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Mirrors > Home > MPE Home > Th. List > evls1rhm | Structured version Visualization version GIF version |
Description: Polynomial evaluation is a homomorphism (into the product ring). (Contributed by AV, 11-Sep-2019.) |
Ref | Expression |
---|---|
evls1rhm.q | ⊢ 𝑄 = (𝑆 evalSub1 𝑅) |
evls1rhm.b | ⊢ 𝐵 = (Base‘𝑆) |
evls1rhm.t | ⊢ 𝑇 = (𝑆 ↑s 𝐵) |
evls1rhm.u | ⊢ 𝑈 = (𝑆 ↾s 𝑅) |
evls1rhm.w | ⊢ 𝑊 = (Poly1‘𝑈) |
Ref | Expression |
---|---|
evls1rhm | ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑄 ∈ (𝑊 RingHom 𝑇)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | evls1rhm.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝑆) | |
2 | 1 | subrgss 19173 | . . . . 5 ⊢ (𝑅 ∈ (SubRing‘𝑆) → 𝑅 ⊆ 𝐵) |
3 | 2 | adantl 475 | . . . 4 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑅 ⊆ 𝐵) |
4 | elpwg 4387 | . . . . 5 ⊢ (𝑅 ∈ (SubRing‘𝑆) → (𝑅 ∈ 𝒫 𝐵 ↔ 𝑅 ⊆ 𝐵)) | |
5 | 4 | adantl 475 | . . . 4 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (𝑅 ∈ 𝒫 𝐵 ↔ 𝑅 ⊆ 𝐵)) |
6 | 3, 5 | mpbird 249 | . . 3 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑅 ∈ 𝒫 𝐵) |
7 | evls1rhm.q | . . . 4 ⊢ 𝑄 = (𝑆 evalSub1 𝑅) | |
8 | eqid 2778 | . . . 4 ⊢ (1o evalSub 𝑆) = (1o evalSub 𝑆) | |
9 | 7, 8, 1 | evls1fval 20080 | . . 3 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ 𝒫 𝐵) → 𝑄 = ((𝑥 ∈ (𝐵 ↑𝑚 (𝐵 ↑𝑚 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘ ((1o evalSub 𝑆)‘𝑅))) |
10 | 6, 9 | syldan 585 | . 2 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑄 = ((𝑥 ∈ (𝐵 ↑𝑚 (𝐵 ↑𝑚 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘ ((1o evalSub 𝑆)‘𝑅))) |
11 | evls1rhm.t | . . . . 5 ⊢ 𝑇 = (𝑆 ↑s 𝐵) | |
12 | eqid 2778 | . . . . 5 ⊢ (𝑥 ∈ (𝐵 ↑𝑚 (𝐵 ↑𝑚 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) = (𝑥 ∈ (𝐵 ↑𝑚 (𝐵 ↑𝑚 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) | |
13 | 1, 11, 12 | evls1rhmlem 20082 | . . . 4 ⊢ (𝑆 ∈ CRing → (𝑥 ∈ (𝐵 ↑𝑚 (𝐵 ↑𝑚 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∈ ((𝑆 ↑s (𝐵 ↑𝑚 1o)) RingHom 𝑇)) |
14 | 13 | adantr 474 | . . 3 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (𝑥 ∈ (𝐵 ↑𝑚 (𝐵 ↑𝑚 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∈ ((𝑆 ↑s (𝐵 ↑𝑚 1o)) RingHom 𝑇)) |
15 | 1on 7850 | . . . . 5 ⊢ 1o ∈ On | |
16 | eqid 2778 | . . . . . 6 ⊢ ((1o evalSub 𝑆)‘𝑅) = ((1o evalSub 𝑆)‘𝑅) | |
17 | eqid 2778 | . . . . . 6 ⊢ (1o mPoly 𝑈) = (1o mPoly 𝑈) | |
18 | evls1rhm.u | . . . . . 6 ⊢ 𝑈 = (𝑆 ↾s 𝑅) | |
19 | eqid 2778 | . . . . . 6 ⊢ (𝑆 ↑s (𝐵 ↑𝑚 1o)) = (𝑆 ↑s (𝐵 ↑𝑚 1o)) | |
20 | 16, 17, 18, 19, 1 | evlsrhm 19917 | . . . . 5 ⊢ ((1o ∈ On ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → ((1o evalSub 𝑆)‘𝑅) ∈ ((1o mPoly 𝑈) RingHom (𝑆 ↑s (𝐵 ↑𝑚 1o)))) |
