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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 2824 | . . 3 ⊢ (1o eval 𝑅) = (1o eval 𝑅) | |
3 | evl1rhm.b | . . 3 ⊢ 𝐵 = (Base‘𝑅) | |
4 | 1, 2, 3 | evl1fval 20494 | . 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 2824 | . . . 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 20487 | . . 3 ⊢ (𝑅 ∈ CRing → (𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∈ ((𝑅 ↑s (𝐵 ↑m 1o)) RingHom 𝑇)) |
8 | 1on 8112 | . . . . 5 ⊢ 1o ∈ On | |
9 | eqid 2824 | . . . . . 6 ⊢ (1o mPoly 𝑅) = (1o mPoly 𝑅) | |
10 | eqid 2824 | . . . . . 6 ⊢ (𝑅 ↑s (𝐵 ↑m 1o)) = (𝑅 ↑s (𝐵 ↑m 1o)) | |
11 | 2, 3, 9, 10 | evlrhm 20312 | . . . . 5 ⊢ ((1o ∈ On ∧ 𝑅 ∈ CRing) → (1o eval 𝑅) ∈ ((1o mPoly 𝑅) RingHom (𝑅 ↑s (𝐵 ↑m 1o)))) |
12 | 8, 11 | mpan 688 | . . . 4 ⊢ (𝑅 ∈ CRing → (1o eval 𝑅) ∈ ((1o mPoly 𝑅) RingHom (𝑅 ↑s (𝐵 ↑m 1o)))) |
13 | eqidd 2825 | . . . . 5 ⊢ (𝑅 ∈ CRing → (Base‘𝑃) = (Base‘𝑃)) | |
14 | eqidd 2825 | . . . . 5 ⊢ (𝑅 ∈ CRing → (Base‘(𝑅 ↑s (𝐵 ↑m 1o))) = (Base‘(𝑅 ↑s (𝐵 ↑m 1o)))) | |
15 | evl1rhm.w | . . . . . . 7 ⊢ 𝑃 = (Poly1‘𝑅) | |
16 | eqid 2824 | . . . . . . 7 ⊢ (PwSer1‘𝑅) = (PwSer1‘𝑅) | |
17 | eqid 2824 | . . . . . . 7 ⊢ (Base‘𝑃) = (Base‘𝑃) | |
18 | 15, 16, 17 | ply1bas 20366 | . . . . . 6 ⊢ (Base‘𝑃) = (Base‘(1o mPoly 𝑅)) |
19 | 18 | a1i 11 | . . . . 5 ⊢ (𝑅 ∈ CRing → (Base‘𝑃) = (Base‘(1o mPoly 𝑅))) |
20 | eqid 2824 | . . . . . . . 8 ⊢ (+g‘𝑃) = (+g‘𝑃) | |
21 | 15, 9, 20 | ply1plusg 20396 | . . . . . . 7 ⊢ (+g‘𝑃) = (+g‘(1o mPoly 𝑅)) |
22 | 21 | a1i 11 | . . . . . 6 ⊢ (𝑅 ∈ CRing → (+g‘𝑃) = (+g‘(1o mPoly 𝑅))) |
23 | 22 | oveqdr 7187 | . . . . 5 ⊢ ((𝑅 ∈ CRing ∧ (𝑥 ∈ (Base‘𝑃) ∧ 𝑦 ∈ (Base‘𝑃))) → (𝑥(+g‘𝑃)𝑦) = (𝑥(+g‘(1o mPoly 𝑅))𝑦)) |
24 | eqidd 2825 | . . . . 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 2824 | . . . . . . . 8 ⊢ (.r‘𝑃) = (.r‘𝑃) | |
26 | 15, 9, 25 | ply1mulr 20398 | . . . . . . 7 ⊢ (.r‘𝑃) = (.r‘(1o mPoly 𝑅)) |
27 | 26 | a1i 11 | . . . . . 6 ⊢ (𝑅 ∈ CRing → (.r‘𝑃) = (.r‘(1o mPoly 𝑅))) |
28 | 27 | oveqdr 7187 | . . . . 5 ⊢ ((𝑅 ∈ CRing ∧ (𝑥 ∈ (Base‘𝑃) ∧ 𝑦 ∈ (Base‘𝑃))) → (𝑥(.r‘𝑃)𝑦) = (𝑥(.r‘(1o mPoly 𝑅))𝑦)) |
29 | eqidd 2825 | . . . . 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 19574 | . . . 4 ⊢ (𝑅 ∈ CRing → (𝑃 RingHom (𝑅 ↑s (𝐵 ↑m 1o))) = ((1o mPoly 𝑅) RingHom (𝑅 ↑s (𝐵 ↑m 1o)))) |
31 | 12, 30 | eleqtrrd 2919 | . . 3 ⊢ (𝑅 ∈ CRing → (1o eval 𝑅) ∈ (𝑃 RingHom (𝑅 ↑s (𝐵 ↑m 1o)))) |
