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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 2798 | . . 3 ⊢ (1o eval 𝑅) = (1o eval 𝑅) | |
3 | evl1rhm.b | . . 3 ⊢ 𝐵 = (Base‘𝑅) | |
4 | 1, 2, 3 | evl1fval 20952 | . 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 2798 | . . . 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 20945 | . . 3 ⊢ (𝑅 ∈ CRing → (𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∈ ((𝑅 ↑s (𝐵 ↑m 1o)) RingHom 𝑇)) |
8 | 1on 8092 | . . . . 5 ⊢ 1o ∈ On | |
9 | eqid 2798 | . . . . . 6 ⊢ (1o mPoly 𝑅) = (1o mPoly 𝑅) | |
10 | eqid 2798 | . . . . . 6 ⊢ (𝑅 ↑s (𝐵 ↑m 1o)) = (𝑅 ↑s (𝐵 ↑m 1o)) | |
11 | 2, 3, 9, 10 | evlrhm 20768 | . . . . 5 ⊢ ((1o ∈ On ∧ 𝑅 ∈ CRing) → (1o eval 𝑅) ∈ ((1o mPoly 𝑅) RingHom (𝑅 ↑s (𝐵 ↑m 1o)))) |
12 | 8, 11 | mpan 689 | . . . 4 ⊢ (𝑅 ∈ CRing → (1o eval 𝑅) ∈ ((1o mPoly 𝑅) RingHom (𝑅 ↑s (𝐵 ↑m 1o)))) |
13 | eqidd 2799 | . . . . 5 ⊢ (𝑅 ∈ CRing → (Base‘𝑃) = (Base‘𝑃)) | |
14 | eqidd 2799 | . . . . 5 ⊢ (𝑅 ∈ CRing → (Base‘(𝑅 ↑s (𝐵 ↑m 1o))) = (Base‘(𝑅 ↑s (𝐵 ↑m 1o)))) | |
15 | evl1rhm.w | . . . . . . 7 ⊢ 𝑃 = (Poly1‘𝑅) | |
16 | eqid 2798 | . . . . . . 7 ⊢ (PwSer1‘𝑅) = (PwSer1‘𝑅) | |
17 | eqid 2798 | . . . . . . 7 ⊢ (Base‘𝑃) = (Base‘𝑃) | |
18 | 15, 16, 17 | ply1bas 20824 | . . . . . 6 ⊢ (Base‘𝑃) = (Base‘(1o mPoly 𝑅)) |
19 | 18 | a1i 11 | . . . . 5 ⊢ (𝑅 ∈ CRing → (Base‘𝑃) = (Base‘(1o mPoly 𝑅))) |
20 | eqid 2798 | . . . . . . . 8 ⊢ (+g‘𝑃) = (+g‘𝑃) | |
21 | 15, 9, 20 | ply1plusg 20854 | . . . . . . 7 ⊢ (+g‘𝑃) = (+g‘(1o mPoly 𝑅)) |
22 | 21 | a1i 11 | . . . . . 6 ⊢ (𝑅 ∈ CRing → (+g‘𝑃) = (+g‘(1o mPoly 𝑅))) |
23 | 22 | oveqdr 7163 | . . . . 5 ⊢ ((𝑅 ∈ CRing ∧ (𝑥 ∈ (Base‘𝑃) ∧ 𝑦 ∈ (Base‘𝑃))) → (𝑥(+g‘𝑃)𝑦) = (𝑥(+g‘(1o mPoly 𝑅))𝑦)) |
24 | eqidd 2799 | . . . . 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 2798 | . . . . . . . 8 ⊢ (.r‘𝑃) = (.r‘𝑃) | |
26 | 15, 9, 25 | ply1mulr 20856 | . . . . . . 7 ⊢ (.r‘𝑃) = (.r‘(1o mPoly 𝑅)) |
27 | 26 | a1i 11 | . . . . . 6 ⊢ (𝑅 ∈ CRing → (.r‘𝑃) = (.r‘(1o mPoly 𝑅))) |
28 | 27 | oveqdr 7163 | . . . . 5 ⊢ ((𝑅 ∈ CRing ∧ (𝑥 ∈ (Base‘𝑃) ∧ 𝑦 ∈ (Base‘𝑃))) → (𝑥(.r‘𝑃)𝑦) = (𝑥(.r‘(1o mPoly 𝑅))𝑦)) |
29 | eqidd 2799 | . . . . 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 19564 | . . . 4 ⊢ (𝑅 ∈ CRing → (𝑃 RingHom (𝑅 ↑s (𝐵 ↑m 1o))) = ((1o mPoly 𝑅) RingHom (𝑅 ↑s (𝐵 ↑m 1o)))) |
31 | 12, 30 | eleqtrrd 2893 | . . 3 ⊢ (𝑅 ∈ CRing → (1o eval 𝑅) ∈ (𝑃 RingHom (𝑅 ↑s (𝐵 ↑m 1o)))) |
32 | rhmco 19485 | . . 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 2894 | 1 ⊢ (𝑅 ∈ CRing → 𝑂 ∈ (𝑃 RingHom 𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 {csn 4525 ↦ cmpt 5110 × cxp 5517 ∘ ccom 5523 Oncon0 6159 ‘cfv 6324 (class class class)co 7135 1oc1o 8078 ↑m cmap 8389 Basecbs 16475 +gcplusg 16557 .rcmulr 16558 ↑s cpws 16712 CRingccrg 19291 RingHom crh 19460 mPoly cmpl 20591 eval cevl 20744 PwSer1cps1 20804 Poly1cpl1 20806 eval1ce1 20938 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-ofr 7390 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-map 8391 df-pm 8392 df-ixp 8445 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-sup 8890 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-fz 12886 df-fzo 13029 df-seq 13365 df-hash 13687 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-hom 16581 df-cco 16582 df-0g 16707 df-gsum 16708 df-prds 16713 df-pws 16715 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-mhm 17948 df-submnd 17949 df-grp 18098 df-minusg 18099 df-sbg 18100 df-mulg 18217 df-subg 18268 df-ghm 18348 df-cntz 18439 df-cmn 18900 df-abl 18901 df-mgp 19233 df-ur 19245 df-srg 19249 df-ring 19292 df-cring 19293 df-rnghom 19463 df-subrg 19526 df-lmod 19629 df-lss 19697 df-lsp 19737 df-assa 20542 df-asp 20543 df-ascl 20544 df-psr 20594 df-mvr 20595 df-mpl 20596 df-opsr 20598 df-evls 20745 df-evl 20746 df-psr1 20809 df-ply1 20811 df-evl1 20940 |
This theorem is referenced by: fveval1fvcl 20957 evl1addd 20965 evl1subd 20966 evl1muld 20967 evl1expd 20969 pf1const 20970 pf1id 20971 pf1subrg 20972 mpfpf1 20975 pf1mpf 20976 evl1gsummul 20984 evl1scvarpw 20987 ply1remlem 24763 ply1rem 24764 fta1glem1 24766 fta1glem2 24767 fta1g 24768 fta1blem 24769 plypf1 24809 lgsqrlem2 25931 lgsqrlem3 25932 pl1cn 31308 idomrootle 40139 |
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