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Mirrors > Home > MPE Home > Th. List > evl1subd | Structured version Visualization version GIF version |
Description: Polynomial evaluation builder for subtraction of polynomials. (Contributed by Mario Carneiro, 4-Jul-2015.) |
Ref | Expression |
---|---|
evl1addd.q | ⊢ 𝑂 = (eval1‘𝑅) |
evl1addd.p | ⊢ 𝑃 = (Poly1‘𝑅) |
evl1addd.b | ⊢ 𝐵 = (Base‘𝑅) |
evl1addd.u | ⊢ 𝑈 = (Base‘𝑃) |
evl1addd.1 | ⊢ (𝜑 → 𝑅 ∈ CRing) |
evl1addd.2 | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
evl1addd.3 | ⊢ (𝜑 → (𝑀 ∈ 𝑈 ∧ ((𝑂‘𝑀)‘𝑌) = 𝑉)) |
evl1addd.4 | ⊢ (𝜑 → (𝑁 ∈ 𝑈 ∧ ((𝑂‘𝑁)‘𝑌) = 𝑊)) |
evl1subd.s | ⊢ − = (-g‘𝑃) |
evl1subd.d | ⊢ 𝐷 = (-g‘𝑅) |
Ref | Expression |
---|---|
evl1subd | ⊢ (𝜑 → ((𝑀 − 𝑁) ∈ 𝑈 ∧ ((𝑂‘(𝑀 − 𝑁))‘𝑌) = (𝑉𝐷𝑊))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | evl1addd.1 | . . . . . 6 ⊢ (𝜑 → 𝑅 ∈ CRing) | |
2 | evl1addd.q | . . . . . . 7 ⊢ 𝑂 = (eval1‘𝑅) | |
3 | evl1addd.p | . . . . . . 7 ⊢ 𝑃 = (Poly1‘𝑅) | |
4 | eqid 2797 | . . . . . . 7 ⊢ (𝑅 ↑s 𝐵) = (𝑅 ↑s 𝐵) | |
5 | evl1addd.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝑅) | |
6 | 2, 3, 4, 5 | evl1rhm 20014 | . . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑂 ∈ (𝑃 RingHom (𝑅 ↑s 𝐵))) |
7 | 1, 6 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑂 ∈ (𝑃 RingHom (𝑅 ↑s 𝐵))) |
8 | rhmghm 19039 | . . . . 5 ⊢ (𝑂 ∈ (𝑃 RingHom (𝑅 ↑s 𝐵)) → 𝑂 ∈ (𝑃 GrpHom (𝑅 ↑s 𝐵))) | |
9 | 7, 8 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑂 ∈ (𝑃 GrpHom (𝑅 ↑s 𝐵))) |
10 | ghmgrp1 17971 | . . . 4 ⊢ (𝑂 ∈ (𝑃 GrpHom (𝑅 ↑s 𝐵)) → 𝑃 ∈ Grp) | |
11 | 9, 10 | syl 17 | . . 3 ⊢ (𝜑 → 𝑃 ∈ Grp) |
12 | evl1addd.3 | . . . 4 ⊢ (𝜑 → (𝑀 ∈ 𝑈 ∧ ((𝑂‘𝑀)‘𝑌) = 𝑉)) | |
13 | 12 | simpld 489 | . . 3 ⊢ (𝜑 → 𝑀 ∈ 𝑈) |
14 | evl1addd.4 | . . . 4 ⊢ (𝜑 → (𝑁 ∈ 𝑈 ∧ ((𝑂‘𝑁)‘𝑌) = 𝑊)) | |
15 | 14 | simpld 489 | . . 3 ⊢ (𝜑 → 𝑁 ∈ 𝑈) |
16 | evl1addd.u | . . . 4 ⊢ 𝑈 = (Base‘𝑃) | |
17 | evl1subd.s | . . . 4 ⊢ − = (-g‘𝑃) | |
