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Theorem plymul0or 26337
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 26287 . . . . . . 7 (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ0)
2 dgrcl 26287 . . . . . . 7 (𝐺 ∈ (Poly‘𝑆) → (deg‘𝐺) ∈ ℕ0)
3 nn0addcl 12559 . . . . . . 7 (((deg‘𝐹) ∈ ℕ0 ∧ (deg‘𝐺) ∈ ℕ0) → ((deg‘𝐹) + (deg‘𝐺)) ∈ ℕ0)
41, 2, 3syl2an 596 . . . . . 6 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((deg‘𝐹) + (deg‘𝐺)) ∈ ℕ0)
5 c0ex 11253 . . . . . . 7 0 ∈ V
65fvconst2 7224 . . . . . 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 6907 . . . . . . . 8 ((𝐹f · 𝐺) = 0𝑝 → (coeff‘(𝐹f · 𝐺)) = (coeff‘0𝑝))
9 coe0 26310 . . . . . . . 8 (coeff‘0𝑝) = (ℕ0 × {0})
108, 9eqtrdi 2791 . . . . . . 7 ((𝐹f · 𝐺) = 0𝑝 → (coeff‘(𝐹f · 𝐺)) = (ℕ0 × {0}))
1110fveq1d 6909 . . . . . 6 ((𝐹f · 𝐺) = 0𝑝 → ((coeff‘(𝐹f · 𝐺))‘((deg‘𝐹) + (deg‘𝐺))) = ((ℕ0 × {0})‘((deg‘𝐹) + (deg‘𝐺))))
1211eqeq1d 2737 . . . . 5 ((𝐹f · 𝐺) = 0𝑝 → (((coeff‘(𝐹f · 𝐺))‘((deg‘𝐹) + (deg‘𝐺))) = 0 ↔ ((ℕ0 × {0})‘((deg‘𝐹) + (deg‘𝐺))) = 0))
137, 12syl5ibrcom 247 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝐹f · 𝐺) = 0𝑝 → ((coeff‘(𝐹f · 𝐺))‘((deg‘𝐹) + (deg‘𝐺))) = 0))
14 eqid 2735 . . . . . . 7 (coeff‘𝐹) = (coeff‘𝐹)
15 eqid 2735 . . . . . . 7 (coeff‘𝐺) = (coeff‘𝐺)
16 eqid 2735 . . . . . . 7 (deg‘𝐹) = (deg‘𝐹)
17 eqid 2735 . . . . . . 7 (deg‘𝐺) = (deg‘𝐺)
1814, 15, 16, 17coemulhi 26308 . . . . . 6 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((coeff‘(𝐹f · 𝐺))‘((deg‘𝐹) + (deg‘𝐺))) = (((coeff‘𝐹)‘(deg‘𝐹)) · ((coeff‘𝐺)‘(deg‘𝐺))))
1918eqeq1d 2737 . . . . 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 26286 . . . . . . . 8 (𝐹 ∈ (Poly‘𝑆) → (coeff‘𝐹):ℕ0⟶ℂ)
2120adantr 480 . . . . . . 7 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (coeff‘𝐹):ℕ0⟶ℂ)
221adantr 480 . . . . . . 7 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (deg‘𝐹) ∈ ℕ0)
2321, 22ffvelcdmd 7105 . . . . . 6 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((coeff‘𝐹)‘(deg‘𝐹)) ∈ ℂ)
2415coef3 26286 . . . . . . . 8 (𝐺 ∈ (Poly‘𝑆) → (coeff‘𝐺):ℕ0⟶ℂ)
2524adantl 481 . . . . . . 7 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (coeff‘𝐺):ℕ0⟶ℂ)
262adantl 481 . . . . . . 7 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (deg‘𝐺) ∈ ℕ0)
2725, 26ffvelcdmd 7105 . . . . . 6 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((coeff‘𝐺)‘(deg‘𝐺)) ∈ ℂ)
2823, 27mul0ord 11911 . . . . 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 279 . . . 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 239 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝐹f · 𝐺) = 0𝑝 → (((coeff‘𝐹)‘(deg‘𝐹)) = 0 ∨ ((coeff‘𝐺)‘(deg‘𝐺)) = 0)))
3116, 14dgreq0 26320 . . . . 5 (𝐹 ∈ (Poly‘𝑆) → (𝐹 = 0𝑝 ↔ ((coeff‘𝐹)‘(deg‘𝐹)) = 0))
3231adantr 480 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹 = 0𝑝 ↔ ((coeff‘𝐹)‘(deg‘𝐹)) = 0))
3317, 15dgreq0 26320 . . . . 5 (𝐺 ∈ (Poly‘𝑆) → (𝐺 = 0𝑝 ↔ ((coeff‘𝐺)‘(deg‘𝐺)) = 0))
3433adantl 481 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐺 = 0𝑝 ↔ ((coeff‘𝐺)‘(deg‘𝐺)) = 0))
3532, 34orbi12d 918 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝐹 = 0𝑝𝐺 = 0𝑝) ↔ (((coeff‘𝐹)‘(deg‘𝐹)) = 0 ∨ ((coeff‘𝐺)‘(deg‘𝐺)) = 0)))
