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Theorem pf1rcl 19915
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 4063 . 2 (𝑋𝑄 → ¬ 𝑄 = ∅)
2 pf1rcl.q . . . 4 𝑄 = ran (eval1𝑅)
3 eqid 2760 . . . . . 6 (eval1𝑅) = (eval1𝑅)
4 eqid 2760 . . . . . 6 (1𝑜 eval 𝑅) = (1𝑜 eval 𝑅)
5 eqid 2760 . . . . . 6 (Base‘𝑅) = (Base‘𝑅)
63, 4, 5evl1fval 19894 . . . . 5 (eval1𝑅) = ((𝑥 ∈ ((Base‘𝑅) ↑𝑚 ((Base‘𝑅) ↑𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ (Base‘𝑅) ↦ (1𝑜 × {𝑦})))) ∘ (1𝑜 eval 𝑅))
76rneqi 5507 . . . 4 ran (eval1𝑅) = ran ((𝑥 ∈ ((Base‘𝑅) ↑𝑚 ((Base‘𝑅) ↑𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ (Base‘𝑅) ↦ (1𝑜 × {𝑦})))) ∘ (1𝑜 eval 𝑅))
8 rnco2 5803 . . . 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 3977 . . . . 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 4073 . . . . . . 7 (¬ ran (1𝑜 eval 𝑅) = ∅ ↔ ∃𝑥 𝑥 ∈ ran (1𝑜 eval 𝑅))
124, 5evlval 19726 . . . . . . . . . . 11 (1𝑜 eval 𝑅) = ((1𝑜 evalSub 𝑅)‘(Base‘𝑅))
1312rneqi 5507 . . . . . . . . . 10 ran (1𝑜 eval 𝑅) = ran ((1𝑜 evalSub 𝑅)‘(Base‘𝑅))
1413mpfrcl 19720 . . . . . . . . 9 (𝑥 ∈ ran (1𝑜 eval 𝑅) → (1𝑜 ∈ V ∧ 𝑅 ∈ CRing ∧ (Base‘𝑅) ∈ (SubRing‘𝑅)))
1514simp2d 1138 . . . . . . . 8 (𝑥 ∈ ran (1𝑜 eval 𝑅) → 𝑅 ∈ CRing)
1615exlimiv 2007 . . . . . . 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 4118 . . . . 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 698 . . . 4 𝑅 ∈ CRing → (dom (𝑥 ∈ ((Base‘𝑅) ↑𝑚 ((Base‘𝑅) ↑𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ (Base‘𝑅) ↦ (1𝑜 × {𝑦})))) ∩ ran (1𝑜 eval 𝑅)) = ∅)
21 imadisj 5642 . . . 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 2806 . 2 𝑅 ∈ CRing → 𝑄 = ∅)
241, 23nsyl2 142 1 (𝑋𝑄𝑅 ∈ CRing)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1632  wex 1853  wcel 2139  Vcvv 3340  cin 3714  wss 3715  c0 4058  {csn 4321  cmpt 4881   × cxp 5264  dom cdm 5266  ran crn 5267  cima 5269  ccom 5270  cfv 6049  (class class class)co 6813  1𝑜c1o 7722  𝑚 cmap 8023  Basecbs 16059  CRingccrg 18748  SubRingcsubrg 18978   evalSub ces 19706   eval cevl 19707  eval1ce1 19881
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-evls 19708  df-evl 19709  df-evl1 19883
This theorem is referenced by:  pf1f  19916  pf1mpf  19918  pf1addcl  19919  pf1mulcl  19920  pf1ind  19921
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