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Mirrors > Home > MPE Home > Th. List > ply1mulr | Structured version Visualization version GIF version |
Description: Value of multiplication in a univariate polynomial ring. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by Mario Carneiro, 4-Jul-2015.) |
Ref | Expression |
---|---|
ply1plusg.y | ⊢ 𝑌 = (Poly1‘𝑅) |
ply1plusg.s | ⊢ 𝑆 = (1o mPoly 𝑅) |
ply1mulr.n | ⊢ · = (.r‘𝑌) |
Ref | Expression |
---|---|
ply1mulr | ⊢ · = (.r‘𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ply1mulr.n | . 2 ⊢ · = (.r‘𝑌) | |
2 | ply1plusg.s | . . . 4 ⊢ 𝑆 = (1o mPoly 𝑅) | |
3 | eqid 2731 | . . . 4 ⊢ (1o mPwSer 𝑅) = (1o mPwSer 𝑅) | |
4 | eqid 2731 | . . . 4 ⊢ (.r‘𝑆) = (.r‘𝑆) | |
5 | 2, 3, 4 | mplmulr 21674 | . . 3 ⊢ (.r‘𝑆) = (.r‘(1o mPwSer 𝑅)) |
6 | eqid 2731 | . . . 4 ⊢ (PwSer1‘𝑅) = (PwSer1‘𝑅) | |
7 | eqid 2731 | . . . 4 ⊢ (.r‘(PwSer1‘𝑅)) = (.r‘(PwSer1‘𝑅)) | |
8 | 6, 3, 7 | psr1mulr 21677 | . . 3 ⊢ (.r‘(PwSer1‘𝑅)) = (.r‘(1o mPwSer 𝑅)) |
9 | fvex 6891 | . . . 4 ⊢ (Base‘(1o mPoly 𝑅)) ∈ V | |
10 | ply1plusg.y | . . . . . 6 ⊢ 𝑌 = (Poly1‘𝑅) | |
11 | 10, 6 | ply1val 21647 | . . . . 5 ⊢ 𝑌 = ((PwSer1‘𝑅) ↾s (Base‘(1o mPoly 𝑅))) |
12 | 11, 7 | ressmulr 17234 | . . . 4 ⊢ ((Base‘(1o mPoly 𝑅)) ∈ V → (.r‘(PwSer1‘𝑅)) = (.r‘𝑌)) |
13 | 9, 12 | ax-mp 5 | . . 3 ⊢ (.r‘(PwSer1‘𝑅)) = (.r‘𝑌) |
14 | 5, 8, 13 | 3eqtr2i 2765 | . 2 ⊢ (.r‘𝑆) = (.r‘𝑌) |
15 | 1, 14 | eqtr4i 2762 | 1 ⊢ · = (.r‘𝑆) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 ∈ wcel 2106 Vcvv 3473 ‘cfv 6532 (class class class)co 7393 1oc1o 8441 Basecbs 17126 .rcmulr 17180 mPwSer cmps 21388 mPoly cmpl 21390 PwSer1cps1 21628 Poly1cpl1 21630 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 ax-cnex 11148 ax-resscn 11149 ax-1cn 11150 ax-icn 11151 ax-addcl 11152 ax-addrcl 11153 ax-mulcl 11154 ax-mulrcl 11155 ax-mulcom 11156 ax-addass 11157 ax-mulass 11158 ax-distr 11159 ax-i2m1 11160 ax-1ne0 11161 ax-1rid 11162 ax-rnegex 11163 ax-rrecex 11164 ax-cnre 11165 ax-pre-lttri 11166 ax-pre-lttrn 11167 ax-pre-ltadd 11168 ax-pre-mulgt0 11169 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6289 df-ord 6356 df-on 6357 df-lim 6358 df-suc 6359 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-riota 7349 df-ov 7396 df-oprab 7397 df-mpo 7398 df-om 7839 df-2nd 7958 df-frecs 8248 df-wrecs 8279 df-recs 8353 df-rdg 8392 df-er 8686 df-en 8923 df-dom 8924 df-sdom 8925 df-pnf 11232 df-mnf 11233 df-xr 11234 df-ltxr 11235 df-le 11236 df-sub 11428 df-neg 11429 df-nn 12195 df-2 12257 df-3 12258 df-4 12259 df-5 12260 df-6 12261 df-7 12262 df-8 12263 df-9 12264 df-dec 12660 df-sets 17079 df-slot 17097 df-ndx 17109 df-base 17127 df-ress 17156 df-mulr 17193 df-ple 17199 df-psr 21393 df-mpl 21395 df-opsr 21397 df-psr1 21633 df-ply1 21635 |
This theorem is referenced by: ressply1mul 21684 subrgply1 21686 ply1opprmul 21692 ply1mpl1 21710 coe1mul 21723 coe1tm 21726 ply1coe 21749 evls1rhm 21770 evl1rhm 21780 deg1mulle2 25556 ply1ass23l 46709 |
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