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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 2610 | . . . 4 ⊢ (1𝑜 mPwSer 𝑅) = (1𝑜 mPwSer 𝑅) | |
4 | eqid 2610 | . . . 4 ⊢ ( ·𝑠 ‘𝑆) = ( ·𝑠 ‘𝑆) | |
5 | 2, 3, 4 | mplvsca2 19267 | . . 3 ⊢ ( ·𝑠 ‘𝑆) = ( ·𝑠 ‘(1𝑜 mPwSer 𝑅)) |
6 | eqid 2610 | . . . 4 ⊢ (PwSer1‘𝑅) = (PwSer1‘𝑅) | |
7 | eqid 2610 | . . . 4 ⊢ ( ·𝑠 ‘(PwSer1‘𝑅)) = ( ·𝑠 ‘(PwSer1‘𝑅)) | |
8 | 6, 3, 7 | psr1vsca 19414 | . . 3 ⊢ ( ·𝑠 ‘(PwSer1‘𝑅)) = ( ·𝑠 ‘(1𝑜 mPwSer 𝑅)) |
9 | fvex 6113 | . . . 4 ⊢ (Base‘(1𝑜 mPoly 𝑅)) ∈ V | |
10 | ply1plusg.y | . . . . . 6 ⊢ 𝑌 = (Poly1‘𝑅) | |
11 | 10, 6 | ply1val 19385 | . . . . 5 ⊢ 𝑌 = ((PwSer1‘𝑅) ↾s (Base‘(1𝑜 mPoly 𝑅))) |
12 | 11, 7 | ressvsca 15855 | . . . 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 2638 | . 2 ⊢ ( ·𝑠 ‘𝑆) = ( ·𝑠 ‘𝑌) |
15 | 1, 14 | eqtr4i 2635 | 1 ⊢ · = ( ·𝑠 ‘𝑆) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∈ wcel 1977 Vcvv 3173 ‘cfv 5804 (class class class)co 6549 1𝑜c1o 7440 Basecbs 15695 ·𝑠 cvsca 15772 mPwSer cmps 19172 mPoly cmpl 19174 PwSer1cps1 19366 Poly1cpl1 19368 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-dec 11370 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-vsca 15785 df-ple 15788 df-psr 19177 df-mpl 19179 df-opsr 19181 df-psr1 19371 df-ply1 19373 |
This theorem is referenced by: ressply1vsca 19423 ply1ascl 19449 coe1tm 19464 ply1coe 19487 deg1vscale 23668 deg1vsca 23669 ply1ass23l 41964 |
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