18 | 16, 17 | grpsubcl 17807 | . . 3 ⊢ ((𝑃 ∈ Grp ∧ 𝑀 ∈ 𝑈 ∧ 𝑁 ∈ 𝑈) → (𝑀 − 𝑁) ∈ 𝑈) |
19 | 11, 13, 15, 18 | syl3anc 1491 | . 2 ⊢ (𝜑 → (𝑀 − 𝑁) ∈ 𝑈) |
20 | eqid 2797 | . . . . . . 7 ⊢ (-g‘(𝑅 ↑s 𝐵)) = (-g‘(𝑅 ↑s 𝐵)) | |
21 | 16, 17, 20 | ghmsub 17977 | . . . . . 6 ⊢ ((𝑂 ∈ (𝑃 GrpHom (𝑅 ↑s 𝐵)) ∧ 𝑀 ∈ 𝑈 ∧ 𝑁 ∈ 𝑈) → (𝑂‘(𝑀 − 𝑁)) = ((𝑂‘𝑀)(-g‘(𝑅 ↑s 𝐵))(𝑂‘𝑁))) |
22 | 9, 13, 15, 21 | syl3anc 1491 | . . . . 5 ⊢ (𝜑 → (𝑂‘(𝑀 − 𝑁)) = ((𝑂‘𝑀)(-g‘(𝑅 ↑s 𝐵))(𝑂‘𝑁))) |
23 | crngring 18870 | . . . . . . 7 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
24 | ringgrp 18864 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
25 | 1, 23, 24 | 3syl 18 | . . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Grp) |
26 | 5 | fvexi 6423 | . . . . . . 7 ⊢ 𝐵 ∈ V |
27 | 26 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ V) |
28 | eqid 2797 | . . . . . . . . 9 ⊢ (Base‘(𝑅 ↑s 𝐵)) = (Base‘(𝑅 ↑s 𝐵)) | |
29 | 16, 28 | rhmf 19040 | . . . . . . . 8 ⊢ (𝑂 ∈ (𝑃 RingHom (𝑅 ↑s 𝐵)) → 𝑂:𝑈⟶(Base‘(𝑅 ↑s 𝐵))) |
30 | 7, 29 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑂:𝑈⟶(Base‘(𝑅 ↑s 𝐵))) |
31 | 30, 13 | ffvelrnd 6584 | . . . . . 6 ⊢ (𝜑 → (𝑂‘𝑀) ∈ (Base‘(𝑅 ↑s 𝐵))) |
32 | 30, 15 | ffvelrnd 6584 | . . . . . 6 ⊢ (𝜑 → (𝑂‘𝑁) ∈ (Base‘(𝑅 ↑s 𝐵))) |
33 | evl1subd.d | . . . . . . 7 ⊢ 𝐷 = (-g‘𝑅) | |
34 | 4, 28, 33, 20 | pwssub 17841 | . . . . . 6 ⊢ (((𝑅 ∈ Grp ∧ 𝐵 ∈ V) ∧ ((𝑂‘𝑀) ∈ (Base‘(𝑅 ↑s 𝐵)) ∧ (𝑂‘𝑁) ∈ (Base‘(𝑅 ↑s 𝐵)))) → ((𝑂‘𝑀)(-g‘(𝑅 ↑s 𝐵))(𝑂‘𝑁)) = ((𝑂‘𝑀) ∘𝑓 𝐷(𝑂‘𝑁))) |
35 | 25, 27, 31, 32, 34 | syl22anc 868 | . . . . 5 ⊢ (𝜑 → ((𝑂‘𝑀)(-g‘(𝑅 ↑s 𝐵))(𝑂‘𝑁)) = ((𝑂‘𝑀) ∘𝑓 𝐷(𝑂‘𝑁))) |
36 | 22, 35 | eqtrd 2831 | . . . 4 ⊢ (𝜑 → (𝑂‘(𝑀 − 𝑁)) = ((𝑂‘𝑀) ∘𝑓 𝐷(𝑂‘𝑁))) |
37 | 36 | fveq1d 6411 | . . 3 ⊢ (𝜑 → ((𝑂‘(𝑀 − 𝑁))‘𝑌) = (((𝑂‘𝑀) ∘𝑓 𝐷(𝑂‘𝑁))‘𝑌)) |
38 | 4, 5, 28, 1, 27, 31 | pwselbas 16460 | . . . . 5 ⊢ (𝜑 → (𝑂‘𝑀):𝐵⟶𝐵) |
39 | 38 | ffnd 6255 | . . . 4 ⊢ (𝜑 → (𝑂‘𝑀) Fn 𝐵) |
40 | 4, 5, 28, 1, 27, 32 | pwselbas 16460 | . . . . 5 ⊢ (𝜑 → (𝑂‘𝑁):𝐵⟶𝐵) |
41 | 40 | ffnd 6255 | . . . 4 ⊢ (𝜑 → (𝑂‘𝑁) Fn 𝐵) |
