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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 23701 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 2610 | . . 3 ⊢ (Poly1‘(oppr‘𝑅)) = (Poly1‘(oppr‘𝑅)) | |
2 | ply1divalg.d | . . . 4 ⊢ 𝐷 = ( deg1 ‘𝑅) | |
3 | eqidd 2611 | . . . . . 6 ⊢ (⊤ → (Base‘𝑅) = (Base‘𝑅)) | |
4 | eqid 2610 | . . . . . . . 8 ⊢ (oppr‘𝑅) = (oppr‘𝑅) | |
5 | eqid 2610 | . . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
6 | 4, 5 | opprbas 18452 | . . . . . . 7 ⊢ (Base‘𝑅) = (Base‘(oppr‘𝑅)) |
7 | 6 | a1i 11 | . . . . . 6 ⊢ (⊤ → (Base‘𝑅) = (Base‘(oppr‘𝑅))) |
8 | eqid 2610 | . . . . . . . . 9 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
9 | 4, 8 | oppradd 18453 | . . . . . . . 8 ⊢ (+g‘𝑅) = (+g‘(oppr‘𝑅)) |
10 | 9 | oveqi 6562 | . . . . . . 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 23650 | . . . . 5 ⊢ (⊤ → ( deg1 ‘𝑅) = ( deg1 ‘(oppr‘𝑅))) |
13 | 12 | trud 1484 | . . . 4 ⊢ ( deg1 ‘𝑅) = ( deg1 ‘(oppr‘𝑅)) |
14 | 2, 13 | eqtri 2632 | . . 3 ⊢ 𝐷 = ( deg1 ‘(oppr‘𝑅)) |
15 | ply1divalg.b | . . . 4 ⊢ 𝐵 = (Base‘𝑃) | |
16 | ply1divalg.p | . . . . . 6 ⊢ 𝑃 = (Poly1‘𝑅) | |
17 | 16 | fveq2i 6106 | . . . . 5 ⊢ (Base‘𝑃) = (Base‘(Poly1‘𝑅)) |
18 | 3, 7, 11 | ply1baspropd 19434 | . . . . . 6 ⊢ (⊤ → (Base‘(Poly1‘𝑅)) = (Base‘(Poly1‘(oppr‘𝑅)))) |
19 | 18 | trud 1484 | . . . . 5 ⊢ (Base‘(Poly1‘𝑅)) = (Base‘(Poly1‘(oppr‘𝑅))) |
20 | 17, 19 | eqtri 2632 | . . . 4 ⊢ (Base‘𝑃) = (Base‘(Poly1‘(oppr‘𝑅))) |
21 | 15, 20 | eqtri 2632 | . . 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 6106 | . . . . . . . 8 ⊢ (+g‘𝑃) = (+g‘(Poly1‘𝑅)) |
25 | 3, 7, 11 | ply1plusgpropd 19435 | . . . . . . . . 9 ⊢ (⊤ → (+g‘(Poly1‘𝑅)) = (+g‘(Poly1‘(oppr‘𝑅)))) |
26 | 25 | trud 1484 | . . . . . . . 8 ⊢ (+g‘(Poly1‘𝑅)) = (+g‘(Poly1‘(oppr‘𝑅))) |
27 | 24, 26 | eqtri 2632 | . . . . . . 7 ⊢ (+g‘𝑃) = (+g‘(Poly1‘(oppr‘𝑅))) |
28 | 27 | a1i 11 | . . . . . 6 ⊢ (⊤ → (+g‘𝑃) = (+g‘(Poly1‘(oppr‘𝑅)))) |
29 | 23, 28 | grpsubpropd 17343 | . . . . 5 ⊢ (⊤ → (-g‘𝑃) = (-g‘(Poly1‘(oppr‘𝑅)))) |
30 | 29 | trud 1484 | . . . 4 ⊢ (-g‘𝑃) = (-g‘(Poly1‘(oppr‘𝑅))) |
31 | 22, 30 | eqtri 2632 | . . 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 6562 | . . . . . . 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 17084 | . . . . 5 ⊢ (⊤ → (0g‘𝑃) = (0g‘(Poly1‘(oppr‘𝑅)))) |
