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Mirrors > Home > MPE Home > Th. List > ply1opprmul | Structured version Visualization version GIF version |
Description: Reversing multiplication in a ring reverses multiplication in the univariate polynomial ring. (Contributed by Stefan O'Rear, 27-Mar-2015.) |
Ref | Expression |
---|---|
ply1opprmul.y | ⊢ 𝑌 = (Poly1‘𝑅) |
ply1opprmul.s | ⊢ 𝑆 = (oppr‘𝑅) |
ply1opprmul.z | ⊢ 𝑍 = (Poly1‘𝑆) |
ply1opprmul.t | ⊢ · = (.r‘𝑌) |
ply1opprmul.u | ⊢ ∙ = (.r‘𝑍) |
ply1opprmul.b | ⊢ 𝐵 = (Base‘𝑌) |
Ref | Expression |
---|---|
ply1opprmul | ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝐹 ∙ 𝐺) = (𝐺 · 𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . 2 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Ring) | |
2 | ply1opprmul.y | . . . 4 ⊢ 𝑌 = (Poly1‘𝑅) | |
3 | ply1opprmul.b | . . . 4 ⊢ 𝐵 = (Base‘𝑌) | |
4 | 2, 3 | ply1bascl 20299 | . . 3 ⊢ (𝐹 ∈ 𝐵 → 𝐹 ∈ (Base‘(PwSer1‘𝑅))) |
5 | eqid 2818 | . . . 4 ⊢ (PwSer1‘𝑅) = (PwSer1‘𝑅) | |
6 | eqid 2818 | . . . 4 ⊢ (Base‘(PwSer1‘𝑅)) = (Base‘(PwSer1‘𝑅)) | |
7 | 5, 6 | psr1bascl 20296 | . . 3 ⊢ (𝐹 ∈ (Base‘(PwSer1‘𝑅)) → 𝐹 ∈ (Base‘(1o mPwSer 𝑅))) |
8 | 4, 7 | syl 17 | . 2 ⊢ (𝐹 ∈ 𝐵 → 𝐹 ∈ (Base‘(1o mPwSer 𝑅))) |
9 | 2, 3 | ply1bascl 20299 | . . 3 ⊢ (𝐺 ∈ 𝐵 → 𝐺 ∈ (Base‘(PwSer1‘𝑅))) |
10 | 5, 6 | psr1bascl 20296 | . . 3 ⊢ (𝐺 ∈ (Base‘(PwSer1‘𝑅)) → 𝐺 ∈ (Base‘(1o mPwSer 𝑅))) |
11 | 9, 10 | syl 17 | . 2 ⊢ (𝐺 ∈ 𝐵 → 𝐺 ∈ (Base‘(1o mPwSer 𝑅))) |
12 | eqid 2818 | . . 3 ⊢ (1o mPwSer 𝑅) = (1o mPwSer 𝑅) | |
13 | ply1opprmul.s | . . 3 ⊢ 𝑆 = (oppr‘𝑅) | |
14 | eqid 2818 | . . 3 ⊢ (1o mPwSer 𝑆) = (1o mPwSer 𝑆) | |
15 | eqid 2818 | . . . 4 ⊢ (1o mPoly 𝑅) = (1o mPoly 𝑅) | |
16 | ply1opprmul.t | . . . . 5 ⊢ · = (.r‘𝑌) | |
17 | 2, 15, 16 | ply1mulr 20323 | . . . 4 ⊢ · = (.r‘(1o mPoly 𝑅)) |
18 | 15, 12, 17 | mplmulr 20317 | . . 3 ⊢ · = (.r‘(1o mPwSer 𝑅)) |
19 | eqid 2818 | . . . 4 ⊢ (1o mPoly 𝑆) = (1o mPoly 𝑆) | |
20 | ply1opprmul.z | . . . . 5 ⊢ 𝑍 = (Poly1‘𝑆) | |
21 | ply1opprmul.u | . . . . 5 ⊢ ∙ = (.r‘𝑍) | |
22 | 20, 19, 21 | ply1mulr 20323 | . . . 4 ⊢ ∙ = (.r‘(1o mPoly 𝑆)) |
23 | 19, 14, 22 | mplmulr 20317 | . . 3 ⊢ ∙ = (.r‘(1o mPwSer 𝑆)) |
24 | eqid 2818 | . . 3 ⊢ (Base‘(1o mPwSer 𝑅)) = (Base‘(1o mPwSer 𝑅)) | |
25 | 12, 13, 14, 18, 23, 24 | psropprmul 20334 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ (Base‘(1o mPwSer 𝑅)) ∧ 𝐺 ∈ (Base‘(1o mPwSer 𝑅))) → (𝐹 ∙ 𝐺) = (𝐺 · 𝐹)) |
26 | 1, 8, 11, 25 | syl3an 1152 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝐹 ∙ 𝐺) = (𝐺 · 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ‘cfv 6348 (class class class)co 7145 1oc1o 8084 Basecbs 16471 .rcmulr 16554 Ringcrg 19226 opprcoppr 19301 mPwSer cmps 20059 mPoly cmpl 20061 PwSer1cps1 20271 Poly1cpl1 20273 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-ofr 7399 df-om 7570 df-1st 7678 df-2nd 7679 df-supp 7820 df-tpos 7881 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-2o 8092 df-oadd 8095 df-er 8278 df-map 8397 df-pm 8398 df-ixp 8450 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-fsupp 8822 df-oi 8962 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-fz 12881 df-fzo 13022 df-seq 13358 df-hash 13679 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-sca 16569 df-vsca 16570 df-tset 16572 df-ple 16573 df-0g 16703 df-gsum 16704 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-grp 18044 df-minusg 18045 df-cntz 18385 df-cmn 18837 df-abl 18838 df-mgp 19169 df-ur 19181 df-ring 19228 df-oppr 19302 df-psr 20064 df-mpl 20066 df-opsr 20068 df-psr1 20276 df-ply1 20278 |
This theorem is referenced by: ply1divalg2 24659 |
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