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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > r1pvsca | Structured version Visualization version GIF version |
Description: Scalar multiplication property of the polynomial remainder operation. (Contributed by Thierry Arnoux, 2-Apr-2025.) |
Ref | Expression |
---|---|
r1padd1.p | ⊢ 𝑃 = (Poly1‘𝑅) |
r1padd1.u | ⊢ 𝑈 = (Base‘𝑃) |
r1padd1.n | ⊢ 𝑁 = (Unic1p‘𝑅) |
r1padd1.e | ⊢ 𝐸 = (rem1p‘𝑅) |
r1pvsca.6 | ⊢ (𝜑 → 𝑅 ∈ Ring) |
r1pvsca.7 | ⊢ (𝜑 → 𝐴 ∈ 𝑈) |
r1pvsca.10 | ⊢ (𝜑 → 𝐷 ∈ 𝑁) |
r1pvsca.1 | ⊢ × = ( ·𝑠 ‘𝑃) |
r1pvsca.k | ⊢ 𝐾 = (Base‘𝑅) |
r1pvsca.2 | ⊢ (𝜑 → 𝐵 ∈ 𝐾) |
Ref | Expression |
---|---|
r1pvsca | ⊢ (𝜑 → ((𝐵 × 𝐴)𝐸𝐷) = (𝐵 × (𝐴𝐸𝐷))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | r1pvsca.6 | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
2 | r1pvsca.2 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝐾) | |
3 | r1pvsca.7 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑈) | |
4 | r1pvsca.10 | . . . . . 6 ⊢ (𝜑 → 𝐷 ∈ 𝑁) | |
5 | eqid 2735 | . . . . . . 7 ⊢ (quot1p‘𝑅) = (quot1p‘𝑅) | |
6 | r1padd1.p | . . . . . . 7 ⊢ 𝑃 = (Poly1‘𝑅) | |
7 | r1padd1.u | . . . . . . 7 ⊢ 𝑈 = (Base‘𝑃) | |
8 | r1padd1.n | . . . . . . 7 ⊢ 𝑁 = (Unic1p‘𝑅) | |
9 | 5, 6, 7, 8 | q1pcl 26211 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐴 ∈ 𝑈 ∧ 𝐷 ∈ 𝑁) → (𝐴(quot1p‘𝑅)𝐷) ∈ 𝑈) |
10 | 1, 3, 4, 9 | syl3anc 1370 | . . . . 5 ⊢ (𝜑 → (𝐴(quot1p‘𝑅)𝐷) ∈ 𝑈) |
11 | 6, 7, 8 | uc1pcl 26198 | . . . . . 6 ⊢ (𝐷 ∈ 𝑁 → 𝐷 ∈ 𝑈) |
12 | 4, 11 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ 𝑈) |
13 | eqid 2735 | . . . . . 6 ⊢ (.r‘𝑃) = (.r‘𝑃) | |
14 | r1pvsca.k | . . . . . 6 ⊢ 𝐾 = (Base‘𝑅) | |
15 | r1pvsca.1 | . . . . . 6 ⊢ × = ( ·𝑠 ‘𝑃) | |
16 | 6, 13, 7, 14, 15 | ply1ass23l 22244 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (𝐵 ∈ 𝐾 ∧ (𝐴(quot1p‘𝑅)𝐷) ∈ 𝑈 ∧ 𝐷 ∈ 𝑈)) → ((𝐵 × (𝐴(quot1p‘𝑅)𝐷))(.r‘𝑃)𝐷) = (𝐵 × ((𝐴(quot1p‘𝑅)𝐷)(.r‘𝑃)𝐷))) |
17 | 1, 2, 10, 12, 16 | syl13anc 1371 | . . . 4 ⊢ (𝜑 → ((𝐵 × (𝐴(quot1p‘𝑅)𝐷))(.r‘𝑃)𝐷) = (𝐵 × ((𝐴(quot1p‘𝑅)𝐷)(.r‘𝑃)𝐷))) |
18 | 17 | oveq2d 7447 | . . 3 ⊢ (𝜑 → ((𝐵 × 𝐴)(-g‘𝑃)((𝐵 × (𝐴(quot1p‘𝑅)𝐷))(.r‘𝑃)𝐷)) = ((𝐵 × 𝐴)(-g‘𝑃)(𝐵 × ((𝐴(quot1p‘𝑅)𝐷)(.r‘𝑃)𝐷)))) |
19 | 6, 7, 8, 5, 1, 3, 4, 15, 14, 2 | q1pvsca 33604 | . . . . 5 ⊢ (𝜑 → ((𝐵 × 𝐴)(quot1p‘𝑅)𝐷) = (𝐵 × (𝐴(quot1p‘𝑅)𝐷))) |
20 | 19 | oveq1d 7446 | . . . 4 ⊢ (𝜑 → (((𝐵 × 𝐴)(quot1p‘𝑅)𝐷)(.r‘𝑃)𝐷) = ((𝐵 × (𝐴(quot1p‘𝑅)𝐷))(.r‘𝑃)𝐷)) |
21 | 20 | oveq2d 7447 | . . 3 ⊢ (𝜑 → ((𝐵 × 𝐴)(-g‘𝑃)(((𝐵 × 𝐴)(quot1p‘𝑅)𝐷)(.r‘𝑃)𝐷)) = ((𝐵 × 𝐴)(-g‘𝑃)((𝐵 × (𝐴(quot1p‘𝑅)𝐷))(.r‘𝑃)𝐷))) |
22 | eqid 2735 | . . . 4 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃) | |
23 | eqid 2735 | . . . 4 ⊢ (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃)) | |
24 | eqid 2735 | . . . 4 ⊢ (-g‘𝑃) = (-g‘𝑃) | |
25 | 6 | ply1lmod 22269 | . . . . 5 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ LMod) |
26 | 1, 25 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑃 ∈ LMod) |
27 | 6 | ply1sca 22270 | . . . . . . . 8 ⊢ (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑃)) |
28 | 1, 27 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑅 = (Scalar‘𝑃)) |
