Theorem ssfzunsn 12257

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, 8-Jun-2021.)

Assertion

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

Proof of Theorem ssfzunsn

Step	Hyp	Ref	Expression
1		simp1 1054	. . 3 ⊢ ((𝑆 ⊆ (𝑀...𝑁) ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ (ℤ≥‘𝑀)) → 𝑆 ⊆ (𝑀...𝑁))
2		simpl 472	. . . . . 6 ⊢ ((𝑁 ∈ ℤ ∧ 𝐼 ∈ (ℤ≥‘𝑀)) → 𝑁 ∈ ℤ)
3		eluzelz 11573	. . . . . . . 8 ⊢ (𝐼 ∈ (ℤ≥‘𝑀) → 𝐼 ∈ ℤ)
4	3	adantl 481	. . . . . . 7 ⊢ ((𝑁 ∈ ℤ ∧ 𝐼 ∈ (ℤ≥‘𝑀)) → 𝐼 ∈ ℤ)
5	2, 4	ifcld 4081	. . . . . 6 ⊢ ((𝑁 ∈ ℤ ∧ 𝐼 ∈ (ℤ≥‘𝑀)) → if(𝐼 ≤ 𝑁, 𝑁, 𝐼) ∈ ℤ)
6		zre 11258	. . . . . . . . 9 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ)
7		eluzelre 11574	. . . . . . . . 9 ⊢ (𝐼 ∈ (ℤ≥‘𝑀) → 𝐼 ∈ ℝ)
8	6, 7	anim12i 588	. . . . . . . 8 ⊢ ((𝑁 ∈ ℤ ∧ 𝐼 ∈ (ℤ≥‘𝑀)) → (𝑁 ∈ ℝ ∧ 𝐼 ∈ ℝ))
9	8	ancomd 466	. . . . . . 7 ⊢ ((𝑁 ∈ ℤ ∧ 𝐼 ∈ (ℤ≥‘𝑀)) → (𝐼 ∈ ℝ ∧ 𝑁 ∈ ℝ))
10		max2 11892	. . . . . . 7 ⊢ ((𝐼 ∈ ℝ ∧ 𝑁 ∈ ℝ) → 𝑁 ≤ if(𝐼 ≤ 𝑁, 𝑁, 𝐼))
11	9, 10	syl 17	. . . . . 6 ⊢ ((𝑁 ∈ ℤ ∧ 𝐼 ∈ (ℤ≥‘𝑀)) → 𝑁 ≤ if(𝐼 ≤ 𝑁, 𝑁, 𝐼))
12		eluz2 11569	. . . . . 6 ⊢ (if(𝐼 ≤ 𝑁, 𝑁, 𝐼) ∈ (ℤ≥‘𝑁) ↔ (𝑁 ∈ ℤ ∧ if(𝐼 ≤ 𝑁, 𝑁, 𝐼) ∈ ℤ ∧ 𝑁 ≤ if(𝐼 ≤ 𝑁, 𝑁, 𝐼)))
13	2, 5, 11, 12	syl3anbrc 1239	. . . . 5 ⊢ ((𝑁 ∈ ℤ ∧ 𝐼 ∈ (ℤ≥‘𝑀)) → if(𝐼 ≤ 𝑁, 𝑁, 𝐼) ∈ (ℤ≥‘𝑁))
14		fzss2 12252	. . . . 5 ⊢ (if(𝐼 ≤ 𝑁, 𝑁, 𝐼) ∈ (ℤ≥‘𝑁) → (𝑀...𝑁) ⊆ (𝑀...if(𝐼 ≤ 𝑁, 𝑁, 𝐼)))
15	13, 14	syl 17	. . . 4 ⊢ ((𝑁 ∈ ℤ ∧ 𝐼 ∈ (ℤ≥‘𝑀)) → (𝑀...𝑁) ⊆ (𝑀...if(𝐼 ≤ 𝑁, 𝑁, 𝐼)))
16	15	3adant1 1072	. . 3 ⊢ ((𝑆 ⊆ (𝑀...𝑁) ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ (ℤ≥‘𝑀)) → (𝑀...𝑁) ⊆ (𝑀...if(𝐼 ≤ 𝑁, 𝑁, 𝐼)))
17	1, 16	sstrd 3578	. 2 ⊢ ((𝑆 ⊆ (𝑀...𝑁) ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ (ℤ≥‘𝑀)) → 𝑆 ⊆ (𝑀...if(𝐼 ≤ 𝑁, 𝑁, 𝐼)))
18		eluzel2 11568	. . . . . 6 ⊢ (𝐼 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ)
19	18	3ad2ant3 1077	. . . . 5 ⊢ ((𝑆 ⊆ (𝑀...𝑁) ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ (ℤ≥‘𝑀)) → 𝑀 ∈ ℤ)
20	5	3adant1 1072	. . . . 5 ⊢ ((𝑆 ⊆ (𝑀...𝑁) ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ (ℤ≥‘𝑀)) → if(𝐼 ≤ 𝑁, 𝑁, 𝐼) ∈ ℤ)
