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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 1136 | . . 3 ⊢ ((𝑆 ⊆ (𝑀...𝑁) ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ (ℤ≥‘𝑀)) → 𝑆 ⊆ (𝑀...𝑁)) | |
2 | eluzel2 12831 | . . . 4 ⊢ (𝐼 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ) | |
3 | 2 | 3ad2ant3 1135 | . . 3 ⊢ ((𝑆 ⊆ (𝑀...𝑁) ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ (ℤ≥‘𝑀)) → 𝑀 ∈ ℤ) |
4 | simp2 1137 | . . 3 ⊢ ((𝑆 ⊆ (𝑀...𝑁) ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ (ℤ≥‘𝑀)) → 𝑁 ∈ ℤ) | |
5 | eluzelz 12836 | . . . 4 ⊢ (𝐼 ∈ (ℤ≥‘𝑀) → 𝐼 ∈ ℤ) | |
6 | 5 | 3ad2ant3 1135 | . . 3 ⊢ ((𝑆 ⊆ (𝑀...𝑁) ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ (ℤ≥‘𝑀)) → 𝐼 ∈ ℤ) |
7 | ssfzunsnext 13550 | . . 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 1372 | . 2 ⊢ ((𝑆 ⊆ (𝑀...𝑁) ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ (ℤ≥‘𝑀)) → (𝑆 ∪ {𝐼}) ⊆ (if(𝐼 ≤ 𝑀, 𝐼, 𝑀)...if(𝐼 ≤ 𝑁, 𝑁, 𝐼))) |
9 | eluz2 12832 | . . . . 5 ⊢ (𝐼 ∈ (ℤ≥‘𝑀) ↔ (𝑀 ∈ ℤ ∧ 𝐼 ∈ ℤ ∧ 𝑀 ≤ 𝐼)) | |
10 | zre 12566 | . . . . . . . . 9 ⊢ (𝐼 ∈ ℤ → 𝐼 ∈ ℝ) | |
11 | 10 | rexrd 11268 | . . . . . . . 8 ⊢ (𝐼 ∈ ℤ → 𝐼 ∈ ℝ*) |
12 | 11 | 3ad2ant2 1134 | . . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝐼 ∈ ℤ ∧ 𝑀 ≤ 𝐼) → 𝐼 ∈ ℝ*) |
13 | zre 12566 | . . . . . . . . 9 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ) | |
14 | 13 | rexrd 11268 | . . . . . . . 8 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ*) |
15 | 14 | 3ad2ant1 1133 | . . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝐼 ∈ ℤ ∧ 𝑀 ≤ 𝐼) → 𝑀 ∈ ℝ*) |
16 | simp3 1138 | . . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝐼 ∈ ℤ ∧ 𝑀 ≤ 𝐼) → 𝑀 ≤ 𝐼) | |
17 | xrmineq 13163 | . . . . . . 7 ⊢ ((𝐼 ∈ ℝ* ∧ 𝑀 ∈ ℝ* ∧ 𝑀 ≤ 𝐼) → if(𝐼 ≤ 𝑀, 𝐼, 𝑀) = 𝑀) | |
18 | 12, 15, 16, 17 | syl3anc 1371 | . . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝐼 ∈ ℤ ∧ 𝑀 ≤ 𝐼) → if(𝐼 ≤ 𝑀, 𝐼, 𝑀) = 𝑀) |
19 | 18 | eqcomd 2738 | . . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝐼 ∈ ℤ ∧ 𝑀 ≤ 𝐼) → 𝑀 = if(𝐼 ≤ 𝑀, 𝐼, 𝑀)) |
20 | 9, 19 | sylbi 216 | . . . 4 ⊢ (𝐼 ∈ (ℤ≥‘𝑀) → 𝑀 = if(𝐼 ≤ 𝑀, 𝐼, 𝑀)) |
21 | 20 | 3ad2ant3 1135 | . . 3 ⊢ ((𝑆 ⊆ (𝑀...𝑁) ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ (ℤ≥‘𝑀)) → 𝑀 = if(𝐼 ≤ 𝑀, 𝐼, 𝑀)) |
22 | 21 | oveq1d 7426 | . 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 4023 | 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 1087 = wceq 1541 ∈ wcel 2106 ∪ cun 3946 ⊆ wss 3948 ifcif 4528 {csn 4628 class class class wbr 5148 ‘cfv 6543 (class class class)co 7411 ℝ*cxr 11251 ≤ cle 11253 ℤcz 12562 ℤ≥cuz 12826 ...cfz 13488 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-pre-lttri 11186 ax-pre-lttrn 11187 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-po 5588 df-so 5589 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7414 df-oprab 7415 df-mpo 7416 df-1st 7977 df-2nd 7978 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-neg 11451 df-z 12563 df-uz 12827 df-fz 13489 |
This theorem is referenced by: (None) |
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