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Mirrors > Home > MPE Home > Th. List > r1pid | Structured version Visualization version GIF version |
Description: Express the original polynomial 𝐹 as 𝐹 = (𝑞 · 𝐺) + 𝑟 using the quotient and remainder functions for 𝑞 and 𝑟. (Contributed by Mario Carneiro, 5-Jun-2015.) |
Ref | Expression |
---|---|
r1pid.p | ⊢ 𝑃 = (Poly1‘𝑅) |
r1pid.b | ⊢ 𝐵 = (Base‘𝑃) |
r1pid.c | ⊢ 𝐶 = (Unic1p‘𝑅) |
r1pid.q | ⊢ 𝑄 = (quot1p‘𝑅) |
r1pid.e | ⊢ 𝐸 = (rem1p‘𝑅) |
r1pid.t | ⊢ · = (.r‘𝑃) |
r1pid.m | ⊢ + = (+g‘𝑃) |
Ref | Expression |
---|---|
r1pid | ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → 𝐹 = (((𝐹𝑄𝐺) · 𝐺) + (𝐹𝐸𝐺))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | r1pid.p | . . . . . 6 ⊢ 𝑃 = (Poly1‘𝑅) | |
2 | r1pid.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝑃) | |
3 | r1pid.c | . . . . . 6 ⊢ 𝐶 = (Unic1p‘𝑅) | |
4 | 1, 2, 3 | uc1pcl 23707 | . . . . 5 ⊢ (𝐺 ∈ 𝐶 → 𝐺 ∈ 𝐵) |
5 | r1pid.e | . . . . . 6 ⊢ 𝐸 = (rem1p‘𝑅) | |
6 | r1pid.q | . . . . . 6 ⊢ 𝑄 = (quot1p‘𝑅) | |
7 | r1pid.t | . . . . . 6 ⊢ · = (.r‘𝑃) | |
8 | eqid 2610 | . . . . . 6 ⊢ (-g‘𝑃) = (-g‘𝑃) | |
9 | 5, 1, 2, 6, 7, 8 | r1pval 23720 | . . . . 5 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝐹𝐸𝐺) = (𝐹(-g‘𝑃)((𝐹𝑄𝐺) · 𝐺))) |
10 | 4, 9 | sylan2 490 | . . . 4 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → (𝐹𝐸𝐺) = (𝐹(-g‘𝑃)((𝐹𝑄𝐺) · 𝐺))) |
11 | 10 | 3adant1 1072 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → (𝐹𝐸𝐺) = (𝐹(-g‘𝑃)((𝐹𝑄𝐺) · 𝐺))) |
12 | 11 | oveq2d 6565 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → (((𝐹𝑄𝐺) · 𝐺) + (𝐹𝐸𝐺)) = (((𝐹𝑄𝐺) · 𝐺) + (𝐹(-g‘𝑃)((𝐹𝑄𝐺) · 𝐺)))) |
13 | 1 | ply1ring 19439 | . . . . 5 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring) |
14 | 13 | 3ad2ant1 1075 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → 𝑃 ∈ Ring) |
15 | ringabl 18403 | . . . 4 ⊢ (𝑃 ∈ Ring → 𝑃 ∈ Abel) | |
16 | 14, 15 | syl 17 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → 𝑃 ∈ Abel) |
17 | 6, 1, 2, 3 | q1pcl 23719 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → (𝐹𝑄𝐺) ∈ 𝐵) |
18 | 4 | 3ad2ant3 1077 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → 𝐺 ∈ 𝐵) |
19 | 2, 7 | ringcl 18384 | . . . 4 ⊢ ((𝑃 ∈ Ring ∧ (𝐹𝑄𝐺) ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → ((𝐹𝑄𝐺) · 𝐺) ∈ 𝐵) |
