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Mirrors > Home > MPE Home > Th. List > ssfzunsn | Structured version Visualization version GIF version |
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.) (Proof shortened by AV, 13-Nov-2021.) |
Ref | Expression |
---|---|
ssfzunsn | ⊢ ((𝑆 ⊆ (𝑀...𝑁) ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ (ℤ≥‘𝑀)) → (𝑆 ∪ {𝐼}) ⊆ (𝑀...if(𝐼 ≤ 𝑁, 𝑁, 𝐼))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1132 | . . 3 ⊢ ((𝑆 ⊆ (𝑀...𝑁) ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ (ℤ≥‘𝑀)) → 𝑆 ⊆ (𝑀...𝑁)) | |
2 | eluzel2 12242 | . . . 4 ⊢ (𝐼 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ) | |
3 | 2 | 3ad2ant3 1131 | . . 3 ⊢ ((𝑆 ⊆ (𝑀...𝑁) ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ (ℤ≥‘𝑀)) → 𝑀 ∈ ℤ) |
4 | simp2 1133 | . . 3 ⊢ ((𝑆 ⊆ (𝑀...𝑁) ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ (ℤ≥‘𝑀)) → 𝑁 ∈ ℤ) | |
5 | eluzelz 12247 | . . . 4 ⊢ (𝐼 ∈ (ℤ≥‘𝑀) → 𝐼 ∈ ℤ) | |
6 | 5 | 3ad2ant3 1131 | . . 3 ⊢ ((𝑆 ⊆ (𝑀...𝑁) ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ (ℤ≥‘𝑀)) → 𝐼 ∈ ℤ) |
7 | ssfzunsnext 12946 | . . 3 ⊢ ((𝑆 ⊆ (𝑀...𝑁) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ)) → (𝑆 ∪ {𝐼}) ⊆ (if(𝐼 ≤ 𝑀, 𝐼, 𝑀)...if(𝐼 ≤ 𝑁, 𝑁, 𝐼))) | |
8 | 1, 3, 4, 6, 7 | syl13anc 1368 | . 2 ⊢ ((𝑆 ⊆ (𝑀...𝑁) ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ (ℤ≥‘𝑀)) → (𝑆 ∪ {𝐼}) ⊆ (if(𝐼 ≤ 𝑀, 𝐼, 𝑀)...if(𝐼 ≤ 𝑁, 𝑁, 𝐼))) |
9 | eluz2 12243 | . . . . 5 ⊢ (𝐼 ∈ (ℤ≥‘𝑀) ↔ (𝑀 ∈ ℤ ∧ 𝐼 ∈ ℤ ∧ 𝑀 ≤ 𝐼)) | |
10 | zre 11979 | . . . . . . . . 9 ⊢ (𝐼 ∈ ℤ → 𝐼 ∈ ℝ) | |
11 | 10 | rexrd 10685 | . . . . . . . 8 ⊢ (𝐼 ∈ ℤ → 𝐼 ∈ ℝ*) |
12 | 11 | 3ad2ant2 1130 | . . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝐼 ∈ ℤ ∧ 𝑀 ≤ 𝐼) → 𝐼 ∈ ℝ*) |
13 | zre 11979 | . . . . . . . . 9 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ) | |
14 | 13 | rexrd 10685 | . . . . . . . 8 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ*) |
15 | 14 | 3ad2ant1 1129 | . . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝐼 ∈ ℤ ∧ 𝑀 ≤ 𝐼) → 𝑀 ∈ ℝ*) |
16 | simp3 1134 | . . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝐼 ∈ ℤ ∧ 𝑀 ≤ 𝐼) → 𝑀 ≤ 𝐼) | |
17 | xrmineq 12567 | . . . . . . 7 ⊢ ((𝐼 ∈ ℝ* ∧ 𝑀 ∈ ℝ* ∧ 𝑀 ≤ 𝐼) → if(𝐼 ≤ 𝑀, 𝐼, 𝑀) = 𝑀) | |
18 | 12, 15, 16, 17 | syl3anc 1367 | . . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝐼 ∈ ℤ ∧ 𝑀 ≤ 𝐼) → if(𝐼 ≤ 𝑀, 𝐼, 𝑀) = 𝑀) |
19 | 18 | eqcomd 2827 | . . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝐼 ∈ ℤ ∧ 𝑀 ≤ 𝐼) → 𝑀 = if(𝐼 ≤ 𝑀, 𝐼, 𝑀)) |
20 | 9, 19 | sylbi 219 | . . . 4 ⊢ (𝐼 ∈ (ℤ≥‘𝑀) → 𝑀 = if(𝐼 ≤ 𝑀, 𝐼, 𝑀)) |
21 | 20 | 3ad2ant3 1131 | . . 3 ⊢ ((𝑆 ⊆ (𝑀...𝑁) ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ (ℤ≥‘𝑀)) → 𝑀 = if(𝐼 ≤ 𝑀, 𝐼, 𝑀)) |
22 | 21 | oveq1d 7165 | . 2 ⊢ ((𝑆 ⊆ (𝑀...𝑁) ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ (ℤ≥‘𝑀)) → (𝑀...if(𝐼 ≤ 𝑁, 𝑁, 𝐼)) = (if(𝐼 ≤ 𝑀, 𝐼, 𝑀)...if(𝐼 ≤ 𝑁, 𝑁, 𝐼))) |
23 | 8, 22 | sseqtrrd 4008 | 1 ⊢ ((𝑆 ⊆ (𝑀...𝑁) ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ (ℤ≥‘𝑀)) → (𝑆 ∪ {𝐼}) ⊆ (𝑀...if(𝐼 ≤ 𝑁, 𝑁, 𝐼))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ∪ cun 3934 ⊆ wss 3936 ifcif 4467 {csn 4561 class class class wbr 5059 ‘cfv 6350 (class class class)co 7150 ℝ*cxr 10668 ≤ cle 10670 ℤcz 11975 ℤ≥cuz 12237 ...cfz 12886 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-pre-lttri 10605 ax-pre-lttrn 10606 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-po 5469 df-so 5470 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-ov 7153 df-oprab 7154 df-mpo 7155 df-1st 7683 df-2nd 7684 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-neg 10867 df-z 11976 df-uz 12238 df-fz 12887 |
This theorem is referenced by: (None) |
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