Theorem ssfzunsnext 13537

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 4538	. . . . . . . 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 4538	. . . . . . . 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 13492	. . . . . . . 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 12645	. . . . . . . . 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 12540	. . . . . . . . . 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 12645	. . . . . . . . 9 ⊢ (𝑘 ∈ (𝑀...𝑁) → 𝑘 ∈ ℝ)
17	16	adantl 481	. . . . . . . 8 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝑘 ∈ ℝ)
18		zre 12540	. . . . . . . . . . . . 13 ⊢ (𝐼 ∈ ℤ → 𝐼 ∈ ℝ)
19	13, 18	anim12i 613	. . . . . . . . . . . 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 13157	. . . . . . . . 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 13495	. . . . . . . . 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 11338	. . . . . . 7 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) ∧ 𝑘 ∈ (𝑀...𝑁)) → if(𝐼 ≤ 𝑀, 𝐼, 𝑀) ≤ 𝑘)
28		zre 12540	. . . . . . . . . 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 12645	. . . . . . . . 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 13496	. . . . . . . . 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 613	. . . . . . . . . . . 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 13154	. . . . . . . . . 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 11338	. . . . . . 7 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ) ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝑘 ≤ if(𝐼 ≤ 𝑁, 𝑁, 𝐼))
42	5, 8, 10, 27, 41	elfzd 13483	. . . . . 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 3955	. . . 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 3960	. 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 13156	. . . . 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 13152	. . . . 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 13483	. . 3 ⊢ ((𝑆 ⊆ (𝑀...𝑁) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ)) → 𝐼 ∈ (if(𝐼 ≤ 𝑀, 𝐼, 𝑀)...if(𝐼 ≤ 𝑁, 𝑁, 𝐼)))
60	59	snssd 4776	. 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 4158	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 1086   ∈ wcel 2109   ∪ cun 3915   ⊆ wss 3917  ifcif 4491  {csn 4592   class class class wbr 5110  (class class class)co 7390  ℝcr 11074   ≤ cle 11216  ℤcz 12536  ...cfz 13475

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-cnex 11131  ax-resscn 11132  ax-pre-lttri 11149  ax-pre-lttrn 11150

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-po 5549  df-so 5550  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-1st 7971  df-2nd 7972  df-er 8674  df-en 8922  df-dom 8923  df-sdom 8924  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-neg 11415  df-z 12537  df-uz 12801  df-fz 13476

This theorem is referenced by:  ssfzunsn  13538  setsstruct2  17151