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Theorem pf1rcl 22354
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 4339 . 2 (𝑋𝑄 → ¬ 𝑄 = ∅)
2 pf1rcl.q . . . 4 𝑄 = ran (eval1𝑅)
3 eqid 2736 . . . . . 6 (eval1𝑅) = (eval1𝑅)
4 eqid 2736 . . . . . 6 (1o eval 𝑅) = (1o eval 𝑅)
5 eqid 2736 . . . . . 6 (Base‘𝑅) = (Base‘𝑅)
63, 4, 5evl1fval 22333 . . . . 5 (eval1𝑅) = ((𝑥 ∈ ((Base‘𝑅) ↑m ((Base‘𝑅) ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ (Base‘𝑅) ↦ (1o × {𝑦})))) ∘ (1o eval 𝑅))
76rneqi 5947 . . . 4 ran (eval1𝑅) = ran ((𝑥 ∈ ((Base‘𝑅) ↑m ((Base‘𝑅) ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ (Base‘𝑅) ↦ (1o × {𝑦})))) ∘ (1o eval 𝑅))
8 rnco2 6272 . . . 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 2768 . . 3 𝑄 = ((𝑥 ∈ ((Base‘𝑅) ↑m ((Base‘𝑅) ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ (Base‘𝑅) ↦ (1o × {𝑦})))) “ ran (1o eval 𝑅))
10 inss2 4237 . . . . 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 4351 . . . . . . 7 (¬ ran (1o eval 𝑅) = ∅ ↔ ∃𝑥 𝑥 ∈ ran (1o eval 𝑅))
124, 5evlval 22120 . . . . . . . . . . 11 (1o eval 𝑅) = ((1o evalSub 𝑅)‘(Base‘𝑅))
1312rneqi 5947 . . . . . . . . . 10 ran (1o eval 𝑅) = ran ((1o evalSub 𝑅)‘(Base‘𝑅))
1413mpfrcl 22110 . . . . . . . . 9 (𝑥 ∈ ran (1o eval 𝑅) → (1o ∈ V ∧ 𝑅 ∈ CRing ∧ (Base‘𝑅) ∈ (SubRing‘𝑅)))
1514simp2d 1143 . . . . . . . 8 (𝑥 ∈ ran (1o eval 𝑅) → 𝑅 ∈ CRing)
1615exlimiv 1929 . . . . . . 7 (∃𝑥 𝑥 ∈ ran (1o eval 𝑅) → 𝑅 ∈ CRing)
1711, 16sylbi 217 . . . . . 6 (¬ ran (1o eval 𝑅) = ∅ → 𝑅 ∈ CRing)
1817con1i 147 . . . . 5 𝑅 ∈ CRing → ran (1o eval 𝑅) = ∅)
19 sseq0 4402 . . . . 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 587 . . . 4 𝑅 ∈ CRing → (dom (𝑥 ∈ ((Base‘𝑅) ↑m ((Base‘𝑅) ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ (Base‘𝑅) ↦ (1o × {𝑦})))) ∩ ran (1o eval 𝑅)) = ∅)
21 imadisj 6097 . . . 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 234 . . 3 𝑅 ∈ CRing → ((𝑥 ∈ ((Base‘𝑅) ↑m ((Base‘𝑅) ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ (Base‘𝑅) ↦ (1o × {𝑦})))) “ ran (1o eval 𝑅)) = ∅)
239, 22eqtrid 2788 . 2 𝑅 ∈ CRing → 𝑄 = ∅)
241, 23nsyl2 141 1 (𝑋𝑄𝑅 ∈ CRing)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1539  wex 1778  wcel 2107  Vcvv 3479  cin 3949  wss 3950  c0 4332  {csn 4625  cmpt 5224   × cxp 5682  dom cdm 5684  ran crn 5685  cima 5687  ccom 5688  cfv 6560  (class class class)co 7432  1oc1o 8500  m cmap 8867  Basecbs 17248  CRingccrg 20232  SubRingcsubrg 20570   evalSub ces 22097   eval cevl 22098  eval1ce1 22319
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  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 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-evls 22099  df-evl 22100  df-evl1 22321
This theorem is referenced by:  pf1f  22355  pf1mpf  22357  pf1addcl  22358  pf1mulcl  22359  pf1ind  22360
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