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Theorem pf1rcl 21867
Description: Reverse closure for the set of polynomial functions. (Contributed by Mario Carneiro, 12-Jun-2015.)
Hypothesis
Ref Expression
pf1rcl.q 𝑄 = ran (eval1β€˜π‘…)
Assertion
Ref Expression
pf1rcl (𝑋 ∈ 𝑄 β†’ 𝑅 ∈ CRing)

Proof of Theorem pf1rcl
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 n0i 4333 . 2 (𝑋 ∈ 𝑄 β†’ Β¬ 𝑄 = βˆ…)
2 pf1rcl.q . . . 4 𝑄 = ran (eval1β€˜π‘…)
3 eqid 2732 . . . . . 6 (eval1β€˜π‘…) = (eval1β€˜π‘…)
4 eqid 2732 . . . . . 6 (1o eval 𝑅) = (1o eval 𝑅)
5 eqid 2732 . . . . . 6 (Baseβ€˜π‘…) = (Baseβ€˜π‘…)
63, 4, 5evl1fval 21846 . . . . 5 (eval1β€˜π‘…) = ((π‘₯ ∈ ((Baseβ€˜π‘…) ↑m ((Baseβ€˜π‘…) ↑m 1o)) ↦ (π‘₯ ∘ (𝑦 ∈ (Baseβ€˜π‘…) ↦ (1o Γ— {𝑦})))) ∘ (1o eval 𝑅))
76rneqi 5936 . . . 4 ran (eval1β€˜π‘…) = ran ((π‘₯ ∈ ((Baseβ€˜π‘…) ↑m ((Baseβ€˜π‘…) ↑m 1o)) ↦ (π‘₯ ∘ (𝑦 ∈ (Baseβ€˜π‘…) ↦ (1o Γ— {𝑦})))) ∘ (1o eval 𝑅))
8 rnco2 6252 . . . 4 ran ((π‘₯ ∈ ((Baseβ€˜π‘…) ↑m ((Baseβ€˜π‘…) ↑m 1o)) ↦ (π‘₯ ∘ (𝑦 ∈ (Baseβ€˜π‘…) ↦ (1o Γ— {𝑦})))) ∘ (1o eval 𝑅)) = ((π‘₯ ∈ ((Baseβ€˜π‘…) ↑m ((Baseβ€˜π‘…) ↑m 1o)) ↦ (π‘₯ ∘ (𝑦 ∈ (Baseβ€˜π‘…) ↦ (1o Γ— {𝑦})))) β€œ ran (1o eval 𝑅))
92, 7, 83eqtri 2764 . . 3 𝑄 = ((π‘₯ ∈ ((Baseβ€˜π‘…) ↑m ((Baseβ€˜π‘…) ↑m 1o)) ↦ (π‘₯ ∘ (𝑦 ∈ (Baseβ€˜π‘…) ↦ (1o Γ— {𝑦})))) β€œ ran (1o eval 𝑅))
10 inss2 4229 . . . . 5 (dom (π‘₯ ∈ ((Baseβ€˜π‘…) ↑m ((Baseβ€˜π‘…) ↑m 1o)) ↦ (π‘₯ ∘ (𝑦 ∈ (Baseβ€˜π‘…) ↦ (1o Γ— {𝑦})))) ∩ ran (1o eval 𝑅)) βŠ† ran (1o eval 𝑅)
11 neq0 4345 . . . . . . 7 (Β¬ ran (1o eval 𝑅) = βˆ… ↔ βˆƒπ‘₯ π‘₯ ∈ ran (1o eval 𝑅))
124, 5evlval 21657 . . . . . . . . . . 11 (1o eval 𝑅) = ((1o evalSub 𝑅)β€˜(Baseβ€˜π‘…))
1312rneqi 5936 . . . . . . . . . 10 ran (1o eval 𝑅) = ran ((1o evalSub 𝑅)β€˜(Baseβ€˜π‘…))
1413mpfrcl 21647 . . . . . . . . 9 (π‘₯ ∈ ran (1o eval 𝑅) β†’ (1o ∈ V ∧ 𝑅 ∈ CRing ∧ (Baseβ€˜π‘…) ∈ (SubRingβ€˜π‘…)))
1514simp2d 1143 . . . . . . . 8 (π‘₯ ∈ ran (1o eval 𝑅) β†’ 𝑅 ∈ CRing)
