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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 2610 | . . . . . 6 ⊢ (𝑅 ↑s 𝐵) = (𝑅 ↑s 𝐵) | |
5 | evl1addd.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝑅) | |
6 | 2, 3, 4, 5 | evl1rhm 19517 | . . . . 5 ⊢ (𝑅 ∈ CRing → 𝑂 ∈ (𝑃 RingHom (𝑅 ↑s 𝐵))) |
7 | 1, 6 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑂 ∈ (𝑃 RingHom (𝑅 ↑s 𝐵))) |
8 | rhmrcl1 18542 | . . . 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 474 | . . 3 ⊢ (𝜑 → 𝑀 ∈ 𝑈) |
12 | evl1addd.4 | . . . 4 ⊢ (𝜑 → (𝑁 ∈ 𝑈 ∧ ((𝑂‘𝑁)‘𝑌) = 𝑊)) | |
13 | 12 | simpld 474 | . . 3 ⊢ (𝜑 → 𝑁 ∈ 𝑈) |
14 | evl1addd.u | . . . 4 ⊢ 𝑈 = (Base‘𝑃) | |
15 | evl1muld.t | . . . 4 ⊢ ∙ = (.r‘𝑃) | |
16 | 14, 15 | ringcl 18384 | . . 3 ⊢ ((𝑃 ∈ Ring ∧ 𝑀 ∈ 𝑈 ∧ 𝑁 ∈ 𝑈) → (𝑀 ∙ 𝑁) ∈ 𝑈) |
17 | 9, 11, 13, 16 | syl3anc 1318 | . 2 ⊢ (𝜑 → (𝑀 ∙ 𝑁) ∈ 𝑈) |
18 | eqid 2610 | . . . . . . 7 ⊢ (.r‘(𝑅 ↑s 𝐵)) = (.r‘(𝑅 ↑s 𝐵)) | |
19 | 14, 15, 18 | rhmmul 18550 | . . . . . 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 1318 | . . . . 5 ⊢ (𝜑 → (𝑂‘(𝑀 ∙ 𝑁)) = ((𝑂‘𝑀)(.r‘(𝑅 ↑s 𝐵))(𝑂‘𝑁))) |
21 | eqid 2610 | . . . . . 6 ⊢ (Base‘(𝑅 ↑s 𝐵)) = (Base‘(𝑅 ↑s 𝐵)) | |
22 | fvex 6113 | . . . . . . . 8 ⊢ (Base‘𝑅) ∈ V | |
23 | 5, 22 | eqeltri 2684 | . . . . . . 7 ⊢ 𝐵 ∈ V |
24 | 23 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ V) |
25 | 14, 21 | rhmf 18549 | . . . . . . . 8 ⊢ (𝑂 ∈ (𝑃 RingHom (𝑅 ↑s 𝐵)) → 𝑂:𝑈⟶(Base‘(𝑅 ↑s 𝐵))) |
26 | 7, 25 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑂:𝑈⟶(Base‘(𝑅 ↑s 𝐵))) |
27 | 26, 11 | ffvelrnd 6268 | . . . . . 6 ⊢ (𝜑 → (𝑂‘𝑀) ∈ (Base‘(𝑅 ↑s 𝐵))) |
28 | 26, 13 | ffvelrnd 6268 | . . . . . 6 ⊢ (𝜑 → (𝑂‘𝑁) ∈ (Base‘(𝑅 ↑s 𝐵))) |
29 | evl1muld.s | . . . . . 6 ⊢ · = (.r‘𝑅) | |
30 | 4, 21, 1, 24, 27, 28, 29, 18 | pwsmulrval 15974 | . . . . 5 ⊢ (𝜑 → ((𝑂‘𝑀)(.r‘(𝑅 ↑s 𝐵))(𝑂‘𝑁)) = ((𝑂‘𝑀) ∘𝑓 · (𝑂‘𝑁))) |
31 | 20, 30 | eqtrd 2644 | . . . 4 ⊢ (𝜑 → (𝑂‘(𝑀 ∙ 𝑁)) = ((𝑂‘𝑀) ∘𝑓 · (𝑂‘𝑁))) |
32 | 31 | fveq1d 6105 | . . 3 ⊢ (𝜑 → ((𝑂‘(𝑀 ∙ 𝑁))‘𝑌) = (((𝑂‘𝑀) ∘𝑓 · (𝑂‘𝑁))‘𝑌)) |
33 | 4, 5, 21, 1, 24, 27 | pwselbas 15972 | . . . . 5 ⊢ (𝜑 → (𝑂‘𝑀):𝐵⟶𝐵) |
34 | ffn 5958 | . . . . 5 ⊢ ((𝑂‘𝑀):𝐵⟶𝐵 → (𝑂‘𝑀) Fn 𝐵) | |
35 | 33, 34 | syl 17 | . . . 4 ⊢ (𝜑 → (𝑂‘𝑀) Fn 𝐵) |
36 | 4, 5, 21, 1, 24, 28 | pwselbas 15972 | . . . . 5 ⊢ (𝜑 → (𝑂‘𝑁):𝐵⟶𝐵) |
37 | ffn 5958 | . . . . 5 ⊢ ((𝑂‘𝑁):𝐵⟶𝐵 → (𝑂‘𝑁) Fn 𝐵) | |
38 | 36, 37 | syl 17 | . . . 4 ⊢ (𝜑 → (𝑂‘𝑁) Fn 𝐵) |
39 | evl1addd.2 | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
40 | fnfvof 6809 | . . . 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))) | |
41 | 35, 38, 24, 39, 40 | syl22anc 1319 | . . 3 ⊢ (𝜑 → (((𝑂‘𝑀) ∘𝑓 · (𝑂‘𝑁))‘𝑌) = (((𝑂‘𝑀)‘𝑌) · ((𝑂‘𝑁)‘𝑌))) |
42 | 10 | simprd 478 | . . . 4 ⊢ (𝜑 → ((𝑂‘𝑀)‘𝑌) = 𝑉) |
43 | 12 | simprd 478 | . . . 4 ⊢ (𝜑 → ((𝑂‘𝑁)‘𝑌) = 𝑊) |
44 | 42, 43 | oveq12d 6567 | . . 3 ⊢ (𝜑 → (((𝑂‘𝑀)‘𝑌) · ((𝑂‘𝑁)‘𝑌)) = (𝑉 · 𝑊)) |
45 | 32, 41, 44 | 3eqtrd 2648 | . 2 ⊢ (𝜑 → ((𝑂‘(𝑀 ∙ 𝑁))‘𝑌) = (𝑉 · 𝑊)) |
46 | 17, 45 | jca 553 | 1 ⊢ (𝜑 → ((𝑀 ∙ 𝑁) ∈ 𝑈 ∧ ((𝑂‘(𝑀 ∙ 𝑁))‘𝑌) = (𝑉 · 𝑊))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 Fn wfn 5799 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ∘𝑓 cof 6793 Basecbs 15695 .rcmulr 15769 ↑s cpws 15930 Ringcrg 18370 CRingccrg 18371 RingHom crh 18535 Poly1cpl1 19368 eval1ce1 19500 |
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-evl 19328 df-psr1 19371 df-ply1 19373 df-evl1 19502 |
This theorem is referenced by: evl1vsd 19529 |
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