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Theorem plymul0or 24869
 Description: Polynomial multiplication has no zero divisors. (Contributed by Mario Carneiro, 26-Jul-2014.)
Assertion
Ref Expression
plymul0or ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝐹f · 𝐺) = 0𝑝 ↔ (𝐹 = 0𝑝𝐺 = 0𝑝)))

Proof of Theorem plymul0or
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 dgrcl 24822 . . . . . . 7 (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ0)
2 dgrcl 24822 . . . . . . 7 (𝐺 ∈ (Poly‘𝑆) → (deg‘𝐺) ∈ ℕ0)
3 nn0addcl 11931 . . . . . . 7 (((deg‘𝐹) ∈ ℕ0 ∧ (deg‘𝐺) ∈ ℕ0) → ((deg‘𝐹) + (deg‘𝐺)) ∈ ℕ0)
41, 2, 3syl2an 597 . . . . . 6 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((deg‘𝐹) + (deg‘𝐺)) ∈ ℕ0)
5 c0ex 10634 . . . . . . 7 0 ∈ V
65fvconst2 6965 . . . . . 6 (((deg‘𝐹) + (deg‘𝐺)) ∈ ℕ0 → ((ℕ0 × {0})‘((deg‘𝐹) + (deg‘𝐺))) = 0)
74, 6syl 17 . . . . 5 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((ℕ0 × {0})‘((deg‘𝐹) + (deg‘𝐺))) = 0)
8 fveq2 6669 . . . . . . . 8 ((𝐹f · 𝐺) = 0𝑝 → (coeff‘(𝐹f · 𝐺)) = (coeff‘0𝑝))
9 coe0 24845 . . . . . . . 8 (coeff‘0𝑝) = (ℕ0 × {0})
108, 9syl6eq 2872 . . . . . . 7 ((𝐹f · 𝐺) = 0𝑝 → (coeff‘(𝐹f · 𝐺)) = (ℕ0 × {0}))
1110fveq1d 6671 . . . . . 6 ((𝐹f · 𝐺) = 0𝑝 → ((coeff‘(𝐹f · 𝐺))‘((deg‘𝐹) + (deg‘𝐺))) = ((ℕ0 × {0})‘((deg‘𝐹) + (deg‘𝐺))))
1211eqeq1d 2823 . . . . 5 ((𝐹f · 𝐺) = 0𝑝 → (((coeff‘(𝐹f · 𝐺))‘((deg‘𝐹) + (deg‘𝐺))) = 0 ↔ ((ℕ0 × {0})‘((deg‘𝐹) + (deg‘𝐺))) = 0))
137, 12syl5ibrcom 249 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝐹f · 𝐺) = 0𝑝 → ((coeff‘(𝐹f · 𝐺))‘((deg‘𝐹) + (deg‘𝐺))) = 0))
14 eqid 2821 . . . . . . 7 (coeff‘𝐹) = (coeff‘𝐹)
15 eqid 2821 . . . . . . 7 (coeff‘𝐺) = (coeff‘𝐺)
16 eqid 2821 . . . . . . 7 (deg‘𝐹) = (deg‘𝐹)
17 eqid 2821 . . . . . . 7 (deg‘𝐺) = (deg‘𝐺)
1814, 15, 16, 17coemulhi 24843 . . . . . 6 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((coeff‘(𝐹f · 𝐺))‘((deg‘𝐹) + (deg‘𝐺))) = (((coeff‘𝐹)‘(deg‘𝐹)) · ((coeff‘𝐺)‘(deg‘𝐺))))
1918eqeq1d 2823 . . . . 5 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (((coeff‘(𝐹f · 𝐺))‘((deg‘𝐹) + (deg‘𝐺))) = 0 ↔ (((coeff‘𝐹)‘(deg‘𝐹)) · ((coeff‘𝐺)‘(deg‘𝐺))) = 0))
2014coef3 24821 . . . . . . . 8 (𝐹 ∈ (Poly‘𝑆) → (coeff‘𝐹):ℕ0⟶ℂ)
2120adantr 483 . . . . . . 7 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (coeff‘𝐹):ℕ0⟶ℂ)
221adantr 483 . . . . . . 7 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (deg‘𝐹) ∈ ℕ0)
2321, 22ffvelrnd 6851 . . . . . 6 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((coeff‘𝐹)‘(deg‘𝐹)) ∈ ℂ)
2415coef3 24821 . . . . . . . 8 (𝐺 ∈ (Poly‘𝑆) → (coeff‘𝐺):ℕ0⟶ℂ)
2524adantl 484 . . . . . . 7 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (coeff‘𝐺):ℕ0⟶ℂ)
262adantl 484 . . . . . . 7 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (deg‘𝐺) ∈ ℕ0)
