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Mirrors > Home > MPE Home > Th. List > evl1muld | Structured version Visualization version GIF version |
Description: Polynomial evaluation builder for multiplication 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)) |
evl1muld.t | ⊢ ∙ = (.r‘𝑃) |
evl1muld.s | ⊢ · = (.r‘𝑅) |
Ref | Expression |
---|---|
evl1muld | ⊢ (𝜑 → ((𝑀 ∙ 𝑁) ∈ 𝑈 ∧ ((𝑂‘(𝑀 ∙ 𝑁))‘𝑌) = (𝑉 · 𝑊))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | evl1addd.1 | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ CRing) | |
2 | evl1addd.q | . . . . . 6 ⊢ 𝑂 = (eval1‘𝑅) | |
3 | evl1addd.p | . . . . . 6 ⊢ 𝑃 = (Poly1‘𝑅) | |
4 | eqid 2821 | . . . . . 6 ⊢ (𝑅 ↑s 𝐵) = (𝑅 ↑s 𝐵) | |
5 | evl1addd.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝑅) | |
6 | 2, 3, 4, 5 | evl1rhm 20489 | . . . . 5 ⊢ (𝑅 ∈ CRing → 𝑂 ∈ (𝑃 RingHom (𝑅 ↑s 𝐵))) |
7 | 1, 6 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑂 ∈ (𝑃 RingHom (𝑅 ↑s 𝐵))) |
8 | rhmrcl1 19465 | . . . 4 ⊢ (𝑂 ∈ (𝑃 RingHom (𝑅 ↑s 𝐵)) → 𝑃 ∈ Ring) | |
9 | 7, 8 | syl 17 | . . 3 ⊢ (𝜑 → 𝑃 ∈ Ring) |
10 | evl1addd.3 | . . . 4 ⊢ (𝜑 → (𝑀 ∈ 𝑈 ∧ ((𝑂‘𝑀)‘𝑌) = 𝑉)) | |
11 | 10 | simpld 497 | . . 3 ⊢ (𝜑 → 𝑀 ∈ 𝑈) |
12 | evl1addd.4 | . . . 4 ⊢ (𝜑 → (𝑁 ∈ 𝑈 ∧ ((𝑂‘𝑁)‘𝑌) = 𝑊)) | |
13 | 12 | simpld 497 | . . 3 ⊢ (𝜑 → 𝑁 ∈ 𝑈) |
14 | evl1addd.u | . . . 4 ⊢ 𝑈 = (Base‘𝑃) | |
15 | evl1muld.t | . . . 4 ⊢ ∙ = (.r‘𝑃) | |
16 | 14, 15 | ringcl 19305 | . . 3 ⊢ ((𝑃 ∈ Ring ∧ 𝑀 ∈ 𝑈 ∧ 𝑁 ∈ 𝑈) → (𝑀 ∙ 𝑁) ∈ 𝑈) |
17 | 9, 11, 13, 16 | syl3anc 1367 | . 2 ⊢ (𝜑 → (𝑀 ∙ 𝑁) ∈ 𝑈) |
18 | eqid 2821 | . . . . . . 7 ⊢ (.r‘(𝑅 ↑s 𝐵)) = (.r‘(𝑅 ↑s 𝐵)) | |
19 | 14, 15, 18 | rhmmul 19473 | . . . . . 6 ⊢ ((𝑂 ∈ (𝑃 RingHom (𝑅 ↑s 𝐵)) ∧ 𝑀 ∈ 𝑈 ∧ 𝑁 ∈ 𝑈) → (𝑂‘(𝑀 ∙ 𝑁)) = ((𝑂‘𝑀)(.r‘(𝑅 ↑s 𝐵))(𝑂‘𝑁))) |
20 | 7, 11, 13, 19 | syl3anc 1367 | . . . . 5 ⊢ (𝜑 → (𝑂‘(𝑀 ∙ 𝑁)) = ((𝑂‘𝑀)(.r‘(𝑅 ↑s 𝐵))(𝑂‘𝑁))) |
21 | eqid 2821 | . . . . . 6 ⊢ (Base‘(𝑅 ↑s 𝐵)) = (Base‘(𝑅 ↑s 𝐵)) | |
22 | 5 | fvexi 6679 | . . . . . . 7 ⊢ 𝐵 ∈ V |
23 | 22 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ V) |
24 | 14, 21 | rhmf 19472 | . . . . . . . 8 ⊢ (𝑂 ∈ (𝑃 RingHom (𝑅 ↑s 𝐵)) → 𝑂:𝑈⟶(Base‘(𝑅 ↑s 𝐵))) |
25 | 7, 24 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑂:𝑈⟶(Base‘(𝑅 ↑s 𝐵))) |
26 | 25, 11 | ffvelrnd 6847 | . . . . . 6 ⊢ (𝜑 → (𝑂‘𝑀) ∈ (Base‘(𝑅 ↑s 𝐵))) |
27 | 25, 13 | ffvelrnd 6847 | . . . . . 6 ⊢ (𝜑 → (𝑂‘𝑁) ∈ (Base‘(𝑅 ↑s 𝐵))) |
28 | evl1muld.s | . . . . . 6 ⊢ · = (.r‘𝑅) | |
29 | 4, 21, 1, 23, 26, 27, 28, 18 | pwsmulrval 16758 | . . . . 5 ⊢ (𝜑 → ((𝑂‘𝑀)(.r‘(𝑅 ↑s 𝐵))(𝑂‘𝑁)) = ((𝑂‘𝑀) ∘f · (𝑂‘𝑁))) |
30 | 20, 29 | eqtrd 2856 | . . . 4 ⊢ (𝜑 → (𝑂‘(𝑀 ∙ 𝑁)) = ((𝑂‘𝑀) ∘f · (𝑂‘𝑁))) |
31 | 30 | fveq1d 6667 | . . 3 ⊢ (𝜑 → ((𝑂‘(𝑀 ∙ 𝑁))‘𝑌) = (((𝑂‘𝑀) ∘f · (𝑂‘𝑁))‘𝑌)) |
