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 20619 | . . 3 ⊢ ℤring ∈ CRing | |
2 | linevalexample.p | . . . 4 ⊢ 𝑃 = (Poly1‘ℤring) | |
3 | linevalexample.b | . . . 4 ⊢ 𝐵 = (Base‘𝑃) | |
4 | zringbas 20623 | . . . 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 2821 | . . . 4 ⊢ (𝑋 − (𝐴‘3)) = (𝑋 − (𝐴‘3)) | |
9 | 3z 12016 | . . . . 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 11918 | . . . . . 6 ⊢ 5 ∈ ℕ0 | |
14 | 13 | nn0zi 12008 | . . . . 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 44497 | . . 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 2821 | . . . 4 ⊢ (-g‘ℤring) = (-g‘ℤring) | |
19 | 18 | zringsubgval 44498 | . . 3 ⊢ ((5 ∈ ℤ ∧ 3 ∈ ℤ) → (5 − 3) = (5(-g‘ℤring)3)) |
20 | 14, 9, 19 | mp2an 690 | . 2 ⊢ (5 − 3) = (5(-g‘ℤring)3) |
21 | 5cn 11726 | . . 3 ⊢ 5 ∈ ℂ | |
22 | 3cn 11719 | . . 3 ⊢ 3 ∈ ℂ | |
23 | 2cn 11713 | . . 3 ⊢ 2 ∈ ℂ | |
24 | 3p2e5 11789 | . . 3 ⊢ (3 + 2) = 5 | |
25 | 21, 22, 23, 24 | subaddrii 10975 | . 2 ⊢ (5 − 3) = 2 |
26 | 17, 20, 25 | 3eqtr2i 2850 | 1 ⊢ ((𝑂‘(𝑋 − (𝐴‘3)))‘5) = 2 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 ‘cfv 6355 (class class class)co 7156 − cmin 10870 2c2 11693 3c3 11694 5c5 11696 ℤcz 11982 Basecbs 16483 -gcsg 18105 CRingccrg 19298 algSccascl 20084 var1cv1 20344 Poly1cpl1 20345 eval1ce1 20477 ℤringzring 20617 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-addf 10616 ax-mulf 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-ofr 7410 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-er 8289 df-map 8408 df-pm 8409 df-ixp 8462 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-sup 8906 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-fz 12894 df-fzo 13035 df-seq 13371 df-hash 13692 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-starv 16580 df-sca 16581 df-vsca 16582 df-ip 16583 df-tset 16584 df-ple 16585 df-ds 16587 df-unif 16588 df-hom 16589 df-cco 16590 df-0g 16715 df-gsum 16716 df-prds 16721 df-pws 16723 df-mre 16857 df-mrc 16858 df-acs 16860 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-mhm 17956 df-submnd 17957 df-grp 18106 df-minusg 18107 df-sbg 18108 df-mulg 18225 df-subg 18276 df-ghm 18356 df-cntz 18447 df-cmn 18908 df-abl 18909 df-mgp 19240 df-ur 19252 df-srg 19256 df-ring 19299 df-cring 19300 df-rnghom 19467 df-subrg 19533 df-lmod 19636 df-lss 19704 df-lsp 19744 df-assa 20085 df-asp 20086 df-ascl 20087 df-psr 20136 df-mvr 20137 df-mpl 20138 df-opsr 20140 df-evls 20286 df-evl 20287 df-psr1 20348 df-vr1 20349 df-ply1 20350 df-evl1 20479 df-cnfld 20546 df-zring 20618 |
This theorem is referenced by: (None) |
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