29 | 28 | fveq2d 6911 | . . . . . 6 ⊢ (𝜑 → (Base‘𝑅) = (Base‘(Scalar‘𝑃))) |
30 | 14, 29 | eqtrid 2787 | . . . . 5 ⊢ (𝜑 → 𝐾 = (Base‘(Scalar‘𝑃))) |
31 | 2, 30 | eleqtrd 2841 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ (Base‘(Scalar‘𝑃))) |
32 | 6 | ply1ring 22265 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring) |
33 | 1, 32 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑃 ∈ Ring) |
34 | 7, 13, 33, 10, 12 | ringcld 20277 | . . . 4 ⊢ (𝜑 → ((𝐴(quot1p‘𝑅)𝐷)(.r‘𝑃)𝐷) ∈ 𝑈) |
35 | 7, 15, 22, 23, 24, 26, 31, 3, 34 | lmodsubdi 20934 | . . 3 ⊢ (𝜑 → (𝐵 × (𝐴(-g‘𝑃)((𝐴(quot1p‘𝑅)𝐷)(.r‘𝑃)𝐷))) = ((𝐵 × 𝐴)(-g‘𝑃)(𝐵 × ((𝐴(quot1p‘𝑅)𝐷)(.r‘𝑃)𝐷)))) |
36 | 18, 21, 35 | 3eqtr4d 2785 | . 2 ⊢ (𝜑 → ((𝐵 × 𝐴)(-g‘𝑃)(((𝐵 × 𝐴)(quot1p‘𝑅)𝐷)(.r‘𝑃)𝐷)) = (𝐵 × (𝐴(-g‘𝑃)((𝐴(quot1p‘𝑅)𝐷)(.r‘𝑃)𝐷)))) |
37 | 7, 22, 15, 23, 26, 31, 3 | lmodvscld 20894 | . . 3 ⊢ (𝜑 → (𝐵 × 𝐴) ∈ 𝑈) |
38 | r1padd1.e | . . . 4 ⊢ 𝐸 = (rem1p‘𝑅) | |
39 | 38, 6, 7, 5, 13, 24 | r1pval 26212 | . . 3 ⊢ (((𝐵 × 𝐴) ∈ 𝑈 ∧ 𝐷 ∈ 𝑈) → ((𝐵 × 𝐴)𝐸𝐷) = ((𝐵 × 𝐴)(-g‘𝑃)(((𝐵 × 𝐴)(quot1p‘𝑅)𝐷)(.r‘𝑃)𝐷))) |
40 | 37, 12, 39 | syl2anc 584 | . 2 ⊢ (𝜑 → ((𝐵 × 𝐴)𝐸𝐷) = ((𝐵 × 𝐴)(-g‘𝑃)(((𝐵 × 𝐴)(quot1p‘𝑅)𝐷)(.r‘𝑃)𝐷))) |
41 | 38, 6, 7, 5, 13, 24 | r1pval 26212 | . . . 4 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐷 ∈ 𝑈) → (𝐴𝐸𝐷) = (𝐴(-g‘𝑃)((𝐴(quot1p‘𝑅)𝐷)(.r‘𝑃)𝐷))) |
42 | 3, 12, 41 | syl2anc 584 | . . 3 ⊢ (𝜑 → (𝐴𝐸𝐷) = (𝐴(-g‘𝑃)((𝐴(quot1p‘𝑅)𝐷)(.r‘𝑃)𝐷))) |
43 | 42 | oveq2d 7447 | . 2 ⊢ (𝜑 → (𝐵 × (𝐴𝐸𝐷)) = (𝐵 × (𝐴(-g‘𝑃)((𝐴(quot1p‘𝑅)𝐷)(.r‘𝑃)𝐷)))) |
44 | 36, 40, 43 | 3eqtr4d 2785 | 1 ⊢ (𝜑 → ((𝐵 × 𝐴)𝐸𝐷) = (𝐵 × (𝐴𝐸𝐷))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 .rcmulr 17299 Scalarcsca 17301 ·𝑠 cvsca 17302 -gcsg 18966 Ringcrg 20251 LModclmod 20875 Poly1cpl1 22194 Unic1pcuc1p 26181 quot1pcq1p 26182 rem1pcr1p 26183 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 ax-addf 11232 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-ofr 7698 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-tpos 8250 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-pm 8868 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-sup 9480 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-fz 13545 df-fzo 13692 df-seq 14040 df-hash 14367 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-0g 17488 df-gsum 17489 df-prds 17494 df-pws 17496 df-mre 17631 df-mrc 17632 df-acs 17634 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-mhm 18809 df-submnd 18810 df-grp 18967 df-minusg 18968 df-sbg 18969 df-mulg 19099 df-subg 19154 df-ghm 19244 df-cntz 19348 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-cring 20254 df-oppr 20351 df-dvdsr 20374 df-unit 20375 df-invr 20405 df-subrng 20563 df-subrg 20587 df-rlreg 20711 df-lmod 20877 df-lss 20948 df-cnfld 21383 df-psr 21947 df-mvr 21948 df-mpl 21949 df-opsr 21951 df-psr1 22197 df-vr1 22198 df-ply1 22199 df-coe1 22200 df-mdeg 26109 df-deg1 26110 df-uc1p 26186 df-q1p 26187 df-r1p 26188 |
This theorem is referenced by: r1p0 33606 r1plmhm 33610 |
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