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Theorem plymul0or 24799
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 24752 . . . . . . 7 (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ0)
2 dgrcl 24752 . . . . . . 7 (𝐺 ∈ (Poly‘𝑆) → (deg‘𝐺) ∈ ℕ0)
3 nn0addcl 11921 . . . . . . 7 (((deg‘𝐹) ∈ ℕ0 ∧ (deg‘𝐺) ∈ ℕ0) → ((deg‘𝐹) + (deg‘𝐺)) ∈ ℕ0)
41, 2, 3syl2an 595 . . . . . 6 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((deg‘𝐹) + (deg‘𝐺)) ∈ ℕ0)
5 c0ex 10624 . . . . . . 7 0 ∈ V
65fvconst2 6959 . . . . . 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 6664 . . . . . . . 8 ((𝐹f · 𝐺) = 0𝑝 → (coeff‘(𝐹f · 𝐺)) = (coeff‘0𝑝))
9 coe0 24775 . . . . . . . 8 (coeff‘0𝑝) = (ℕ0 × {0})
108, 9syl6eq 2872 . . . . . . 7 ((𝐹f · 𝐺) = 0𝑝 → (coeff‘(𝐹f · 𝐺)) = (ℕ0 × {0}))
1110fveq1d 6666 . . . . . 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 248 . . . 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 24773 . . . . . 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 24751 . . . . . . . 8 (𝐹 ∈ (Poly‘𝑆) → (coeff‘𝐹):ℕ0⟶ℂ)
2120adantr 481 . . . . . . 7 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (coeff‘𝐹):ℕ0⟶ℂ)
221adantr 481 . . . . . . 7 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (deg‘𝐹) ∈ ℕ0)
2321, 22ffvelrnd 6845 . . . . . 6 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((coeff‘𝐹)‘(deg‘𝐹)) ∈ ℂ)
2415coef3 24751 . . . . . . . 8 (𝐺 ∈ (Poly‘𝑆) → (coeff‘𝐺):ℕ0⟶ℂ)
2524adantl 482 . . . . . . 7 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (coeff‘𝐺):ℕ0⟶ℂ)
262adantl 482 . . . . . . 7 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (deg‘𝐺) ∈ ℕ0)
2725, 26ffvelrnd 6845 . . . . . 6 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((coeff‘𝐺)‘(deg‘𝐺)) ∈ ℂ)
2823, 27mul0ord 11279 . . . . 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 280 . . . 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 240 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝐹f · 𝐺) = 0𝑝 → (((coeff‘𝐹)‘(deg‘𝐹)) = 0 ∨ ((coeff‘𝐺)‘(deg‘𝐺)) = 0)))
3116, 14dgreq0 24784 . . . . 5 (𝐹 ∈ (Poly‘𝑆) → (𝐹 = 0𝑝 ↔ ((coeff‘𝐹)‘(deg‘𝐹)) = 0))
3231adantr 481 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹 = 0𝑝 ↔ ((coeff‘𝐹)‘(deg‘𝐹)) = 0))
3317, 15dgreq0 24784 . . . . 5 (𝐺 ∈ (Poly‘𝑆) → (𝐺 = 0𝑝 ↔ ((coeff‘𝐺)‘(deg‘𝐺)) = 0))
3433adantl 482 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐺 = 0𝑝 ↔ ((coeff‘𝐺)‘(deg‘𝐺)) = 0))
3532, 34orbi12d 912 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝐹 = 0𝑝𝐺 = 0𝑝) ↔ (((coeff‘𝐹)‘(deg‘𝐹)) = 0 ∨ ((coeff‘𝐺)‘(deg‘𝐺)) = 0)))
3630, 35sylibrd 260 . 2 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝐹f · 𝐺) = 0𝑝 → (𝐹 = 0𝑝𝐺 = 0𝑝)))
37 cnex 10607 . . . . . . 7 ℂ ∈ V
3837a1i 11 . . . . . 6 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ℂ ∈ V)
39 plyf 24717 . . . . . . 7 (𝐺 ∈ (Poly‘𝑆) → 𝐺:ℂ⟶ℂ)
4039adantl 482 . . . . . 6 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐺:ℂ⟶ℂ)
41 0cnd 10623 . . . . . 6 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 0 ∈ ℂ)
42 mul02 10807 . . . . . . 7 (𝑥 ∈ ℂ → (0 · 𝑥) = 0)
4342adantl 482 . . . . . 6 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑥 ∈ ℂ) → (0 · 𝑥) = 0)
