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Mirrors > Home > MPE Home > Th. List > ply1divalg2 | Structured version Visualization version GIF version |
Description: Reverse the order of multiplication in ply1divalg 25035 via the opposite ring. (Contributed by Stefan O'Rear, 28-Mar-2015.) |
Ref | Expression |
---|---|
ply1divalg.p | ⊢ 𝑃 = (Poly1‘𝑅) |
ply1divalg.d | ⊢ 𝐷 = ( deg1 ‘𝑅) |
ply1divalg.b | ⊢ 𝐵 = (Base‘𝑃) |
ply1divalg.m | ⊢ − = (-g‘𝑃) |
ply1divalg.z | ⊢ 0 = (0g‘𝑃) |
ply1divalg.t | ⊢ ∙ = (.r‘𝑃) |
ply1divalg.r1 | ⊢ (𝜑 → 𝑅 ∈ Ring) |
ply1divalg.f | ⊢ (𝜑 → 𝐹 ∈ 𝐵) |
ply1divalg.g1 | ⊢ (𝜑 → 𝐺 ∈ 𝐵) |
ply1divalg.g2 | ⊢ (𝜑 → 𝐺 ≠ 0 ) |
ply1divalg.g3 | ⊢ (𝜑 → ((coe1‘𝐺)‘(𝐷‘𝐺)) ∈ 𝑈) |
ply1divalg.u | ⊢ 𝑈 = (Unit‘𝑅) |
Ref | Expression |
---|---|
ply1divalg2 | ⊢ (𝜑 → ∃!𝑞 ∈ 𝐵 (𝐷‘(𝐹 − (𝑞 ∙ 𝐺))) < (𝐷‘𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2737 | . . 3 ⊢ (Poly1‘(oppr‘𝑅)) = (Poly1‘(oppr‘𝑅)) | |
2 | ply1divalg.d | . . . 4 ⊢ 𝐷 = ( deg1 ‘𝑅) | |
3 | eqidd 2738 | . . . . . 6 ⊢ (⊤ → (Base‘𝑅) = (Base‘𝑅)) | |
4 | eqid 2737 | . . . . . . . 8 ⊢ (oppr‘𝑅) = (oppr‘𝑅) | |
5 | eqid 2737 | . . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
6 | 4, 5 | opprbas 19647 | . . . . . . 7 ⊢ (Base‘𝑅) = (Base‘(oppr‘𝑅)) |
7 | 6 | a1i 11 | . . . . . 6 ⊢ (⊤ → (Base‘𝑅) = (Base‘(oppr‘𝑅))) |
8 | eqid 2737 | . . . . . . . . 9 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
9 | 4, 8 | oppradd 19648 | . . . . . . . 8 ⊢ (+g‘𝑅) = (+g‘(oppr‘𝑅)) |
10 | 9 | oveqi 7226 | . . . . . . 7 ⊢ (𝑞(+g‘𝑅)𝑟) = (𝑞(+g‘(oppr‘𝑅))𝑟) |
11 | 10 | a1i 11 | . . . . . 6 ⊢ ((⊤ ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → (𝑞(+g‘𝑅)𝑟) = (𝑞(+g‘(oppr‘𝑅))𝑟)) |
12 | 3, 7, 11 | deg1propd 24984 | . . . . 5 ⊢ (⊤ → ( deg1 ‘𝑅) = ( deg1 ‘(oppr‘𝑅))) |
13 | 12 | mptru 1550 | . . . 4 ⊢ ( deg1 ‘𝑅) = ( deg1 ‘(oppr‘𝑅)) |
14 | 2, 13 | eqtri 2765 | . . 3 ⊢ 𝐷 = ( deg1 ‘(oppr‘𝑅)) |
15 | ply1divalg.b | . . . 4 ⊢ 𝐵 = (Base‘𝑃) | |
16 | ply1divalg.p | . . . . . 6 ⊢ 𝑃 = (Poly1‘𝑅) | |
17 | 16 | fveq2i 6720 | . . . . 5 ⊢ (Base‘𝑃) = (Base‘(Poly1‘𝑅)) |
18 | 3, 7, 11 | ply1baspropd 21164 | . . . . . 6 ⊢ (⊤ → (Base‘(Poly1‘𝑅)) = (Base‘(Poly1‘(oppr‘𝑅)))) |
19 | 18 | mptru 1550 | . . . . 5 ⊢ (Base‘(Poly1‘𝑅)) = (Base‘(Poly1‘(oppr‘𝑅))) |
20 | 17, 19 | eqtri 2765 | . . . 4 ⊢ (Base‘𝑃) = (Base‘(Poly1‘(oppr‘𝑅))) |
21 | 15, 20 | eqtri 2765 | . . 3 ⊢ 𝐵 = (Base‘(Poly1‘(oppr‘𝑅))) |
22 | ply1divalg.m | . . . 4 ⊢ − = (-g‘𝑃) | |
23 | 20 | a1i 11 | . . . . . 6 ⊢ (⊤ → (Base‘𝑃) = (Base‘(Poly1‘(oppr‘𝑅)))) |
24 | 16 | fveq2i 6720 | . . . . . . . 8 ⊢ (+g‘𝑃) = (+g‘(Poly1‘𝑅)) |
25 | 3, 7, 11 | ply1plusgpropd 21165 | . . . . . . . . 9 ⊢ (⊤ → (+g‘(Poly1‘𝑅)) = (+g‘(Poly1‘(oppr‘𝑅)))) |
26 | 25 | mptru 1550 | . . . . . . . 8 ⊢ (+g‘(Poly1‘𝑅)) = (+g‘(Poly1‘(oppr‘𝑅))) |
27 | 24, 26 | eqtri 2765 | . . . . . . 7 ⊢ (+g‘𝑃) = (+g‘(Poly1‘(oppr‘𝑅))) |
28 | 27 | a1i 11 | . . . . . 6 ⊢ (⊤ → (+g‘𝑃) = (+g‘(Poly1‘(oppr‘𝑅)))) |
29 | 23, 28 | grpsubpropd 18468 | . . . . 5 ⊢ (⊤ → (-g‘𝑃) = (-g‘(Poly1‘(oppr‘𝑅)))) |
30 | 29 | mptru 1550 | . . . 4 ⊢ (-g‘𝑃) = (-g‘(Poly1‘(oppr‘𝑅))) |
31 | 22, 30 | eqtri 2765 | . . 3 ⊢ − = (-g‘(Poly1‘(oppr‘𝑅))) |
32 | ply1divalg.z | . . . 4 ⊢ 0 = (0g‘𝑃) | |
33 | 15 | a1i 11 | . . . . . 6 ⊢ (⊤ → 𝐵 = (Base‘𝑃)) |
34 | 21 | a1i 11 | . . . . . 6 ⊢ (⊤ → 𝐵 = (Base‘(Poly1‘(oppr‘𝑅)))) |
35 | 27 | oveqi 7226 | . . . . . . 7 ⊢ (𝑞(+g‘𝑃)𝑟) = (𝑞(+g‘(Poly1‘(oppr‘𝑅)))𝑟) |
36 | 35 | a1i 11 | . . . . . 6 ⊢ ((⊤ ∧ (𝑞 ∈ 𝐵 ∧ 𝑟 ∈ 𝐵)) → (𝑞(+g‘𝑃)𝑟) = (𝑞(+g‘(Poly1‘(oppr‘𝑅)))𝑟)) |
37 | 33, 34, 36 | grpidpropd 18134 | . . . . 5 ⊢ (⊤ → (0g‘𝑃) = (0g‘(Poly1‘(oppr‘𝑅)))) |
38 | 37 | mptru 1550 | . . . 4 ⊢ (0g‘𝑃) = (0g‘(Poly1‘(oppr‘𝑅))) |
39 | 32, 38 | eqtri 2765 | . . 3 ⊢ 0 = (0g‘(Poly1‘(oppr‘𝑅))) |
40 | eqid 2737 | . . 3 ⊢ (.r‘(Poly1‘(oppr‘𝑅))) = (.r‘(Poly1‘(oppr‘𝑅))) | |
41 | ply1divalg.r1 | . . . 4 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
42 | 4 | opprring 19649 | . . . 4 ⊢ (𝑅 ∈ Ring → (oppr‘𝑅) ∈ Ring) |
43 | 41, 42 | syl 17 | . . 3 ⊢ (𝜑 → (oppr‘𝑅) ∈ Ring) |
44 | ply1divalg.f | . . 3 ⊢ (𝜑 → 𝐹 ∈ 𝐵) | |
45 | ply1divalg.g1 | . . 3 ⊢ (𝜑 → 𝐺 ∈ 𝐵) | |
46 | ply1divalg.g2 | . . 3 ⊢ (𝜑 → 𝐺 ≠ 0 ) | |
47 | ply1divalg.g3 | . . 3 ⊢ (𝜑 → ((coe1‘𝐺)‘(𝐷‘𝐺)) ∈ 𝑈) | |
48 | ply1divalg.u | . . . 4 ⊢ 𝑈 = (Unit‘𝑅) | |
49 | 48, 4 | opprunit 19679 | . . 3 ⊢ 𝑈 = (Unit‘(oppr‘𝑅)) |
50 | 1, 14, 21, 31, 39, 40, 43, 44, 45, 46, 47, 49 | ply1divalg 25035 | . 2 ⊢ (𝜑 → ∃!𝑞 ∈ 𝐵 (𝐷‘(𝐹 − (𝐺(.r‘(Poly1‘(oppr‘𝑅)))𝑞))) < (𝐷‘𝐺)) |
51 | 41 | adantr 484 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐵) → 𝑅 ∈ Ring) |
52 | 45 | adantr 484 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐵) → 𝐺 ∈ 𝐵) |
53 | simpr 488 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐵) → 𝑞 ∈ 𝐵) | |
54 | ply1divalg.t | . . . . . . . . 9 ⊢ ∙ = (.r‘𝑃) | |
55 | 16, 4, 1, 54, 40, 15 | ply1opprmul 21160 | . . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝐺 ∈ 𝐵 ∧ 𝑞 ∈ 𝐵) → (𝐺(.r‘(Poly1‘(oppr‘𝑅)))𝑞) = (𝑞 ∙ 𝐺)) |
56 | 51, 52, 53, 55 | syl3anc 1373 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐵) → (𝐺(.r‘(Poly1‘(oppr‘𝑅)))𝑞) = (𝑞 ∙ 𝐺)) |
57 | 56 | eqcomd 2743 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐵) → (𝑞 ∙ 𝐺) = (𝐺(.r‘(Poly1‘(oppr‘𝑅)))𝑞)) |
58 | 57 | oveq2d 7229 | . . . . 5 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐵) → (𝐹 − (𝑞 ∙ 𝐺)) = (𝐹 − (𝐺(.r‘(Poly1‘(oppr‘𝑅)))𝑞))) |
