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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 109 | . . . . . 6 ⊢ (((𝑁 ∈ 𝑍 ∧ 𝑛 ∈ 𝑍) ∧ 𝑛 ≤ 𝑁) → 𝑛 ≤ 𝑁) | |
2 | 1 | iftrued 3533 | . . . . 5 ⊢ (((𝑁 ∈ 𝑍 ∧ 𝑛 ∈ 𝑍) ∧ 𝑛 ≤ 𝑁) → if(𝑛 ≤ 𝑁, ((𝑍 × {0})‘𝑛), 0) = ((𝑍 × {0})‘𝑛)) |
3 | c0ex 7914 | . . . . . . 7 ⊢ 0 ∈ V | |
4 | 3 | fvconst2 5712 | . . . . . 6 ⊢ (𝑛 ∈ 𝑍 → ((𝑍 × {0})‘𝑛) = 0) |
5 | 4 | ad2antlr 486 | . . . . 5 ⊢ (((𝑁 ∈ 𝑍 ∧ 𝑛 ∈ 𝑍) ∧ 𝑛 ≤ 𝑁) → ((𝑍 × {0})‘𝑛) = 0) |
6 | 2, 5 | eqtrd 2203 | . . . 4 ⊢ (((𝑁 ∈ 𝑍 ∧ 𝑛 ∈ 𝑍) ∧ 𝑛 ≤ 𝑁) → if(𝑛 ≤ 𝑁, ((𝑍 × {0})‘𝑛), 0) = 0) |
7 | simpr 109 | . . . . 5 ⊢ (((𝑁 ∈ 𝑍 ∧ 𝑛 ∈ 𝑍) ∧ ¬ 𝑛 ≤ 𝑁) → ¬ 𝑛 ≤ 𝑁) | |
8 | 7 | iffalsed 3536 | . . . 4 ⊢ (((𝑁 ∈ 𝑍 ∧ 𝑛 ∈ 𝑍) ∧ ¬ 𝑛 ≤ 𝑁) → if(𝑛 ≤ 𝑁, ((𝑍 × {0})‘𝑛), 0) = 0) |
9 | eluzelz 9496 | . . . . . . 7 ⊢ (𝑛 ∈ (ℤ≥‘𝑀) → 𝑛 ∈ ℤ) | |
10 | fser0const.z | . . . . . . 7 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
11 | 9, 10 | eleq2s 2265 | . . . . . 6 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ ℤ) |
12 | eluzelz 9496 | . . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℤ) | |
13 | 12, 10 | eleq2s 2265 | . . . . . 6 ⊢ (𝑁 ∈ 𝑍 → 𝑁 ∈ ℤ) |
14 | zdcle 9288 | . . . . . 6 ⊢ ((𝑛 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID 𝑛 ≤ 𝑁) | |
15 | 11, 13, 14 | syl2anr 288 | . . . . 5 ⊢ ((𝑁 ∈ 𝑍 ∧ 𝑛 ∈ 𝑍) → DECID 𝑛 ≤ 𝑁) |
16 | exmiddc 831 | . . . . 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 793 | . . 3 ⊢ ((𝑁 ∈ 𝑍 ∧ 𝑛 ∈ 𝑍) → if(𝑛 ≤ 𝑁, ((𝑍 × {0})‘𝑛), 0) = 0) |
19 | 18 | mpteq2dva 4079 | . 2 ⊢ (𝑁 ∈ 𝑍 → (𝑛 ∈ 𝑍 ↦ if(𝑛 ≤ 𝑁, ((𝑍 × {0})‘𝑛), 0)) = (𝑛 ∈ 𝑍 ↦ 0)) |
20 | fconstmpt 4658 | . 2 ⊢ (𝑍 × {0}) = (𝑛 ∈ 𝑍 ↦ 0) | |
21 | 19, 20 | eqtr4di 2221 | 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 103 ∨ wo 703 DECID wdc 829 = wceq 1348 ∈ wcel 2141 ifcif 3526 {csn 3583 class class class wbr 3989 ↦ cmpt 4050 × cxp 4609 ‘cfv 5198 0cc0 7774 ≤ cle 7955 ℤcz 9212 ℤ≥cuz 9487 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-addcom 7874 ax-addass 7876 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-0id 7882 ax-rnegex 7883 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-ltadd 7890 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-if 3527 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-br 3990 df-opab 4051 df-mpt 4052 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-fv 5206 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-inn 8879 df-n0 9136 df-z 9213 df-uz 9488 |
This theorem is referenced by: isumz 11352 |
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