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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > linevalexample | Structured version Visualization version GIF version |
Description: The polynomial 𝑥 − 3 over ℤ evaluated for 𝑥 = 5 results in 2. (Contributed by AV, 3-Jul-2019.) |
Ref | Expression |
---|---|
linevalexample.p | ⊢ 𝑃 = (Poly1‘ℤring) |
linevalexample.b | ⊢ 𝐵 = (Base‘𝑃) |
linevalexample.x | ⊢ 𝑋 = (var1‘ℤring) |
linevalexample.m | ⊢ − = (-g‘𝑃) |
linevalexample.a | ⊢ 𝐴 = (algSc‘𝑃) |
linevalexample.g | ⊢ 𝐺 = (𝑋 − (𝐴‘3)) |
linevalexample.o | ⊢ 𝑂 = (eval1‘ℤring) |
Ref | Expression |
---|---|
linevalexample | ⊢ ((𝑂‘(𝑋 − (𝐴‘3)))‘5) = 2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zringcrng 20018 | . . 3 ⊢ ℤring ∈ CRing | |
2 | linevalexample.p | . . . 4 ⊢ 𝑃 = (Poly1‘ℤring) | |
3 | linevalexample.b | . . . 4 ⊢ 𝐵 = (Base‘𝑃) | |
4 | zringbas 20022 | . . . 4 ⊢ ℤ = (Base‘ℤring) | |
5 | linevalexample.x | . . . 4 ⊢ 𝑋 = (var1‘ℤring) | |
6 | linevalexample.m | . . . 4 ⊢ − = (-g‘𝑃) | |
7 | linevalexample.a | . . . 4 ⊢ 𝐴 = (algSc‘𝑃) | |
8 | eqid 2756 | . . . 4 ⊢ (𝑋 − (𝐴‘3)) = (𝑋 − (𝐴‘3)) | |
9 | 3z 11598 | . . . . 5 ⊢ 3 ∈ ℤ | |
10 | 9 | a1i 11 | . . . 4 ⊢ (ℤring ∈ CRing → 3 ∈ ℤ) |
11 | linevalexample.o | . . . 4 ⊢ 𝑂 = (eval1‘ℤring) | |
12 | id 22 | . . . 4 ⊢ (ℤring ∈ CRing → ℤring ∈ CRing) | |
13 | 5nn0 11500 | . . . . . 6 ⊢ 5 ∈ ℕ0 | |
14 | 13 | nn0zi 11590 | . . . . 5 ⊢ 5 ∈ ℤ |
15 | 14 | a1i 11 | . . . 4 ⊢ (ℤring ∈ CRing → 5 ∈ ℤ) |
16 | 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 15 | lineval 42688 | . . 3 ⊢ (ℤring ∈ CRing → ((𝑂‘(𝑋 − (𝐴‘3)))‘5) = (5(-g‘ℤring)3)) |
17 | 1, 16 | ax-mp 5 | . 2 ⊢ ((𝑂‘(𝑋 − (𝐴‘3)))‘5) = (5(-g‘ℤring)3) |
18 | eqid 2756 | . . . 4 ⊢ (-g‘ℤring) = (-g‘ℤring) | |
19 | 18 | zringsubgval 42689 | . . 3 ⊢ ((5 ∈ ℤ ∧ 3 ∈ ℤ) → (5 − 3) = (5(-g‘ℤring)3)) |
20 | 14, 9, 19 | mp2an 710 | . 2 ⊢ (5 − 3) = (5(-g‘ℤring)3) |
21 | 5cn 11288 | . . 3 ⊢ 5 ∈ ℂ | |
22 | 3cn 11283 | . . 3 ⊢ 3 ∈ ℂ | |
23 | 2cn 11279 | . . 3 ⊢ 2 ∈ ℂ | |
24 | 3p2e5 11348 | . . 3 ⊢ (3 + 2) = 5 | |
25 | 21, 22, 23, 24 | subaddrii 10558 | . 2 ⊢ (5 − 3) = 2 |
26 | 17, 20, 25 | 3eqtr2i 2784 | 1 ⊢ ((𝑂‘(𝑋 − (𝐴‘3)))‘5) = 2 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1628 ∈ wcel 2135 ‘cfv 6045 (class class class)co 6809 − cmin 10454 2c2 11258 3c3 11259 5c5 11261 ℤcz 11565 Basecbs 16055 -gcsg 17621 CRingccrg 18744 algSccascl 19509 var1cv1 19744 Poly1cpl1 19745 eval1ce1 19877 ℤringzring 20016 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1867 ax-4 1882 ax-5 1984 ax-6 2050 ax-7 2086 ax-8 2137 ax-9 2144 ax-10 2164 ax-11 2179 ax-12 2192 ax-13 2387 ax-ext 2736 ax-rep 4919 ax-sep 4929 ax-nul 4937 ax-pow 4988 ax-pr 5051 ax-un 7110 ax-inf2 8707 ax-cnex 10180 ax-resscn 10181 ax-1cn 10182 ax-icn 10183 ax-addcl 10184 