![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
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 26252 | . . . . 5 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹:ℂ⟶ℂ) | |
2 | 1 | ffvelcdmda 7104 | . . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑥 ∈ ℂ) → (𝐹‘𝑥) ∈ ℂ) |
3 | 2 | mul02d 11457 | . . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑥 ∈ ℂ) → (0 · (𝐹‘𝑥)) = 0) |
4 | 3 | mpteq2dva 5248 | . 2 ⊢ (𝐹 ∈ (Poly‘𝑆) → (𝑥 ∈ ℂ ↦ (0 · (𝐹‘𝑥))) = (𝑥 ∈ ℂ ↦ 0)) |
5 | c0ex 11253 | . . . . . 6 ⊢ 0 ∈ V | |
6 | 5 | fconst 6795 | . . . . 5 ⊢ (ℂ × {0}):ℂ⟶{0} |
7 | df-0p 25719 | . . . . . 6 ⊢ 0𝑝 = (ℂ × {0}) | |
8 | 7 | feq1i 6728 | . . . . 5 ⊢ (0𝑝:ℂ⟶{0} ↔ (ℂ × {0}):ℂ⟶{0}) |
9 | 6, 8 | mpbir 231 | . . . 4 ⊢ 0𝑝:ℂ⟶{0} |
10 | ffn 6737 | . . . 4 ⊢ (0𝑝:ℂ⟶{0} → 0𝑝 Fn ℂ) | |
11 | 9, 10 | mp1i 13 | . . 3 ⊢ (𝐹 ∈ (Poly‘𝑆) → 0𝑝 Fn ℂ) |
12 | 1 | ffnd 6738 | . . 3 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹 Fn ℂ) |
13 | cnex 11234 | . . . 4 ⊢ ℂ ∈ V | |
14 | 13 | a1i 11 | . . 3 ⊢ (𝐹 ∈ (Poly‘𝑆) → ℂ ∈ V) |
15 | inidm 4235 | . . 3 ⊢ (ℂ ∩ ℂ) = ℂ | |
16 | 0pval 25720 | . . . 4 ⊢ (𝑥 ∈ ℂ → (0𝑝‘𝑥) = 0) | |
17 | 16 | adantl 481 | . . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑥 ∈ ℂ) → (0𝑝‘𝑥) = 0) |
18 | eqidd 2736 | . . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑥 ∈ ℂ) → (𝐹‘𝑥) = (𝐹‘𝑥)) | |
19 | 11, 12, 14, 14, 15, 17, 18 | offval 7706 | . 2 ⊢ (𝐹 ∈ (Poly‘𝑆) → (0𝑝 ∘f · 𝐹) = (𝑥 ∈ ℂ ↦ (0 · (𝐹‘𝑥)))) |
20 | fconstmpt 5751 | . . . 4 ⊢ (ℂ × {0}) = (𝑥 ∈ ℂ ↦ 0) | |
21 | 7, 20 | eqtri 2763 | . . 3 ⊢ 0𝑝 = (𝑥 ∈ ℂ ↦ 0) |
22 | 21 | a1i 11 | . 2 ⊢ (𝐹 ∈ (Poly‘𝑆) → 0𝑝 = (𝑥 ∈ ℂ ↦ 0)) |
23 | 4, 19, 22 | 3eqtr4d 2785 | 1 ⊢ (𝐹 ∈ (Poly‘𝑆) → (0𝑝 ∘f · 𝐹) = 0𝑝) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 Vcvv 3478 {csn 4631 ↦ cmpt 5231 × cxp 5687 Fn wfn 6558 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ∘f cof 7695 ℂcc 11151 0cc0 11153 · cmul 11158 0𝑝c0p 25718 Polycply 26238 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12612 df-uz 12877 df-rp 13033 df-fz 13545 df-fzo 13692 df-seq 14040 df-exp 14100 df-hash 14367 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-clim 15521 df-sum 15720 df-0p 25719 df-ply 26242 |
This theorem is referenced by: plymulx 34542 |
Copyright terms: Public domain | W3C validator |