42 | evl1addd.2 | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
43 | fnfvof 7143 | . . . 4 ⊢ ((((𝑂‘𝑀) Fn 𝐵 ∧ (𝑂‘𝑁) Fn 𝐵) ∧ (𝐵 ∈ V ∧ 𝑌 ∈ 𝐵)) → (((𝑂‘𝑀) ∘𝑓 𝐷(𝑂‘𝑁))‘𝑌) = (((𝑂‘𝑀)‘𝑌)𝐷((𝑂‘𝑁)‘𝑌))) | |
44 | 39, 41, 27, 42, 43 | syl22anc 868 | . . 3 ⊢ (𝜑 → (((𝑂‘𝑀) ∘𝑓 𝐷(𝑂‘𝑁))‘𝑌) = (((𝑂‘𝑀)‘𝑌)𝐷((𝑂‘𝑁)‘𝑌))) |
45 | 12 | simprd 490 | . . . 4 ⊢ (𝜑 → ((𝑂‘𝑀)‘𝑌) = 𝑉) |
46 | 14 | simprd 490 | . . . 4 ⊢ (𝜑 → ((𝑂‘𝑁)‘𝑌) = 𝑊) |
47 | 45, 46 | oveq12d 6894 | . . 3 ⊢ (𝜑 → (((𝑂‘𝑀)‘𝑌)𝐷((𝑂‘𝑁)‘𝑌)) = (𝑉𝐷𝑊)) |
48 | 37, 44, 47 | 3eqtrd 2835 | . 2 ⊢ (𝜑 → ((𝑂‘(𝑀 − 𝑁))‘𝑌) = (𝑉𝐷𝑊)) |
49 | 19, 48 | jca 508 | 1 ⊢ (𝜑 → ((𝑀 − 𝑁) ∈ 𝑈 ∧ ((𝑂‘(𝑀 − 𝑁))‘𝑌) = (𝑉𝐷𝑊))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 = wceq 1653 ∈ wcel 2157 Vcvv 3383 Fn wfn 6094 ⟶wf 6095 ‘cfv 6099 (class class class)co 6876 ∘𝑓 cof 7127 Basecbs 16180 ↑s cpws 16418 Grpcgrp 17734 -gcsg 17736 GrpHom cghm 17966 Ringcrg 18859 CRingccrg 18860 RingHom crh 19026 Poly1cpl1 19865 eval1ce1 19997 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2375 ax-ext 2775 ax-rep 4962 ax-sep 4973 ax-nul 4981 ax-pow 5033 ax-pr 5095 ax-un 7181 ax-inf2 8786 ax-cnex 10278 ax-resscn 10279 ax-1cn 10280 ax-icn 10281 ax-addcl 10282 ax-addrcl 10283 ax-mulcl 10284 ax-mulrcl 10285 ax-mulcom 10286 ax-addass 10287 ax-mulass 10288 ax-distr 10289 ax-i2m1 10290 ax-1ne0 10291 ax-1rid 10292 ax-rnegex 10293 ax-rrecex 10294 ax-cnre 10295 ax-pre-lttri 10296 ax-pre-lttrn 10297 ax-pre-ltadd 10298 ax-pre-mulgt0 10299 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2590 df-eu 2607 df-clab 2784 df-cleq 2790 df-clel 2793 df-nfc 2928 df-ne 2970 df-nel 3073 df-ral 3092 df-rex 3093 df-reu 3094 df-rmo 3095 df-rab 3096 df-v 3385 df-sbc 3632 df-csb 3727 df-dif 3770 df-un 3772 df-in 3774 df-ss 3781 df-pss 3783 df-nul 4114 df-if 4276 df-pw 4349 df-sn 4367 df-pr 4369 df-tp 4371 df-op 4373 df-uni 4627 df-int 4666 df-iun 4710 df-iin 4711 df-br 4842 df-opab 4904 df-mpt 4921 df-tr 4944 