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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 25502 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 2736 | . . 3 ⊢ (Poly1‘(oppr‘𝑅)) = (Poly1‘(oppr‘𝑅)) | |
2 | ply1divalg.d | . . . 4 ⊢ 𝐷 = ( deg1 ‘𝑅) | |
3 | eqidd 2737 | . . . . . 6 ⊢ (⊤ → (Base‘𝑅) = (Base‘𝑅)) | |
4 | eqid 2736 | . . . . . . . 8 ⊢ (oppr‘𝑅) = (oppr‘𝑅) | |
5 | eqid 2736 | . . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
6 | 4, 5 | opprbas 20056 | . . . . . . 7 ⊢ (Base‘𝑅) = (Base‘(oppr‘𝑅)) |
7 | 6 | a1i 11 | . . . . . 6 ⊢ (⊤ → (Base‘𝑅) = (Base‘(oppr‘𝑅))) |
8 | eqid 2736 | . . . . . . . . 9 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
9 | 4, 8 | oppradd 20058 | . . . . . . . 8 ⊢ (+g‘𝑅) = (+g‘(oppr‘𝑅)) |
10 | 9 | oveqi 7370 | . . . . . . 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 25451 | . . . . 5 ⊢ (⊤ → ( deg1 ‘𝑅) = ( deg1 ‘(oppr‘𝑅))) |
13 | 12 | mptru 1548 | . . . 4 ⊢ ( deg1 ‘𝑅) = ( deg1 ‘(oppr‘𝑅)) |
14 | 2, 13 | eqtri 2764 | . . 3 ⊢ 𝐷 = ( deg1 ‘(oppr‘𝑅)) |
15 | ply1divalg.b | . . . 4 ⊢ 𝐵 = (Base‘𝑃) | |
16 | ply1divalg.p | . . . . . 6 ⊢ 𝑃 = (Poly1‘𝑅) | |
17 | 16 | fveq2i 6845 | . . . . 5 ⊢ (Base‘𝑃) = (Base‘(Poly1‘𝑅)) |
18 | 3, 7, 11 | ply1baspropd 21614 | . . . . . 6 ⊢ (⊤ → (Base‘(Poly1‘𝑅)) = (Base‘(Poly1‘(oppr‘𝑅)))) |
19 | 18 | mptru 1548 | . . . . 5 ⊢ (Base‘(Poly1‘𝑅)) = (Base‘(Poly1‘(oppr‘𝑅))) |
20 | 17, 19 | eqtri 2764 | . . . 4 ⊢ (Base‘𝑃) = (Base‘(Poly1‘(oppr‘𝑅))) |
21 | 15, 20 | eqtri 2764 | . . 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 6845 | . . . . . . . 8 ⊢ (+g‘𝑃) = (+g‘(Poly1‘𝑅)) |
25 | 3, 7, 11 | ply1plusgpropd 21615 | . . . . . . . . 9 ⊢ (⊤ → (+g‘(Poly1‘𝑅)) = (+g‘(Poly1‘(oppr‘𝑅)))) |
26 | 25 | mptru 1548 | . . . . . . . 8 ⊢ (+g‘(Poly1‘𝑅)) = (+g‘(Poly1‘(oppr‘𝑅))) |
27 | 24, 26 | eqtri 2764 | . . . . . . 7 ⊢ (+g‘𝑃) = (+g‘(Poly1‘(oppr‘𝑅))) |
28 | 27 | a1i 11 | . . . . . 6 ⊢ (⊤ → (+g‘𝑃) = (+g‘(Poly1‘(oppr‘𝑅)))) |
29 | 23, 28 | grpsubpropd 18852 | . . . . 5 ⊢ (⊤ → (-g‘𝑃) = (-g‘(Poly1‘(oppr‘𝑅)))) |
30 | 29 | mptru 1548 | . . . 4 ⊢ (-g‘𝑃) = (-g‘(Poly1‘(oppr‘𝑅))) |
31 | 22, 30 | eqtri 2764 | . . 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 7370 | . . . . . . 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 18517 | . . . . 5 ⊢ (⊤ → (0g‘𝑃) = (0g‘(Poly1‘(oppr‘𝑅)))) |
38 | 37 | mptru 1548 | . . . 4 ⊢ (0g‘𝑃) = (0g‘(Poly1‘(oppr‘𝑅))) |
39 | 32, 38 | eqtri 2764 | . . 3 ⊢ 0 = (0g‘(Poly1‘(oppr‘𝑅))) |
40 | eqid 2736 | . . 3 ⊢ (.r‘(Poly1‘(oppr‘𝑅))) = (.r‘(Poly1‘(oppr‘𝑅))) | |
41 | ply1divalg.r1 | . . . 4 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
42 | 4 | opprring 20060 | . . . 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 20090 | . . 3 ⊢ 𝑈 = (Unit‘(oppr‘𝑅)) |
50 | 1, 14, 21, 31, 39, 40, 43, 44, 45, 46, 47, 49 | ply1divalg 25502 | . 2 ⊢ (𝜑 → ∃!𝑞 ∈ 𝐵 (𝐷‘(𝐹 − (𝐺(.r‘(Poly1‘(oppr‘𝑅)))𝑞))) < (𝐷‘𝐺)) |
51 | 41 | adantr 481 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐵) → 𝑅 ∈ Ring) |