21 | 15, 20 | mp3an1 1521 | . . . 4 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → ((1o evalSub 𝑆)‘𝑅) ∈ ((1o mPoly 𝑈) RingHom (𝑆 ↑s (𝐵 ↑𝑚 1o)))) |
22 | eqidd 2779 | . . . . 5 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (Base‘𝑊) = (Base‘𝑊)) | |
23 | eqidd 2779 | . . . . 5 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (Base‘(𝑆 ↑s (𝐵 ↑𝑚 1o))) = (Base‘(𝑆 ↑s (𝐵 ↑𝑚 1o)))) | |
24 | evls1rhm.w | . . . . . . 7 ⊢ 𝑊 = (Poly1‘𝑈) | |
25 | eqid 2778 | . . . . . . 7 ⊢ (PwSer1‘𝑈) = (PwSer1‘𝑈) | |
26 | eqid 2778 | . . . . . . 7 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
27 | 24, 25, 26 | ply1bas 19961 | . . . . . 6 ⊢ (Base‘𝑊) = (Base‘(1o mPoly 𝑈)) |
28 | 27 | a1i 11 | . . . . 5 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (Base‘𝑊) = (Base‘(1o mPoly 𝑈))) |
29 | eqid 2778 | . . . . . . . 8 ⊢ (+g‘𝑊) = (+g‘𝑊) | |
30 | 24, 17, 29 | ply1plusg 19991 | . . . . . . 7 ⊢ (+g‘𝑊) = (+g‘(1o mPoly 𝑈)) |
31 | 30 | a1i 11 | . . . . . 6 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (+g‘𝑊) = (+g‘(1o mPoly 𝑈))) |
32 | 31 | oveqdr 6950 | . . . . 5 ⊢ (((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → (𝑥(+g‘𝑊)𝑦) = (𝑥(+g‘(1o mPoly 𝑈))𝑦)) |
33 | eqidd 2779 | . . . . 5 ⊢ (((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ∧ (𝑥 ∈ (Base‘(𝑆 ↑s (𝐵 ↑𝑚 1o))) ∧ 𝑦 ∈ (Base‘(𝑆 ↑s (𝐵 ↑𝑚 1o))))) → (𝑥(+g‘(𝑆 ↑s (𝐵 ↑𝑚 1o)))𝑦) = (𝑥(+g‘(𝑆 ↑s (𝐵 ↑𝑚 1o)))𝑦)) | |
34 | eqid 2778 | . . . . . . . 8 ⊢ (.r‘𝑊) = (.r‘𝑊) | |
35 | 24, 17, 34 | ply1mulr 19993 | . . . . . . 7 ⊢ (.r‘𝑊) = (.r‘(1o mPoly 𝑈)) |
36 | 35 | a1i 11 | . . . . . 6 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (.r‘𝑊) = (.r‘(1o mPoly 𝑈))) |
37 | 36 | oveqdr 6950 | . . . . 5 ⊢ (((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → (𝑥(.r‘𝑊)𝑦) = (𝑥(.r‘(1o mPoly 𝑈))𝑦)) |
38 | eqidd 2779 | . . . . 5 ⊢ (((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ∧ (𝑥 ∈ (Base‘(𝑆 ↑s (𝐵 ↑𝑚 1o))) ∧ 𝑦 ∈ (Base‘(𝑆 ↑s (𝐵 ↑𝑚 1o))))) → (𝑥(.r‘(𝑆 ↑s (𝐵 ↑𝑚 1o)))𝑦) = (𝑥(.r‘(𝑆 ↑s (𝐵 ↑𝑚 1o)))𝑦)) | |
39 | 22, 23, 28, 23, 32, 33, 37, 38 | rhmpropd 19207 | . . . 4 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (𝑊 RingHom (𝑆 ↑s (𝐵 ↑𝑚 1o))) = ((1o mPoly 𝑈) RingHom (𝑆 ↑s (𝐵 ↑𝑚 1o)))) |
40 | 21, 39 | eleqtrrd 2862 | . . 3 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → ((1o evalSub 𝑆)‘𝑅) ∈ (𝑊 RingHom (𝑆 ↑s (𝐵 ↑𝑚 1o)))) |
41 | rhmco 19126 | . . 3 ⊢ (((𝑥 ∈ (𝐵 ↑𝑚 (𝐵 ↑𝑚 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∈ ((𝑆 ↑s (𝐵 ↑𝑚 1o)) RingHom 𝑇) ∧ ((1o evalSub 𝑆)‘𝑅) ∈ (𝑊 RingHom (𝑆 ↑s (𝐵 ↑𝑚 1o)))) → ((𝑥 ∈ (𝐵 ↑𝑚 (𝐵 ↑𝑚 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘ ((1o evalSub 𝑆)‘𝑅)) ∈ (𝑊 RingHom 𝑇)) | |