df-id 5218 df-eprel 5223 df-po 5231 df-so 5232 df-fr 5269 df-se 5270 df-we 5271 df-xp 5316 df-rel 5317 df-cnv 5318 df-co 5319 df-dm 5320 df-rn 5321 df-res 5322 df-ima 5323 df-pred 5896 df-ord 5942 df-on 5943 df-lim 5944 df-suc 5945 df-iota 6062 df-fun 6101 df-fn 6102 df-f 6103 df-f1 6104 df-fo 6105 df-f1o 6106 df-fv 6107 df-isom 6108 df-riota 6837 df-ov 6879 df-oprab 6880 df-mpt2 6881 df-of 7129 df-ofr 7130 df-om 7298 df-1st 7399 df-2nd 7400 df-supp 7531 df-wrecs 7643 df-recs 7705 df-rdg 7743 df-1o 7797 df-2o 7798 df-oadd 7801 df-er 7980 df-map 8095 df-pm 8096 df-ixp 8147 df-en 8194 df-dom 8195 df-sdom 8196 df-fin 8197 df-fsupp 8516 df-sup 8588 df-oi 8655 df-card 9049 df-pnf 10363 df-mnf 10364 df-xr 10365 df-ltxr 10366 df-le 10367 df-sub 10556 df-neg 10557 df-nn 11311 df-2 11372 df-3 11373 df-4 11374 df-5 11375 df-6 11376 df-7 11377 df-8 11378 df-9 11379 df-n0 11577 df-z 11663 df-dec 11780 df-uz 11927 df-fz 12577 df-fzo 12717 df-seq 13052 df-hash 13367 df-struct 16182 df-ndx 16183 df-slot 16184 df-base 16186 df-sets 16187 df-ress 16188 df-plusg 16276 df-mulr 16277 df-sca 16279 df-vsca 16280 df-ip 16281 df-tset 16282 df-ple 16283 df-ds 16285 df-hom 16287 df-cco 16288 df-0g 16413 df-gsum 16414 df-prds 16419 df-pws 16421 df-mre 16557 df-mrc 16558 df-acs 16560 df-mgm 17553 df-sgrp 17595 df-mnd 17606 df-mhm 17646 df-submnd 17647 df-grp 17737 df-minusg 17738 df-sbg 17739 df-mulg 17853 df-subg 17900 df-ghm 17967 df-cntz 18058 df-cmn 18506 df-abl 18507 df-mgp 18802 df-ur 18814 df-srg 18818 df-ring 18861 df-cring 18862 df-rnghom 19029 df-subrg 19092 df-lmod 19179 df-lss 19247 df-lsp 19289 df-assa 19631 df-asp 19632 df-ascl 19633 df-psr 19675 df-mvr 19676 df-mpl 19677 df-opsr 19679 df-evls 19824 df-evl 19825 df-psr1 19868 df-ply1 19870 df-evl1 19999 |
This theorem is referenced by: ply1remlem 24259 lgsqrlem1 25419 idomrootle 38545 lineval 42968 |
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