21	3	3ad2ant3 1077	. . . . 5 ⊢ ((𝑆 ⊆ (𝑀...𝑁) ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ (ℤ≥‘𝑀)) → 𝐼 ∈ ℤ)
22	19, 20, 21	3jca 1235	. . . 4 ⊢ ((𝑆 ⊆ (𝑀...𝑁) ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ (ℤ≥‘𝑀)) → (𝑀 ∈ ℤ ∧ if(𝐼 ≤ 𝑁, 𝑁, 𝐼) ∈ ℤ ∧ 𝐼 ∈ ℤ))
23		eluzle 11576	. . . . . 6 ⊢ (𝐼 ∈ (ℤ≥‘𝑀) → 𝑀 ≤ 𝐼)
24	23	3ad2ant3 1077	. . . . 5 ⊢ ((𝑆 ⊆ (𝑀...𝑁) ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ (ℤ≥‘𝑀)) → 𝑀 ≤ 𝐼)
25	8	3adant1 1072	. . . . . . 7 ⊢ ((𝑆 ⊆ (𝑀...𝑁) ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ (ℤ≥‘𝑀)) → (𝑁 ∈ ℝ ∧ 𝐼 ∈ ℝ))
26	25	ancomd 466	. . . . . 6 ⊢ ((𝑆 ⊆ (𝑀...𝑁) ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ (ℤ≥‘𝑀)) → (𝐼 ∈ ℝ ∧ 𝑁 ∈ ℝ))
27		max1 11890	. . . . . 6 ⊢ ((𝐼 ∈ ℝ ∧ 𝑁 ∈ ℝ) → 𝐼 ≤ if(𝐼 ≤ 𝑁, 𝑁, 𝐼))
28	26, 27	syl 17	. . . . 5 ⊢ ((𝑆 ⊆ (𝑀...𝑁) ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ (ℤ≥‘𝑀)) → 𝐼 ≤ if(𝐼 ≤ 𝑁, 𝑁, 𝐼))
29	24, 28	jca 553	. . . 4 ⊢ ((𝑆 ⊆ (𝑀...𝑁) ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ (ℤ≥‘𝑀)) → (𝑀 ≤ 𝐼 ∧ 𝐼 ≤ if(𝐼 ≤ 𝑁, 𝑁, 𝐼)))
30		elfz2 12204	. . . 4 ⊢ (𝐼 ∈ (𝑀...if(𝐼 ≤ 𝑁, 𝑁, 𝐼)) ↔ ((𝑀 ∈ ℤ ∧ if(𝐼 ≤ 𝑁, 𝑁, 𝐼) ∈ ℤ ∧ 𝐼 ∈ ℤ) ∧ (𝑀 ≤ 𝐼 ∧ 𝐼 ≤ if(𝐼 ≤ 𝑁, 𝑁, 𝐼))))
31	22, 29, 30	sylanbrc 695	. . 3 ⊢ ((𝑆 ⊆ (𝑀...𝑁) ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ (ℤ≥‘𝑀)) → 𝐼 ∈ (𝑀...if(𝐼 ≤ 𝑁, 𝑁, 𝐼)))
32	31	snssd 4281	. 2 ⊢ ((𝑆 ⊆ (𝑀...𝑁) ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ (ℤ≥‘𝑀)) → {𝐼} ⊆ (𝑀...if(𝐼 ≤ 𝑁, 𝑁, 𝐼)))
33	17, 32	unssd 3751	1 ⊢ ((𝑆 ⊆ (𝑀...𝑁) ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ (ℤ≥‘𝑀)) → (𝑆 ∪ {𝐼}) ⊆ (𝑀...if(𝐼 ≤ 𝑁, 𝑁, 𝐼)))

Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   ∈ wcel 1977   ∪ cun 3538   ⊆ wss 3540  ifcif 4036  {csn 4125   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  ℝcr 9814   ≤ cle 9954  ℤcz 11254  ℤ_≥cuz 11563  ...cfz 12197

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-pre-lttri 9889  ax-pre-lttrn 9890

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-po 4959  df-so 4960  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-neg 10148  df-z 11255  df-uz 11564  df-fz 12198

This theorem is referenced by:  setsstruct  15727