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Theorem mon1pval 24748
Description: Value of the set of monic polynomials. (Contributed by Stefan O'Rear, 28-Mar-2015.)
Hypotheses
Ref Expression
uc1pval.p 𝑃 = (Poly1𝑅)
uc1pval.b 𝐵 = (Base‘𝑃)
uc1pval.z 0 = (0g𝑃)
uc1pval.d 𝐷 = ( deg1𝑅)
mon1pval.m 𝑀 = (Monic1p𝑅)
mon1pval.o 1 = (1r𝑅)
Assertion
Ref Expression
mon1pval 𝑀 = {𝑓𝐵 ∣ (𝑓0 ∧ ((coe1𝑓)‘(𝐷𝑓)) = 1 )}
Distinct variable groups:   𝐵,𝑓   𝐷,𝑓   1 ,𝑓   𝑅,𝑓   0 ,𝑓
Allowed substitution hints:   𝑃(𝑓)   𝑀(𝑓)

Proof of Theorem mon1pval
Dummy variable 𝑟 is distinct from all other variables.
StepHypRef Expression
1 mon1pval.m . 2 𝑀 = (Monic1p𝑅)
2 fveq2 6661 . . . . . . . 8 (𝑟 = 𝑅 → (Poly1𝑟) = (Poly1𝑅))
3 uc1pval.p . . . . . . . 8 𝑃 = (Poly1𝑅)
42, 3syl6eqr 2877 . . . . . . 7 (𝑟 = 𝑅 → (Poly1𝑟) = 𝑃)
54fveq2d 6665 . . . . . 6 (𝑟 = 𝑅 → (Base‘(Poly1𝑟)) = (Base‘𝑃))
6 uc1pval.b . . . . . 6 𝐵 = (Base‘𝑃)
75, 6syl6eqr 2877 . . . . 5 (𝑟 = 𝑅 → (Base‘(Poly1𝑟)) = 𝐵)
84fveq2d 6665 . . . . . . . 8 (𝑟 = 𝑅 → (0g‘(Poly1𝑟)) = (0g𝑃))
9 uc1pval.z . . . . . . . 8 0 = (0g𝑃)
108, 9syl6eqr 2877 . . . . . . 7 (𝑟 = 𝑅 → (0g‘(Poly1𝑟)) = 0 )
1110neeq2d 3074 . . . . . 6 (𝑟 = 𝑅 → (𝑓 ≠ (0g‘(Poly1𝑟)) ↔ 𝑓0 ))
12 fveq2 6661 . . . . . . . . . 10 (𝑟 = 𝑅 → ( deg1𝑟) = ( deg1𝑅))
13 uc1pval.d . . . . . . . . . 10 𝐷 = ( deg1𝑅)
1412, 13syl6eqr 2877 . . . . . . . . 9 (𝑟 = 𝑅 → ( deg1𝑟) = 𝐷)
1514fveq1d 6663 . . . . . . . 8 (𝑟 = 𝑅 → (( deg1𝑟)‘𝑓) = (𝐷𝑓))
1615fveq2d 6665 . . . . . . 7 (𝑟 = 𝑅 → ((coe1𝑓)‘(( deg1𝑟)‘𝑓)) = ((coe1𝑓)‘(𝐷𝑓)))
17 fveq2 6661 . . . . . . . 8 (𝑟 = 𝑅 → (1r𝑟) = (1r𝑅))
18 mon1pval.o . . . . . . . 8 1 = (1r𝑅)
1917, 18syl6eqr 2877 . . . . . . 7 (𝑟 = 𝑅 → (1r𝑟) = 1 )
2016, 19eqeq12d 2840 . . . . . 6 (𝑟 = 𝑅 → (((coe1𝑓)‘(( deg1𝑟)‘𝑓)) = (1r𝑟) ↔ ((coe1𝑓)‘(𝐷𝑓)) = 1 ))
2111, 20anbi12d 633 . . . . 5 (𝑟 = 𝑅 → ((𝑓 ≠ (0g‘(Poly1𝑟)) ∧ ((coe1𝑓)‘(( deg1𝑟)‘𝑓)) = (1r𝑟)) ↔ (𝑓0 ∧ ((coe1𝑓)‘(𝐷𝑓)) = 1 )))