2725, 26ffvelrnd 6851 . . . . . 6 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((coeff‘𝐺)‘(deg‘𝐺)) ∈ ℂ)
2823, 27mul0ord 11289 . . . . 5 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((((coeff‘𝐹)‘(deg‘𝐹)) · ((coeff‘𝐺)‘(deg‘𝐺))) = 0 ↔ (((coeff‘𝐹)‘(deg‘𝐹)) = 0 ∨ ((coeff‘𝐺)‘(deg‘𝐺)) = 0)))
2919, 28bitrd 281 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (((coeff‘(𝐹f · 𝐺))‘((deg‘𝐹) + (deg‘𝐺))) = 0 ↔ (((coeff‘𝐹)‘(deg‘𝐹)) = 0 ∨ ((coeff‘𝐺)‘(deg‘𝐺)) = 0)))
3013, 29sylibd 241 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝐹f · 𝐺) = 0𝑝 → (((coeff‘𝐹)‘(deg‘𝐹)) = 0 ∨ ((coeff‘𝐺)‘(deg‘𝐺)) = 0)))
3116, 14dgreq0 24854 . . . . 5 (𝐹 ∈ (Poly‘𝑆) → (𝐹 = 0𝑝 ↔ ((coeff‘𝐹)‘(deg‘𝐹)) = 0))
3231adantr 483 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹 = 0𝑝 ↔ ((coeff‘𝐹)‘(deg‘𝐹)) = 0))
3317, 15dgreq0 24854 . . . . 5 (𝐺 ∈ (Poly‘𝑆) → (𝐺 = 0𝑝 ↔ ((coeff‘𝐺)‘(deg‘𝐺)) = 0))
3433adantl 484 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐺 = 0𝑝 ↔ ((coeff‘𝐺)‘(deg‘𝐺)) = 0))
3532, 34orbi12d 915 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝐹 = 0𝑝𝐺 = 0𝑝) ↔ (((coeff‘𝐹)‘(deg‘𝐹)) = 0 ∨ ((coeff‘𝐺)‘(deg‘𝐺)) = 0)))
3630, 35sylibrd 261 . 2 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝐹f · 𝐺) = 0𝑝 → (𝐹 = 0𝑝𝐺 = 0𝑝)))
37 cnex 10617 . . . . . . 7 ℂ ∈ V
3837a1i 11 . . . . . 6 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ℂ ∈ V)
39 plyf 24787 . . . . . . 7 (𝐺 ∈ (Poly‘𝑆) → 𝐺:ℂ⟶ℂ)
4039adantl 484 . . . . . 6 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐺:ℂ⟶ℂ)
41 0cnd 10633 . . . . . 6 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 0 ∈ ℂ)
42 mul02 10817 . . . . . . 7 (𝑥 ∈ ℂ → (0 · 𝑥) = 0)
4342adantl 484 . . . . . 6 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑥 ∈ ℂ) → (0 · 𝑥) = 0)
4438, 40, 41, 41, 43caofid2 7439 . . . . 5 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((ℂ × {0}) ∘f · 𝐺) = (ℂ × {0}))
45 id 22 . . . . . . . 8 (𝐹 = 0𝑝𝐹 = 0𝑝)
46 df-0p 24270 . . . . . . . 8 0𝑝 = (ℂ × {0})
4745, 46syl6eq 2872 . . . . . . 7 (𝐹 = 0𝑝𝐹 = (ℂ × {0}))
4847oveq1d 7170 . . . . . 6 (𝐹 = 0𝑝 → (𝐹f · 𝐺) = ((ℂ × {0}) ∘f · 𝐺))
4948eqeq1d 2823 . . . . 5 (𝐹 = 0𝑝 → ((𝐹f · 𝐺) = (ℂ × {0}) ↔ ((ℂ × {0}) ∘f · 𝐺) = (ℂ × {0})))
5044, 49syl5ibrcom 249 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹 = 0𝑝 → (𝐹f · 𝐺) = (ℂ × {0})))
51 plyf 24787 . . . . . . 7 (𝐹 ∈ (Poly‘𝑆) → 𝐹:ℂ⟶ℂ)
5251adantr 483 . . . . . 6 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐹:ℂ⟶ℂ)
53 mul01 10818 . . . . . . 7 (𝑥 ∈ ℂ → (𝑥 · 0) = 0)
5453adantl 484 . . . . . 6 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑥 ∈ ℂ) → (𝑥 · 0) = 0)
5538, 52, 41, 41, 54caofid1 7438 . . . . 5 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹f · (ℂ × {0})) = (ℂ × {0}))
56 id 22 . . . . . . . 8 (𝐺 = 0𝑝𝐺 = 0𝑝)
5756, 46syl6eq 2872 . . . . . . 7 (𝐺 = 0𝑝𝐺 = (ℂ × {0}))
