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Theorem pf1rcl 21083
 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 4235 . 2 (𝑋𝑄 → ¬ 𝑄 = ∅)
2 pf1rcl.q . . . 4 𝑄 = ran (eval1𝑅)
3 eqid 2759 . . . . . 6 (eval1𝑅) = (eval1𝑅)
4 eqid 2759 . . . . . 6 (1o eval 𝑅) = (1o eval 𝑅)
5 eqid 2759 . . . . . 6 (Base‘𝑅) = (Base‘𝑅)
63, 4, 5evl1fval 21062 . . . . 5 (eval1𝑅) = ((𝑥 ∈ ((Base‘𝑅) ↑m ((Base‘𝑅) ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ (Base‘𝑅) ↦ (1o × {𝑦})))) ∘ (1o eval 𝑅))
76rneqi 5784 . . . 4 ran (eval1𝑅) = ran ((𝑥 ∈ ((Base‘𝑅) ↑m ((Base‘𝑅) ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ (Base‘𝑅) ↦ (1o × {𝑦})))) ∘ (1o eval 𝑅))
8 rnco2 6089 . . . 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 2786 . . 3 𝑄 = ((𝑥 ∈ ((Base‘𝑅) ↑m ((Base‘𝑅) ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ (Base‘𝑅) ↦ (1o × {𝑦})))) “ ran (1o eval 𝑅))
10 inss2 4137 . . . . 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 4247 . . . . . . 7 (¬ ran (1o eval 𝑅) = ∅ ↔ ∃𝑥 𝑥 ∈ ran (1o eval 𝑅))
124, 5evlval 20873 . . . . . . . . . . 11 (1o eval 𝑅) = ((1o evalSub 𝑅)‘(Base‘𝑅))
1312rneqi 5784 . . . . . . . . . 10 ran (1o eval 𝑅) = ran ((1o evalSub 𝑅)‘(Base‘𝑅))
1413mpfrcl 20863 . . . . . . . . 9 (𝑥 ∈ ran (1o eval 𝑅) → (1o ∈ V ∧ 𝑅 ∈ CRing ∧ (Base‘𝑅) ∈ (SubRing‘𝑅)))
1514simp2d 1141 . . . . . . . 8 (𝑥 ∈ ran (1o eval 𝑅) → 𝑅 ∈ CRing)
1615exlimiv 1932 . . . . . . 7 (∃𝑥 𝑥 ∈ ran (1o eval 𝑅) → 𝑅 ∈ CRing)
1711, 16sylbi 220 . . . . . 6 (¬ ran (1o eval 𝑅) = ∅ → 𝑅 ∈ CRing)
1817con1i 149 . . . . 5 𝑅 ∈ CRing → ran (1o eval 𝑅) = ∅)
19 sseq0 4299 . . . . 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 590 . . . 4 𝑅 ∈ CRing → (dom (𝑥 ∈ ((Base‘𝑅) ↑m ((Base‘𝑅) ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ (Base‘𝑅) ↦ (1o × {𝑦})))) ∩ ran (1o eval 𝑅)) = ∅)
21 imadisj 5926 . . . 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 237 . . 3 𝑅 ∈ CRing → ((𝑥 ∈ ((Base‘𝑅) ↑m ((Base‘𝑅) ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ (Base‘𝑅) ↦ (1o × {𝑦})))) “ ran (1o eval 𝑅)) = ∅)
239, 22syl5eq 2806 . 2 𝑅 ∈ CRing → 𝑄 = ∅)
241, 23nsyl2 143 1 (𝑋𝑄𝑅 ∈ CRing)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1539  ∃wex 1782   ∈ wcel 2112  Vcvv 3410   ∩ cin 3860   ⊆ wss 3861  ∅c0 4228  {csn 4526   ↦ cmpt 5117   × cxp 5527  dom cdm 5529  ran crn 5530   “ cima 5532   ∘ ccom 5533  ‘cfv 6341  (class class class)co 7157  1oc1o 8112   ↑m cmap 8423  Basecbs 16556  CRingccrg 19381  SubRingcsubrg 19614   evalSub ces 20848   eval cevl 20849  eval1ce1 21048 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-rep 5161  ax-sep 5174  ax-nul 5181  ax-pow 5239  ax-pr 5303  ax-un 7466 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-reu 3078  df-rab 3080  df-v 3412  df-sbc 3700  df-csb 3809  df-dif 3864  df-un 3866  df-in 3868  df-ss 3878  df-nul 4229  df-if 4425  df-pw 4500  df-sn 4527  df-pr 4529  df-op 4533  df-uni 4803  df-iun 4889  df-br 5038  df-opab 5100  df-mpt 5118  df-id 5435  df-xp 5535  df-rel 5536  df-cnv 5537  df-co 5538  df-dm 5539  df-rn 5540  df-res 5541  df-ima 5542  df-iota 6300  df-fun 6343  df-fn 6344  df-f 6345  df-f1 6346  df-fo 6347  df-f1o 6348  df-fv 6349  df-riota 7115  df-ov 7160  df-oprab 7161  df-mpo 7162  df-evls 20850  df-evl 20851  df-evl1 21050 This theorem is referenced by:  pf1f  21084  pf1mpf  21086  pf1addcl  21087  pf1mulcl  21088  pf1ind  21089
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