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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 26177 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 20341 | . . . . . . 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 20343 | . . . . . . . 8 ⊢ (+g‘𝑅) = (+g‘(oppr‘𝑅)) | 
| 10 | 9 | oveqi 7444 | . . . . . . 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 26125 | . . . . 5 ⊢ (⊤ → (deg1‘𝑅) = (deg1‘(oppr‘𝑅))) | 
| 13 | 12 | mptru 1547 | . . . 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 6909 | . . . . 5 ⊢ (Base‘𝑃) = (Base‘(Poly1‘𝑅)) | 
| 18 | 3, 7, 11 | ply1baspropd 22244 | . . . . . 6 ⊢ (⊤ → (Base‘(Poly1‘𝑅)) = (Base‘(Poly1‘(oppr‘𝑅)))) | 
| 19 | 18 | mptru 1547 | . . . . 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 6909 | . . . . . . . 8 ⊢ (+g‘𝑃) = (+g‘(Poly1‘𝑅)) | 
| 25 | 3, 7, 11 | ply1plusgpropd 22245 | . . . . . . . . 9 ⊢ (⊤ → (+g‘(Poly1‘𝑅)) = (+g‘(Poly1‘(oppr‘𝑅)))) | 
| 26 | 25 | mptru 1547 | . . . . . . . 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 19063 | . . . . 5 ⊢ (⊤ → (-g‘𝑃) = (-g‘(Poly1‘(oppr‘𝑅)))) | 
| 30 | 29 | mptru 1547 | . . . 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 7444 | . . . . . . 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 18675 | . . . . 5 ⊢ (⊤ → (0g‘𝑃) = (0g‘(Poly1‘(oppr‘𝑅)))) | 
| 38 | 37 | mptru 1547 | . . . 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 20347 | . . . 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 20377 | . . 3 ⊢ 𝑈 = (Unit‘(oppr‘𝑅)) | 
| 50 | 1, 14, 21, 31, 39, 40, 43, 44, 45, 46, 47, 49 | ply1divalg 26177 | . 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 484 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐵) → 𝑞 ∈ 𝐵) | |
| 54 | ply1divalg.t | . . . . . . . . 9 ⊢ ∙ = (.r‘𝑃) | |
| 55 | 16, 4, 1, 54, 40, 15 | ply1opprmul 22240 | . . . . . . . 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 7447 | . . . . 5 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐵) → (𝐹 − (𝑞 ∙ 𝐺)) = (𝐹 − (𝐺(.r‘(Poly1‘(oppr‘𝑅)))𝑞))) | 
| 59 | 58 | fveq2d 6910 | . . . 4 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐵) → (𝐷‘(𝐹 − (𝑞 ∙ 𝐺))) = (𝐷‘(𝐹 − (𝐺(.r‘(Poly1‘(oppr‘𝑅)))𝑞)))) | 
| 60 | 59 | breq1d 5153 | . . 3 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐵) → ((𝐷‘(𝐹 − (𝑞 ∙ 𝐺))) < (𝐷‘𝐺) ↔ (𝐷‘(𝐹 − (𝐺(.r‘(Poly1‘(oppr‘𝑅)))𝑞))) < (𝐷‘𝐺))) | 
| 61 | 60 | reubidva 3396 | . 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 257 | 1 ⊢ (𝜑 → ∃!𝑞 ∈ 𝐵 (𝐷‘(𝐹 − (𝑞 ∙ 𝐺))) < (𝐷‘𝐺)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ⊤wtru 1541 ∈ wcel 2108 ≠ wne 2940 ∃!wreu 3378 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 < clt 11295 Basecbs 17247 +gcplusg 17297 .rcmulr 17298 0gc0g 17484 -gcsg 18953 Ringcrg 20230 opprcoppr 20333 Unitcui 20355 Poly1cpl1 22178 coe1cco1 22179 deg1cdg1 26093 | 
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-ofr 7698 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-tpos 8251 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-pm 8869 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-sup 9482 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-fz 13548 df-fzo 13695 df-seq 14043 df-hash 14370 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-0g 17486 df-gsum 17487 df-prds 17492 df-pws 17494 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-mhm 18796 df-submnd 18797 df-grp 18954 df-minusg 18955 df-sbg 18956 df-mulg 19086 df-subg 19141 df-ghm 19231 df-cntz 19335 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-cring 20233 df-oppr 20334 df-dvdsr 20357 df-unit 20358 df-invr 20388 df-subrng 20546 df-subrg 20570 df-rlreg 20694 df-lmod 20860 df-lss 20930 df-cnfld 21365 df-psr 21929 df-mvr 21930 df-mpl 21931 df-opsr 21933 df-psr1 22181 df-vr1 22182 df-ply1 22183 df-coe1 22184 df-mdeg 26094 df-deg1 26095 | 
| This theorem is referenced by: q1peqb 26195 ply1divalg3 35647 | 
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