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Theorem pf1rcl 22343
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 4336 . 2 (𝑋𝑄 → ¬ 𝑄 = ∅)
2 pf1rcl.q . . . 4 𝑄 = ran (eval1𝑅)
3 eqid 2726 . . . . . 6 (eval1𝑅) = (eval1𝑅)
4 eqid 2726 . . . . . 6 (1o eval 𝑅) = (1o eval 𝑅)
5 eqid 2726 . . . . . 6 (Base‘𝑅) = (Base‘𝑅)
63, 4, 5evl1fval 22322 . . . . 5 (eval1𝑅) = ((𝑥 ∈ ((Base‘𝑅) ↑m ((Base‘𝑅) ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ (Base‘𝑅) ↦ (1o × {𝑦})))) ∘ (1o eval 𝑅))
76rneqi 5945 . . . 4 ran (eval1𝑅) = ran ((𝑥 ∈ ((Base‘𝑅) ↑m ((Base‘𝑅) ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ (Base‘𝑅) ↦ (1o × {𝑦})))) ∘ (1o eval 𝑅))
8 rnco2 6266 . . . 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 2758 . . 3 𝑄 = ((𝑥 ∈ ((Base‘𝑅) ↑m ((Base‘𝑅) ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ (Base‘𝑅) ↦ (1o × {𝑦})))) “ ran (1o eval 𝑅))
10 inss2 4231 . . . . 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 4348 . . . . . . 7 (¬ ran (1o eval 𝑅) = ∅ ↔ ∃𝑥 𝑥 ∈ ran (1o eval 𝑅))
124, 5evlval 22112 . . . . . . . . . . 11 (1o eval 𝑅) = ((1o evalSub 𝑅)‘(Base‘𝑅))
1312rneqi 5945 . . . . . . . . . 10 ran (1o eval 𝑅) = ran ((1o evalSub 𝑅)‘(Base‘𝑅))
1413mpfrcl 22102 . . . . . . . . 9 (𝑥 ∈ ran (1o eval 𝑅) → (1o ∈ V ∧ 𝑅 ∈ CRing ∧ (Base‘𝑅) ∈ (SubRing‘𝑅)))
1514simp2d 1140 . . . . . . . 8 (𝑥 ∈ ran (1o eval 𝑅) → 𝑅 ∈ CRing)
1615exlimiv 1926 . . . . . . 7 (∃𝑥 𝑥 ∈ ran (1o eval 𝑅) → 𝑅 ∈ CRing)
1711, 16sylbi 216 . . . . . 6 (¬ ran (1o eval 𝑅) = ∅ → 𝑅 ∈ CRing)
1817con1i 147 . . . . 5 𝑅 ∈ CRing → ran (1o eval 𝑅) = ∅)
19 sseq0 4404 . . . . 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 585 . . . 4 𝑅 ∈ CRing → (dom (𝑥 ∈ ((Base‘𝑅) ↑m ((Base‘𝑅) ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ (Base‘𝑅) ↦ (1o × {𝑦})))) ∩ ran (1o eval 𝑅)) = ∅)
21 imadisj 6091 . . . 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 233 . . 3 𝑅 ∈ CRing → ((𝑥 ∈ ((Base‘𝑅) ↑m ((Base‘𝑅) ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ (Base‘𝑅) ↦ (1o × {𝑦})))) “ ran (1o eval 𝑅)) = ∅)
239, 22eqtrid 2778 . 2 𝑅 ∈ CRing → 𝑄 = ∅)
241, 23nsyl2 141 1 (𝑋𝑄𝑅 ∈ CRing)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1534  wex 1774  wcel 2099  Vcvv 3462  cin 3946  wss 3947  c0 4325  {csn 4633  cmpt 5238   × cxp 5682  dom cdm 5684  ran crn 5685  cima 5687  ccom 5688  cfv 6556  (class class class)co 7426  1oc1o 8491  m cmap 8857  Basecbs 17215  CRingccrg 20219  SubRingcsubrg 20553   evalSub ces 22087   eval cevl 22088  eval1ce1 22308
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5292  ax-sep 5306  ax-nul 5313  ax-pow 5371  ax-pr 5435  ax-un 7748
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4916  df-iun 5005  df-br 5156  df-opab 5218  df-mpt 5239  df-id 5582  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6508  df-fun 6558  df-fn 6559  df-f 6560  df-f1 6561  df-fo 6562  df-f1o 6563  df-fv 6564  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-evls 22089  df-evl 22090  df-evl1 22310
This theorem is referenced by:  pf1f  22344  pf1mpf  22346  pf1addcl  22347  pf1mulcl  22348  pf1ind  22349
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