52 | 45 | adantr 481 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐵) → 𝐺 ∈ 𝐵) |
53 | simpr 485 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐵) → 𝑞 ∈ 𝐵) | |
54 | ply1divalg.t | . . . . . . . . 9 ⊢ ∙ = (.r‘𝑃) | |
55 | 16, 4, 1, 54, 40, 15 | ply1opprmul 21610 | . . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝐺 ∈ 𝐵 ∧ 𝑞 ∈ 𝐵) → (𝐺(.r‘(Poly1‘(oppr‘𝑅)))𝑞) = (𝑞 ∙ 𝐺)) |
56 | 51, 52, 53, 55 | syl3anc 1371 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐵) → (𝐺(.r‘(Poly1‘(oppr‘𝑅)))𝑞) = (𝑞 ∙ 𝐺)) |
57 | 56 | eqcomd 2742 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐵) → (𝑞 ∙ 𝐺) = (𝐺(.r‘(Poly1‘(oppr‘𝑅)))𝑞)) |
58 | 57 | oveq2d 7373 | . . . . 5 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐵) → (𝐹 − (𝑞 ∙ 𝐺)) = (𝐹 − (𝐺(.r‘(Poly1‘(oppr‘𝑅)))𝑞))) |
59 | 58 | fveq2d 6846 | . . . 4 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐵) → (𝐷‘(𝐹 − (𝑞 ∙ 𝐺))) = (𝐷‘(𝐹 − (𝐺(.r‘(Poly1‘(oppr‘𝑅)))𝑞)))) |
60 | 59 | breq1d 5115 | . . 3 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐵) → ((𝐷‘(𝐹 − (𝑞 ∙ 𝐺))) < (𝐷‘𝐺) ↔ (𝐷‘(𝐹 − (𝐺(.r‘(Poly1‘(oppr‘𝑅)))𝑞))) < (𝐷‘𝐺))) |
61 | 60 | reubidva 3369 | . 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 256 | 1 ⊢ (𝜑 → ∃!𝑞 ∈ 𝐵 (𝐷‘(𝐹 − (𝑞 ∙ 𝐺))) < (𝐷‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ⊤wtru 1542 ∈ wcel 2106 ≠ wne 2943 ∃!wreu 3351 class class class wbr 5105 ‘cfv 6496 (class class class)co 7357 < clt 11189 Basecbs 17083 +gcplusg 17133 .rcmulr 17134 0gc0g 17321 -gcsg 18750 Ringcrg 19964 opprcoppr 20048 Unitcui 20068 Poly1cpl1 21548 coe1cco1 21549 deg1 cdg1 25416 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129 ax-addf 11130 ax-mulf 11131 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7617 df-ofr 7618 df-om 7803 df-1st 7921 df-2nd 7922 df-supp 8093 df-tpos 8157 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-er 8648 df-map 8767 df-pm 8768 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9306 df-sup 9378 df-oi 9446 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-z 12500 df-dec 12619 df-uz 12764 df-fz 13425 df-fzo 13568 df-seq 13907 df-hash 14231 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-starv 17148 df-sca 17149 df-vsca 17150 df-ip 17151 df-tset 17152 df-ple 17153 df-ds 17155 df-unif 17156 df-hom 17157 df-cco 17158 df-0g 17323 df-gsum 17324 df-prds 17329 df-pws 17331 df-mre 17466 df-mrc 17467 df-acs 17469 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-mhm 18601 df-submnd 18602 df-grp 18751 df-minusg 18752 df-sbg 18753 df-mulg 18873 df-subg 18925 df-ghm 19006 df-cntz 19097 df-cmn 19564 df-abl 19565 df-mgp 19897 df-ur 19914 df-ring 19966 df-cring 19967 df-oppr 20049 df-dvdsr 20070 df-unit 20071 df-invr 20101 df-subrg 20220 df-lmod 20324 df-lss 20393 df-rlreg 20753 df-cnfld 20797 df-psr 21311 df-mvr 21312 df-mpl 21313 df-opsr 21315 df-psr1 21551 df-vr1 21552 df-ply1 21553 df-coe1 21554 df-mdeg 25417 df-deg1 25418 |
This theorem is referenced by: q1peqb 25519 |
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