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Theorem plymul0or 24327
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 24280 . . . . . . 7 (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ0)
2 dgrcl 24280 . . . . . . 7 (𝐺 ∈ (Poly‘𝑆) → (deg‘𝐺) ∈ ℕ0)
3 nn0addcl 11575 . . . . . . 7 (((deg‘𝐹) ∈ ℕ0 ∧ (deg‘𝐺) ∈ ℕ0) → ((deg‘𝐹) + (deg‘𝐺)) ∈ ℕ0)
41, 2, 3syl2an 589 . . . . . 6 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((deg‘𝐹) + (deg‘𝐺)) ∈ ℕ0)
5 c0ex 10287 . . . . . . 7 0 ∈ V
65fvconst2 6662 . . . . . 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 6375 . . . . . . . 8 ((𝐹𝑓 · 𝐺) = 0𝑝 → (coeff‘(𝐹𝑓 · 𝐺)) = (coeff‘0𝑝))
9 coe0 24303 . . . . . . . 8 (coeff‘0𝑝) = (ℕ0 × {0})
108, 9syl6eq 2815 . . . . . . 7 ((𝐹𝑓 · 𝐺) = 0𝑝 → (coeff‘(𝐹𝑓 · 𝐺)) = (ℕ0 × {0}))
1110fveq1d 6377 . . . . . 6 ((𝐹𝑓 · 𝐺) = 0𝑝 → ((coeff‘(𝐹𝑓 · 𝐺))‘((deg‘𝐹) + (deg‘𝐺))) = ((ℕ0 × {0})‘((deg‘𝐹) + (deg‘𝐺))))
1211eqeq1d 2767 . . . . 5 ((𝐹𝑓 · 𝐺) = 0𝑝 → (((coeff‘(𝐹𝑓 · 𝐺))‘((deg‘𝐹) + (deg‘𝐺))) = 0 ↔ ((ℕ0 × {0})‘((deg‘𝐹) + (deg‘𝐺))) = 0))
137, 12syl5ibrcom 238 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝐹𝑓 · 𝐺) = 0𝑝 → ((coeff‘(𝐹𝑓 · 𝐺))‘((deg‘𝐹) + (deg‘𝐺))) = 0))
14 eqid 2765 . . . . . . 7 (coeff‘𝐹) = (coeff‘𝐹)
15 eqid 2765 . . . . . . 7 (coeff‘𝐺) = (coeff‘𝐺)
16 eqid 2765 . . . . . . 7 (deg‘𝐹) = (deg‘𝐹)
17 eqid 2765 . . . . . . 7 (deg‘𝐺) = (deg‘𝐺)
1814, 15, 16, 17coemulhi 24301 . . . . . 6 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((coeff‘(𝐹𝑓 · 𝐺))‘((deg‘𝐹) + (deg‘𝐺))) = (((coeff‘𝐹)‘(deg‘𝐹)) · ((coeff‘𝐺)‘(deg‘𝐺))))
1918eqeq1d 2767 . . . . 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 24279 . . . . . . . 8 (𝐹 ∈ (Poly‘𝑆) → (coeff‘𝐹):ℕ0⟶ℂ)
2120adantr 472 . . . . . . 7 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (coeff‘𝐹):ℕ0⟶ℂ)
221adantr 472 . . . . . . 7 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (deg‘𝐹) ∈ ℕ0)
2321, 22ffvelrnd 6550 . . . . . 6 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((coeff‘𝐹)‘(deg‘𝐹)) ∈ ℂ)
2415coef3 24279 . . . . . . . 8 (𝐺 ∈ (Poly‘𝑆) → (coeff‘𝐺):ℕ0⟶ℂ)
2524adantl 473 . . . . . . 7 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (coeff‘𝐺):ℕ0⟶ℂ)
262adantl 473 . . . . . . 7 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (deg‘𝐺) ∈ ℕ0)
2725, 26ffvelrnd 6550 . . . . . 6 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((coeff‘𝐺)‘(deg‘𝐺)) ∈ ℂ)
2823, 27mul0ord 10931 . . . . 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 270 . . . 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 230 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝐹𝑓 · 𝐺) = 0𝑝 → (((coeff‘𝐹)‘(deg‘𝐹)) = 0 ∨ ((coeff‘𝐺)‘(deg‘𝐺)) = 0)))
3116, 14dgreq0 24312 . . . . 5 (𝐹 ∈ (Poly‘𝑆) → (𝐹 = 0𝑝 ↔ ((coeff‘𝐹)‘(deg‘𝐹)) = 0))
3231adantr 472 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹 = 0𝑝 ↔ ((coeff‘𝐹)‘(deg‘𝐹)) = 0))
3317, 15dgreq0 24312 . . . . 5 (𝐺 ∈ (Poly‘𝑆) → (𝐺 = 0𝑝 ↔ ((coeff‘𝐺)‘(deg‘𝐺)) = 0))
3433adantl 473 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐺 = 0𝑝 ↔ ((coeff‘𝐺)‘(deg‘𝐺)) = 0))
3532, 34orbi12d 942 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝐹 = 0𝑝𝐺 = 0𝑝) ↔ (((coeff‘𝐹)‘(deg‘𝐹)) = 0 ∨ ((coeff‘𝐺)‘(deg‘𝐺)) = 0)))
3630, 35sylibrd 250 . 2 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝐹𝑓 · 𝐺) = 0𝑝 → (𝐹 = 0𝑝𝐺 = 0𝑝)))
37 cnex 10270 . . . . . . 7 ℂ ∈ V
3837a1i 11 . . . . . 6 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ℂ ∈ V)
39 plyf 24245 . . . . . . 7 (𝐺 ∈ (Poly‘𝑆) → 𝐺:ℂ⟶ℂ)
