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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 19980 | . . 3 ⊢ (𝐹 ∈ 𝐵 → 𝐹 ∈ (Base‘(PwSer1‘𝑅))) |
5 | eqid 2778 | . . . 4 ⊢ (PwSer1‘𝑅) = (PwSer1‘𝑅) | |
6 | eqid 2778 | . . . 4 ⊢ (Base‘(PwSer1‘𝑅)) = (Base‘(PwSer1‘𝑅)) | |
7 | 5, 6 | psr1bascl 19977 | . . 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 19980 | . . 3 ⊢ (𝐺 ∈ 𝐵 → 𝐺 ∈ (Base‘(PwSer1‘𝑅))) |
10 | 5, 6 | psr1bascl 19977 | . . 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 2778 | . . 3 ⊢ (1o mPwSer 𝑅) = (1o mPwSer 𝑅) | |
13 | ply1opprmul.s | . . 3 ⊢ 𝑆 = (oppr‘𝑅) | |
14 | eqid 2778 | . . 3 ⊢ (1o mPwSer 𝑆) = (1o mPwSer 𝑆) | |
15 | eqid 2778 | . . . 4 ⊢ (1o mPoly 𝑅) = (1o mPoly 𝑅) | |
16 | ply1opprmul.t | . . . . 5 ⊢ · = (.r‘𝑌) | |
17 | 2, 15, 16 | ply1mulr 20004 | . . . 4 ⊢ · = (.r‘(1o mPoly 𝑅)) |
18 | 15, 12, 17 | mplmulr 19998 | . . 3 ⊢ · = (.r‘(1o mPwSer 𝑅)) |
19 | eqid 2778 | . . . 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 20004 | . . . 4 ⊢ ∙ = (.r‘(1o mPoly 𝑆)) |
23 | 19, 14, 22 | mplmulr 19998 | . . 3 ⊢ ∙ = (.r‘(1o mPwSer 𝑆)) |
24 | eqid 2778 | . . 3 ⊢ (Base‘(1o mPwSer 𝑅)) = (Base‘(1o mPwSer 𝑅)) | |
25 | 12, 13, 14, 18, 23, 24 | psropprmul 20015 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ (Base‘(1o mPwSer 𝑅)) ∧ 𝐺 ∈ (Base‘(1o mPwSer 𝑅))) → (𝐹 ∙ 𝐺) = (𝐺 · 𝐹)) |
26 | 1, 8, 11, 25 | syl3an 1160 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝐹 ∙ 𝐺) = (𝐺 · 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1071 = wceq 1601 ∈ wcel 2107 ‘cfv 6137 (class class class)co 6924 1oc1o 7838 Basecbs 16266 .rcmulr 16350 Ringcrg 18945 opprcoppr 19020 mPwSer cmps 19759 mPoly cmpl 19761 PwSer1cps1 19952 Poly1cpl1 19954 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-int 4713 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-se 5317 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-isom 6146 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-of 7176 df-ofr 7177 df-om 7346 df-1st 7447 df-2nd 7448 df-supp 7579 df-tpos 7636 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-1o 7845 df-2o 7846 df-oadd 7849 df-er 8028 df-map 8144 df-pm 8145 df-ixp 8197 df-en 8244 df-dom 8245 df-sdom 8246 df-fin 8247 df-fsupp 8566 df-oi 8706 df-card 9100 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-nn 11380 df-2 11443 df-3 11444 df-4 11445 df-5 11446 df-6 11447 df-7 11448 df-8 11449 df-9 11450 df-n0 11648 df-z 11734 df-dec 11851 df-uz 11998 df-fz 12649 df-fzo 12790 df-seq 13125 df-hash 13442 df-struct 16268 df-ndx 16269 df-slot 16270 df-base 16272 df-sets 16273 df-ress 16274 df-plusg 16362 df-mulr 16363 df-sca 16365 df-vsca 16366 df-tset 16368 df-ple 16369 df-0g 16499 df-gsum 16500 df-mgm 17639 df-sgrp 17681 df-mnd 17692 df-grp 17823 df-minusg 17824 df-cntz 18144 df-cmn 18592 df-abl 18593 df-mgp 18888 df-ur 18900 df-ring 18947 df-oppr 19021 df-psr 19764 df-mpl 19766 df-opsr 19768 df-psr1 19957 df-ply1 19959 |
This theorem is referenced by: ply1divalg2 24346 |
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