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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 2737 | . . . . . . 7 ⊢ (𝑅 ↑s 𝐵) = (𝑅 ↑s 𝐵) | |
5 | evl1addd.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝑅) | |
6 | 2, 3, 4, 5 | evl1rhm 21714 | . . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑂 ∈ (𝑃 RingHom (𝑅 ↑s 𝐵))) |
7 | 1, 6 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑂 ∈ (𝑃 RingHom (𝑅 ↑s 𝐵))) |
8 | rhmghm 20166 | . . . . 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 19017 | . . . 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 496 | . . 3 ⊢ (𝜑 → 𝑀 ∈ 𝑈) |
14 | evl1addd.4 | . . . 4 ⊢ (𝜑 → (𝑁 ∈ 𝑈 ∧ ((𝑂‘𝑁)‘𝑌) = 𝑊)) | |
15 | 14 | simpld 496 | . . 3 ⊢ (𝜑 → 𝑁 ∈ 𝑈) |
16 | evl1addd.u | . . . 4 ⊢ 𝑈 = (Base‘𝑃) | |
17 | evl1subd.s | . . . 4 ⊢ − = (-g‘𝑃) | |
18 | 16, 17 | grpsubcl 18834 | . . 3 ⊢ ((𝑃 ∈ Grp ∧ 𝑀 ∈ 𝑈 ∧ 𝑁 ∈ 𝑈) → (𝑀 − 𝑁) ∈ 𝑈) |
19 | 11, 13, 15, 18 | syl3anc 1372 | . 2 ⊢ (𝜑 → (𝑀 − 𝑁) ∈ 𝑈) |
20 | eqid 2737 | . . . . . . 7 ⊢ (-g‘(𝑅 ↑s 𝐵)) = (-g‘(𝑅 ↑s 𝐵)) | |
21 | 16, 17, 20 | ghmsub 19023 | . . . . . 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 1372 | . . . . 5 ⊢ (𝜑 → (𝑂‘(𝑀 − 𝑁)) = ((𝑂‘𝑀)(-g‘(𝑅 ↑s 𝐵))(𝑂‘𝑁))) |
23 | crngring 19983 | . . . . . . 7 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
24 | ringgrp 19976 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
25 | 1, 23, 24 | 3syl 18 | . . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Grp) |
26 | 5 | fvexi 6861 | . . . . . . 7 ⊢ 𝐵 ∈ V |
27 | 26 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ V) |
28 | eqid 2737 | . . . . . . . . 9 ⊢ (Base‘(𝑅 ↑s 𝐵)) = (Base‘(𝑅 ↑s 𝐵)) | |
29 | 16, 28 | rhmf 20167 | . . . . . . . 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 | ffvelcdmd 7041 | . . . . . 6 ⊢ (𝜑 → (𝑂‘𝑀) ∈ (Base‘(𝑅 ↑s 𝐵))) |
32 | 30, 15 | ffvelcdmd 7041 | . . . . . 6 ⊢ (𝜑 → (𝑂‘𝑁) ∈ (Base‘(𝑅 ↑s 𝐵))) |
33 | evl1subd.d | . . . . . . 7 ⊢ 𝐷 = (-g‘𝑅) | |
34 | 4, 28, 33, 20 | pwssub 18868 | . . . . . 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 838 | . . . . 5 ⊢ (𝜑 → ((𝑂‘𝑀)(-g‘(𝑅 ↑s 𝐵))(𝑂‘𝑁)) = ((𝑂‘𝑀) ∘f 𝐷(𝑂‘𝑁))) |
36 | 22, 35 | eqtrd 2777 | . . . 4 ⊢ (𝜑 → (𝑂‘(𝑀 − 𝑁)) = ((𝑂‘𝑀) ∘f 𝐷(𝑂‘𝑁))) |
37 | 36 | fveq1d 6849 | . . 3 ⊢ (𝜑 → ((𝑂‘(𝑀 − 𝑁))‘𝑌) = (((𝑂‘𝑀) ∘f 𝐷(𝑂‘𝑁))‘𝑌)) |
38 | 4, 5, 28, 1, 27, 31 | pwselbas 17378 | . . . . 5 ⊢ (𝜑 → (𝑂‘𝑀):𝐵⟶𝐵) |
39 | 38 | ffnd 6674 | . . . 4 ⊢ (𝜑 → (𝑂‘𝑀) Fn 𝐵) |
40 | 4, 5, 28, 1, 27, 32 | pwselbas 17378 | . . . . 5 ⊢ (𝜑 → (𝑂‘𝑁):𝐵⟶𝐵) |
41 | 40 | ffnd 6674 | . . . 4 ⊢ (𝜑 → (𝑂‘𝑁) Fn 𝐵) |
