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Mirrors > Home > MPE Home > Th. List > Mathboxes > signlem0 | Structured version Visualization version GIF version |
Description: Adding a zero as the highest coefficient does not change the parity of the sign changes. (Contributed by Thierry Arnoux, 12-Oct-2018.) |
Ref | Expression |
---|---|
signsv.p | ⊢ ⨣ = (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏)) |
signsv.w | ⊢ 𝑊 = {〈(Base‘ndx), {-1, 0, 1}〉, 〈(+g‘ndx), ⨣ 〉} |
signsv.t | ⊢ 𝑇 = (𝑓 ∈ Word ℝ ↦ (𝑛 ∈ (0..^(#‘𝑓)) ↦ (𝑊 Σg (𝑖 ∈ (0...𝑛) ↦ (sgn‘(𝑓‘𝑖)))))) |
signsv.v | ⊢ 𝑉 = (𝑓 ∈ Word ℝ ↦ Σ𝑗 ∈ (1..^(#‘𝑓))if(((𝑇‘𝑓)‘𝑗) ≠ ((𝑇‘𝑓)‘(𝑗 − 1)), 1, 0)) |
Ref | Expression |
---|---|
signlem0 | ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) → (𝑉‘(𝐹 ++ 〈“0”〉)) = (𝑉‘𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0re 10040 | . . 3 ⊢ 0 ∈ ℝ | |
2 | signsv.p | . . . 4 ⊢ ⨣ = (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏)) | |
3 | signsv.w | . . . 4 ⊢ 𝑊 = {〈(Base‘ndx), {-1, 0, 1}〉, 〈(+g‘ndx), ⨣ 〉} | |
4 | signsv.t | . . . 4 ⊢ 𝑇 = (𝑓 ∈ Word ℝ ↦ (𝑛 ∈ (0..^(#‘𝑓)) ↦ (𝑊 Σg (𝑖 ∈ (0...𝑛) ↦ (sgn‘(𝑓‘𝑖)))))) | |
5 | signsv.v | . . . 4 ⊢ 𝑉 = (𝑓 ∈ Word ℝ ↦ Σ𝑗 ∈ (1..^(#‘𝑓))if(((𝑇‘𝑓)‘𝑗) ≠ ((𝑇‘𝑓)‘(𝑗 − 1)), 1, 0)) | |
6 | 2, 3, 4, 5 | signsvfn 30659 | . . 3 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ 0 ∈ ℝ) → (𝑉‘(𝐹 ++ 〈“0”〉)) = ((𝑉‘𝐹) + if((((𝑇‘𝐹)‘((#‘𝐹) − 1)) · 0) < 0, 1, 0))) |
7 | 1, 6 | mpan2 707 | . 2 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) → (𝑉‘(𝐹 ++ 〈“0”〉)) = ((𝑉‘𝐹) + if((((𝑇‘𝐹)‘((#‘𝐹) − 1)) · 0) < 0, 1, 0))) |
8 | 1 | ltnri 10146 | . . . . 5 ⊢ ¬ 0 < 0 |
9 | neg1cn 11124 | . . . . . . . . 9 ⊢ -1 ∈ ℂ | |
10 | ax-1cn 9994 | . . . . . . . . 9 ⊢ 1 ∈ ℂ | |
11 | prssi 4353 | . . . . . . . . 9 ⊢ ((-1 ∈ ℂ ∧ 1 ∈ ℂ) → {-1, 1} ⊆ ℂ) | |
12 | 9, 10, 11 | mp2an 708 | . . . . . . . 8 ⊢ {-1, 1} ⊆ ℂ |
13 | simpl 473 | . . . . . . . . . . 11 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) → 𝐹 ∈ (Word ℝ ∖ {∅})) | |
14 | eldifsn 4317 | . . . . . . . . . . 11 ⊢ (𝐹 ∈ (Word ℝ ∖ {∅}) ↔ (𝐹 ∈ Word ℝ ∧ 𝐹 ≠ ∅)) | |
15 | 13, 14 | sylib 208 | . . . . . . . . . 10 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) → (𝐹 ∈ Word ℝ ∧ 𝐹 ≠ ∅)) |
16 | lennncl 13325 | . . . . . . . . . 10 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐹 ≠ ∅) → (#‘𝐹) ∈ ℕ) | |
17 | fzo0end 12560 | . . . . . . . . . 10 ⊢ ((#‘𝐹) ∈ ℕ → ((#‘𝐹) − 1) ∈ (0..^(#‘𝐹))) | |
18 | 15, 16, 17 | 3syl 18 | . . . . . . . . 9 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) → ((#‘𝐹) − 1) ∈ (0..^(#‘𝐹))) |
19 | 2, 3, 4, 5 | signstfvcl 30650 | . . . . . . . . 9 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ ((#‘𝐹) − 1) ∈ (0..^(#‘𝐹))) → ((𝑇‘𝐹)‘((#‘𝐹) − 1)) ∈ {-1, 1}) |
