Mathbox for Stefan O'Rear |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > mzpmul | Structured version Visualization version GIF version |
Description: The pointwise product of two polynomial functions is a polynomial function. See also mzpmulmpt 39426. (Contributed by Stefan O'Rear, 4-Oct-2014.) |
Ref | Expression |
---|---|
mzpmul | ⊢ ((𝐴 ∈ (mzPoly‘𝑉) ∧ 𝐵 ∈ (mzPoly‘𝑉)) → (𝐴 ∘f · 𝐵) ∈ (mzPoly‘𝑉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfvex 6684 | . . . . 5 ⊢ (𝐴 ∈ (mzPoly‘𝑉) → 𝑉 ∈ V) | |
2 | 1 | adantr 483 | . . . 4 ⊢ ((𝐴 ∈ (mzPoly‘𝑉) ∧ 𝐵 ∈ (mzPoly‘𝑉)) → 𝑉 ∈ V) |
3 | mzpincl 39418 | . . . 4 ⊢ (𝑉 ∈ V → (mzPoly‘𝑉) ∈ (mzPolyCld‘𝑉)) | |
4 | 2, 3 | syl 17 | . . 3 ⊢ ((𝐴 ∈ (mzPoly‘𝑉) ∧ 𝐵 ∈ (mzPoly‘𝑉)) → (mzPoly‘𝑉) ∈ (mzPolyCld‘𝑉)) |
5 | mzpcl34 39415 | . . . 4 ⊢ (((mzPoly‘𝑉) ∈ (mzPolyCld‘𝑉) ∧ 𝐴 ∈ (mzPoly‘𝑉) ∧ 𝐵 ∈ (mzPoly‘𝑉)) → ((𝐴 ∘f + 𝐵) ∈ (mzPoly‘𝑉) ∧ (𝐴 ∘f · 𝐵) ∈ (mzPoly‘𝑉))) | |
6 | 5 | 3expib 1118 | . . 3 ⊢ ((mzPoly‘𝑉) ∈ (mzPolyCld‘𝑉) → ((𝐴 ∈ (mzPoly‘𝑉) ∧ 𝐵 ∈ (mzPoly‘𝑉)) → ((𝐴 ∘f + 𝐵) ∈ (mzPoly‘𝑉) ∧ (𝐴 ∘f · 𝐵) ∈ (mzPoly‘𝑉)))) |
7 | 4, 6 | mpcom 38 | . 2 ⊢ ((𝐴 ∈ (mzPoly‘𝑉) ∧ 𝐵 ∈ (mzPoly‘𝑉)) → ((𝐴 ∘f + 𝐵) ∈ (mzPoly‘𝑉) ∧ (𝐴 ∘f · 𝐵) ∈ (mzPoly‘𝑉))) |
8 | 7 | simprd 498 | 1 ⊢ ((𝐴 ∈ (mzPoly‘𝑉) ∧ 𝐵 ∈ (mzPoly‘𝑉)) → (𝐴 ∘f · 𝐵) ∈ (mzPoly‘𝑉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2114 Vcvv 3481 ‘cfv 6336 (class class class)co 7137 ∘f cof 7388 + caddc 10521 · cmul 10523 mzPolyCldcmzpcl 39405 mzPolycmzp 39406 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2792 ax-rep 5171 ax-sep 5184 ax-nul 5191 ax-pow 5247 ax-pr 5311 ax-un 7442 ax-cnex 10574 ax-resscn 10575 ax-1cn 10576 ax-icn 10577 ax-addcl 10578 ax-addrcl 10579 ax-mulcl 10580 ax-mulrcl 10581 ax-mulcom 10582 ax-addass 10583 ax-mulass 10584 ax-distr 10585 ax-i2m1 10586 ax-1ne0 10587 ax-1rid 10588 ax-rnegex 10589 ax-rrecex 10590 ax-cnre 10591 ax-pre-lttri 10592 ax-pre-lttrn 10593 ax-pre-ltadd 10594 ax-pre-mulgt0 10595 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2891 df-nfc 2959 df-ne 3012 df-nel 3119 df-ral 3138 df-rex 3139 df-reu 3140 df-rab 3142 df-v 3483 df-sbc 3759 df-csb 3867 df-dif 3922 df-un 3924 df-in 3926 df-ss 3935 df-pss 3937 df-nul 4275 df-if 4449 df-pw 4522 df-sn 4549 df-pr 4551 df-tp 4553 df-op 4555 df-uni 4820 df-int 4858 df-iun 4902 df-br 5048 df-opab 5110 df-mpt 5128 df-tr 5154 df-id 5441 df-eprel 5446 df-po 5455 df-so 5456 df-fr 5495 df-we 5497 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 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6295 df-fun 6338 df-fn 6339 df-f 6340 df-f1 6341 df-fo 6342 df-f1o 6343 df-fv 6344 df-riota 7095 df-ov 7140 df-oprab 7141 df-mpo 7142 df-of 7390 df-om 7562 df-wrecs 7928 df-recs 7989 df-rdg 8027 df-er 8270 df-map 8389 df-en 8491 df-dom 8492 df-sdom 8493 df-pnf 10658 df-mnf 10659 df-xr 10660 df-ltxr 10661 df-le 10662 df-sub 10853 df-neg 10854 df-nn 11620 df-n0 11880 df-z 11964 df-mzpcl 39407 df-mzp 39408 |
This theorem is referenced by: mzpmulmpt 39426 mzpmfp 39431 |
Copyright terms: Public domain | W3C validator |