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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 2726 | . . 3 ⊢ (1o eval 𝑅) = (1o eval 𝑅) | |
3 | evl1rhm.b | . . 3 ⊢ 𝐵 = (Base‘𝑅) | |
4 | 1, 2, 3 | evl1fval 22197 | . 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 2726 | . . . 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 22190 | . . 3 ⊢ (𝑅 ∈ CRing → (𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∈ ((𝑅 ↑s (𝐵 ↑m 1o)) RingHom 𝑇)) |
8 | 1on 8476 | . . . . 5 ⊢ 1o ∈ On | |
9 | eqid 2726 | . . . . . 6 ⊢ (1o mPoly 𝑅) = (1o mPoly 𝑅) | |
10 | eqid 2726 | . . . . . 6 ⊢ (𝑅 ↑s (𝐵 ↑m 1o)) = (𝑅 ↑s (𝐵 ↑m 1o)) | |
11 | 2, 3, 9, 10 | evlrhm 21996 | . . . . 5 ⊢ ((1o ∈ On ∧ 𝑅 ∈ CRing) → (1o eval 𝑅) ∈ ((1o mPoly 𝑅) RingHom (𝑅 ↑s (𝐵 ↑m 1o)))) |
12 | 8, 11 | mpan 687 | . . . 4 ⊢ (𝑅 ∈ CRing → (1o eval 𝑅) ∈ ((1o mPoly 𝑅) RingHom (𝑅 ↑s (𝐵 ↑m 1o)))) |
13 | eqidd 2727 | . . . . 5 ⊢ (𝑅 ∈ CRing → (Base‘𝑃) = (Base‘𝑃)) | |
14 | eqidd 2727 | . . . . 5 ⊢ (𝑅 ∈ CRing → (Base‘(𝑅 ↑s (𝐵 ↑m 1o))) = (Base‘(𝑅 ↑s (𝐵 ↑m 1o)))) | |
15 | evl1rhm.w | . . . . . . 7 ⊢ 𝑃 = (Poly1‘𝑅) | |
16 | eqid 2726 | . . . . . . 7 ⊢ (PwSer1‘𝑅) = (PwSer1‘𝑅) | |
17 | eqid 2726 | . . . . . . 7 ⊢ (Base‘𝑃) = (Base‘𝑃) | |
18 | 15, 16, 17 | ply1bas 22064 | . . . . . 6 ⊢ (Base‘𝑃) = (Base‘(1o mPoly 𝑅)) |
19 | 18 | a1i 11 | . . . . 5 ⊢ (𝑅 ∈ CRing → (Base‘𝑃) = (Base‘(1o mPoly 𝑅))) |
20 | eqid 2726 | . . . . . . . 8 ⊢ (+g‘𝑃) = (+g‘𝑃) | |
21 | 15, 9, 20 | ply1plusg 22092 | . . . . . . 7 ⊢ (+g‘𝑃) = (+g‘(1o mPoly 𝑅)) |
22 | 21 | a1i 11 | . . . . . 6 ⊢ (𝑅 ∈ CRing → (+g‘𝑃) = (+g‘(1o mPoly 𝑅))) |
23 | 22 | oveqdr 7432 | . . . . 5 ⊢ ((𝑅 ∈ CRing ∧ (𝑥 ∈ (Base‘𝑃) ∧ 𝑦 ∈ (Base‘𝑃))) → (𝑥(+g‘𝑃)𝑦) = (𝑥(+g‘(1o mPoly 𝑅))𝑦)) |
24 | eqidd 2727 | . . . . 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 2726 | . . . . . . . 8 ⊢ (.r‘𝑃) = (.r‘𝑃) | |
26 | 15, 9, 25 | ply1mulr 22094 | . . . . . . 7 ⊢ (.r‘𝑃) = (.r‘(1o mPoly 𝑅)) |
27 | 26 | a1i 11 | . . . . . 6 ⊢ (𝑅 ∈ CRing → (.r‘𝑃) = (.r‘(1o mPoly 𝑅))) |
28 | 27 | oveqdr 7432 | . . . . 5 ⊢ ((𝑅 ∈ CRing ∧ (𝑥 ∈ (Base‘𝑃) ∧ 𝑦 ∈ (Base‘𝑃))) → (𝑥(.r‘𝑃)𝑦) = (𝑥(.r‘(1o mPoly 𝑅))𝑦)) |
29 | eqidd 2727 | . . . . 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 20508 | . . . 4 ⊢ (𝑅 ∈ CRing → (𝑃 RingHom (𝑅 ↑s (𝐵 ↑m 1o))) = ((1o mPoly 𝑅) RingHom (𝑅 ↑s (𝐵 ↑m 1o)))) |
31 | 12, 30 | eleqtrrd 2830 | . . 3 ⊢ (𝑅 ∈ CRing → (1o eval 𝑅) ∈ (𝑃 RingHom (𝑅 ↑s (𝐵 ↑m 1o)))) |
