ∧ 𝑘 ∈ (𝑀...𝑁)) → if(𝐼 ≤ 𝑁, 𝑁, 𝐼) ∈ ℤ)
9		elfzelz 13469	. . . . . . . 8 ⊢ (𝑘 ∈ (𝑀...𝑁) → 𝑘 ∈ ℤ)
10	9	adantl 482	. . . . . . 7 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝑘 ∈ ℤ)
11	4	zred 12624	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → if(𝐼 ≤ 𝑀, 𝐼, 𝑀) ∈ ℝ)
12	11	adantr 481	. . . . . . . 8 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) ∧ 𝑘 ∈ (𝑀...𝑁)) → if(𝐼 ≤ 𝑀, 𝐼, 𝑀) ∈ ℝ)
13		zre 12519	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ)
14	13	3ad2ant1 1139	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → 𝑀 ∈ ℝ)
15	14	adantr 481	. . . . . . . 8 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝑀 ∈ ℝ)
16	9	zred 12624	. . . . . . . . 9 ⊢ (𝑘 ∈ (𝑀...𝑁) → 𝑘 ∈ ℝ)
17	16	adantl 482	. . . . . . . 8 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝑘 ∈ ℝ)
18		zre 12519	. . . . . . . . . . . . 13 ⊢ (𝐼 ∈ ℤ → 𝐼 ∈ ℝ)
19	13, 18	anim12i 619	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℤ ∧ 𝐼 ∈ ℤ) → (𝑀 ∈ ℝ ∧ 𝐼 ∈ ℝ))
20	19	ancomd 462	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℤ ∧ 𝐼 ∈ ℤ) → (𝐼 ∈ ℝ ∧ 𝑀 ∈ ℝ))
21	20	3adant2 1137	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → (𝐼 ∈ ℝ ∧ 𝑀 ∈ ℝ))
22	21	adantr 481	. . . . . . . . 9 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐼 ∈ ℝ ∧ 𝑀 ∈ ℝ))
23		min2 13133	. . . . . . . . 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 13472	. . . . . . . . 9 ⊢ (𝑘 ∈ (𝑀...𝑁) → 𝑀 ≤ 𝑘)
26	25	adantl 482	. . . . . . . 8 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝑀 ≤ 𝑘)
27	12, 15, 17, 24, 26	letrd 11294	. . . . . . 7 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) ∧ 𝑘 ∈ (𝑀...𝑁)) → if(𝐼 ≤ 𝑀, 𝐼, 𝑀) ≤ 𝑘)
28		zre 12519	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ)
29	28	3ad2ant2 1140	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → 𝑁 ∈ ℝ)
30	29	adantr 481	. . . . . . . 8 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝑁 ∈ ℝ)
31	7	zred 12624	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → if(𝐼 ≤ 𝑁, 𝑁, 𝐼) ∈ ℝ)
32	31	adantr 481	. . . . . . . 8 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) ∧ 𝑘 ∈ (𝑀...𝑁)) → if(𝐼 ≤ 𝑁, 𝑁, 𝐼) ∈ ℝ)
33		elfzle2 13473	. . . . . . . . 9 ⊢ (𝑘 ∈ (𝑀...𝑁) → 𝑘 ≤ 𝑁)
34	33	adantl 482	. . . . . . . 8 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝑘 ≤ 𝑁)
35	28, 18	anim12i 619	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → (𝑁 ∈ ℝ ∧ 𝐼 ∈ ℝ))
36	35	3adant1 1136	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → (𝑁 ∈ ℝ ∧ 𝐼 ∈ ℝ))
37	36	ancomd 462	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → (𝐼 ∈ ℝ ∧ 𝑁 ∈ ℝ))
38		max2 13130	. . . . . . . . . 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 481	. . . . . . . 8 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝑁 ≤ if(𝐼 ≤ 𝑁, 𝑁, 𝐼))
