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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 3628 | . . . . 5 ⊢ (((𝑁 ∈ 𝑍 ∧ 𝑛 ∈ 𝑍) ∧ 𝑛 ≤ 𝑁) → if(𝑛 ≤ 𝑁, ((𝑍 × {0})‘𝑛), 0) = ((𝑍 × {0})‘𝑛)) |
| 3 | c0ex 8264 | . . . . . . 7 ⊢ 0 ∈ V | |
| 4 | 3 | fvconst2 5899 | . . . . . 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 2265 | . . . 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 3631 | . . . 4 ⊢ (((𝑁 ∈ 𝑍 ∧ 𝑛 ∈ 𝑍) ∧ ¬ 𝑛 ≤ 𝑁) → if(𝑛 ≤ 𝑁, ((𝑍 × {0})‘𝑛), 0) = 0) |
| 9 | eluzelz 9859 | . . . . . . 7 ⊢ (𝑛 ∈ (ℤ≥‘𝑀) → 𝑛 ∈ ℤ) | |
| 10 | fser0const.z | . . . . . . 7 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
| 11 | 9, 10 | eleq2s 2327 | . . . . . 6 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ ℤ) |
| 12 | eluzelz 9859 | . . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℤ) | |
| 13 | 12, 10 | eleq2s 2327 | . . . . . 6 ⊢ (𝑁 ∈ 𝑍 → 𝑁 ∈ ℤ) |
| 14 | zdcle 9650 | . . . . . 6 ⊢ ((𝑛 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID 𝑛 ≤ 𝑁) | |
| 15 | 11, 13, 14 | syl2anr 290 | . . . . 5 ⊢ ((𝑁 ∈ 𝑍 ∧ 𝑛 ∈ 𝑍) → DECID 𝑛 ≤ 𝑁) |
| 16 | exmiddc 844 | . . . . 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 806 | . . 3 ⊢ ((𝑁 ∈ 𝑍 ∧ 𝑛 ∈ 𝑍) → if(𝑛 ≤ 𝑁, ((𝑍 × {0})‘𝑛), 0) = 0) |
| 19 | 18 | mpteq2dva 4199 | . 2 ⊢ (𝑁 ∈ 𝑍 → (𝑛 ∈ 𝑍 ↦ if(𝑛 ≤ 𝑁, ((𝑍 × {0})‘𝑛), 0)) = (𝑛 ∈ 𝑍 ↦ 0)) |
| 20 | fconstmpt 4796 | . 2 ⊢ (𝑍 × {0}) = (𝑛 ∈ 𝑍 ↦ 0) | |
| 21 | 19, 20 | eqtr4di 2283 | 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 716 DECID wdc 842 = wceq 1398 ∈ wcel 2203 ifcif 3619 {csn 3688 class class class wbr 4108 ↦ cmpt 4170 × cxp 4746 ‘cfv 5351 0cc0 8123 ≤ cle 8305 ℤcz 9573 ℤ≥cuz 9849 |
| 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 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2205 ax-14 2206 ax-ext 2214 ax-sep 4227 ax-pow 4286 ax-pr 4321 ax-un 4553 ax-setind 4658 ax-cnex 8214 ax-resscn 8215 ax-1cn 8216 ax-1re 8217 ax-icn 8218 ax-addcl 8219 ax-addrcl 8220 ax-mulcl 8221 ax-addcom 8223 ax-addass 8225 ax-distr 8227 ax-i2m1 8228 ax-0lt1 8229 ax-0id 8231 ax-rnegex 8232 ax-cnre 8234 ax-pre-ltirr 8235 ax-pre-ltwlin 8236 ax-pre-lttrn 8237 ax-pre-ltadd 8239 |
| This theorem depends on definitions: df-bi 117 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2083 df-mo 2084 df-clab 2219 df-cleq 2225 df-clel 2228 df-nfc 2373 df-ne 2413 df-nel 2508 df-ral 2525 df-rex 2526 df-reu 2527 df-rab 2529 df-v 2814 df-sbc 3042 df-dif 3212 df-un 3214 df-in 3216 df-ss 3223 df-if 3620 df-pw 3670 df-sn 3694 df-pr 3695 df-op 3697 df-uni 3914 df-int 3949 df-br 4109 df-opab 4171 df-mpt 4172 df-id 4413 df-xp 4754 df-rel 4755 df-cnv 4756 df-co 4757 df-dm 4758 df-rn 4759 df-res 4760 df-ima 4761 df-iota 5311 df-fun 5353 df-fn 5354 df-f 5355 df-fv 5359 df-riota 6002 df-ov 6052 df-oprab 6053 df-mpo 6054 df-pnf 8306 df-mnf 8307 df-xr 8308 df-ltxr 8309 df-le 8310 df-sub 8442 df-neg 8443 df-inn 9234 df-n0 9493 df-z 9574 df-uz 9850 |
| This theorem is referenced by: isumz 12068 |
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