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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 2727 | . . . . . . 7 ⊢ (𝑅 ↑s 𝐵) = (𝑅 ↑s 𝐵) | |
5 | evl1addd.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝑅) | |
6 | 2, 3, 4, 5 | evl1rhm 22256 | . . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑂 ∈ (𝑃 RingHom (𝑅 ↑s 𝐵))) |
7 | 1, 6 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑂 ∈ (𝑃 RingHom (𝑅 ↑s 𝐵))) |
8 | rhmghm 20428 | . . . . 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 19177 | . . . 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 493 | . . 3 ⊢ (𝜑 → 𝑀 ∈ 𝑈) |
14 | evl1addd.4 | . . . 4 ⊢ (𝜑 → (𝑁 ∈ 𝑈 ∧ ((𝑂‘𝑁)‘𝑌) = 𝑊)) | |
15 | 14 | simpld 493 | . . 3 ⊢ (𝜑 → 𝑁 ∈ 𝑈) |
16 | evl1addd.u | . . . 4 ⊢ 𝑈 = (Base‘𝑃) | |
17 | evl1subd.s | . . . 4 ⊢ − = (-g‘𝑃) | |
18 | 16, 17 | grpsubcl 18981 | . . 3 ⊢ ((𝑃 ∈ Grp ∧ 𝑀 ∈ 𝑈 ∧ 𝑁 ∈ 𝑈) → (𝑀 − 𝑁) ∈ 𝑈) |
19 | 11, 13, 15, 18 | syl3anc 1368 | . 2 ⊢ (𝜑 → (𝑀 − 𝑁) ∈ 𝑈) |
20 | eqid 2727 | . . . . . . 7 ⊢ (-g‘(𝑅 ↑s 𝐵)) = (-g‘(𝑅 ↑s 𝐵)) | |
21 | 16, 17, 20 | ghmsub 19183 | . . . . . 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 1368 | . . . . 5 ⊢ (𝜑 → (𝑂‘(𝑀 − 𝑁)) = ((𝑂‘𝑀)(-g‘(𝑅 ↑s 𝐵))(𝑂‘𝑁))) |
23 | crngring 20190 | . . . . . . 7 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
24 | ringgrp 20183 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
25 | 1, 23, 24 | 3syl 18 | . . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Grp) |
26 | 5 | fvexi 6914 | . . . . . . 7 ⊢ 𝐵 ∈ V |
27 | 26 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ V) |
28 | eqid 2727 | . . . . . . . . 9 ⊢ (Base‘(𝑅 ↑s 𝐵)) = (Base‘(𝑅 ↑s 𝐵)) | |
29 | 16, 28 | rhmf 20429 | . . . . . . . 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 7098 | . . . . . 6 ⊢ (𝜑 → (𝑂‘𝑀) ∈ (Base‘(𝑅 ↑s 𝐵))) |
32 | 30, 15 | ffvelcdmd 7098 | . . . . . 6 ⊢ (𝜑 → (𝑂‘𝑁) ∈ (Base‘(𝑅 ↑s 𝐵))) |
33 | evl1subd.d | . . . . . . 7 ⊢ 𝐷 = (-g‘𝑅) | |
34 | 4, 28, 33, 20 | pwssub 19015 | . . . . . 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 837 | . . . . 5 ⊢ (𝜑 → ((𝑂‘𝑀)(-g‘(𝑅 ↑s 𝐵))(𝑂‘𝑁)) = ((𝑂‘𝑀) ∘f 𝐷(𝑂‘𝑁))) |
36 | 22, 35 | eqtrd 2767 | . . . 4 ⊢ (𝜑 → (𝑂‘(𝑀 − 𝑁)) = ((𝑂‘𝑀) ∘f 𝐷(𝑂‘𝑁))) |
37 | 36 | fveq1d 6902 | . . 3 ⊢ (𝜑 → ((𝑂‘(𝑀 − 𝑁))‘𝑌) = (((𝑂‘𝑀) ∘f 𝐷(𝑂‘𝑁))‘𝑌)) |
38 | 4, 5, 28, 1, 27, 31 | pwselbas 17476 | . . . . 5 ⊢ (𝜑 → (𝑂‘𝑀):𝐵⟶𝐵) |
39 | 38 | ffnd 6726 | . . . 4 ⊢ (𝜑 → (𝑂‘𝑀) Fn 𝐵) |
40 | 4, 5, 28, 1, 27, 32 | pwselbas 17476 | . . . . 5 ⊢ (𝜑 → (𝑂‘𝑁):𝐵⟶𝐵) |
41 | 40 | ffnd 6726 | . . . 4 ⊢ (𝜑 → (𝑂‘𝑁) Fn 𝐵) |
42 | evl1addd.2 | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
43 | fnfvof 7706 | . . . 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 837 | . . 3 ⊢ (𝜑 → (((𝑂‘𝑀) ∘f 𝐷(𝑂‘𝑁))‘𝑌) = (((𝑂‘𝑀)‘𝑌)𝐷((𝑂‘𝑁)‘𝑌))) |
45 | 12 | simprd 494 | . . . 4 ⊢ (𝜑 → ((𝑂‘𝑀)‘𝑌) = 𝑉) |
46 | 14 | simprd 494 | . . . 4 ⊢ (𝜑 → ((𝑂‘𝑁)‘𝑌) = 𝑊) |
47 | 45, 46 | oveq12d 7442 | . . 3 ⊢ (𝜑 → (((𝑂‘𝑀)‘𝑌)𝐷((𝑂‘𝑁)‘𝑌)) = (𝑉𝐷𝑊)) |