32 | 4, 5, 21, 1, 23, 26 | pwselbas 16756 | . . . . 5 ⊢ (𝜑 → (𝑂‘𝑀):𝐵⟶𝐵) |
33 | 32 | ffnd 6510 | . . . 4 ⊢ (𝜑 → (𝑂‘𝑀) Fn 𝐵) |
34 | 4, 5, 21, 1, 23, 27 | pwselbas 16756 | . . . . 5 ⊢ (𝜑 → (𝑂‘𝑁):𝐵⟶𝐵) |
35 | 34 | ffnd 6510 | . . . 4 ⊢ (𝜑 → (𝑂‘𝑁) Fn 𝐵) |
36 | evl1addd.2 | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
37 | fnfvof 7417 | . . . 4 ⊢ ((((𝑂‘𝑀) Fn 𝐵 ∧ (𝑂‘𝑁) Fn 𝐵) ∧ (𝐵 ∈ V ∧ 𝑌 ∈ 𝐵)) → (((𝑂‘𝑀) ∘f · (𝑂‘𝑁))‘𝑌) = (((𝑂‘𝑀)‘𝑌) · ((𝑂‘𝑁)‘𝑌))) | |
38 | 33, 35, 23, 36, 37 | syl22anc 836 | . . 3 ⊢ (𝜑 → (((𝑂‘𝑀) ∘f · (𝑂‘𝑁))‘𝑌) = (((𝑂‘𝑀)‘𝑌) · ((𝑂‘𝑁)‘𝑌))) |
39 | 10 | simprd 498 | . . . 4 ⊢ (𝜑 → ((𝑂‘𝑀)‘𝑌) = 𝑉) |
40 | 12 | simprd 498 | . . . 4 ⊢ (𝜑 → ((𝑂‘𝑁)‘𝑌) = 𝑊) |
41 | 39, 40 | oveq12d 7168 | . . 3 ⊢ (𝜑 → (((𝑂‘𝑀)‘𝑌) · ((𝑂‘𝑁)‘𝑌)) = (𝑉 · 𝑊)) |
42 | 31, 38, 41 | 3eqtrd 2860 | . 2 ⊢ (𝜑 → ((𝑂‘(𝑀 ∙ 𝑁))‘𝑌) = (𝑉 · 𝑊)) |
43 | 17, 42 | jca 514 | 1 ⊢ (𝜑 → ((𝑀 ∙ 𝑁) ∈ 𝑈 ∧ ((𝑂‘(𝑀 ∙ 𝑁))‘𝑌) = (𝑉 · 𝑊))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 Vcvv 3495 Fn wfn 6345 ⟶wf 6346 ‘cfv 6350 (class class class)co 7150 ∘f cof 7401 Basecbs 16477 .rcmulr 16560 ↑s cpws 16714 Ringcrg 19291 CRingccrg 19292 RingHom crh 19458 Poly1cpl1 20339 eval1ce1 20471 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-iin 4915 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-se 5510 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-isom 6359 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7403 df-ofr 7404 df-om 7575 df-1st 7683 df-2nd 7684 df-supp 7825 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-2o 8097 df-oadd 8100 df-er 8283 df-map 8402 df-pm 8403 df-ixp 8456 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-fsupp 8828 df-sup 8900 df-oi 8968 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-fz 12887 df-fzo 13028 df-seq 13364 df-hash 13685 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-sca 16575 df-vsca 16576 df-ip 16577 df-tset 16578 df-ple 16579 df-ds 16581 df-hom 16583 df-cco 16584 df-0g 16709 df-gsum 16710 df-prds 16715 df-pws 16717 df-mre 16851 df-mrc 16852 df-acs 16854 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-mhm 17950 df-submnd 17951 df-grp 18100 df-minusg 18101 df-sbg 18102 df-mulg 18219 df-subg 18270 df-ghm 18350 df-cntz 18441 df-cmn 18902 df-abl 18903 df-mgp 19234 df-ur 19246 df-srg 19250 df-ring 19293 df-cring 19294 df-rnghom 19461 df-subrg 19527 df-lmod 19630 df-lss 19698 df-lsp 19738 df-assa 20079 df-asp 20080 df-ascl 20081 df-psr 20130 df-mvr 20131 df-mpl 20132 df-opsr 20134 df-evls 20280 df-evl 20281 df-psr1 20342 df-ply1 20344 df-evl1 20473 |
This theorem is referenced by: evl1vsd 20501 |
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