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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 21484 | . . 3 ⊢ ℤring ∈ CRing | |
2 | linevalexample.p | . . . 4 ⊢ 𝑃 = (Poly1‘ℤring) | |
3 | linevalexample.b | . . . 4 ⊢ 𝐵 = (Base‘𝑃) | |
4 | zringbas 21489 | . . . 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 2740 | . . . 4 ⊢ (𝑋 − (𝐴‘3)) = (𝑋 − (𝐴‘3)) | |
9 | 3z 12678 | . . . . 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 12575 | . . . . . 6 ⊢ 5 ∈ ℕ0 | |
14 | 13 | nn0zi 12670 | . . . . 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 48125 | . . 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 2740 | . . . 4 ⊢ (-g‘ℤring) = (-g‘ℤring) | |
19 | 18 | zringsubgval 21506 | . . 3 ⊢ ((5 ∈ ℤ ∧ 3 ∈ ℤ) → (5 − 3) = (5(-g‘ℤring)3)) |
20 | 14, 9, 19 | mp2an 691 | . 2 ⊢ (5 − 3) = (5(-g‘ℤring)3) |
21 | 5cn 12383 | . . 3 ⊢ 5 ∈ ℂ | |
22 | 3cn 12376 | . . 3 ⊢ 3 ∈ ℂ | |
23 | 2cn 12370 | . . 3 ⊢ 2 ∈ ℂ | |
24 | 3p2e5 12446 | . . 3 ⊢ (3 + 2) = 5 | |
25 | 21, 22, 23, 24 | subaddrii 11627 | . 2 ⊢ (5 − 3) = 2 |
26 | 17, 20, 25 | 3eqtr2i 2774 | 1 ⊢ ((𝑂‘(𝑋 − (𝐴‘3)))‘5) = 2 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2108 ‘cfv 6575 (class class class)co 7450 − cmin 11522 2c2 12350 3c3 12351 5c5 12353 ℤcz 12641 Basecbs 17260 -gcsg 18977 CRingccrg 20263 ℤringczring 21482 algSccascl 21897 var1cv1 22200 Poly1cpl1 22201 eval1ce1 22341 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7772 ax-cnex 11242 ax-resscn 11243 ax-1cn 11244 ax-icn 11245 ax-addcl 11246 ax-addrcl 11247 ax-mulcl 11248 ax-mulrcl 11249 ax-mulcom 11250 ax-addass 11251 ax-mulass 11252 ax-distr 11253 ax-i2m1 11254 ax-1ne0 11255 ax-1rid 11256 ax-rnegex 11257 ax-rrecex 11258 ax-cnre 11259 ax-pre-lttri 11260 ax-pre-lttrn 11261 ax-pre-ltadd 11262 ax-pre-mulgt0 11263 ax-addf 11265 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6334 df-ord 6400 df-on 6401 df-lim 6402 df-suc 6403 df-iota 6527 df-fun 6577 df-fn 6578 df-f 6579 df-f1 6580 df-fo 6581 df-f1o 6582 df-fv 6583 df-isom 6584 df-riota 7406 df-ov 7453 df-oprab 7454 df-mpo 7455 df-of 7716 df-ofr 7717 df-om 7906 df-1st 8032 df-2nd 8033 df-supp 8204 df-frecs 8324 df-wrecs 8355 df-recs 8429 df-rdg 8468 df-1o 8524 df-2o 8525 df-er 8765 df-map 8888 df-pm 8889 df-ixp 8958 df-en 9006 df-dom 9007 df-sdom 9008 df-fin 9009 df-fsupp 9434 df-sup 9513 df-oi 9581 df-card 10010 df-pnf 11328 df-mnf 11329 df-xr 11330 df-ltxr 11331 df-le 11332 df-sub 11524 df-neg 11525 df-nn 12296 df-2 12358 df-3 12359 df-4 12360 df-5 12361 df-6 12362 df-7 12363 df-8 12364 df-9 12365 df-n0 12556 df-z 12642 df-dec 12761 df-uz 12906 df-fz 13570 df-fzo 13714 df-seq 14055 df-hash 14382 df-struct 17196 df-sets 17213 df-slot 17231 df-ndx 17243 df-base 17261 df-ress 17290 df-plusg 17326 df-mulr 17327 df-starv 17328 df-sca 17329 df-vsca 17330 df-ip 17331 df-tset 17332 df-ple 17333 df-ds 17335 df-unif 17336 df-hom 17337 df-cco 17338 df-0g 17503 df-gsum 17504 df-prds 17509 df-pws 17511 df-mre 17646 df-mrc 17647 df-acs 17649 df-mgm 18680 df-sgrp 18759 df-mnd 18775 df-mhm 18820 df-submnd 18821 df-grp 18978 df-minusg 18979 df-sbg 18980 df-mulg 19110 df-subg 19165 df-ghm 19255 df-cntz 19359 df-cmn 19826 df-abl 19827 df-mgp 20164 df-rng 20182 df-ur 20211 df-srg 20216 df-ring 20264 df-cring 20265 df-rhm 20500 df-subrng 20574 df-subrg 20599 df-lmod 20884 df-lss 20955 df-lsp 20995 df-cnfld 21390 df-zring 21483 df-assa 21898 df-asp 21899 df-ascl 21900 df-psr 21954 df-mvr 21955 df-mpl 21956 df-opsr 21958 df-evls 22123 df-evl 22124 df-psr1 22204 df-vr1 22205 df-ply1 22206 df-evl1 22343 |
This theorem is referenced by: (None) |
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