38 | 37 | trud 1484 | . . . 4 ⊢ (0g‘𝑃) = (0g‘(Poly1‘(oppr‘𝑅))) |
39 | 32, 38 | eqtri 2632 | . . 3 ⊢ 0 = (0g‘(Poly1‘(oppr‘𝑅))) |
40 | eqid 2610 | . . 3 ⊢ (.r‘(Poly1‘(oppr‘𝑅))) = (.r‘(Poly1‘(oppr‘𝑅))) | |
41 | ply1divalg.r1 | . . . 4 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
42 | 4 | opprring 18454 | . . . 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 18484 | . . 3 ⊢ 𝑈 = (Unit‘(oppr‘𝑅)) |
50 | 1, 14, 21, 31, 39, 40, 43, 44, 45, 46, 47, 49 | ply1divalg 23701 | . 2 ⊢ (𝜑 → ∃!𝑞 ∈ 𝐵 (𝐷‘(𝐹 − (𝐺(.r‘(Poly1‘(oppr‘𝑅)))𝑞))) < (𝐷‘𝐺)) |
51 | 41 | adantr 480 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐵) → 𝑅 ∈ Ring) |
52 | 45 | adantr 480 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐵) → 𝐺 ∈ 𝐵) |
53 | simpr 476 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐵) → 𝑞 ∈ 𝐵) | |
54 | ply1divalg.t | . . . . . . . . 9 ⊢ ∙ = (.r‘𝑃) | |
55 | 16, 4, 1, 54, 40, 15 | ply1opprmul 19430 | . . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝐺 ∈ 𝐵 ∧ 𝑞 ∈ 𝐵) → (𝐺(.r‘(Poly1‘(oppr‘𝑅)))𝑞) = (𝑞 ∙ 𝐺)) |
56 | 51, 52, 53, 55 | syl3anc 1318 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐵) → (𝐺(.r‘(Poly1‘(oppr‘𝑅)))𝑞) = (𝑞 ∙ 𝐺)) |
57 | 56 | eqcomd 2616 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐵) → (𝑞 ∙ 𝐺) = (𝐺(.r‘(Poly1‘(oppr‘𝑅)))𝑞)) |
58 | 57 | oveq2d 6565 | . . . . 5 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐵) → (𝐹 − (𝑞 ∙ 𝐺)) = (𝐹 − (𝐺(.r‘(Poly1‘(oppr‘𝑅)))𝑞))) |
59 | 58 | fveq2d 6107 | . . . 4 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐵) → (𝐷‘(𝐹 − (𝑞 ∙ 𝐺))) = (𝐷‘(𝐹 − (𝐺(.r‘(Poly1‘(oppr‘𝑅)))𝑞)))) |
60 | 59 | breq1d 4593 | . . 3 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐵) → ((𝐷‘(𝐹 − (𝑞 ∙ 𝐺))) < (𝐷‘𝐺) ↔ (𝐷‘(𝐹 − (𝐺(.r‘(Poly1‘(oppr‘𝑅)))𝑞))) < (𝐷‘𝐺))) |
61 | 60 | reubidva 3102 | . 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 246 | 1 ⊢ (𝜑 → ∃!𝑞 ∈ 𝐵 (𝐷‘(𝐹 − (𝑞 ∙ 𝐺))) < (𝐷‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ⊤wtru 1476 ∈ wcel 1977 ≠ wne 2780 ∃!wreu 2898 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 < clt 9953 Basecbs 15695 +gcplusg 15768 .rcmulr 15769 0gc0g 15923 -gcsg 17247 Ringcrg 18370 opprcoppr 18445 Unitcui 18462 Poly1cpl1 19368 coe1cco1 19369 deg1 cdg1 23618 |
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 |
This theorem is referenced by: q1peqb 23718 |
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