3630, 35sylibrd 259 . 2 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝐹f · 𝐺) = 0𝑝 → (𝐹 = 0𝑝𝐺 = 0𝑝)))
37 cnex 11234 . . . . . . 7 ℂ ∈ V
3837a1i 11 . . . . . 6 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ℂ ∈ V)
39 plyf 26252 . . . . . . 7 (𝐺 ∈ (Poly‘𝑆) → 𝐺:ℂ⟶ℂ)
4039adantl 481 . . . . . 6 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐺:ℂ⟶ℂ)
41 0cnd 11252 . . . . . 6 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 0 ∈ ℂ)
42 mul02 11437 . . . . . . 7 (𝑥 ∈ ℂ → (0 · 𝑥) = 0)
4342adantl 481 . . . . . 6 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑥 ∈ ℂ) → (0 · 𝑥) = 0)
4438, 40, 41, 41, 43caofid2 7733 . . . . 5 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((ℂ × {0}) ∘f · 𝐺) = (ℂ × {0}))
45 id 22 . . . . . . . 8 (𝐹 = 0𝑝𝐹 = 0𝑝)
46 df-0p 25719 . . . . . . . 8 0𝑝 = (ℂ × {0})
4745, 46eqtrdi 2791 . . . . . . 7 (𝐹 = 0𝑝𝐹 = (ℂ × {0}))
4847oveq1d 7446 . . . . . 6 (𝐹 = 0𝑝 → (𝐹f · 𝐺) = ((ℂ × {0}) ∘f · 𝐺))
4948eqeq1d 2737 . . . . 5 (𝐹 = 0𝑝 → ((𝐹f · 𝐺) = (ℂ × {0}) ↔ ((ℂ × {0}) ∘f · 𝐺) = (ℂ × {0})))
5044, 49syl5ibrcom 247 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹 = 0𝑝 → (𝐹f · 𝐺) = (ℂ × {0})))
51 plyf 26252 . . . . . . 7 (𝐹 ∈ (Poly‘𝑆) → 𝐹:ℂ⟶ℂ)
5251adantr 480 . . . . . 6 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐹:ℂ⟶ℂ)
53 mul01 11438 . . . . . . 7 (𝑥 ∈ ℂ → (𝑥 · 0) = 0)
5453adantl 481 . . . . . 6 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑥 ∈ ℂ) → (𝑥 · 0) = 0)
5538, 52, 41, 41, 54caofid1 7732 . . . . 5 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹f · (ℂ × {0})) = (ℂ × {0}))
56 id 22 . . . . . . . 8 (𝐺 = 0𝑝𝐺 = 0𝑝)
5756, 46eqtrdi 2791 . . . . . . 7 (𝐺 = 0𝑝𝐺 = (ℂ × {0}))
5857oveq2d 7447 . . . . . 6 (𝐺 = 0𝑝 → (𝐹f · 𝐺) = (𝐹f · (ℂ × {0})))
5958eqeq1d 2737 . . . . 5 (𝐺 = 0𝑝 → ((𝐹f · 𝐺) = (ℂ × {0}) ↔ (𝐹f · (ℂ × {0})) = (ℂ × {0})))
6055, 59syl5ibrcom 247 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐺 = 0𝑝 → (𝐹f · 𝐺) = (ℂ × {0})))
6150, 60jaod 859 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝐹 = 0𝑝𝐺 = 0𝑝) → (𝐹f · 𝐺) = (ℂ × {0})))
6246eqeq2i 2748 . . 3 ((𝐹f · 𝐺) = 0𝑝 ↔ (𝐹f · 𝐺) = (ℂ × {0}))
6361, 62imbitrrdi 252 . 2 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝐹 = 0𝑝𝐺 = 0𝑝) → (𝐹f · 𝐺) = 0𝑝))
6436, 63impbid 212 1 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝐹f · 𝐺) = 0𝑝 ↔ (𝐹 = 0𝑝𝐺 = 0𝑝)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wo 847   = wceq 1537  wcel 2106  Vcvv 3478  {csn 4631   × cxp 5687  wf 6559  cfv 6563  (class class class)co 7431  f cof 7695  cc 11151  0cc0 11153   + caddc 11156   · cmul 11158  0cn0 12524  0𝑝c0p 25718  Polycply 26238  coeffccoe 26240  degcdgr 26241
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-pm 8868  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-sup 9480  df-inf 9481  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-fl 13829  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-rlim 15522  df-sum 15720  df-0p 25719  df-ply 26242  df-coe 26244  df-dgr 26245
This theorem is referenced by:  plydiveu  26355  quotcan  26366  vieta1lem1  26367  vieta1lem2  26368
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