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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 2819 | . . . 4 ⊢ (1o mPwSer 𝑅) = (1o mPwSer 𝑅) | |
4 | eqid 2819 | . . . 4 ⊢ ( ·𝑠 ‘𝑆) = ( ·𝑠 ‘𝑆) | |
5 | 2, 3, 4 | mplvsca2 20218 | . . 3 ⊢ ( ·𝑠 ‘𝑆) = ( ·𝑠 ‘(1o mPwSer 𝑅)) |
6 | eqid 2819 | . . . 4 ⊢ (PwSer1‘𝑅) = (PwSer1‘𝑅) | |
7 | eqid 2819 | . . . 4 ⊢ ( ·𝑠 ‘(PwSer1‘𝑅)) = ( ·𝑠 ‘(PwSer1‘𝑅)) | |
8 | 6, 3, 7 | psr1vsca 20383 | . . 3 ⊢ ( ·𝑠 ‘(PwSer1‘𝑅)) = ( ·𝑠 ‘(1o mPwSer 𝑅)) |
9 | fvex 6676 | . . . 4 ⊢ (Base‘(1o mPoly 𝑅)) ∈ V | |
10 | ply1plusg.y | . . . . . 6 ⊢ 𝑌 = (Poly1‘𝑅) | |
11 | 10, 6 | ply1val 20354 | . . . . 5 ⊢ 𝑌 = ((PwSer1‘𝑅) ↾s (Base‘(1o mPoly 𝑅))) |
12 | 11, 7 | ressvsca 16643 | . . . 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 2848 | . 2 ⊢ ( ·𝑠 ‘𝑆) = ( ·𝑠 ‘𝑌) |
15 | 1, 14 | eqtr4i 2845 | 1 ⊢ · = ( ·𝑠 ‘𝑆) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1531 ∈ wcel 2108 Vcvv 3493 ‘cfv 6348 (class class class)co 7148 1oc1o 8087 Basecbs 16475 ·𝑠 cvsca 16561 mPwSer cmps 20123 mPoly cmpl 20125 PwSer1cps1 20335 Poly1cpl1 20337 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-cnex 10585 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4831 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-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-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-om 7573 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-er 8281 df-en 8502 df-dom 8503 df-sdom 8504 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-nn 11631 df-2 11692 df-3 11693 df-4 11694 df-5 11695 df-6 11696 df-7 11697 df-8 11698 df-9 11699 df-dec 12091 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-vsca 16574 df-ple 16577 df-psr 20128 df-mpl 20130 df-opsr 20132 df-psr1 20340 df-ply1 20342 |
This theorem is referenced by: ressply1vsca 20392 ply1ascl 20418 coe1tm 20433 ply1coe 20456 deg1vscale 24690 deg1vsca 24691 ply1ass23l 44427 |
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