5857oveq2d 7171 . . . . . 6 (𝐺 = 0𝑝 → (𝐹f · 𝐺) = (𝐹f · (ℂ × {0})))
5958eqeq1d 2823 . . . . 5 (𝐺 = 0𝑝 → ((𝐹f · 𝐺) = (ℂ × {0}) ↔ (𝐹f · (ℂ × {0})) = (ℂ × {0})))
6055, 59syl5ibrcom 249 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐺 = 0𝑝 → (𝐹f · 𝐺) = (ℂ × {0})))
6150, 60jaod 855 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝐹 = 0𝑝𝐺 = 0𝑝) → (𝐹f · 𝐺) = (ℂ × {0})))
6246eqeq2i 2834 . . 3 ((𝐹f · 𝐺) = 0𝑝 ↔ (𝐹f · 𝐺) = (ℂ × {0}))
6361, 62syl6ibr 254 . 2 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝐹 = 0𝑝𝐺 = 0𝑝) → (𝐹f · 𝐺) = 0𝑝))
6436, 63impbid 214 1 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝐹f · 𝐺) = 0𝑝 ↔ (𝐹 = 0𝑝𝐺 = 0𝑝)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∨ wo 843   = wceq 1533   ∈ wcel 2110  Vcvv 3494  {csn 4566   × cxp 5552  ⟶wf 6350  ‘cfv 6354  (class class class)co 7155   ∘f cof 7406  ℂcc 10534  0cc0 10536   + caddc 10539   · cmul 10541  ℕ0cn0 11896  0𝑝c0p 24269  Polycply 24773  coeffccoe 24775  degcdgr 24776 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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5189  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460  ax-inf2 9103  ax-cnex 10592  ax-resscn 10593  ax-1cn 10594  ax-icn 10595  ax-addcl 10596  ax-addrcl 10597  ax-mulcl 10598  ax-mulrcl 10599  ax-mulcom 10600  ax-addass 10601  ax-mulass 10602  ax-distr 10603  ax-i2m1 10604  ax-1ne0 10605  ax-1rid 10606  ax-rnegex 10607  ax-rrecex 10608  ax-cnre 10609  ax-pre-lttri 10610  ax-pre-lttrn 10611  ax-pre-ltadd 10612  ax-pre-mulgt0 10613  ax-pre-sup 10614  ax-addf 10615 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-fal 1546  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4838  df-int 4876  df-iun 4920  df-br 5066  df-opab 5128  df-mpt 5146  df-tr 5172  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-se 5514  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-isom 6363  df-riota 7113  df-ov 7158  df-oprab 7159  df-mpo 7160  df-of 7408  df-om 7580  df-1st 7688  df-2nd 7689  df-wrecs 7946  df-recs 8007  df-rdg 8045  df-1o 8101  df-oadd 8105  df-er 8288  df-map 8407  df-pm 8408  df-en 8509  df-dom 8510  df-sdom 8511  df-fin 8512  df-sup 8905  df-inf 8906  df-oi 8973  df-card 9367  df-pnf 10676  df-mnf 10677  df-xr 10678  df-ltxr 10679  df-le 10680  df-sub 10871  df-neg 10872  df-div 11297  df-nn 11638  df-2 11699  df-3 11700  df-n0 11897  df-z 11981  df-uz 12243  df-rp 12389  df-fz 12892  df-fzo 13033  df-fl 13161  df-seq 13369  df-exp 13429  df-hash 13690  df-cj 14457  df-re 14458  df-im 14459  df-sqrt 14593  df-abs 14594  df-clim 14844  df-rlim 14845  df-sum 15042  df-0p 24270  df-ply 24777  df-coe 24779  df-dgr 24780 This theorem is referenced by:  plydiveu  24886  quotcan  24897  vieta1lem1  24898  vieta1lem2  24899
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