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Mirrors > Home > ILE Home > Th. List > fser0const | GIF version |
Description: Simplifying an expression which turns out just to be a constant zero sequence. (Contributed by Jim Kingdon, 16-Sep-2022.) |
Ref | Expression |
---|---|
fser0const.z | ⊢ 𝑍 = (ℤ≥‘𝑀) |
Ref | Expression |
---|---|
fser0const | ⊢ (𝑁 ∈ 𝑍 → (𝑛 ∈ 𝑍 ↦ if(𝑛 ≤ 𝑁, ((𝑍 × {0})‘𝑛), 0)) = (𝑍 × {0})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 110 | . . . . . 6 ⊢ (((𝑁 ∈ 𝑍 ∧ 𝑛 ∈ 𝑍) ∧ 𝑛 ≤ 𝑁) → 𝑛 ≤ 𝑁) | |
2 | 1 | iftrued 3541 | . . . . 5 ⊢ (((𝑁 ∈ 𝑍 ∧ 𝑛 ∈ 𝑍) ∧ 𝑛 ≤ 𝑁) → if(𝑛 ≤ 𝑁, ((𝑍 × {0})‘𝑛), 0) = ((𝑍 × {0})‘𝑛)) |
3 | c0ex 7939 | . . . . . . 7 ⊢ 0 ∈ V | |
4 | 3 | fvconst2 5728 | . . . . . 6 ⊢ (𝑛 ∈ 𝑍 → ((𝑍 × {0})‘𝑛) = 0) |
5 | 4 | ad2antlr 489 | . . . . 5 ⊢ (((𝑁 ∈ 𝑍 ∧ 𝑛 ∈ 𝑍) ∧ 𝑛 ≤ 𝑁) → ((𝑍 × {0})‘𝑛) = 0) |
6 | 2, 5 | eqtrd 2210 | . . . 4 ⊢ (((𝑁 ∈ 𝑍 ∧ 𝑛 ∈ 𝑍) ∧ 𝑛 ≤ 𝑁) → if(𝑛 ≤ 𝑁, ((𝑍 × {0})‘𝑛), 0) = 0) |
7 | simpr 110 | . . . . 5 ⊢ (((𝑁 ∈ 𝑍 ∧ 𝑛 ∈ 𝑍) ∧ ¬ 𝑛 ≤ 𝑁) → ¬ 𝑛 ≤ 𝑁) | |
8 | 7 | iffalsed 3544 | . . . 4 ⊢ (((𝑁 ∈ 𝑍 ∧ 𝑛 ∈ 𝑍) ∧ ¬ 𝑛 ≤ 𝑁) → if(𝑛 ≤ 𝑁, ((𝑍 × {0})‘𝑛), 0) = 0) |
9 | eluzelz 9523 | . . . . . . 7 ⊢ (𝑛 ∈ (ℤ≥‘𝑀) → 𝑛 ∈ ℤ) | |
10 | fser0const.z | . . . . . . 7 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
11 | 9, 10 | eleq2s 2272 | . . . . . 6 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ ℤ) |
12 | eluzelz 9523 | . . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℤ) | |
13 | 12, 10 | eleq2s 2272 | . . . . . 6 ⊢ (𝑁 ∈ 𝑍 → 𝑁 ∈ ℤ) |
14 | zdcle 9315 | . . . . . 6 ⊢ ((𝑛 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID 𝑛 ≤ 𝑁) | |
15 | 11, 13, 14 | syl2anr 290 | . . . . 5 ⊢ ((𝑁 ∈ 𝑍 ∧ 𝑛 ∈ 𝑍) → DECID 𝑛 ≤ 𝑁) |
16 | exmiddc 836 | . . . . 5 ⊢ (DECID 𝑛 ≤ 𝑁 → (𝑛 ≤ 𝑁 ∨ ¬ 𝑛 ≤ 𝑁)) | |
17 | 15, 16 | syl 14 | . . . 4 ⊢ ((𝑁 ∈ 𝑍 ∧ 𝑛 ∈ 𝑍) → (𝑛 ≤ 𝑁 ∨ ¬ 𝑛 ≤ 𝑁)) |
18 | 6, 8, 17 | mpjaodan 798 | . . 3 ⊢ ((𝑁 ∈ 𝑍 ∧ 𝑛 ∈ 𝑍) → if(𝑛 ≤ 𝑁, ((𝑍 × {0})‘𝑛), 0) = 0) |
19 | 18 | mpteq2dva 4090 | . 2 ⊢ (𝑁 ∈ 𝑍 → (𝑛 ∈ 𝑍 ↦ if(𝑛 ≤ 𝑁, ((𝑍 × {0})‘𝑛), 0)) = (𝑛 ∈ 𝑍 ↦ 0)) |
20 | fconstmpt 4670 | . 2 ⊢ (𝑍 × {0}) = (𝑛 ∈ 𝑍 ↦ 0) | |
21 | 19, 20 | eqtr4di 2228 | 1 ⊢ (𝑁 ∈ 𝑍 → (𝑛 ∈ 𝑍 ↦ if(𝑛 ≤ 𝑁, ((𝑍 × {0})‘𝑛), 0)) = (𝑍 × {0})) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ∨ wo 708 DECID wdc 834 = wceq 1353 ∈ wcel 2148 ifcif 3534 {csn 3591 class class class wbr 4000 ↦ cmpt 4061 × cxp 4621 ‘cfv 5212 0cc0 7799 ≤ cle 7980 ℤcz 9239 ℤ≥cuz 9514 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-setind 4533 ax-cnex 7890 ax-resscn 7891 ax-1cn 7892 ax-1re 7893 ax-icn 7894 ax-addcl 7895 ax-addrcl 7896 ax-mulcl 7897 ax-addcom 7899 ax-addass 7901 ax-distr 7903 ax-i2m1 7904 ax-0lt1 7905 ax-0id 7907 ax-rnegex 7908 ax-cnre 7910 ax-pre-ltirr 7911 ax-pre-ltwlin 7912 ax-pre-lttrn 7913 ax-pre-ltadd 7915 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-if 3535 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-br 4001 df-opab 4062 df-mpt 4063 df-id 4290 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-rn 4634 df-res 4635 df-ima 4636 df-iota 5174 df-fun 5214 df-fn 5215 df-f 5216 df-fv 5220 df-riota 5825 df-ov 5872 df-oprab 5873 df-mpo 5874 df-pnf 7981 df-mnf 7982 df-xr 7983 df-ltxr 7984 df-le 7985 df-sub 8117 df-neg 8118 df-inn 8906 df-n0 9163 df-z 9240 df-uz 9515 |
This theorem is referenced by: isumz 11378 |
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