Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > ply1divalg2 | Structured version Visualization version GIF version |
Description: Reverse the order of multiplication in ply1divalg 24734 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 2824 | . . 3 ⊢ (Poly1‘(oppr‘𝑅)) = (Poly1‘(oppr‘𝑅)) | |
2 | ply1divalg.d | . . . 4 ⊢ 𝐷 = ( deg1 ‘𝑅) | |
3 | eqidd 2825 | . . . . . 6 ⊢ (⊤ → (Base‘𝑅) = (Base‘𝑅)) | |
4 | eqid 2824 | . . . . . . . 8 ⊢ (oppr‘𝑅) = (oppr‘𝑅) | |
5 | eqid 2824 | . . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
6 | 4, 5 | opprbas 19382 | . . . . . . 7 ⊢ (Base‘𝑅) = (Base‘(oppr‘𝑅)) |
7 | 6 | a1i 11 | . . . . . 6 ⊢ (⊤ → (Base‘𝑅) = (Base‘(oppr‘𝑅))) |
8 | eqid 2824 | . . . . . . . . 9 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
9 | 4, 8 | oppradd 19383 | . . . . . . . 8 ⊢ (+g‘𝑅) = (+g‘(oppr‘𝑅)) |
10 | 9 | oveqi 7172 | . . . . . . 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 24683 | . . . . 5 ⊢ (⊤ → ( deg1 ‘𝑅) = ( deg1 ‘(oppr‘𝑅))) |
13 | 12 | mptru 1543 | . . . 4 ⊢ ( deg1 ‘𝑅) = ( deg1 ‘(oppr‘𝑅)) |
14 | 2, 13 | eqtri 2847 | . . 3 ⊢ 𝐷 = ( deg1 ‘(oppr‘𝑅)) |
15 | ply1divalg.b | . . . 4 ⊢ 𝐵 = (Base‘𝑃) | |
16 | ply1divalg.p | . . . . . 6 ⊢ 𝑃 = (Poly1‘𝑅) | |
17 | 16 | fveq2i 6676 | . . . . 5 ⊢ (Base‘𝑃) = (Base‘(Poly1‘𝑅)) |
18 | 3, 7, 11 | ply1baspropd 20414 | . . . . . 6 ⊢ (⊤ → (Base‘(Poly1‘𝑅)) = (Base‘(Poly1‘(oppr‘𝑅)))) |
19 | 18 | mptru 1543 | . . . . 5 ⊢ (Base‘(Poly1‘𝑅)) = (Base‘(Poly1‘(oppr‘𝑅))) |
20 | 17, 19 | eqtri 2847 | . . . 4 ⊢ (Base‘𝑃) = (Base‘(Poly1‘(oppr‘𝑅))) |
21 | 15, 20 | eqtri 2847 | . . 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 6676 | . . . . . . . 8 ⊢ (+g‘𝑃) = (+g‘(Poly1‘𝑅)) |
25 | 3, 7, 11 | ply1plusgpropd 20415 | . . . . . . . . 9 ⊢ (⊤ → (+g‘(Poly1‘𝑅)) = (+g‘(Poly1‘(oppr‘𝑅)))) |
26 | 25 | mptru 1543 | . . . . . . . 8 ⊢ (+g‘(Poly1‘𝑅)) = (+g‘(Poly1‘(oppr‘𝑅))) |
27 | 24, 26 | eqtri 2847 | . . . . . . 7 ⊢ (+g‘𝑃) = (+g‘(Poly1‘(oppr‘𝑅))) |
28 | 27 | a1i 11 | . . . . . 6 ⊢ (⊤ → (+g‘𝑃) = (+g‘(Poly1‘(oppr‘𝑅)))) |
29 | 23, 28 | grpsubpropd 18207 | . . . . 5 ⊢ (⊤ → (-g‘𝑃) = (-g‘(Poly1‘(oppr‘𝑅)))) |
30 | 29 | mptru 1543 | . . . 4 ⊢ (-g‘𝑃) = (-g‘(Poly1‘(oppr‘𝑅))) |
31 | 22, 30 | eqtri 2847 | . . 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 7172 | . . . . . . 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 17875 | . . . . 5 ⊢ (⊤ → (0g‘𝑃) = (0g‘(Poly1‘(oppr‘𝑅)))) |
38 | 37 | mptru 1543 | . . . 4 ⊢ (0g‘𝑃) = (0g‘(Poly1‘(oppr‘𝑅))) |
39 | 32, 38 | eqtri 2847 | . . 3 ⊢ 0 = (0g‘(Poly1‘(oppr‘𝑅))) |
40 | eqid 2824 | . . 3 ⊢ (.r‘(Poly1‘(oppr‘𝑅))) = (.r‘(Poly1‘(oppr‘𝑅))) | |
41 | ply1divalg.r1 | . . . 4 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
42 | 4 | opprring 19384 | . . . 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 19414 | . . 3 ⊢ 𝑈 = (Unit‘(oppr‘𝑅)) |
50 | 1, 14, 21, 31, 39, 40, 43, 44, 45, 46, 47, 49 | ply1divalg 24734 | . 2 ⊢ (𝜑 → ∃!𝑞 ∈ 𝐵 (𝐷‘(𝐹 − (𝐺(.r‘(Poly1‘(oppr‘𝑅)))𝑞))) < (𝐷‘𝐺)) |
51 | 41 | adantr 483 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐵) → 𝑅 ∈ Ring) |
52 | 45 | adantr 483 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐵) → 𝐺 ∈ 𝐵) |