48 | 37, 44, 47 | 3eqtrd 2771 | . 2 ⊢ (𝜑 → ((𝑂‘(𝑀 − 𝑁))‘𝑌) = (𝑉𝐷𝑊)) |
49 | 19, 48 | jca 510 | 1 ⊢ (𝜑 → ((𝑀 − 𝑁) ∈ 𝑈 ∧ ((𝑂‘(𝑀 − 𝑁))‘𝑌) = (𝑉𝐷𝑊))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 Vcvv 3471 Fn wfn 6546 ⟶wf 6547 ‘cfv 6551 (class class class)co 7424 ∘f cof 7687 Basecbs 17185 ↑s cpws 17433 Grpcgrp 18895 -gcsg 18897 GrpHom cghm 19172 Ringcrg 20178 CRingccrg 20179 RingHom crh 20413 Poly1cpl1 22101 eval1ce1 22238 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2698 ax-rep 5287 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 ax-cnex 11200 ax-resscn 11201 ax-1cn 11202 ax-icn 11203 ax-addcl 11204 ax-addrcl 11205 ax-mulcl 11206 ax-mulrcl 11207 ax-mulcom 11208 ax-addass 11209 ax-mulass 11210 ax-distr 11211 ax-i2m1 11212 ax-1ne0 11213 ax-1rid 11214 ax-rnegex 11215 ax-rrecex 11216 ax-cnre 11217 ax-pre-lttri 11218 ax-pre-lttrn 11219 ax-pre-ltadd 11220 ax-pre-mulgt0 11221 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4911 df-int 4952 df-iun 5000 df-iin 5001 df-br 5151 df-opab 5213 df-mpt 5234 df-tr 5268 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5635 df-se 5636 df-we 5637 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-pred 6308 df-ord 6375 df-on 6376 df-lim 6377 df-suc 6378 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-isom 6560 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-of 7689 df-ofr 7690 df-om 7875 df-1st 7997 df-2nd 7998 df-supp 8170 df-frecs 8291 df-wrecs 8322 df-recs 8396 df-rdg 8435 df-1o 8491 df-er 8729 df-map 8851 df-pm 8852 df-ixp 8921 df-en 8969 df-dom 8970 df-sdom 8971 df-fin 8972 df-fsupp 9392 df-sup 9471 df-oi 9539 df-card 9968 df-pnf 11286 df-mnf 11287 df-xr 11288 df-ltxr 11289 df-le 11290 df-sub 11482 df-neg 11483 df-nn 12249 df-2 12311 df-3 12312 df-4 12313 df-5 12314 df-6 12315 df-7 12316 df-8 12317 df-9 12318 df-n0 12509 df-z 12595 df-dec 12714 df-uz 12859 df-fz 13523 df-fzo 13666 df-seq 14005 df-hash 14328 df-struct 17121 df-sets 17138 df-slot 17156 df-ndx 17168 df-base 17186 df-ress 17215 df-plusg 17251 df-mulr 17252 df-sca 17254 df-vsca 17255 df-ip 17256 df-tset 17257 df-ple 17258 df-ds 17260 df-hom 17262 df-cco 17263 df-0g 17428 df-gsum 17429 df-prds 17434 df-pws 17436 df-mre 17571 df-mrc 17572 df-acs 17574 df-mgm 18605 df-sgrp 18684 df-mnd 18700 df-mhm 18745 df-submnd 18746 df-grp 18898 df-minusg 18899 df-sbg 18900 df-mulg 19029 df-subg 19083 df-ghm 19173 df-cntz 19273 df-cmn 19742 df-abl 19743 df-mgp 20080 df-rng 20098 df-ur 20127 df-srg 20132 df-ring 20180 df-cring 20181 df-rhm 20416 df-subrng 20488 df-subrg 20513 df-lmod 20750 df-lss 20821 df-lsp 20861 df-assa 21792 df-asp 21793 df-ascl 21794 df-psr 21847 df-mvr 21848 df-mpl 21849 df-opsr 21851 df-evls 22023 df-evl 22024 df-psr1 22104 df-ply1 22106 df-evl1 22240 |
This theorem is referenced by: ply1remlem 26117 idomrootle 26125 lgsqrlem1 27297 evls1subd 33261 aks6d1c2lem4 41602 aks6d1c6lem2 41647 lineval 47513 |
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