4438, 40, 41, 41, 43caofid2 7429 . . . . 5 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((ℂ × {0}) ∘f · 𝐺) = (ℂ × {0}))
45 id 22 . . . . . . . 8 (𝐹 = 0𝑝𝐹 = 0𝑝)
46 df-0p 24200 . . . . . . . 8 0𝑝 = (ℂ × {0})
4745, 46syl6eq 2872 . . . . . . 7 (𝐹 = 0𝑝𝐹 = (ℂ × {0}))
4847oveq1d 7160 . . . . . 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 248 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹 = 0𝑝 → (𝐹f · 𝐺) = (ℂ × {0})))
51 plyf 24717 . . . . . . 7 (𝐹 ∈ (Poly‘𝑆) → 𝐹:ℂ⟶ℂ)
5251adantr 481 . . . . . 6 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐹:ℂ⟶ℂ)
53 mul01 10808 . . . . . . 7 (𝑥 ∈ ℂ → (𝑥 · 0) = 0)
5453adantl 482 . . . . . 6 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑥 ∈ ℂ) → (𝑥 · 0) = 0)
5538, 52, 41, 41, 54caofid1 7428 . . . . 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 7161 . . . . . 6 (𝐺 = 0𝑝 → (𝐹f · 𝐺) = (𝐹f · (ℂ × {0})))
5958eqeq1d 2823 . . . . 5 (𝐺 = 0𝑝 → ((𝐹f · 𝐺) = (ℂ × {0}) ↔ (𝐹f · (ℂ × {0})) = (ℂ × {0})))
6055, 59syl5ibrcom 248 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐺 = 0𝑝 → (𝐹f · 𝐺) = (ℂ × {0})))
6150, 60jaod 853 . . 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 253 . 2 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝐹 = 0𝑝𝐺 = 0𝑝) → (𝐹f · 𝐺) = 0𝑝))
6436, 63impbid 213 1 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝐹f · 𝐺) = 0𝑝 ↔ (𝐹 = 0𝑝𝐺 = 0𝑝)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wo 841   = wceq 1528  wcel 2105  Vcvv 3495  {csn 4559   × cxp 5547  wf 6345  cfv 6349  (class class class)co 7145  f cof 7396  cc 10524  0cc0 10526   + caddc 10529   · cmul 10531  0cn0 11886  0𝑝c0p 24199  Polycply 24703  coeffccoe 24705  degcdgr 24706
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7450  ax-inf2 9093  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603  ax-pre-sup 10604  ax-addf 10605
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  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 3497  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 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4833  df-int 4870  df-iun 4914  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-se 5509  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142  df-ord 6188  df-on 6189  df-lim 6190  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-isom 6358  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-of 7398  df-om 7569  df-1st 7680  df-2nd 7681  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-1o 8093  df-oadd 8097  df-er 8279  df-map 8398  df-pm 8399  df-en 8499  df-dom 8500  df-sdom 8501  df-fin 8502  df-sup 8895  df-inf 8896  df-oi 8963  df-card 9357  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-div 11287  df-nn 11628  df-2 11689  df-3 11690  df-n0 11887  df-z 11971  df-uz 12233  df-rp 12380  df-fz 12883  df-fzo 13024  df-fl 13152  df-seq 13360  df-exp 13420  df-hash 13681  df-cj 14448  df-re 14449  df-im 14450  df-sqrt 14584  df-abs 14585  df-clim 14835  df-rlim 14836  df-sum 15033  df-0p 24200  df-ply 24707  df-coe 24709  df-dgr 24710
This theorem is referenced by:  plydiveu  24816  quotcan  24827  vieta1lem1  24828  vieta1lem2  24829
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