Theorem pf1rcl 19713

Description: Reverse closure for the set of polynomial functions. (Contributed by Mario Carneiro, 12-Jun-2015.)

Hypothesis

Ref	Expression
pf1rcl.q	⊢ 𝑄 = ran (eval1‘𝑅)

Assertion

Ref	Expression
pf1rcl	⊢ (𝑋 ∈ 𝑄 → 𝑅 ∈ CRing)

Proof of Theorem pf1rcl

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		n0i 3920	. 2 ⊢ (𝑋 ∈ 𝑄 → ¬ 𝑄 = ∅)
2		pf1rcl.q	. . . 4 ⊢ 𝑄 = ran (eval1‘𝑅)
3		eqid 2622	. . . . . 6 ⊢ (eval1‘𝑅) = (eval1‘𝑅)
4		eqid 2622	. . . . . 6 ⊢ (1𝑜 eval 𝑅) = (1𝑜 eval 𝑅)
5		eqid 2622	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
6	3, 4, 5	evl1fval 19692	. . . . 5 ⊢ (eval1‘𝑅) = ((𝑥 ∈ ((Base‘𝑅) ↑𝑚 ((Base‘𝑅) ↑𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ (Base‘𝑅) ↦ (1𝑜 × {𝑦})))) ∘ (1𝑜 eval 𝑅))
7	6	rneqi 5352	. . . 4 ⊢ ran (eval1‘𝑅) = ran ((𝑥 ∈ ((Base‘𝑅) ↑𝑚 ((Base‘𝑅) ↑𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ (Base‘𝑅) ↦ (1𝑜 × {𝑦})))) ∘ (1𝑜 eval 𝑅))
8		rnco2 5642	. . . 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))
9	2, 7, 8	3eqtri 2648	. . 3 ⊢ 𝑄 = ((𝑥 ∈ ((Base‘𝑅) ↑𝑚 ((Base‘𝑅) ↑𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ (Base‘𝑅) ↦ (1𝑜 × {𝑦})))) “ ran (1𝑜 eval 𝑅))
10		inss2 3834	. . . . 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 3930	. . . . . . 7 ⊢ (¬ ran (1𝑜 eval 𝑅) = ∅ ↔ ∃𝑥 𝑥 ∈ ran (1𝑜 eval 𝑅))
12	4, 5	evlval 19524	. . . . . . . . . . 11 ⊢ (1𝑜 eval 𝑅) = ((1𝑜 evalSub 𝑅)‘(Base‘𝑅))
13	12	rneqi 5352	. . . . . . . . . 10 ⊢ ran (1𝑜 eval 𝑅) = ran ((1𝑜 evalSub 𝑅)‘(Base‘𝑅))
14	13	mpfrcl 19518	. . . . . . . . 9 ⊢ (𝑥 ∈ ran (1𝑜 eval 𝑅) → (1𝑜 ∈ V ∧ 𝑅 ∈ CRing ∧ (Base‘𝑅) ∈ (SubRing‘𝑅)))
15	14	simp2d 1074	. . . . . . . 8 ⊢ (𝑥 ∈ ran (1𝑜 eval 𝑅) → 𝑅 ∈ CRing)
16	15	exlimiv 1858	. . . . . . 7 ⊢ (∃𝑥 𝑥 ∈ ran (1𝑜 eval 𝑅) → 𝑅 ∈ CRing)
17	11, 16	sylbi 207	. . . . . 6 ⊢ (¬ ran (1𝑜 eval 𝑅) = ∅ → 𝑅 ∈ CRing)
18	17	con1i 144	. . . . 5 ⊢ (¬ 𝑅 ∈ CRing → ran (1𝑜 eval 𝑅) = ