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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 19529 | . . . . 5 ⊢ (𝑅 ∈ (SubRing‘𝑆) → 𝑅 ⊆ 𝐵) |
3 | 2 | adantl 485 | . . . 4 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑅 ⊆ 𝐵) |
4 | elpwg 4500 | . . . . 5 ⊢ (𝑅 ∈ (SubRing‘𝑆) → (𝑅 ∈ 𝒫 𝐵 ↔ 𝑅 ⊆ 𝐵)) | |
5 | 4 | adantl 485 | . . . 4 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (𝑅 ∈ 𝒫 𝐵 ↔ 𝑅 ⊆ 𝐵)) |
6 | 3, 5 | mpbird 260 | . . 3 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑅 ∈ 𝒫 𝐵) |
7 | evls1rhm.q | . . . 4 ⊢ 𝑄 = (𝑆 evalSub1 𝑅) | |
8 | eqid 2798 | . . . 4 ⊢ (1o evalSub 𝑆) = (1o evalSub 𝑆) | |
9 | 7, 8, 1 | evls1fval 20943 | . . 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 594 | . 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 | . . . 4 ⊢ 𝑇 = (𝑆 ↑s 𝐵) | |
12 | 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})))) | |
13 | 1, 11, 12 | evls1rhmlem 20945 | . . 3 ⊢ (𝑆 ∈ CRing → (𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∈ ((𝑆 ↑s (𝐵 ↑m 1o)) RingHom 𝑇)) |
14 | 1on 8092 | . . . . 5 ⊢ 1o ∈ On | |
15 | eqid 2798 | . . . . . 6 ⊢ ((1o evalSub 𝑆)‘𝑅) = ((1o evalSub 𝑆)‘𝑅) | |
16 | eqid 2798 | . . . . . 6 ⊢ (1o mPoly 𝑈) = (1o mPoly 𝑈) | |
17 | evls1rhm.u | . . . . . 6 ⊢ 𝑈 = (𝑆 ↾s 𝑅) | |
18 | eqid 2798 | . . . . . 6 ⊢ (𝑆 ↑s (𝐵 ↑m 1o)) = (𝑆 ↑s (𝐵 ↑m 1o)) | |
19 | 15, 16, 17, 18, 1 | evlsrhm 20760 | . . . . 5 ⊢ ((1o ∈ On ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → ((1o evalSub 𝑆)‘𝑅) ∈ ((1o mPoly 𝑈) RingHom (𝑆 ↑s (𝐵 ↑m 1o)))) |
20 | 14, 19 | mp3an1 1445 | . . . 4 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → ((1o evalSub 𝑆)‘𝑅) ∈ ((1o mPoly 𝑈) RingHom (𝑆 ↑s (𝐵 ↑m 1o)))) |
21 | eqidd 2799 | . . . . 5 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (Base‘𝑊) = (Base‘𝑊)) | |
22 | eqidd 2799 | . . . . 5 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (Base‘(𝑆 ↑s (𝐵 ↑m 1o))) = (Base‘(𝑆 ↑s (𝐵 ↑m 1o)))) | |
23 | evls1rhm.w | . . . . . . 7 ⊢ 𝑊 = (Poly1‘𝑈) | |
24 | eqid 2798 | . . . . . . 7 ⊢ (PwSer1‘𝑈) = (PwSer1‘𝑈) | |
25 | eqid 2798 | . . . . . . 7 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
26 | 23, 24, 25 | ply1bas 20824 | . . . . . 6 ⊢ (Base‘𝑊) = (Base‘(1o mPoly 𝑈)) |
27 | 26 | a1i 11 | . . . . 5 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (Base‘𝑊) = (Base‘(1o mPoly 𝑈))) |
28 | eqid 2798 | . . . . . . . 8 ⊢ (+g‘𝑊) = (+g‘𝑊) | |
29 | 23, 16, 28 | ply1plusg 20854 | . . . . . . 7 ⊢ (+g‘𝑊) = (+g‘(1o mPoly 𝑈)) |
30 | 29 | a1i 11 | . . . . . 6 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (+g‘𝑊) = (+g‘(1o mPoly 𝑈))) |
31 | 30 | oveqdr 7163 | . . . . 5 ⊢ (((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → (𝑥(+g‘𝑊)𝑦) = (𝑥(+g‘(1o mPoly 𝑈))𝑦)) |
32 | eqidd 2799 | . . . . 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)) | |
33 | eqid 2798 | . . . . . . . 8 ⊢ (.r‘𝑊) = (.r‘𝑊) | |
34 | 23, 16, 33 | ply1mulr 20856 | . . . . . . 7 ⊢ (.r‘𝑊) = (.r‘(1o mPoly 𝑈)) |
35 | 34 | a1i 11 | . . . . . 6 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (.r‘𝑊) = (.r‘(1o mPoly 𝑈))) |
36 | 35 | oveqdr 7163 | . . . . 5 ⊢ (((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → (𝑥(.r‘𝑊)𝑦) = (𝑥(.r‘(1o mPoly 𝑈))𝑦)) |
37 | eqidd 2799 | . . . . 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)) | |
38 | 21, 22, 27, 22, 31, 32, 36, 37 | rhmpropd 19564 | . . . 4 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (𝑊 RingHom (𝑆 ↑s (𝐵 ↑m 1o))) = ((1o mPoly 𝑈) RingHom (𝑆 ↑s (𝐵 ↑m 1o)))) |
39 | 20, 38 | eleqtrrd 2893 | . . 3 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → ((1o evalSub 𝑆)‘𝑅) ∈ (𝑊 RingHom (𝑆 ↑s (𝐵 ↑m 1o)))) |
40 | rhmco 19485 | . . 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)) | |
41 | 13, 39, 40 | syl2an2r 684 | . 2 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → ((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘ ((1o evalSub 𝑆)‘𝑅)) ∈ (𝑊 RingHom 𝑇)) |
42 | 10, 41 | eqeltrd 2890 | 1 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑄 ∈ (𝑊 RingHom 𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ⊆ wss 3881 𝒫 cpw 4497 {csn 4525 ↦ cmpt 5110 × cxp 5517 ∘ ccom 5523 Oncon0 6159 ‘cfv 6324 (class class class)co 7135 1oc1o 8078 ↑m cmap 8389 Basecbs 16475 ↾s cress 16476 +gcplusg 16557 .rcmulr 16558 ↑s cpws 16712 CRingccrg 19291 RingHom crh 19460 SubRingcsubrg 19524 mPoly cmpl 20591 evalSub ces 20743 PwSer1cps1 20804 Poly1cpl1 20806 evalSub1 ces1 20937 |
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-psr1 20809 df-ply1 20811 df-evls1 20939 |
This theorem is referenced by: evls1gsumadd 20948 evls1gsummul 20949 evls1pw 20950 |
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