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Theorem plymulx 31435
Description: Coefficients of a polynomial multiplied by Xp. (Contributed by Thierry Arnoux, 25-Sep-2018.)
Assertion
Ref Expression
plymulx (𝐹 ∈ (Poly‘ℝ) → (coeff‘(𝐹𝑓 · Xp)) = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 0, ((coeff‘𝐹)‘(𝑛 − 1)))))
Distinct variable group:   𝑛,𝐹

Proof of Theorem plymulx
Dummy variable 𝑚 is distinct from all other variables.
StepHypRef Expression
1 ax-resscn 10440 . . . . . . 7 ℝ ⊆ ℂ
2 1re 10487 . . . . . . 7 1 ∈ ℝ
3 plyid 24482 . . . . . . 7 ((ℝ ⊆ ℂ ∧ 1 ∈ ℝ) → Xp ∈ (Poly‘ℝ))
41, 2, 3mp2an 688 . . . . . 6 Xp ∈ (Poly‘ℝ)
5 plymul02 31433 . . . . . . 7 (Xp ∈ (Poly‘ℝ) → (0𝑝𝑓 · Xp) = 0𝑝)
65fveq2d 6542 . . . . . 6 (Xp ∈ (Poly‘ℝ) → (coeff‘(0𝑝𝑓 · Xp)) = (coeff‘0𝑝))
74, 6ax-mp 5 . . . . 5 (coeff‘(0𝑝𝑓 · Xp)) = (coeff‘0𝑝)
8 fconstmpt 5500 . . . . . 6 (ℕ0 × {0}) = (𝑛 ∈ ℕ0 ↦ 0)
9 coe0 24529 . . . . . 6 (coeff‘0𝑝) = (ℕ0 × {0})
10 eqidd 2796 . . . . . . . 8 ((𝑛 ∈ ℕ0𝑛 = 0) → 0 = 0)
11 elnnne0 11759 . . . . . . . . . . 11 (𝑛 ∈ ℕ ↔ (𝑛 ∈ ℕ0𝑛 ≠ 0))
12 df-ne 2985 . . . . . . . . . . . 12 (𝑛 ≠ 0 ↔ ¬ 𝑛 = 0)
1312anbi2i 622 . . . . . . . . . . 11 ((𝑛 ∈ ℕ0𝑛 ≠ 0) ↔ (𝑛 ∈ ℕ0 ∧ ¬ 𝑛 = 0))
1411, 13bitr2i 277 . . . . . . . . . 10 ((𝑛 ∈ ℕ0 ∧ ¬ 𝑛 = 0) ↔ 𝑛 ∈ ℕ)
15 nnm1nn0 11786 . . . . . . . . . 10 (𝑛 ∈ ℕ → (𝑛 − 1) ∈ ℕ0)
1614, 15sylbi 218 . . . . . . . . 9 ((𝑛 ∈ ℕ0 ∧ ¬ 𝑛 = 0) → (𝑛 − 1) ∈ ℕ0)
17 eqidd 2796 . . . . . . . . . 10 (𝑚 = (𝑛 − 1) → 0 = 0)
18 fconstmpt 5500 . . . . . . . . . . 11 (ℕ0 × {0}) = (𝑚 ∈ ℕ0 ↦ 0)
199, 18eqtri 2819 . . . . . . . . . 10 (coeff‘0𝑝) = (𝑚 ∈ ℕ0 ↦ 0)
20 c0ex 10481 . . . . . . . . . 10 0 ∈ V
2117, 19, 20fvmpt 6635 . . . . . . . . 9 ((𝑛 − 1) ∈ ℕ0 → ((coeff‘0𝑝)‘(𝑛 − 1)) = 0)
2216, 21syl 17 . . . . . . . 8 ((𝑛 ∈ ℕ0 ∧ ¬ 𝑛 = 0) → ((coeff‘0𝑝)‘(𝑛 − 1)) = 0)
2310, 22ifeqda 4416 . . . . . . 7 (𝑛 ∈ ℕ0 → if(𝑛 = 0, 0, ((coeff‘0𝑝)‘(𝑛 − 1))) = 0)
2423mpteq2ia 5051 . . . . . 6 (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 0, ((coeff‘0𝑝)‘(𝑛 − 1)))) = (𝑛 ∈ ℕ0 ↦ 0)
258, 9, 243eqtr4ri 2830 . . . . 5 (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 0, ((coeff‘0𝑝)‘(𝑛 − 1)))) = (coeff‘0𝑝)
267, 25eqtr4i 2822 . . . 4 (coeff‘(0𝑝𝑓 · Xp)) = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 0, ((coeff‘0𝑝)‘(𝑛 − 1))))
27 fvoveq1 7039 . . . 4 (𝐹 = 0𝑝 → (coeff‘(𝐹𝑓 · Xp)) = (coeff‘(0𝑝𝑓 · Xp)))
28 simpl 483 . . . . . . . 8 ((𝐹 = 0𝑝𝑛 ∈ ℕ0) → 𝐹 = 0𝑝)
2928fveq2d 6542 . . . . . . 7 ((𝐹 = 0𝑝𝑛 ∈ ℕ0) → (coeff‘𝐹) = (coeff‘0𝑝))
3029fveq1d 6540 . . . . . 6 ((𝐹 = 0𝑝𝑛 ∈ ℕ0) → ((coeff‘𝐹)‘(𝑛 − 1)) = ((coeff‘0𝑝)‘(𝑛 − 1)))
3130ifeq2d 4400 . . . . 5 ((𝐹 = 0𝑝𝑛 ∈ ℕ0) → if(𝑛 = 0, 0, ((coeff‘𝐹)‘(𝑛 − 1))) = if(𝑛 = 0, 0, ((coeff‘0𝑝)‘(𝑛 − 1))))
3231mpteq2dva 5055 . . . 4 (𝐹 = 0𝑝 → (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 0, ((coeff‘𝐹)‘(𝑛 − 1)))) = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 0, ((coeff‘0𝑝)‘(𝑛 − 1)))))
3326, 27, 323eqtr4a 2857 . . 3 (𝐹 = 0𝑝 → (coeff‘(𝐹𝑓 · Xp)) = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 0, ((coeff‘𝐹)‘(𝑛 − 1)))))
3433adantl 482 . 2 ((𝐹 ∈ (Poly‘ℝ) ∧ 𝐹 = 0𝑝) → (coeff‘(𝐹𝑓 · Xp)) = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 0, ((coeff‘𝐹)‘(𝑛 − 1)))))
