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Theorem pf1rcl 21731
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 4298 . 2 (𝑋 ∈ 𝑄 β†’ Β¬ 𝑄 = βˆ…)
2 pf1rcl.q . . . 4 𝑄 = ran (eval1β€˜π‘…)
3 eqid 2737 . . . . . 6 (eval1β€˜π‘…) = (eval1β€˜π‘…)
4 eqid 2737 . . . . . 6 (1o eval 𝑅) = (1o eval 𝑅)
5 eqid 2737 . . . . . 6 (Baseβ€˜π‘…) = (Baseβ€˜π‘…)
63, 4, 5evl1fval 21710 . . . . 5 (eval1β€˜π‘…) = ((π‘₯ ∈ ((Baseβ€˜π‘…) ↑m ((Baseβ€˜π‘…) ↑m 1o)) ↦ (π‘₯ ∘ (𝑦 ∈ (Baseβ€˜π‘…) ↦ (1o Γ— {𝑦})))) ∘ (1o eval 𝑅))
76rneqi 5897 . . . 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 6210 . . . 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 2769 . . 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 4194 . . . . 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 4310 . . . . . . 7 (Β¬ ran (1o eval 𝑅) = βˆ… ↔ βˆƒπ‘₯ π‘₯ ∈ ran (1o eval 𝑅))
124, 5evlval 21521 . . . . . . . . . . 11 (1o eval 𝑅) = ((1o evalSub 𝑅)β€˜(Baseβ€˜π‘…))
1312rneqi 5897 . . . . . . . . . 10 ran (1o eval 𝑅) = ran ((1o evalSub 𝑅)β€˜(Baseβ€˜π‘…))
1413mpfrcl 21511 . . . . . . . . 9 (π‘₯ ∈ ran (1o eval 𝑅) β†’ (1o ∈ V ∧ 𝑅 ∈ CRing ∧ (Baseβ€˜π‘…) ∈ (SubRingβ€˜π‘…)))
1514simp2d 1144 . . . . . . . 8 (π‘₯ ∈ ran (1o eval 𝑅) β†’ 𝑅 ∈ CRing)
1615exlimiv 1934 . . . . . . 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 4364 . . . . 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 588 . . . 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 6037 . . . 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 2789 . 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 1542  βˆƒwex 1782   ∈ wcel 2107  Vcvv 3448   ∩ cin 3914   βŠ† wss 3915  βˆ…c0 4287  {csn 4591   ↦ cmpt 5193   Γ— cxp 5636  dom cdm 5638  ran crn 5639   β€œ cima 5641   ∘ ccom 5642  β€˜cfv 6501  (class class class)co 7362  1oc1o 8410   ↑m cmap 8772  Basecbs 17090  CRingccrg 19972  SubRingcsubrg 20234   evalSub ces 21496   eval cevl 21497  eval1ce1 21696
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-evls 21498  df-evl 21499  df-evl1 21698
This theorem is referenced by:  pf1f  21732  pf1mpf  21734  pf1addcl  21735  pf1mulcl  21736  pf1ind  21737
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