20 | 18, 19 | mpdan 702 | . . . . . . . 8 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) → ((𝑇‘𝐹)‘((#‘𝐹) − 1)) ∈ {-1, 1}) |
21 | 12, 20 | sseldi 3601 | . . . . . . 7 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) → ((𝑇‘𝐹)‘((#‘𝐹) − 1)) ∈ ℂ) |
22 | 21 | mul01d 10235 | . . . . . 6 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) → (((𝑇‘𝐹)‘((#‘𝐹) − 1)) · 0) = 0) |
23 | 22 | breq1d 4663 | . . . . 5 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) → ((((𝑇‘𝐹)‘((#‘𝐹) − 1)) · 0) < 0 ↔ 0 < 0)) |
24 | 8, 23 | mtbiri 317 | . . . 4 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) → ¬ (((𝑇‘𝐹)‘((#‘𝐹) − 1)) · 0) < 0) |
25 | 24 | iffalsed 4097 | . . 3 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) → if((((𝑇‘𝐹)‘((#‘𝐹) − 1)) · 0) < 0, 1, 0) = 0) |
26 | 25 | oveq2d 6666 | . 2 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) → ((𝑉‘𝐹) + if((((𝑇‘𝐹)‘((#‘𝐹) − 1)) · 0) < 0, 1, 0)) = ((𝑉‘𝐹) + 0)) |
27 | 2, 3, 4, 5 | signsvvf 30656 | . . . . . 6 ⊢ 𝑉:Word ℝ⟶ℕ0 |
28 | 27 | a1i 11 | . . . . 5 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) → 𝑉:Word ℝ⟶ℕ0) |
29 | 13 | eldifad 3586 | . . . . 5 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) → 𝐹 ∈ Word ℝ) |
30 | 28, 29 | ffvelrnd 6360 | . . . 4 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) → (𝑉‘𝐹) ∈ ℕ0) |
31 | 30 | nn0cnd 11353 | . . 3 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) → (𝑉‘𝐹) ∈ ℂ) |
32 | 31 | addid1d 10236 | . 2 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) → ((𝑉‘𝐹) + 0) = (𝑉‘𝐹)) |
33 | 7, 26, 32 | 3eqtrd 2660 | 1 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) → (𝑉‘(𝐹 ++ 〈“0”〉)) = (𝑉‘𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∖ cdif 3571 ⊆ wss 3574 ∅c0 3915 ifcif 4086 {csn 4177 {cpr 4179 {ctp 4181 〈cop 4183 class class class wbr 4653 ↦ cmpt 4729 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ↦ cmpt2 6652 ℂcc 9934 ℝcr 9935 0cc0 9936 1c1 9937 + caddc 9939 · cmul 9941 < clt 10074 − cmin 10266 -cneg 10267 ℕcn 11020 ℕ0cn0 11292 ...cfz 12326 ..^cfzo 12465 #chash 13117 Word cword 13291 ++ cconcat 13293 〈“cs1 13294 sgncsgn 13826 Σcsu 14416 ndxcnx 15854 Basecbs 15857 +gcplusg 15941 Σg cgsu 16101 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-oi 8415 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-xnn0 11364 df-z 11378 df-uz 11688 df-rp 11833 df-fz 12327 df-fzo 12466 df-seq 12802 df-exp 12861 df-hash 13118 df-word 13299 df-lsw 13300 df-concat 13301 df-s1 13302 df-substr 13303 df-sgn 13827 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 df-sum 14417 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-plusg 15954 df-0g 16102 df-gsum 16103 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-mulg 17541 df-cntz 17750 |
This theorem is referenced by: (None) |
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