32 | rhmco 19492 | . . 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 586 | . 2 ⊢ (𝑅 ∈ CRing → ((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘ (1o eval 𝑅)) ∈ (𝑃 RingHom 𝑇)) |
34 | 4, 33 | eqeltrid 2920 | 1 ⊢ (𝑅 ∈ CRing → 𝑂 ∈ (𝑃 RingHom 𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 {csn 4570 ↦ cmpt 5149 × cxp 5556 ∘ ccom 5562 Oncon0 6194 ‘cfv 6358 (class class class)co 7159 1oc1o 8098 ↑m cmap 8409 Basecbs 16486 +gcplusg 16568 .rcmulr 16569 ↑s cpws 16723 CRingccrg 19301 RingHom crh 19467 mPoly cmpl 20136 eval cevl 20288 PwSer1cps1 20346 Poly1cpl1 20348 eval1ce1 20480 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-iin 4925 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-se 5518 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-of 7412 df-ofr 7413 df-om 7584 df-1st 7692 df-2nd 7693 df-supp 7834 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-2o 8106 df-oadd 8109 df-er 8292 df-map 8411 df-pm 8412 df-ixp 8465 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-fsupp 8837 df-sup 8909 df-oi 8977 df-card 9371 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-fz 12896 df-fzo 13037 df-seq 13373 df-hash 13694 df-struct 16488 df-ndx 16489 df-slot 16490 df-base 16492 df-sets 16493 df-ress 16494 df-plusg 16581 df-mulr 16582 df-sca 16584 df-vsca 16585 df-ip 16586 df-tset 16587 df-ple 16588 df-ds 16590 df-hom 16592 df-cco 16593 df-0g 16718 df-gsum 16719 df-prds 16724 df-pws 16726 df-mre 16860 df-mrc 16861 df-acs 16863 df-mgm 17855 df-sgrp 17904 df-mnd 17915 df-mhm 17959 df-submnd 17960 df-grp 18109 df-minusg 18110 df-sbg 18111 df-mulg 18228 df-subg 18279 df-ghm 18359 df-cntz 18450 df-cmn 18911 df-abl 18912 df-mgp 19243 df-ur 19255 df-srg 19259 df-ring 19302 df-cring 19303 df-rnghom 19470 df-subrg 19536 df-lmod 19639 df-lss 19707 df-lsp 19747 df-assa 20088 df-asp 20089 df-ascl 20090 df-psr 20139 df-mvr 20140 df-mpl 20141 df-opsr 20143 df-evls 20289 df-evl 20290 df-psr1 20351 df-ply1 20353 df-evl1 20482 |
This theorem is referenced by: fveval1fvcl 20499 evl1addd 20507 evl1subd 20508 evl1muld 20509 evl1expd 20511 pf1const 20512 pf1id 20513 pf1subrg 20514 mpfpf1 20517 pf1mpf 20518 evl1gsummul 20526 evl1scvarpw 20529 ply1remlem 24759 ply1rem 24760 fta1glem1 24762 fta1glem2 24763 fta1g 24764 fta1blem 24765 plypf1 24805 lgsqrlem2 25926 lgsqrlem3 25927 pl1cn 31202 idomrootle 39801 |
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