35 simpl 483 . . . 4 ((𝐹 ∈ (Poly‘ℝ) ∧ ¬ 𝐹 = 0𝑝) → 𝐹 ∈ (Poly‘ℝ))
36 elsng 4486 . . . . . 6 (𝐹 ∈ (Poly‘ℝ) → (𝐹 ∈ {0𝑝} ↔ 𝐹 = 0𝑝))
3736notbid 319 . . . . 5 (𝐹 ∈ (Poly‘ℝ) → (¬ 𝐹 ∈ {0𝑝} ↔ ¬ 𝐹 = 0𝑝))
3837biimpar 478 . . . 4 ((𝐹 ∈ (Poly‘ℝ) ∧ ¬ 𝐹 = 0𝑝) → ¬ 𝐹 ∈ {0𝑝})
3935, 38eldifd 3870 . . 3 ((𝐹 ∈ (Poly‘ℝ) ∧ ¬ 𝐹 = 0𝑝) → 𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}))
40 plymulx0 31434 . . 3 (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → (coeff‘(𝐹𝑓 · Xp)) = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 0, ((coeff‘𝐹)‘(𝑛 − 1)))))
4139, 40syl 17 . 2 ((𝐹 ∈ (Poly‘ℝ) ∧ ¬ 𝐹 = 0𝑝) → (coeff‘(𝐹𝑓 · Xp)) = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 0, ((coeff‘𝐹)‘(𝑛 − 1)))))
4234, 41pm2.61dan 809 1 (𝐹 ∈ (Poly‘ℝ) → (coeff‘(𝐹𝑓 · Xp)) = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 0, ((coeff‘𝐹)‘(𝑛 − 1)))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1522  wcel 2081  wne 2984  cdif 3856  wss 3859  ifcif 4381  {csn 4472  cmpt 5041   × cxp 5441  cfv 6225  (class class class)co 7016  𝑓 cof 7265  cc 10381  cr 10382  0cc0 10383  1c1 10384   · cmul 10388  cmin 10717  cn 11486  0cn0 11745  0𝑝c0p 23953  Polycply 24457  Xpcidp 24458  coeffccoe 24459
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319  ax-inf2 8950  ax-cnex 10439  ax-resscn 10440  ax-1cn 10441  ax-icn 10442  ax-addcl 10443  ax-addrcl 10444  ax-mulcl 10445  ax-mulrcl 10446  ax-mulcom 10447  ax-addass 10448  ax-mulass 10449  ax-distr 10450  ax-i2m1 10451  ax-1ne0 10452  ax-1rid 10453  ax-rnegex 10454  ax-rrecex 10455  ax-cnre 10456  ax-pre-lttri 10457  ax-pre-lttrn 10458  ax-pre-ltadd 10459  ax-pre-mulgt0 10460  ax-pre-sup 10461  ax-addf 10462
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-fal 1535  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-nel 3091  df-ral 3110  df-rex 3111  df-reu 3112  df-rmo 3113  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-uni 4746  df-int 4783  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-tr 5064  df-id 5348  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-se 5403  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-pred 6023  df-ord 6069  df-on 6070  df-lim 6071  df-suc 6072  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-isom 6234  df-riota 6977  df-ov 7019  df-oprab 7020  df-mpo 7021  df-of 7267  df-om 7437  df-1st 7545  df-2nd 7546  df-wrecs 7798  df-recs 7860  df-rdg 7898  df-1o 7953  df-oadd 7957  df-er 8139  df-map 8258  df-pm 8259  df-en 8358  df-dom 8359  df-sdom 8360  df-fin 8361  df-sup 8752  df-inf 8753  df-oi 8820  df-card 9214  df-pnf 10523  df-mnf 10524  df-xr 10525  df-ltxr 10526  df-le 10527  df-sub 10719  df-neg 10720  df-div 11146  df-nn 11487  df-2 11548  df-3 11549  df-n0 11746  df-z 11830  df-uz 12094  df-rp 12240  df-fz 12743  df-fzo 12884  df-fl 13012  df-seq 13220  df-exp 13280  df-hash 13541  df-cj 14292  df-re 14293  df-im 14294  df-sqrt 14428  df-abs 14429  df-clim 14679  df-rlim 14680  df-sum 14877  df-0p 23954  df-ply 24461  df-idp 24462  df-coe 24463  df-dgr 24464
This theorem is referenced by: (None)
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