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Mirrors > Home > MPE Home > Th. List > evl1rhm | Structured version Visualization version GIF version |
Description: Polynomial evaluation is a homomorphism (into the product ring). (Contributed by Mario Carneiro, 12-Jun-2015.) (Proof shortened by AV, 13-Sep-2019.) |
Ref | Expression |
---|---|
evl1rhm.q | ⊢ 𝑂 = (eval1‘𝑅) |
evl1rhm.w | ⊢ 𝑃 = (Poly1‘𝑅) |
evl1rhm.t | ⊢ 𝑇 = (𝑅 ↑s 𝐵) |
evl1rhm.b | ⊢ 𝐵 = (Base‘𝑅) |
Ref | Expression |
---|---|
evl1rhm | ⊢ (𝑅 ∈ CRing → 𝑂 ∈ (𝑃 RingHom 𝑇)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | evl1rhm.q | . . 3 ⊢ 𝑂 = (eval1‘𝑅) | |
2 | eqid 2738 | . . 3 ⊢ (1o eval 𝑅) = (1o eval 𝑅) | |
3 | evl1rhm.b | . . 3 ⊢ 𝐵 = (Base‘𝑅) | |
4 | 1, 2, 3 | evl1fval 21098 | . 2 ⊢ 𝑂 = ((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘ (1o eval 𝑅)) |
5 | evl1rhm.t | . . . 4 ⊢ 𝑇 = (𝑅 ↑s 𝐵) | |
6 | eqid 2738 | . . . 4 ⊢ (𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) = (𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) | |
7 | 3, 5, 6 | evls1rhmlem 21091 | . . 3 ⊢ (𝑅 ∈ CRing → (𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∈ ((𝑅 ↑s (𝐵 ↑m 1o)) RingHom 𝑇)) |
8 | 1on 8138 | . . . . 5 ⊢ 1o ∈ On | |
9 | eqid 2738 | . . . . . 6 ⊢ (1o mPoly 𝑅) = (1o mPoly 𝑅) | |
10 | eqid 2738 | . . . . . 6 ⊢ (𝑅 ↑s (𝐵 ↑m 1o)) = (𝑅 ↑s (𝐵 ↑m 1o)) | |
11 | 2, 3, 9, 10 | evlrhm 20910 | . . . . 5 ⊢ ((1o ∈ On ∧ 𝑅 ∈ CRing) → (1o eval 𝑅) ∈ ((1o mPoly 𝑅) RingHom (𝑅 ↑s (𝐵 ↑m 1o)))) |
12 | 8, 11 | mpan 690 | . . . 4 ⊢ (𝑅 ∈ CRing → (1o eval 𝑅) ∈ ((1o mPoly 𝑅) RingHom (𝑅 ↑s (𝐵 ↑m 1o)))) |
13 | eqidd 2739 | . . . . 5 ⊢ (𝑅 ∈ CRing → (Base‘𝑃) = (Base‘𝑃)) | |
14 | eqidd 2739 | . . . . 5 ⊢ (𝑅 ∈ CRing → (Base‘(𝑅 ↑s (𝐵 ↑m 1o))) = (Base‘(𝑅 ↑s (𝐵 ↑m 1o)))) | |
15 | evl1rhm.w | . . . . . . 7 ⊢ 𝑃 = (Poly1‘𝑅) | |
16 | eqid 2738 | . . . . . . 7 ⊢ (PwSer1‘𝑅) = (PwSer1‘𝑅) | |
17 | eqid 2738 | . . . . . . 7 ⊢ (Base‘𝑃) = (Base‘𝑃) | |
18 | 15, 16, 17 | ply1bas 20970 | . . . . . 6 ⊢ (Base‘𝑃) = (Base‘(1o mPoly 𝑅)) |
19 | 18 | a1i 11 | . . . . 5 ⊢ (𝑅 ∈ CRing → (Base‘𝑃) = (Base‘(1o mPoly 𝑅))) |
20 | eqid 2738 | . . . . . . . 8 ⊢ (+g‘𝑃) = (+g‘𝑃) | |
21 | 15, 9, 20 | ply1plusg 21000 | . . . . . . 7 ⊢ (+g‘𝑃) = (+g‘(1o mPoly 𝑅)) |
22 | 21 | a1i 11 | . . . . . 6 ⊢ (𝑅 ∈ CRing → (+g‘𝑃) = (+g‘(1o mPoly 𝑅))) |
23 | 22 | oveqdr 7198 | . . . . 5 ⊢ ((𝑅 ∈ CRing ∧ (𝑥 ∈ (Base‘𝑃) ∧ 𝑦 ∈ (Base‘𝑃))) → (𝑥(+g‘𝑃)𝑦) = (𝑥(+g‘(1o mPoly 𝑅))𝑦)) |
24 | eqidd 2739 | . . . . 5 ⊢ ((𝑅 ∈ CRing ∧ (𝑥 ∈ (Base‘(𝑅 ↑s (𝐵 ↑m 1o))) ∧ 𝑦 ∈ (Base‘(𝑅 ↑s (𝐵 ↑m 1o))))) → (𝑥(+g‘(𝑅 ↑s (𝐵 ↑m 1o)))𝑦) = (𝑥(+g‘(𝑅 ↑s (𝐵 ↑m 1o)))𝑦)) | |
25 | eqid 2738 | . . . . . . . 8 ⊢ (.r‘𝑃) = (.r‘𝑃) | |
26 | 15, 9, 25 | ply1mulr 21002 | . . . . . . 7 ⊢ (.r‘𝑃) = (.r‘(1o mPoly 𝑅)) |
27 | 26 | a1i 11 | . . . . . 6 ⊢ (𝑅 ∈ CRing → (.r‘𝑃) = (.r‘(1o mPoly 𝑅))) |
28 | 27 | oveqdr 7198 | . . . . 5 ⊢ ((𝑅 ∈ CRing ∧ (𝑥 ∈ (Base‘𝑃) ∧ 𝑦 ∈ (Base‘𝑃))) → (𝑥(.r‘𝑃)𝑦) = (𝑥(.r‘(1o mPoly 𝑅))𝑦)) |
29 | eqidd 2739 | . . . . 5 ⊢ ((𝑅 ∈ CRing ∧ (𝑥 ∈ (Base‘(𝑅 ↑s (𝐵 ↑m 1o))) ∧ 𝑦 ∈ (Base‘(𝑅 ↑s (𝐵 ↑m 1o))))) → (𝑥(.r‘(𝑅 ↑s (𝐵 ↑m 1o)))𝑦) = (𝑥(.r‘(𝑅 ↑s (𝐵 ↑m 1o)))𝑦)) | |