1615exlimiv 1933 . . . . . . 7 (βˆƒπ‘₯ π‘₯ ∈ ran (1o eval 𝑅) β†’ 𝑅 ∈ CRing)
1711, 16sylbi 216 . . . . . 6 (Β¬ ran (1o eval 𝑅) = βˆ… β†’ 𝑅 ∈ CRing)
1817con1i 147 . . . . 5 (Β¬ 𝑅 ∈ CRing β†’ ran (1o eval 𝑅) = βˆ…)
19 sseq0 4399 . . . . 5 (((dom (π‘₯ ∈ ((Baseβ€˜π‘…) ↑m ((Baseβ€˜π‘…) ↑m 1o)) ↦ (π‘₯ ∘ (𝑦 ∈ (Baseβ€˜π‘…) ↦ (1o Γ— {𝑦})))) ∩ ran (1o eval 𝑅)) βŠ† ran (1o eval 𝑅) ∧ ran (1o eval 𝑅) = βˆ…) β†’ (dom (π‘₯ ∈ ((Baseβ€˜π‘…) ↑m ((Baseβ€˜π‘…) ↑m 1o)) ↦ (π‘₯ ∘ (𝑦 ∈ (Baseβ€˜π‘…) ↦ (1o Γ— {𝑦})))) ∩ ran (1o eval 𝑅)) = βˆ…)
2010, 18, 19sylancr 587 . . . 4 (Β¬ 𝑅 ∈ CRing β†’ (dom (π‘₯ ∈ ((Baseβ€˜π‘…) ↑m ((Baseβ€˜π‘…) ↑m 1o)) ↦ (π‘₯ ∘ (𝑦 ∈ (Baseβ€˜π‘…) ↦ (1o Γ— {𝑦})))) ∩ ran (1o eval 𝑅)) = βˆ…)
21 imadisj 6079 . . . 4 (((π‘₯ ∈ ((Baseβ€˜π‘…) ↑m ((Baseβ€˜π‘…) ↑m 1o)) ↦ (π‘₯ ∘ (𝑦 ∈ (Baseβ€˜π‘…) ↦ (1o Γ— {𝑦})))) β€œ ran (1o eval 𝑅)) = βˆ… ↔ (dom (π‘₯ ∈ ((Baseβ€˜π‘…) ↑m ((Baseβ€˜π‘…) ↑m 1o)) ↦ (π‘₯ ∘ (𝑦 ∈ (Baseβ€˜π‘…) ↦ (1o Γ— {𝑦})))) ∩ ran (1o eval 𝑅)) = βˆ…)
2220, 21sylibr 233 . . 3 (Β¬ 𝑅 ∈ CRing β†’ ((π‘₯ ∈ ((Baseβ€˜π‘…) ↑m ((Baseβ€˜π‘…) ↑m 1o)) ↦ (π‘₯ ∘ (𝑦 ∈ (Baseβ€˜π‘…) ↦ (1o Γ— {𝑦})))) β€œ ran (1o eval 𝑅)) = βˆ…)
239, 22eqtrid 2784 . 2 (Β¬ 𝑅 ∈ CRing β†’ 𝑄 = βˆ…)
241, 23nsyl2 141 1 (𝑋 ∈ 𝑄 β†’ 𝑅 ∈ CRing)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   = wceq 1541  βˆƒwex 1781   ∈ wcel 2106  Vcvv 3474   ∩ cin 3947   βŠ† wss 3948  βˆ…c0 4322  {csn 4628   ↦ cmpt 5231   Γ— cxp 5674  dom cdm 5676  ran crn 5677   β€œ cima 5679   ∘ ccom 5680  β€˜cfv 6543  (class class class)co 7408  1oc1o 8458   ↑m cmap 8819  Basecbs 17143  CRingccrg 20056  SubRingcsubrg 20314   evalSub ces 21632   eval cevl 21633  eval1ce1 21832
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-evls 21634  df-evl 21635  df-evl1 21834
This theorem is referenced by:  pf1f  21868  pf1mpf  21870  pf1addcl  21871  pf1mulcl  21872  pf1ind  21873
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