4039adantl 473 . . . . . 6 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐺:ℂ⟶ℂ)
41 0cnd 10286 . . . . . 6 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 0 ∈ ℂ)
42 mul02 10468 . . . . . . 7 (𝑥 ∈ ℂ → (0 · 𝑥) = 0)
4342adantl 473 . . . . . 6 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑥 ∈ ℂ) → (0 · 𝑥) = 0)
4438, 40, 41, 41, 43caofid2 7126 . . . . 5 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((ℂ × {0}) ∘𝑓 · 𝐺) = (ℂ × {0}))
45 id 22 . . . . . . . 8 (𝐹 = 0𝑝𝐹 = 0𝑝)
46 df-0p 23728 . . . . . . . 8 0𝑝 = (ℂ × {0})
4745, 46syl6eq 2815 . . . . . . 7 (𝐹 = 0𝑝𝐹 = (ℂ × {0}))
4847oveq1d 6857 . . . . . 6 (𝐹 = 0𝑝 → (𝐹𝑓 · 𝐺) = ((ℂ × {0}) ∘𝑓 · 𝐺))
4948eqeq1d 2767 . . . . 5 (𝐹 = 0𝑝 → ((𝐹𝑓 · 𝐺) = (ℂ × {0}) ↔ ((ℂ × {0}) ∘𝑓 · 𝐺) = (ℂ × {0})))
5044, 49syl5ibrcom 238 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹 = 0𝑝 → (𝐹𝑓 · 𝐺) = (ℂ × {0})))
51 plyf 24245 . . . . . . 7 (𝐹 ∈ (Poly‘𝑆) → 𝐹:ℂ⟶ℂ)
5251adantr 472 . . . . . 6 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐹:ℂ⟶ℂ)
53 mul01 10469 . . . . . . 7 (𝑥 ∈ ℂ → (𝑥 · 0) = 0)
5453adantl 473 . . . . . 6 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑥 ∈ ℂ) → (𝑥 · 0) = 0)
5538, 52, 41, 41, 54caofid1 7125 . . . . 5 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹𝑓 · (ℂ × {0})) = (ℂ × {0}))
56 id 22 . . . . . . . 8 (𝐺 = 0𝑝𝐺 = 0𝑝)
5756, 46syl6eq 2815 . . . . . . 7 (𝐺 = 0𝑝𝐺 = (ℂ × {0}))
5857oveq2d 6858 . . . . . 6 (𝐺 = 0𝑝 → (𝐹𝑓 · 𝐺) = (𝐹𝑓 · (ℂ × {0})))
5958eqeq1d 2767 . . . . 5 (𝐺 = 0𝑝 → ((𝐹𝑓 · 𝐺) = (ℂ × {0}) ↔ (𝐹𝑓 · (ℂ × {0})) = (ℂ × {0})))
6055, 59syl5ibrcom 238 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐺 = 0𝑝 → (𝐹𝑓 · 𝐺) = (ℂ × {0})))
6150, 60jaod 885 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝐹 = 0𝑝𝐺 = 0𝑝) → (𝐹𝑓 · 𝐺) = (ℂ × {0})))
6246eqeq2i 2777 . . 3 ((𝐹𝑓 · 𝐺) = 0𝑝 ↔ (𝐹𝑓 · 𝐺) = (ℂ × {0}))
6361, 62syl6ibr 243 . 2 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝐹 = 0𝑝𝐺 = 0𝑝) → (𝐹𝑓 · 𝐺) = 0𝑝))
6436, 63impbid 203 1 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝐹𝑓 · 𝐺) = 0𝑝 ↔ (𝐹 = 0𝑝𝐺 = 0𝑝)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  wo 873   = wceq 1652  wcel 2155  Vcvv 3350  {csn 4334   × cxp 5275  wf 6064  cfv 6068  (class class class)co 6842  𝑓 cof 7093  cc 10187  0cc0 10189   + caddc 10192   · cmul 10194  0cn0 11538  0𝑝c0p 23727  Polycply 24231  coeffccoe 24233  degcdgr 24234
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-inf2 8753  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266  ax-pre-sup 10267  ax-addf 10268
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-se 5237  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-isom 6077  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-of 7095  df-om 7264  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-oadd 7768  df-er 7947  df-map 8062  df-pm 8063  df-en 8161  df-dom 8162  df-sdom 8163  df-fin 8164  df-sup 8555  df-inf 8556  df-oi 8622  df-card 9016  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-div 10939  df-nn 11275  df-2 11335  df-3 11336  df-n0 11539  df-z 11625  df-uz 11887  df-rp 12029  df-fz 12534  df-fzo 12674  df-fl 12801  df-seq 13009  df-exp 13068  df-hash 13322  df-cj 14126  df-re 14127  df-im 14128  df-sqrt 14262  df-abs 14263  df-clim 14506  df-rlim 14507  df-sum 14704  df-0p 23728  df-ply 24235  df-coe 24237  df-dgr 24238
This theorem is referenced by:  plydiveu  24344  quotcan  24355  vieta1lem1  24356  vieta1lem2  24357
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