227, 21rabeqbidv 3471 . . . 4 (𝑟 = 𝑅 → {𝑓 ∈ (Base‘(Poly1𝑟)) ∣ (𝑓 ≠ (0g‘(Poly1𝑟)) ∧ ((coe1𝑓)‘(( deg1𝑟)‘𝑓)) = (1r𝑟))} = {𝑓𝐵 ∣ (𝑓0 ∧ ((coe1𝑓)‘(𝐷𝑓)) = 1 )})
23 df-mon1 24737 . . . 4 Monic1p = (𝑟 ∈ V ↦ {𝑓 ∈ (Base‘(Poly1𝑟)) ∣ (𝑓 ≠ (0g‘(Poly1𝑟)) ∧ ((coe1𝑓)‘(( deg1𝑟)‘𝑓)) = (1r𝑟))})
246fvexi 6675 . . . . 5 𝐵 ∈ V
2524rabex 5221 . . . 4 {𝑓𝐵 ∣ (𝑓0 ∧ ((coe1𝑓)‘(𝐷𝑓)) = 1 )} ∈ V
2622, 23, 25fvmpt 6759 . . 3 (𝑅 ∈ V → (Monic1p𝑅) = {𝑓𝐵 ∣ (𝑓0 ∧ ((coe1𝑓)‘(𝐷𝑓)) = 1 )})
27 fvprc 6654 . . . 4 𝑅 ∈ V → (Monic1p𝑅) = ∅)
28 ssrab2 4042 . . . . . 6 {𝑓𝐵 ∣ (𝑓0 ∧ ((coe1𝑓)‘(𝐷𝑓)) = 1 )} ⊆ 𝐵
29 fvprc 6654 . . . . . . . . . 10 𝑅 ∈ V → (Poly1𝑅) = ∅)
303, 29syl5eq 2871 . . . . . . . . 9 𝑅 ∈ V → 𝑃 = ∅)
3130fveq2d 6665 . . . . . . . 8 𝑅 ∈ V → (Base‘𝑃) = (Base‘∅))
326, 31syl5eq 2871 . . . . . . 7 𝑅 ∈ V → 𝐵 = (Base‘∅))
33 base0 16536 . . . . . . 7 ∅ = (Base‘∅)
3432, 33syl6eqr 2877 . . . . . 6 𝑅 ∈ V → 𝐵 = ∅)
3528, 34sseqtrid 4005 . . . . 5 𝑅 ∈ V → {𝑓𝐵 ∣ (𝑓0 ∧ ((coe1𝑓)‘(𝐷𝑓)) = 1 )} ⊆ ∅)
36 ss0 4335 . . . . 5 ({𝑓𝐵 ∣ (𝑓0 ∧ ((coe1𝑓)‘(𝐷𝑓)) = 1 )} ⊆ ∅ → {𝑓𝐵 ∣ (𝑓0 ∧ ((coe1𝑓)‘(𝐷𝑓)) = 1 )} = ∅)
3735, 36syl 17 . . . 4 𝑅 ∈ V → {𝑓𝐵 ∣ (𝑓0 ∧ ((coe1𝑓)‘(𝐷𝑓)) = 1 )} = ∅)
3827, 37eqtr4d 2862 . . 3 𝑅 ∈ V → (Monic1p𝑅) = {𝑓𝐵 ∣ (𝑓0 ∧ ((coe1𝑓)‘(𝐷𝑓)) = 1 )})
3926, 38pm2.61i 185 . 2 (Monic1p𝑅) = {𝑓𝐵 ∣ (𝑓0 ∧ ((coe1𝑓)‘(𝐷𝑓)) = 1 )}
401, 39eqtri 2847 1 𝑀 = {𝑓𝐵 ∣ (𝑓0 ∧ ((coe1𝑓)‘(𝐷𝑓)) = 1 )}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 399   = wceq 1538  wcel 2115  wne 3014  {crab 3137  Vcvv 3480  wss 3919  c0 4276  cfv 6343  Basecbs 16483  0gc0g 16713  1rcur 19251  Poly1cpl1 20813  coe1cco1 20814   deg1 cdg1 24661  Monic1pcmn1 24732
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-iota 6302  df-fun 6345  df-fv 6351  df-slot 16487  df-base 16489  df-mon1 24737
This theorem is referenced by:  ismon1p  24749
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