42 | 14, 40, 41 | syl2anc 579 | . 2 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → ((𝑥 ∈ (𝐵 ↑𝑚 (𝐵 ↑𝑚 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘ ((1o evalSub 𝑆)‘𝑅)) ∈ (𝑊 RingHom 𝑇)) |
43 | 10, 42 | eqeltrd 2859 | 1 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑄 ∈ (𝑊 RingHom 𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1601 ∈ wcel 2107 ⊆ wss 3792 𝒫 cpw 4379 {csn 4398 ↦ cmpt 4965 × cxp 5353 ∘ ccom 5359 Oncon0 5976 ‘cfv 6135 (class class class)co 6922 1oc1o 7836 ↑𝑚 cmap 8140 Basecbs 16255 ↾s cress 16256 +gcplusg 16338 .rcmulr 16339 ↑s cpws 16493 CRingccrg 18935 RingHom crh 19101 SubRingcsubrg 19168 mPoly cmpl 19750 evalSub ces 19900 PwSer1cps1 19941 Poly1cpl1 19943 evalSub1 ces1 20074 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-inf2 8835 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-int 4711 df-iun 4755 df-iin 4756 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-se 5315 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-isom 6144 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-of 7174 df-ofr 7175 df-om 7344 df-1st 7445 df-2nd 7446 df-supp 7577 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-2o 7844 df-oadd 7847 df-er 8026 df-map 8142 df-pm 8143 df-ixp 8195 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-fsupp 8564 df-sup 8636 df-oi 8704 df-card 9098 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-7 11443 df-8 11444 df-9 11445 df-n0 11643 df-z 11729 df-dec 11846 df-uz 11993 df-fz 12644 df-fzo 12785 df-seq 13120 df-hash 13436 df-struct 16257 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-ress 16263 df-plusg 16351 df-mulr 16352 df-sca 16354 df-vsca 16355 df-ip 16356 df-tset 16357 df-ple 16358 df-ds 16360 df-hom 16362 df-cco 16363 df-0g 16488 df-gsum 16489 df-prds 16494 df-pws 16496 df-mre 16632 df-mrc 16633 df-acs 16635 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-mhm 17721 df-submnd 17722 df-grp 17812 df-minusg 17813 df-sbg 17814 df-mulg 17928 df-subg 17975 df-ghm 18042 df-cntz 18133 df-cmn 18581 df-abl 18582 df-mgp 18877 df-ur 18889 df-srg 18893 df-ring 18936 df-cring 18937 df-rnghom 19104 df-subrg 19170 df-lmod 19257 df-lss 19325 df-lsp 19367 df-assa 19709 df-asp 19710 df-ascl 19711 df-psr 19753 df-mvr 19754 df-mpl 19755 df-opsr 19757 df-evls 19902 df-psr1 19946 df-ply1 19948 df-evls1 20076 |
This theorem is referenced by: evls1gsumadd 20085 evls1gsummul 20086 evls1varpw 20087 |
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