42 | evl1addd.2 | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
43 | fnfvof 7639 | . . . 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 838 | . . 3 ⊢ (𝜑 → (((𝑂‘𝑀) ∘f 𝐷(𝑂‘𝑁))‘𝑌) = (((𝑂‘𝑀)‘𝑌)𝐷((𝑂‘𝑁)‘𝑌))) |
45 | 12 | simprd 497 | . . . 4 ⊢ (𝜑 → ((𝑂‘𝑀)‘𝑌) = 𝑉) |
46 | 14 | simprd 497 | . . . 4 ⊢ (𝜑 → ((𝑂‘𝑁)‘𝑌) = 𝑊) |
47 | 45, 46 | oveq12d 7380 | . . 3 ⊢ (𝜑 → (((𝑂‘𝑀)‘𝑌)𝐷((𝑂‘𝑁)‘𝑌)) = (𝑉𝐷𝑊)) |
48 | 37, 44, 47 | 3eqtrd 2781 | . 2 ⊢ (𝜑 → ((𝑂‘(𝑀 − 𝑁))‘𝑌) = (𝑉𝐷𝑊)) |
49 | 19, 48 | jca 513 | 1 ⊢ (𝜑 → ((𝑀 − 𝑁) ∈ 𝑈 ∧ ((𝑂‘(𝑀 − 𝑁))‘𝑌) = (𝑉𝐷𝑊))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 Vcvv 3448 Fn wfn 6496 ⟶wf 6497 ‘cfv 6501 (class class class)co 7362 ∘f cof 7620 Basecbs 17090 ↑s cpws 17335 Grpcgrp 18755 -gcsg 18757 GrpHom cghm 19012 Ringcrg 19971 CRingccrg 19972 RingHom crh 20152 Poly1cpl1 21564 eval1ce1 21696 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-iin 4962 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-se 5594 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-isom 6510 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-of 7622 df-ofr 7623 df-om 7808 df-1st 7926 df-2nd 7927 df-supp 8098 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-er 8655 df-map 8774 df-pm 8775 df-ixp 8843 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-fsupp 9313 df-sup 9385 df-oi 9453 df-card 9882 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-nn 12161 df-2 12223 df-3 12224 df-4 12225 df-5 12226 df-6 12227 df-7 12228 df-8 12229 df-9 12230 df-n0 12421 df-z 12507 df-dec 12626 df-uz 12771 df-fz 13432 df-fzo 13575 df-seq 13914 df-hash 14238 df-struct 17026 df-sets 17043 df-slot 17061 df-ndx 17073 df-base 17091 df-ress 17120 df-plusg 17153 df-mulr 17154 df-sca 17156 df-vsca 17157 df-ip 17158 df-tset 17159 df-ple 17160 df-ds 17162 df-hom 17164 df-cco 17165 df-0g 17330 df-gsum 17331 df-prds 17336 df-pws 17338 df-mre 17473 df-mrc 17474 df-acs 17476 df-mgm 18504 df-sgrp 18553 df-mnd 18564 df-mhm 18608 df-submnd 18609 df-grp 18758 df-minusg 18759 df-sbg 18760 df-mulg 18880 df-subg 18932 df-ghm 19013 df-cntz 19104 df-cmn 19571 df-abl 19572 df-mgp 19904 df-ur 19921 df-srg 19925 df-ring 19973 df-cring 19974 df-rnghom 20155 df-subrg 20236 df-lmod 20340 df-lss 20409 df-lsp 20449 df-assa 21275 df-asp 21276 df-ascl 21277 df-psr 21327 df-mvr 21328 df-mpl 21329 df-opsr 21331 df-evls 21498 df-evl 21499 df-psr1 21567 df-ply1 21569 df-evl1 21698 |
This theorem is referenced by: ply1remlem 25543 lgsqrlem1 26710 idomrootle 41551 lineval 46549 |
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