30 | 13, 14, 19, 14, 23, 24, 28, 29 | rhmpropd 19690 | . . . 4 ⊢ (𝑅 ∈ CRing → (𝑃 RingHom (𝑅 ↑s (𝐵 ↑m 1o))) = ((1o mPoly 𝑅) RingHom (𝑅 ↑s (𝐵 ↑m 1o)))) |
31 | 12, 30 | eleqtrrd 2836 | . . 3 ⊢ (𝑅 ∈ CRing → (1o eval 𝑅) ∈ (𝑃 RingHom (𝑅 ↑s (𝐵 ↑m 1o)))) |
32 | rhmco 19611 | . . 3 ⊢ (((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∈ ((𝑅 ↑s (𝐵 ↑m 1o)) RingHom 𝑇) ∧ (1o eval 𝑅) ∈ (𝑃 RingHom (𝑅 ↑s (𝐵 ↑m 1o)))) → ((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘ (1o eval 𝑅)) ∈ (𝑃 RingHom 𝑇)) | |
33 | 7, 31, 32 | syl2anc 587 | . 2 ⊢ (𝑅 ∈ CRing → ((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘ (1o eval 𝑅)) ∈ (𝑃 RingHom 𝑇)) |
34 | 4, 33 | eqeltrid 2837 | 1 ⊢ (𝑅 ∈ CRing → 𝑂 ∈ (𝑃 RingHom 𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2114 {csn 4516 ↦ cmpt 5110 × cxp 5523 ∘ ccom 5529 Oncon0 6172 ‘cfv 6339 (class class class)co 7170 1oc1o 8124 ↑m cmap 8437 Basecbs 16586 +gcplusg 16668 .rcmulr 16669 ↑s cpws 16823 CRingccrg 19417 RingHom crh 19586 mPoly cmpl 20719 eval cevl 20885 PwSer1cps1 20950 Poly1cpl1 20952 eval1ce1 21084 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-int 4837 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-se 5484 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-isom 6348 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-of 7425 df-ofr 7426 df-om 7600 df-1st 7714 df-2nd 7715 df-supp 7857 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-1o 8131 df-er 8320 df-map 8439 df-pm 8440 df-ixp 8508 df-en 8556 df-dom 8557 df-sdom 8558 df-fin 8559 df-fsupp 8907 df-sup 8979 df-oi 9047 df-card 9441 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-nn 11717 df-2 11779 df-3 11780 df-4 11781 df-5 11782 df-6 11783 df-7 11784 df-8 11785 df-9 11786 df-n0 11977 df-z 12063 df-dec 12180 df-uz 12325 df-fz 12982 df-fzo 13125 df-seq 13461 df-hash 13783 df-struct 16588 df-ndx 16589 df-slot 16590 df-base 16592 df-sets 16593 df-ress 16594 df-plusg 16681 df-mulr 16682 df-sca 16684 df-vsca 16685 df-ip 16686 df-tset 16687 df-ple 16688 df-ds 16690 df-hom 16692 df-cco 16693 df-0g 16818 df-gsum 16819 df-prds 16824 df-pws 16826 df-mre 16960 df-mrc 16961 df-acs 16963 df-mgm 17968 df-sgrp 18017 df-mnd 18028 df-mhm 18072 df-submnd 18073 df-grp 18222 df-minusg 18223 df-sbg 18224 df-mulg 18343 df-subg 18394 df-ghm 18474 df-cntz 18565 df-cmn 19026 df-abl 19027 df-mgp 19359 df-ur 19371 df-srg 19375 df-ring 19418 df-cring 19419 df-rnghom 19589 df-subrg 19652 df-lmod 19755 df-lss 19823 df-lsp 19863 df-assa 20669 df-asp 20670 df-ascl 20671 df-psr 20722 df-mvr 20723 df-mpl 20724 df-opsr 20726 df-evls 20886 df-evl 20887 df-psr1 20955 df-ply1 20957 df-evl1 21086 |
This theorem is referenced by: fveval1fvcl 21103 evl1addd 21111 evl1subd 21112 evl1muld 21113 evl1expd 21115 pf1const 21116 pf1id 21117 pf1subrg 21118 mpfpf1 21121 pf1mpf 21122 evl1gsummul 21130 evl1scvarpw 21133 ply1remlem 24915 ply1rem 24916 fta1glem1 24918 fta1glem2 24919 fta1g 24920 fta1blem 24921 plypf1 24961 lgsqrlem2 26083 lgsqrlem3 26084 pl1cn 31477 idomrootle 40592 |
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