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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 2820 | . . . . . 6 ⊢ (𝑅 ↑s 𝐵) = (𝑅 ↑s 𝐵) | |
5 | evl1addd.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝑅) | |
6 | 2, 3, 4, 5 | evl1rhm 20490 | . . . . 5 ⊢ (𝑅 ∈ CRing → 𝑂 ∈ (𝑃 RingHom (𝑅 ↑s 𝐵))) |
7 | 1, 6 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑂 ∈ (𝑃 RingHom (𝑅 ↑s 𝐵))) |
8 | rhmrcl1 19466 | . . . 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 19306 | . . 3 ⊢ ((𝑃 ∈ Ring ∧ 𝑀 ∈ 𝑈 ∧ 𝑁 ∈ 𝑈) → (𝑀 ∙ 𝑁) ∈ 𝑈) |
17 | 9, 11, 13, 16 | syl3anc 1366 | . 2 ⊢ (𝜑 → (𝑀 ∙ 𝑁) ∈ 𝑈) |
18 | eqid 2820 | . . . . . . 7 ⊢ (.r‘(𝑅 ↑s 𝐵)) = (.r‘(𝑅 ↑s 𝐵)) | |
19 | 14, 15, 18 | rhmmul 19474 | . . . . . 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 1366 | . . . . 5 ⊢ (𝜑 → (𝑂‘(𝑀 ∙ 𝑁)) = ((𝑂‘𝑀)(.r‘(𝑅 ↑s 𝐵))(𝑂‘𝑁))) |
21 | eqid 2820 | . . . . . 6 ⊢ (Base‘(𝑅 ↑s 𝐵)) = (Base‘(𝑅 ↑s 𝐵)) | |
22 | 5 | fvexi 6677 | . . . . . . 7 ⊢ 𝐵 ∈ V |
23 | 22 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ V) |
24 | 14, 21 | rhmf 19473 | . . . . . . . 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 6845 | . . . . . 6 ⊢ (𝜑 → (𝑂‘𝑀) ∈ (Base‘(𝑅 ↑s 𝐵))) |
27 | 25, 13 | ffvelrnd 6845 | . . . . . 6 ⊢ (𝜑 → (𝑂‘𝑁) ∈ (Base‘(𝑅 ↑s 𝐵))) |
28 | evl1muld.s | . . . . . 6 ⊢ · = (.r‘𝑅) | |
29 | 4, 21, 1, 23, 26, 27, 28, 18 | pwsmulrval 16759 | . . . . 5 ⊢ (𝜑 → ((𝑂‘𝑀)(.r‘(𝑅 ↑s 𝐵))(𝑂‘𝑁)) = ((𝑂‘𝑀) ∘f · (𝑂‘𝑁))) |
30 | 20, 29 | eqtrd 2855 | . . . 4 ⊢ (𝜑 → (𝑂‘(𝑀 ∙ 𝑁)) = ((𝑂‘𝑀) ∘f · (𝑂‘𝑁))) |
31 | 30 | fveq1d 6665 | . . 3 ⊢ (𝜑 → ((𝑂‘(𝑀 ∙ 𝑁))‘𝑌) = (((𝑂‘𝑀) ∘f · (𝑂‘𝑁))‘𝑌)) |
32 | 4, 5, 21, 1, 23, 26 | pwselbas 16757 | . . . . 5 ⊢ (𝜑 → (𝑂‘𝑀):𝐵⟶𝐵) |
33 | 32 | ffnd 6508 | . . . 4 ⊢ (𝜑 → (𝑂‘𝑀) Fn 𝐵) |
34 | 4, 5, 21, 1, 23, 27 | pwselbas 16757 | . . . . 5 ⊢ (𝜑 → (𝑂‘𝑁):𝐵⟶𝐵) |
35 | 34 | ffnd 6508 | . . . 4 ⊢ (𝜑 → (𝑂‘𝑁) Fn 𝐵) |
36 | evl1addd.2 | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
37 | fnfvof 7416 | . . . 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 7167 | . . 3 ⊢ (𝜑 → (((𝑂‘𝑀)‘𝑌) · ((𝑂‘𝑁)‘𝑌)) = (𝑉 · 𝑊)) |
42 | 31, 38, 41 | 3eqtrd 2859 | . 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 1536 ∈ wcel 2113 Vcvv 3491 Fn wfn 6343 ⟶wf 6344 ‘cfv 6348 (class class class)co 7149 ∘f cof 7400 Basecbs 16478 .rcmulr 16561 ↑s cpws 16715 Ringcrg 19292 CRingccrg 19293 RingHom crh 19459 Poly1cpl1 20340 eval1ce1 20472 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-cnex 10586 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 ax-pre-mulgt0 10607 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4870 df-iun 4914 df-iin 4915 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-of 7402 df-ofr 7403 df-om 7574 df-1st 7682 df-2nd 7683 df-supp 7824 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-1o 8095 df-2o 8096 df-oadd 8099 df-er 8282 df-map 8401 df-pm 8402 df-ixp 8455 df-en 8503 df-dom 8504 df-sdom 8505 df-fin 8506 df-fsupp 8827 df-sup 8899 df-oi 8967 df-card 9361 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 df-nn 11632 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 12890 df-fzo 13031 df-seq 13367 df-hash 13688 df-struct 16480 df-ndx 16481 df-slot 16482 df-base 16484 df-sets 16485 df-ress 16486 df-plusg 16573 df-mulr 16574 df-sca 16576 df-vsca 16577 df-ip 16578 df-tset 16579 df-ple 16580 df-ds 16582 df-hom 16584 df-cco 16585 df-0g 16710 df-gsum 16711 df-prds 16716 df-pws 16718 df-mre 16852 df-mrc 16853 df-acs 16855 df-mgm 17847 df-sgrp 17896 df-mnd 17907 df-mhm 17951 df-submnd 17952 df-grp 18101 df-minusg 18102 df-sbg 18103 df-mulg 18220 df-subg 18271 df-ghm 18351 df-cntz 18442 df-cmn 18903 df-abl 18904 df-mgp 19235 df-ur 19247 df-srg 19251 df-ring 19294 df-cring 19295 df-rnghom 19462 df-subrg 19528 df-lmod 19631 df-lss 19699 df-lsp 19739 df-assa 20080 df-asp 20081 df-ascl 20082 df-psr 20131 df-mvr 20132 df-mpl 20133 df-opsr 20135 df-evls 20281 df-evl 20282 df-psr1 20343 df-ply1 20345 df-evl1 20474 |
This theorem is referenced by: evl1vsd 20502 |
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