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Theorem pf1rcl 19761
Description: Reverse closure for the set of polynomial functions. (Contributed by Mario Carneiro, 12-Jun-2015.)
Hypothesis
Ref Expression
pf1rcl.q 𝑄 = ran (eval1𝑅)
Assertion
Ref Expression
pf1rcl (𝑋𝑄𝑅 ∈ CRing)

Proof of Theorem pf1rcl
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 n0i 3953 . 2 (𝑋𝑄 → ¬ 𝑄 = ∅)
2 pf1rcl.q . . . 4 𝑄 = ran (eval1𝑅)
3 eqid 2651 . . . . . 6 (eval1𝑅) = (eval1𝑅)
4 eqid 2651 . . . . . 6 (1𝑜 eval 𝑅) = (1𝑜 eval 𝑅)
5 eqid 2651 . . . . . 6 (Base‘𝑅) = (Base‘𝑅)
63, 4, 5evl1fval 19740 . . . . 5 (eval1𝑅) = ((𝑥 ∈ ((Base‘𝑅) ↑𝑚 ((Base‘𝑅) ↑𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ (Base‘𝑅) ↦ (1𝑜 × {𝑦})))) ∘ (1𝑜 eval 𝑅))
76rneqi 5384 . . . 4 ran (eval1𝑅) = ran ((𝑥 ∈ ((Base‘𝑅) ↑𝑚 ((Base‘𝑅) ↑𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ (Base‘𝑅) ↦ (1𝑜 × {𝑦})))) ∘ (1𝑜 eval 𝑅))
8 rnco2 5680 . . . 4 ran ((𝑥 ∈ ((Base‘𝑅) ↑𝑚 ((Base‘𝑅) ↑𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ (Base‘𝑅) ↦ (1𝑜 × {𝑦})))) ∘ (1𝑜 eval 𝑅)) = ((𝑥 ∈ ((Base‘𝑅) ↑𝑚 ((Base‘𝑅) ↑𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ (Base‘𝑅) ↦ (1𝑜 × {𝑦})))) “ ran (1𝑜 eval 𝑅))
92, 7, 83eqtri 2677 . . 3 𝑄 = ((𝑥 ∈ ((Base‘𝑅) ↑𝑚 ((Base‘𝑅) ↑𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ (Base‘𝑅) ↦ (1𝑜 × {𝑦})))) “ ran (1𝑜 eval 𝑅))
10 inss2 3867 . . . . 5 (dom (𝑥 ∈ ((Base‘𝑅) ↑𝑚 ((Base‘𝑅) ↑𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ (Base‘𝑅) ↦ (1𝑜 × {𝑦})))) ∩ ran (1𝑜 eval 𝑅)) ⊆ ran (1𝑜 eval 𝑅)
11 neq0 3963 . . . . . . 7 (¬ ran (1𝑜 eval 𝑅) = ∅ ↔ ∃𝑥 𝑥 ∈ ran (1𝑜 eval 𝑅))
124, 5evlval 19572 . . . . . . . . . . 11 (1𝑜 eval 𝑅) = ((1𝑜 evalSub 𝑅)‘(Base‘𝑅))
1312rneqi 5384 . . . . . . . . . 10 ran (1𝑜 eval 𝑅) = ran ((1𝑜 evalSub 𝑅)‘(Base‘𝑅))
1413mpfrcl 19566 . . . . . . . . 9 (𝑥 ∈ ran (1𝑜 eval 𝑅) → (1𝑜 ∈ V ∧ 𝑅 ∈ CRing ∧ (Base‘𝑅) ∈ (SubRing‘𝑅)))
1514simp2d 1094 . . . . . . . 8 (𝑥 ∈ ran (1𝑜 eval 𝑅) → 𝑅 ∈ CRing)
1615exlimiv 1898 . . . . . . 7 (∃𝑥 𝑥 ∈ ran (1𝑜 eval 𝑅) → 𝑅 ∈ CRing)
1711, 16sylbi 207 . . . . . 6 (¬ ran (1𝑜 eval 𝑅) = ∅ → 𝑅 ∈ CRing)
1817con1i 144 . . . . 5 𝑅 ∈ CRing → ran (1𝑜 eval 𝑅) = ∅)
19 sseq0 4008 . . . . 5 (((dom (𝑥 ∈ ((Base‘𝑅) ↑𝑚 ((Base‘𝑅) ↑𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ (Base‘𝑅) ↦ (1𝑜 × {𝑦})))) ∩ ran (1𝑜 eval 𝑅)) ⊆ ran (1𝑜 eval 𝑅) ∧ ran (1𝑜 eval 𝑅) = ∅) → (dom (𝑥 ∈ ((Base‘𝑅) ↑𝑚 ((Base‘𝑅) ↑𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ (Base‘𝑅) ↦ (1𝑜 × {𝑦})))) ∩ ran (1𝑜 eval 𝑅)) = ∅)
2010, 18, 19sylancr 696 . . . 4 𝑅 ∈ CRing → (dom (𝑥 ∈ ((Base‘𝑅) ↑𝑚 ((Base‘𝑅) ↑𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ (Base‘𝑅) ↦ (1𝑜 × {𝑦})))) ∩ ran (1𝑜 eval 𝑅)) = ∅)
21 imadisj 5519 . . . 4 (((𝑥 ∈ ((Base‘𝑅) ↑𝑚 ((Base‘𝑅) ↑𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ (Base‘𝑅) ↦ (1𝑜 × {𝑦})))) “ ran (1𝑜 eval 𝑅)) = ∅ ↔ (dom (𝑥 ∈ ((Base‘𝑅) ↑𝑚 ((Base‘𝑅) ↑𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ (Base‘𝑅) ↦ (1𝑜 × {𝑦})))) ∩ ran (1𝑜 eval 𝑅)) = ∅)
2220, 21sylibr 224 . . 3 𝑅 ∈ CRing → ((𝑥 ∈ ((Base‘𝑅) ↑𝑚 ((Base‘𝑅) ↑𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ (Base‘𝑅) ↦ (1𝑜 × {𝑦})))) “ ran (1𝑜 eval 𝑅)) = ∅)
239, 22syl5eq 2697 . 2 𝑅 ∈ CRing → 𝑄 = ∅)
241, 23nsyl2 142 1 (𝑋𝑄𝑅 ∈ CRing)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1523  wex 1744  wcel 2030  Vcvv 3231  cin 3606  wss 3607  c0 3948  {csn 4210  cmpt 4762   × cxp 5141  dom cdm 5143  ran crn 5144  cima 5146  ccom 5147  cfv 5926  (class class class)co 6690  1𝑜c1o 7598  𝑚 cmap 7899  Basecbs 15904  CRingccrg 18594  SubRingcsubrg 18824   evalSub ces 19552   eval cevl 19553  eval1ce1 19727
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-evls 19554  df-evl 19555  df-evl1 19729
This theorem is referenced by:  pf1f  19762  pf1mpf  19764  pf1addcl  19765  pf1mulcl  19766  pf1ind  19767
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