59 | 58 | fveq2d 6721 | . . . 4 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐵) → (𝐷‘(𝐹 − (𝑞 ∙ 𝐺))) = (𝐷‘(𝐹 − (𝐺(.r‘(Poly1‘(oppr‘𝑅)))𝑞)))) |
60 | 59 | breq1d 5063 | . . 3 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐵) → ((𝐷‘(𝐹 − (𝑞 ∙ 𝐺))) < (𝐷‘𝐺) ↔ (𝐷‘(𝐹 − (𝐺(.r‘(Poly1‘(oppr‘𝑅)))𝑞))) < (𝐷‘𝐺))) |
61 | 60 | reubidva 3300 | . 2 ⊢ (𝜑 → (∃!𝑞 ∈ 𝐵 (𝐷‘(𝐹 − (𝑞 ∙ 𝐺))) < (𝐷‘𝐺) ↔ ∃!𝑞 ∈ 𝐵 (𝐷‘(𝐹 − (𝐺(.r‘(Poly1‘(oppr‘𝑅)))𝑞))) < (𝐷‘𝐺))) |
62 | 50, 61 | mpbird 260 | 1 ⊢ (𝜑 → ∃!𝑞 ∈ 𝐵 (𝐷‘(𝐹 − (𝑞 ∙ 𝐺))) < (𝐷‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ⊤wtru 1544 ∈ wcel 2110 ≠ wne 2940 ∃!wreu 3063 class class class wbr 5053 ‘cfv 6380 (class class class)co 7213 < clt 10867 Basecbs 16760 +gcplusg 16802 .rcmulr 16803 0gc0g 16944 -gcsg 18367 Ringcrg 19562 opprcoppr 19640 Unitcui 19657 Poly1cpl1 21098 coe1cco1 21099 deg1 cdg1 24949 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 ax-pre-sup 10807 ax-addf 10808 ax-mulf 10809 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-iin 4907 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-se 5510 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-isom 6389 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-of 7469 df-ofr 7470 df-om 7645 df-1st 7761 df-2nd 7762 df-supp 7904 df-tpos 7968 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-er 8391 df-map 8510 df-pm 8511 df-ixp 8579 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-fsupp 8986 df-sup 9058 df-oi 9126 df-card 9555 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-nn 11831 df-2 11893 df-3 11894 df-4 11895 df-5 11896 df-6 11897 df-7 11898 df-8 11899 df-9 11900 df-n0 12091 df-z 12177 df-dec 12294 df-uz 12439 df-fz 13096 df-fzo 13239 df-seq 13575 df-hash 13897 df-struct 16700 df-sets 16717 df-slot 16735 df-ndx 16745 df-base 16761 df-ress 16785 df-plusg 16815 df-mulr 16816 df-starv 16817 df-sca 16818 df-vsca 16819 df-tset 16821 df-ple 16822 df-ds 16824 df-unif 16825 df-0g 16946 df-gsum 16947 df-mre 17089 df-mrc 17090 df-acs 17092 df-mgm 18114 df-sgrp 18163 df-mnd 18174 df-mhm 18218 df-submnd 18219 df-grp 18368 df-minusg 18369 df-sbg 18370 df-mulg 18489 df-subg 18540 df-ghm 18620 df-cntz 18711 df-cmn 19172 df-abl 19173 df-mgp 19505 df-ur 19517 df-ring 19564 df-cring 19565 df-oppr 19641 df-dvdsr 19659 df-unit 19660 df-invr 19690 df-subrg 19798 df-lmod 19901 df-lss 19969 df-rlreg 20321 df-cnfld 20364 df-psr 20868 df-mvr 20869 df-mpl 20870 df-opsr 20872 df-psr1 21101 df-vr1 21102 df-ply1 21103 df-coe1 21104 df-mdeg 24950 df-deg1 24951 |
This theorem is referenced by: q1peqb 25052 |
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