32 | rhmco 20400 | . . 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 583 | . 2 ⊢ (𝑅 ∈ CRing → ((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘ (1o eval 𝑅)) ∈ (𝑃 RingHom 𝑇)) |
34 | 4, 33 | eqeltrid 2831 | 1 ⊢ (𝑅 ∈ CRing → 𝑂 ∈ (𝑃 RingHom 𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 {csn 4623 ↦ cmpt 5224 × cxp 5667 ∘ ccom 5673 Oncon0 6357 ‘cfv 6536 (class class class)co 7404 1oc1o 8457 ↑m cmap 8819 Basecbs 17150 +gcplusg 17203 .rcmulr 17204 ↑s cpws 17398 CRingccrg 20136 RingHom crh 20368 mPoly cmpl 21795 eval cevl 21971 PwSer1cps1 22044 Poly1cpl1 22046 eval1ce1 22183 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-of 7666 df-ofr 7667 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8144 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-er 8702 df-map 8821 df-pm 8822 df-ixp 8891 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-fsupp 9361 df-sup 9436 df-oi 9504 df-card 9933 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-nn 12214 df-2 12276 df-3 12277 df-4 12278 df-5 12279 df-6 12280 df-7 12281 df-8 12282 df-9 12283 df-n0 12474 df-z 12560 df-dec 12679 df-uz 12824 df-fz 13488 df-fzo 13631 df-seq 13970 df-hash 14293 df-struct 17086 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17151 df-ress 17180 df-plusg 17216 df-mulr 17217 df-sca 17219 df-vsca 17220 df-ip 17221 df-tset 17222 df-ple 17223 df-ds 17225 df-hom 17227 df-cco 17228 df-0g 17393 df-gsum 17394 df-prds 17399 df-pws 17401 df-mre 17536 df-mrc 17537 df-acs 17539 df-mgm 18570 df-sgrp 18649 df-mnd 18665 df-mhm 18710 df-submnd 18711 df-grp 18863 df-minusg 18864 df-sbg 18865 df-mulg 18993 df-subg 19047 df-ghm 19136 df-cntz 19230 df-cmn 19699 df-abl 19700 df-mgp 20037 df-rng 20055 df-ur 20084 df-srg 20089 df-ring 20137 df-cring 20138 df-rhm 20371 df-subrng 20443 df-subrg 20468 df-lmod 20705 df-lss 20776 df-lsp 20816 df-assa 21743 df-asp 21744 df-ascl 21745 df-psr 21798 df-mvr 21799 df-mpl 21800 df-opsr 21802 df-evls 21972 df-evl 21973 df-psr1 22049 df-ply1 22051 df-evl1 22185 |
This theorem is referenced by: fveval1fvcl 22202 evl1addd 22210 evl1subd 22211 evl1muld 22212 evl1expd 22214 pf1const 22215 pf1id 22216 pf1subrg 22217 mpfpf1 22220 pf1mpf 22221 evl1gsummul 22229 evl1scvarpw 22232 ply1remlem 26049 ply1rem 26050 fta1glem1 26052 fta1glem2 26053 fta1g 26054 fta1blem 26055 idomrootle 26057 plypf1 26096 lgsqrlem2 27230 lgsqrlem3 27231 ressply1evl 33155 irngss 33269 pl1cn 33464 |
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