ax-addrcl 10185 ax-mulcl 10186 ax-mulrcl 10187 ax-mulcom 10188 ax-addass 10189 ax-mulass 10190 ax-distr 10191 ax-i2m1 10192 ax-1ne0 10193 ax-1rid 10194 ax-rnegex 10195 ax-rrecex 10196 ax-cnre 10197 ax-pre-lttri 10198 ax-pre-lttrn 10199 ax-pre-ltadd 10200 ax-pre-mulgt0 10201 ax-addf 10203 ax-mulf 10204 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1631 df-ex 1850 df-nf 1855 df-sb 2043 df-eu 2607 df-mo 2608 df-clab 2743 df-cleq 2749 df-clel 2752 df-nfc 2887 df-ne 2929 df-nel 3032 df-ral 3051 df-rex 3052 df-reu 3053 df-rmo 3054 df-rab 3055 df-v 3338 df-sbc 3573 df-csb 3671 df-dif 3714 df-un 3716 df-in 3718 df-ss 3725 df-pss 3727 df-nul 4055 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-tp 4322 df-op 4324 df-uni 4585 df-int 4624 df-iun 4670 df-iin 4671 df-br 4801 df-opab 4861 df-mpt 4878 df-tr 4901 df-id 5170 df-eprel 5175 df-po 5183 df-so 5184 df-fr 5221 df-se 5222 df-we 5223 df-xp 5268 df-rel 5269 df-cnv 5270 df-co 5271 df-dm 5272 df-rn 5273 df-res 5274 df-ima 5275 df-pred 5837 df-ord 5883 df-on 5884 df-lim 5885 df-suc 5886 df-iota 6008 df-fun 6047 df-fn 6048 df-f 6049 df-f1 6050 df-fo 6051 df-f1o 6052 df-fv 6053 df-isom 6054 df-riota 6770 df-ov 6812 df-oprab 6813 df-mpt2 6814 df-of 7058 df-ofr 7059 df-om 7227 df-1st 7329 df-2nd 7330 df-supp 7460 df-wrecs 7572 df-recs 7633 df-rdg 7671 df-1o 7725 df-2o 7726 df-oadd 7729 df-er 7907 df-map 8021 df-pm 8022 df-ixp 8071 df-en 8118 df-dom 8119 df-sdom 8120 df-fin 8121 df-fsupp 8437 df-sup 8509 df-oi 8576 df-card 8951 df-pnf 10264 df-mnf 10265 df-xr 10266 df-ltxr 10267 df-le 10268 df-sub 10456 df-neg 10457 df-nn 11209 df-2 11267 df-3 11268 df-4 11269 df-5 11270 df-6 11271 df-7 11272 df-8 11273 df-9 11274 df-n0 11481 df-z 11566 df-dec 11682 df-uz 11876 df-fz 12516 df-fzo 12656 df-seq 12992 df-hash 13308 df-struct 16057 df-ndx 16058 df-slot 16059 df-base 16061 df-sets 16062 df-ress 16063 df-plusg 16152 df-mulr 16153 df-starv 16154 df-sca 16155 df-vsca 16156 df-ip 16157 df-tset 16158 df-ple 16159 df-ds 16162 df-unif 16163 df-hom 16164 df-cco 16165 df-0g 16300 df-gsum 16301 df-prds 16306 df-pws 16308 df-mre 16444 df-mrc 16445 df-acs 16447 df-mgm 17439 df-sgrp 17481 df-mnd 17492 df-mhm 17532 df-submnd 17533 df-grp 17622 df-minusg 17623 df-sbg 17624 df-mulg 17738 df-subg 17788 df-ghm 17855 df-cntz 17946 df-cmn 18391 df-abl 18392 df-mgp 18686 df-ur 18698 df-srg 18702 df-ring 18745 df-cring 18746 df-rnghom 18913 df-subrg 18976 df-lmod 19063 df-lss 19131 df-lsp 19170 df-assa 19510 df-asp 19511 df-ascl 19512 df-psr 19554 df-mvr 19555 df-mpl 19556 df-opsr 19558 df-evls 19704 df-evl 19705 df-psr1 19748 df-vr1 19749 df-ply1 19750 df-evl1 19879 df-cnfld 19945 df-zring 20017 |
This theorem is referenced by: (None) |
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