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Mirrors > Home > MPE Home > Th. List > ply1vsca | Structured version Visualization version GIF version |
Description: Value of scalar 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) |
ply1vscafval.n | ⊢ · = ( ·𝑠 ‘𝑌) |
Ref | Expression |
---|---|
ply1vsca | ⊢ · = ( ·𝑠 ‘𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ply1vscafval.n | . 2 ⊢ · = ( ·𝑠 ‘𝑌) | |
2 | ply1plusg.s | . . . 4 ⊢ 𝑆 = (1o mPoly 𝑅) | |
3 | eqid 2736 | . . . 4 ⊢ (1o mPwSer 𝑅) = (1o mPwSer 𝑅) | |
4 | eqid 2736 | . . . 4 ⊢ ( ·𝑠 ‘𝑆) = ( ·𝑠 ‘𝑆) | |
5 | 2, 3, 4 | mplvsca2 20928 | . . 3 ⊢ ( ·𝑠 ‘𝑆) = ( ·𝑠 ‘(1o mPwSer 𝑅)) |
6 | eqid 2736 | . . . 4 ⊢ (PwSer1‘𝑅) = (PwSer1‘𝑅) | |
7 | eqid 2736 | . . . 4 ⊢ ( ·𝑠 ‘(PwSer1‘𝑅)) = ( ·𝑠 ‘(PwSer1‘𝑅)) | |
8 | 6, 3, 7 | psr1vsca 21098 | . . 3 ⊢ ( ·𝑠 ‘(PwSer1‘𝑅)) = ( ·𝑠 ‘(1o mPwSer 𝑅)) |
9 | fvex 6708 | . . . 4 ⊢ (Base‘(1o mPoly 𝑅)) ∈ V | |
10 | ply1plusg.y | . . . . . 6 ⊢ 𝑌 = (Poly1‘𝑅) | |
11 | 10, 6 | ply1val 21069 | . . . . 5 ⊢ 𝑌 = ((PwSer1‘𝑅) ↾s (Base‘(1o mPoly 𝑅))) |
12 | 11, 7 | ressvsca 16835 | . . . 4 ⊢ ((Base‘(1o mPoly 𝑅)) ∈ V → ( ·𝑠 ‘(PwSer1‘𝑅)) = ( ·𝑠 ‘𝑌)) |
13 | 9, 12 | ax-mp 5 | . . 3 ⊢ ( ·𝑠 ‘(PwSer1‘𝑅)) = ( ·𝑠 ‘𝑌) |
14 | 5, 8, 13 | 3eqtr2i 2765 | . 2 ⊢ ( ·𝑠 ‘𝑆) = ( ·𝑠 ‘𝑌) |
15 | 1, 14 | eqtr4i 2762 | 1 ⊢ · = ( ·𝑠 ‘𝑆) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1543 ∈ wcel 2112 Vcvv 3398 ‘cfv 6358 (class class class)co 7191 1oc1o 8173 Basecbs 16666 ·𝑠 cvsca 16753 mPwSer cmps 20817 mPoly cmpl 20819 PwSer1cps1 21050 Poly1cpl1 21052 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-nn 11796 df-2 11858 df-3 11859 df-4 11860 df-5 11861 df-6 11862 df-7 11863 df-8 11864 df-9 11865 df-dec 12259 df-ndx 16669 df-slot 16670 df-base 16672 df-sets 16673 df-ress 16674 df-vsca 16766 df-ple 16769 df-psr 20822 df-mpl 20824 df-opsr 20826 df-psr1 21055 df-ply1 21057 |
This theorem is referenced by: ressply1vsca 21107 ply1ascl 21133 coe1tm 21148 ply1coe 21171 deg1vscale 24956 deg1vsca 24957 ply1ass23l 45339 |
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