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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 18604 | . . . . 5 ⊢ (𝑅 ∈ (SubRing‘𝑆) → 𝑅 ⊆ 𝐵) |
3 | 2 | adantl 481 | . . . 4 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑅 ⊆ 𝐵) |
4 | elpwg 4116 | . . . . 5 ⊢ (𝑅 ∈ (SubRing‘𝑆) → (𝑅 ∈ 𝒫 𝐵 ↔ 𝑅 ⊆ 𝐵)) | |
5 | 4 | adantl 481 | . . . 4 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (𝑅 ∈ 𝒫 𝐵 ↔ 𝑅 ⊆ 𝐵)) |
6 | 3, 5 | mpbird 246 | . . 3 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑅 ∈ 𝒫 𝐵) |
7 | evls1rhm.q | . . . 4 ⊢ 𝑄 = (𝑆 evalSub1 𝑅) | |
8 | eqid 2610 | . . . 4 ⊢ (1𝑜 evalSub 𝑆) = (1𝑜 evalSub 𝑆) | |
9 | 7, 8, 1 | evls1fval 19505 | . . 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 486 | . 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 2610 | . . . . 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 19507 | . . . 4 ⊢ (𝑆 ∈ CRing → (𝑥 ∈ (𝐵 ↑𝑚 (𝐵 ↑𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1𝑜 × {𝑦})))) ∈ ((𝑆 ↑s (𝐵 ↑𝑚 1𝑜)) RingHom 𝑇)) |
14 | 13 | adantr 480 | . . 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 7454 | . . . . 5 ⊢ 1𝑜 ∈ On | |
16 | eqid 2610 | . . . . . 6 ⊢ ((1𝑜 evalSub 𝑆)‘𝑅) = ((1𝑜 evalSub 𝑆)‘𝑅) | |
17 | eqid 2610 | . . . . . 6 ⊢ (1𝑜 mPoly 𝑈) = (1𝑜 mPoly 𝑈) | |
18 | evls1rhm.u | . . . . . 6 ⊢ 𝑈 = (𝑆 ↾s 𝑅) | |
19 | eqid 2610 | . . . . . 6 ⊢ (𝑆 ↑s (𝐵 ↑𝑚 1𝑜)) = (𝑆 ↑s (𝐵 ↑𝑚 1𝑜)) | |
20 | 16, 17, 18, 19, 1 | evlsrhm 19342 | . . . . 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 1403 | . . . 4 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → ((1𝑜 evalSub 𝑆)‘𝑅) ∈ ((1𝑜 mPoly 𝑈) RingHom (𝑆 ↑s (𝐵 ↑𝑚 1𝑜)))) |
22 | eqidd 2611 | . . . . 5 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (Base‘𝑊) = (Base‘𝑊)) | |
23 | eqidd 2611 | . . . . 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 2610 | . . . . . . 7 ⊢ (PwSer1‘𝑈) = (PwSer1‘𝑈) | |
26 | eqid 2610 | . . . . . . 7 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
27 | 24, 25, 26 | ply1bas 19386 | . . . . . 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 2610 | . . . . . . . 8 ⊢ (+g‘𝑊) = (+g‘𝑊) | |
30 | 24, 17, 29 | ply1plusg 19416 | . . . . . . 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 6573 | . . . . 5 ⊢ (((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → (𝑥(+g‘𝑊)𝑦) = (𝑥(+g‘(1𝑜 mPoly 𝑈))𝑦)) |
33 | eqidd 2611 | . . . . 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 2610 | . . . . . . . 8 ⊢ (.r‘𝑊) = (.r‘𝑊) | |
35 | 24, 17, 34 | ply1mulr 19418 | . . . . . . 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 6573 | . . . . 5 ⊢ (((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → (𝑥(.r‘𝑊)𝑦) = (𝑥(.r‘(1𝑜 mPoly 𝑈))𝑦)) |
38 | eqidd 2611 | . . . . 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 18638 | . . . 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 2691 | . . 3 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → ((1𝑜 evalSub 𝑆)‘𝑅) ∈ (𝑊 RingHom (𝑆 ↑s (𝐵 ↑𝑚 1𝑜)))) |
41 | rhmco 18560 | . . 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 691 | . 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 2688 | 1 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑄 ∈ (𝑊 RingHom 𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ⊆ wss 3540 𝒫 cpw 4108 {csn 4125 ↦ cmpt 4643 × cxp 5036 ∘ ccom 5042 Oncon0 5640 ‘cfv 5804 (class class class)co 6549 1𝑜c1o 7440 ↑𝑚 cmap 7744 Basecbs 15695 ↾s cress 15696 +gcplusg 15768 .rcmulr 15769 ↑s cpws 15930 CRingccrg 18371 RingHom crh 18535 SubRingcsubrg 18599 mPoly cmpl 19174 evalSub ces 19325 PwSer1cps1 19366 Poly1cpl1 19368 evalSub1 ces1 19499 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-ofr 6796 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-sup 8231 df-oi 8298 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-fz 12198 df-fzo 12335 df-seq 12664 df-hash 12980 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-hom 15793 df-cco 15794 df-0g 15925 df-gsum 15926 df-prds 15931 df-pws 15933 df-mre 16069 df-mrc 16070 df-acs 16072 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-mhm 17158 df-submnd 17159 df-grp 17248 df-minusg 17249 df-sbg 17250 df-mulg 17364 df-subg 17414 df-ghm 17481 df-cntz 17573 df-cmn 18018 df-abl 18019 df-mgp 18313 df-ur 18325 df-srg 18329 df-ring 18372 df-cring 18373 df-rnghom 18538 df-subrg 18601 df-lmod 18688 df-lss 18754 df-lsp 18793 df-assa 19133 df-asp 19134 df-ascl 19135 df-psr 19177 df-mvr 19178 df-mpl 19179 df-opsr 19181 df-evls 19327 df-psr1 19371 df-ply1 19373 df-evls1 19501 |
This theorem is referenced by: evls1gsumadd 19510 evls1gsummul 19511 evls1varpw 19512 |
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