20 | 14, 17, 18, 19 | syl3anc 1318 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → ((𝐹𝑄𝐺) · 𝐺) ∈ 𝐵) |
21 | ringgrp 18375 | . . . . 5 ⊢ (𝑃 ∈ Ring → 𝑃 ∈ Grp) | |
22 | 14, 21 | syl 17 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → 𝑃 ∈ Grp) |
23 | simp2 1055 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → 𝐹 ∈ 𝐵) | |
24 | 2, 8 | grpsubcl 17318 | . . . 4 ⊢ ((𝑃 ∈ Grp ∧ 𝐹 ∈ 𝐵 ∧ ((𝐹𝑄𝐺) · 𝐺) ∈ 𝐵) → (𝐹(-g‘𝑃)((𝐹𝑄𝐺) · 𝐺)) ∈ 𝐵) |
25 | 22, 23, 20, 24 | syl3anc 1318 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → (𝐹(-g‘𝑃)((𝐹𝑄𝐺) · 𝐺)) ∈ 𝐵) |
26 | r1pid.m | . . . 4 ⊢ + = (+g‘𝑃) | |
27 | 2, 26 | ablcom 18033 | . . 3 ⊢ ((𝑃 ∈ Abel ∧ ((𝐹𝑄𝐺) · 𝐺) ∈ 𝐵 ∧ (𝐹(-g‘𝑃)((𝐹𝑄𝐺) · 𝐺)) ∈ 𝐵) → (((𝐹𝑄𝐺) · 𝐺) + (𝐹(-g‘𝑃)((𝐹𝑄𝐺) · 𝐺))) = ((𝐹(-g‘𝑃)((𝐹𝑄𝐺) · 𝐺)) + ((𝐹𝑄𝐺) · 𝐺))) |
28 | 16, 20, 25, 27 | syl3anc 1318 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → (((𝐹𝑄𝐺) · 𝐺) + (𝐹(-g‘𝑃)((𝐹𝑄𝐺) · 𝐺))) = ((𝐹(-g‘𝑃)((𝐹𝑄𝐺) · 𝐺)) + ((𝐹𝑄𝐺) · 𝐺))) |
29 | 2, 26, 8 | grpnpcan 17330 | . . 3 ⊢ ((𝑃 ∈ Grp ∧ 𝐹 ∈ 𝐵 ∧ ((𝐹𝑄𝐺) · 𝐺) ∈ 𝐵) → ((𝐹(-g‘𝑃)((𝐹𝑄𝐺) · 𝐺)) + ((𝐹𝑄𝐺) · 𝐺)) = 𝐹) |
30 | 22, 23, 20, 29 | syl3anc 1318 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → ((𝐹(-g‘𝑃)((𝐹𝑄𝐺) · 𝐺)) + ((𝐹𝑄𝐺) · 𝐺)) = 𝐹) |
31 | 12, 28, 30 | 3eqtrrd 2649 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → 𝐹 = (((𝐹𝑄𝐺) · 𝐺) + (𝐹𝐸𝐺))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 +gcplusg 15768 .rcmulr 15769 Grpcgrp 17245 -gcsg 17247 Abelcabl 18017 Ringcrg 18370 Poly1cpl1 19368 Unic1pcuc1p 23690 quot1pcq1p 23691 rem1pcr1p 23692 |
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 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895 |
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-tpos 7239 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-starv 15783 df-sca 15784 df-vsca 15785 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-0g 15925 df-gsum 15926 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-ring 18372 df-cring 18373 df-oppr 18446 df-dvdsr 18464 df-unit 18465 df-invr 18495 df-subrg 18601 df-lmod 18688 df-lss 18754 df-rlreg 19104 df-psr 19177 df-mvr 19178 df-mpl 19179 df-opsr 19181 df-psr1 19371 df-vr1 19372 df-ply1 19373 df-coe1 19374 df-cnfld 19568 df-mdeg 23619 df-deg1 23620 df-uc1p 23695 df-q1p 23696 df-r1p 23697 |
This theorem is referenced by: ply1rem 23727 |
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