41	17, 30, 32, 34, 40	letrd 11294	. . . . . . 7 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝑘 ≤ if(𝐼 ≤ 𝑁, 𝑁, 𝐼))
42	5, 8, 10, 27, 41	elfzd 13460	. . . . . 6 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝑘 ∈ (if(𝐼 ≤ 𝑀, 𝐼, 𝑀)...if(𝐼 ≤ 𝑁, 𝑁, 𝐼)))
43	42	ex 413	. . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → (𝑘 ∈ (𝑀...𝑁) → 𝑘 ∈ (if(𝐼 ≤ 𝑀, 𝐼, 𝑀)...if(𝐼 ≤ 𝑁, 𝑁, 𝐼))))
44	43	ssrdv 3921	. . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → (𝑀...𝑁) ⊆ (if(𝐼 ≤ 𝑀, 𝐼, 𝑀)...if(𝐼 ≤ 𝑁, 𝑁, 𝐼)))
45	44	adantl 482	. . 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 3925	. 2 ⊢ ((𝑆 ⊆ (𝑀...𝑁) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ)) → 𝑆 ⊆ (if(𝐼 ≤ 𝑀, 𝐼, 𝑀)...if(𝐼 ≤ 𝑁, 𝑁, 𝐼)))
47	4	adantl 482	. . . 4 ⊢ ((𝑆 ⊆ (𝑀...𝑁) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ)) → if(𝐼 ≤ 𝑀, 𝐼, 𝑀) ∈ ℤ)
48	7	adantl 482	. . . 4 ⊢ ((𝑆 ⊆ (𝑀...𝑁) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ)) → if(𝐼 ≤ 𝑁, 𝑁, 𝐼) ∈ ℤ)
49	2	adantl 482	. . . 4 ⊢ ((𝑆 ⊆ (𝑀...𝑁) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ)) → 𝐼 ∈ ℤ)
50	19	3adant2 1137	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) → (𝑀 ∈ ℝ ∧ 𝐼 ∈ ℝ))
51	50	adantl 482	. . . . . 6 ⊢ ((𝑆 ⊆ (𝑀...𝑁) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ)) → (𝑀 ∈ ℝ ∧ 𝐼 ∈ ℝ))
52	51	ancomd 462	. . . . 5 ⊢ ((𝑆 ⊆ (𝑀...𝑁) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ)) → (𝐼 ∈ ℝ ∧ 𝑀 ∈ ℝ))
53		min1 13132	. . . . 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 482	. . . . . 6 ⊢ ((𝑆 ⊆ (𝑀...𝑁) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ)) → (𝑁 ∈ ℝ ∧ 𝐼 ∈ ℝ))
56	55	ancomd 462	. . . . 5 ⊢ ((𝑆 ⊆ (𝑀...𝑁) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ)) → (𝐼 ∈ ℝ ∧ 𝑁 ∈ ℝ))
57		max1 13128	. . . . 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 13460	. . 3 ⊢ ((𝑆 ⊆ (𝑀...𝑁) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ)) → 𝐼 ∈ (if(𝐼 ≤ 𝑀, 𝐼, 𝑀)...if(𝐼 ≤ 𝑁, 𝑁, 𝐼)))
60	59	snssd 4718	. 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 4121	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 396   ∧ w3a 1092   ∈ wcel 2119   ∪ cun 3881   ⊆ wss 3883  ifcif 4454  {csn 4555   class class class wbr 5072  (class class class)co 7356  ℝcr 11028   ≤ cle 11171  ℤcz 12515  ...cfz 13452

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678  ax-cnex 11085  ax-resscn 11086  ax-pre-lttri 11103  ax-pre-lttrn 11104

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-po 5526  df-so 5527  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7931  df-2nd 7932  df-er 8633  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-neg 11371  df-z 12516  df-uz 12780  df-fz 13453

This theorem is referenced by:  ssfzunsn  13515  setsstruct2  17135