53 | simpr 487 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐵) → 𝑞 ∈ 𝐵) | |
54 | ply1divalg.t | . . . . . . . . 9 ⊢ ∙ = (.r‘𝑃) | |
55 | 16, 4, 1, 54, 40, 15 | ply1opprmul 20410 | . . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝐺 ∈ 𝐵 ∧ 𝑞 ∈ 𝐵) → (𝐺(.r‘(Poly1‘(oppr‘𝑅)))𝑞) = (𝑞 ∙ 𝐺)) |
56 | 51, 52, 53, 55 | syl3anc 1367 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐵) → (𝐺(.r‘(Poly1‘(oppr‘𝑅)))𝑞) = (𝑞 ∙ 𝐺)) |
57 | 56 | eqcomd 2830 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐵) → (𝑞 ∙ 𝐺) = (𝐺(.r‘(Poly1‘(oppr‘𝑅)))𝑞)) |
58 | 57 | oveq2d 7175 | . . . . 5 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐵) → (𝐹 − (𝑞 ∙ 𝐺)) = (𝐹 − (𝐺(.r‘(Poly1‘(oppr‘𝑅)))𝑞))) |
59 | 58 | fveq2d 6677 | . . . 4 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐵) → (𝐷‘(𝐹 − (𝑞 ∙ 𝐺))) = (𝐷‘(𝐹 − (𝐺(.r‘(Poly1‘(oppr‘𝑅)))𝑞)))) |
60 | 59 | breq1d 5079 | . . 3 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐵) → ((𝐷‘(𝐹 − (𝑞 ∙ 𝐺))) < (𝐷‘𝐺) ↔ (𝐷‘(𝐹 − (𝐺(.r‘(Poly1‘(oppr‘𝑅)))𝑞))) < (𝐷‘𝐺))) |
61 | 60 | reubidva 3391 | . 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 259 | 1 ⊢ (𝜑 → ∃!𝑞 ∈ 𝐵 (𝐷‘(𝐹 − (𝑞 ∙ 𝐺))) < (𝐷‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ⊤wtru 1537 ∈ wcel 2113 ≠ wne 3019 ∃!wreu 3143 class class class wbr 5069 ‘cfv 6358 (class class class)co 7159 < clt 10678 Basecbs 16486 +gcplusg 16568 .rcmulr 16569 0gc0g 16716 -gcsg 18108 Ringcrg 19300 opprcoppr 19375 Unitcui 19392 Poly1cpl1 20348 coe1cco1 20349 deg1 cdg1 24651 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 ax-pre-sup 10618 ax-addf 10619 ax-mulf 10620 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-iin 4925 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-se 5518 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-of 7412 df-ofr 7413 df-om 7584 df-1st 7692 df-2nd 7693 df-supp 7834 df-tpos 7895 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-2o 8106 df-oadd 8109 df-er 8292 df-map 8411 df-pm 8412 df-ixp 8465 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-fsupp 8837 df-sup 8909 df-oi 8977 df-card 9371 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-fz 12896 df-fzo 13037 df-seq 13373 df-hash 13694 df-struct 16488 df-ndx 16489 df-slot 16490 df-base 16492 df-sets 16493 df-ress 16494 df-plusg 16581 df-mulr 16582 df-starv 16583 df-sca 16584 df-vsca 16585 df-tset 16587 df-ple 16588 df-ds 16590 df-unif 16591 df-0g 16718 df-gsum 16719 df-mre 16860 df-mrc 16861 df-acs 16863 df-mgm 17855 df-sgrp 17904 df-mnd 17915 df-mhm 17959 df-submnd 17960 df-grp 18109 df-minusg 18110 df-sbg 18111 df-mulg 18228 df-subg 18279 df-ghm 18359 df-cntz 18450 df-cmn 18911 df-abl 18912 df-mgp 19243 df-ur 19255 df-ring 19302 df-cring 19303 df-oppr 19376 df-dvdsr 19394 df-unit 19395 df-invr 19425 df-subrg 19536 df-lmod 19639 df-lss 19707 df-rlreg 20059 df-psr 20139 df-mvr 20140 df-mpl 20141 df-opsr 20143 df-psr1 20351 df-vr1 20352 df-ply1 20353 df-coe1 20354 df-cnfld 20549 df-mdeg 24652 df-deg1 24653 |
This theorem is referenced by: q1peqb 24751 |
Copyright terms: Public domain | W3C validator |