Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  mon1pval Structured version   Visualization version   GIF version

Theorem mon1pval 23705
 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 6103 . . . . . . . 8 (𝑟 = 𝑅 → (Poly1𝑟) = (Poly1𝑅))
3 uc1pval.p . . . . . . . 8 𝑃 = (Poly1𝑅)
42, 3syl6eqr 2662 . . . . . . 7 (𝑟 = 𝑅 → (Poly1𝑟) = 𝑃)
54fveq2d 6107 . . . . . 6 (𝑟 = 𝑅 → (Base‘(Poly1𝑟)) = (Base‘𝑃))
6 uc1pval.b . . . . . 6 𝐵 = (Base‘𝑃)
75, 6syl6eqr 2662 . . . . 5 (𝑟 = 𝑅 → (Base‘(Poly1𝑟)) = 𝐵)
84fveq2d 6107 . . . . . . . 8 (𝑟 = 𝑅 → (0g‘(Poly1𝑟)) = (0g𝑃))
9 uc1pval.z . . . . . . . 8 0 = (0g𝑃)
108, 9syl6eqr 2662 . . . . . . 7 (𝑟 = 𝑅 → (0g‘(Poly1𝑟)) = 0 )
1110neeq2d 2842 . . . . . 6 (𝑟 = 𝑅 → (𝑓 ≠ (0g‘(Poly1𝑟)) ↔ 𝑓0 ))
12 fveq2 6103 . . . . . . . . . 10 (𝑟 = 𝑅 → ( deg1𝑟) = ( deg1𝑅))
13 uc1pval.d . . . . . . . . . 10 𝐷 = ( deg1𝑅)
1412, 13syl6eqr 2662 . . . . . . . . 9 (𝑟 = 𝑅 → ( deg1𝑟) = 𝐷)
1514fveq1d 6105 . . . . . . . 8 (𝑟 = 𝑅 → (( deg1𝑟)‘𝑓) = (𝐷𝑓))
1615fveq2d 6107 . . . . . . 7 (𝑟 = 𝑅 → ((coe1𝑓)‘(( deg1𝑟)‘𝑓)) = ((coe1𝑓)‘(𝐷𝑓)))
17 fveq2 6103 . . . . . . . 8 (𝑟 = 𝑅 → (1r𝑟) = (1r𝑅))
18 mon1pval.o . . . . . . . 8 1 = (1r𝑅)
1917, 18syl6eqr 2662 . . . . . . 7 (𝑟 = 𝑅 → (1r𝑟) = 1 )
2016, 19eqeq12d 2625 . . . . . 6 (𝑟 = 𝑅 → (((coe1𝑓)‘(( deg1𝑟)‘𝑓)) = (1r𝑟) ↔ ((coe1𝑓)‘(𝐷𝑓)) = 1 ))
2111, 20anbi12d 743 . . . . 5 (𝑟 = 𝑅 → ((𝑓 ≠ (0g‘(Poly1𝑟)) ∧ ((coe1𝑓)‘(( deg1𝑟)‘𝑓)) = (1r𝑟)) ↔ (𝑓0 ∧ ((coe1𝑓)‘(𝐷𝑓)) = 1 )))
227, 21rabeqbidv 3168 . . . 4 (𝑟 = 𝑅 → {𝑓 ∈ (Base‘(Poly1𝑟)) ∣ (𝑓 ≠ (0g‘(Poly1𝑟)) ∧ ((coe1𝑓)‘(( deg1𝑟)‘𝑓)) = (1r𝑟))} = {𝑓𝐵 ∣ (𝑓0 ∧ ((coe1𝑓)‘(𝐷𝑓)) = 1 )})
23 df-mon1 23694 . . . 4 Monic1p = (𝑟 ∈ V ↦ {𝑓 ∈ (Base‘(Poly1𝑟)) ∣ (𝑓 ≠ (0g‘(Poly1𝑟)) ∧ ((coe1𝑓)‘(( deg1𝑟)‘𝑓)) = (1r𝑟))})
24 fvex 6113 . . . . . 6 (Base‘𝑃) ∈ V
256, 24eqeltri 2684 . . . . 5 𝐵 ∈ V
2625rabex 4740 . . . 4 {𝑓𝐵 ∣ (𝑓0 ∧ ((coe1𝑓)‘(𝐷𝑓)) = 1 )} ∈ V
2722, 23, 26fvmpt 6191 . . 3 (𝑅 ∈ V → (Monic1p𝑅) = {𝑓𝐵 ∣ (𝑓0 ∧ ((coe1𝑓)‘(𝐷𝑓)) = 1 )})
28 fvprc 6097 . . . 4 𝑅 ∈ V → (Monic1p𝑅) = ∅)
29 ssrab2 3650 . . . . . 6 {𝑓𝐵 ∣ (𝑓0 ∧ ((coe1𝑓)‘(𝐷𝑓)) = 1 )} ⊆ 𝐵
30 fvprc 6097 . . . . . . . . . 10 𝑅 ∈ V → (Poly1𝑅) = ∅)
313, 30syl5eq 2656 . . . . . . . . 9 𝑅 ∈ V → 𝑃 = ∅)
3231fveq2d 6107 . . . . . . . 8 𝑅 ∈ V → (Base‘𝑃) = (Base‘∅))
336, 32syl5eq 2656 . . . . . . 7 𝑅 ∈ V → 𝐵 = (Base‘∅))
34 base0 15740 . . . . . . 7 ∅ = (Base‘∅)
3533, 34syl6eqr 2662 . . . . . 6 𝑅 ∈ V → 𝐵 = ∅)
3629, 35syl5sseq 3616 . . . . 5 𝑅 ∈ V → {𝑓𝐵 ∣ (𝑓0 ∧ ((coe1𝑓)‘(𝐷𝑓)) = 1 )} ⊆ ∅)
37 ss0 3926 . . . . 5 ({𝑓𝐵 ∣ (𝑓0 ∧ ((coe1𝑓)‘(𝐷𝑓)) = 1 )} ⊆ ∅ → {𝑓𝐵 ∣ (𝑓0 ∧ ((coe1𝑓)‘(𝐷𝑓)) = 1 )} = ∅)
3836, 37syl 17 . . . 4 𝑅 ∈ V → {𝑓𝐵 ∣ (𝑓0 ∧ ((coe1𝑓)‘(𝐷𝑓)) = 1 )} = ∅)
3928, 38eqtr4d 2647 . . 3 𝑅 ∈ V → (Monic1p𝑅) = {𝑓𝐵 ∣ (𝑓0 ∧ ((coe1𝑓)‘(𝐷𝑓)) = 1 )})
4027, 39pm2.61i 175 . 2 (Monic1p𝑅) = {𝑓𝐵 ∣ (𝑓0 ∧ ((coe1𝑓)‘(𝐷𝑓)) = 1 )}
411, 40eqtri 2632 1 𝑀 = {𝑓𝐵 ∣ (𝑓0 ∧ ((coe1𝑓)‘(𝐷𝑓)) = 1 )}
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  {crab 2900  Vcvv 3173   ⊆ wss 3540  ∅c0 3874  ‘cfv 5804  Basecbs 15695  0gc0g 15923  1rcur 18324  Poly1cpl1 19368  coe1cco1 19369   deg1 cdg1 23618  Monic1pcmn1 23689 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-slot 15699  df-base 15700  df-mon1 23694 This theorem is referenced by:  ismon1p  23706
 Copyright terms: Public domain W3C validator