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Mirrors > Home > MPE Home > Th. List > Mathboxes > plymul02 | Structured version Visualization version GIF version |
Description: Product of a polynomial with the zero polynomial. (Contributed by Thierry Arnoux, 26-Sep-2018.) |
Ref | Expression |
---|---|
plymul02 | ⊢ (𝐹 ∈ (Poly‘𝑆) → (0𝑝 ∘f · 𝐹) = 0𝑝) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | plyf 25510 | . . . . 5 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹:ℂ⟶ℂ) | |
2 | 1 | ffvelcdmda 7031 | . . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑥 ∈ ℂ) → (𝐹‘𝑥) ∈ ℂ) |
3 | 2 | mul02d 11311 | . . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑥 ∈ ℂ) → (0 · (𝐹‘𝑥)) = 0) |
4 | 3 | mpteq2dva 5203 | . 2 ⊢ (𝐹 ∈ (Poly‘𝑆) → (𝑥 ∈ ℂ ↦ (0 · (𝐹‘𝑥))) = (𝑥 ∈ ℂ ↦ 0)) |
5 | c0ex 11107 | . . . . . 6 ⊢ 0 ∈ V | |
6 | 5 | fconst 6725 | . . . . 5 ⊢ (ℂ × {0}):ℂ⟶{0} |
7 | df-0p 24985 | . . . . . 6 ⊢ 0𝑝 = (ℂ × {0}) | |
8 | 7 | feq1i 6656 | . . . . 5 ⊢ (0𝑝:ℂ⟶{0} ↔ (ℂ × {0}):ℂ⟶{0}) |
9 | 6, 8 | mpbir 230 | . . . 4 ⊢ 0𝑝:ℂ⟶{0} |
10 | ffn 6665 | . . . 4 ⊢ (0𝑝:ℂ⟶{0} → 0𝑝 Fn ℂ) | |
11 | 9, 10 | mp1i 13 | . . 3 ⊢ (𝐹 ∈ (Poly‘𝑆) → 0𝑝 Fn ℂ) |
12 | 1 | ffnd 6666 | . . 3 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹 Fn ℂ) |
13 | cnex 11090 | . . . 4 ⊢ ℂ ∈ V | |
14 | 13 | a1i 11 | . . 3 ⊢ (𝐹 ∈ (Poly‘𝑆) → ℂ ∈ V) |
15 | inidm 4176 | . . 3 ⊢ (ℂ ∩ ℂ) = ℂ | |
16 | 0pval 24986 | . . . 4 ⊢ (𝑥 ∈ ℂ → (0𝑝‘𝑥) = 0) | |
17 | 16 | adantl 482 | . . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑥 ∈ ℂ) → (0𝑝‘𝑥) = 0) |
18 | eqidd 2738 | . . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑥 ∈ ℂ) → (𝐹‘𝑥) = (𝐹‘𝑥)) | |
19 | 11, 12, 14, 14, 15, 17, 18 | offval 7618 | . 2 ⊢ (𝐹 ∈ (Poly‘𝑆) → (0𝑝 ∘f · 𝐹) = (𝑥 ∈ ℂ ↦ (0 · (𝐹‘𝑥)))) |
20 | fconstmpt 5692 | . . . 4 ⊢ (ℂ × {0}) = (𝑥 ∈ ℂ ↦ 0) | |
21 | 7, 20 | eqtri 2765 | . . 3 ⊢ 0𝑝 = (𝑥 ∈ ℂ ↦ 0) |
22 | 21 | a1i 11 | . 2 ⊢ (𝐹 ∈ (Poly‘𝑆) → 0𝑝 = (𝑥 ∈ ℂ ↦ 0)) |
23 | 4, 19, 22 | 3eqtr4d 2787 | 1 ⊢ (𝐹 ∈ (Poly‘𝑆) → (0𝑝 ∘f · 𝐹) = 0𝑝) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 Vcvv 3443 {csn 4584 ↦ cmpt 5186 × cxp 5629 Fn wfn 6488 ⟶wf 6489 ‘cfv 6493 (class class class)co 7351 ∘f cof 7607 ℂcc 11007 0cc0 11009 · cmul 11014 0𝑝c0p 24984 Polycply 25496 |
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 2708 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-inf2 9535 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-pre-sup 11087 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-int 4906 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-se 5587 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-of 7609 df-om 7795 df-1st 7913 df-2nd 7914 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-1o 8404 df-er 8606 df-map 8725 df-en 8842 df-dom 8843 df-sdom 8844 df-fin 8845 df-sup 9336 df-oi 9404 df-card 9833 df-pnf 11149 df-mnf 11150 df-xr 11151 df-ltxr 11152 df-le 11153 df-sub 11345 df-neg 11346 df-div 11771 df-nn 12112 df-2 12174 df-3 12175 df-n0 12372 df-z 12458 df-uz 12722 df-rp 12870 df-fz 13379 df-fzo 13522 df-seq 13861 df-exp 13922 df-hash 14184 df-cj 14943 df-re 14944 df-im 14945 df-sqrt 15079 df-abs 15080 df-clim 15329 df-sum 15530 df-0p 24985 df-ply 25500 |
This theorem is referenced by: plymulx 32963 |
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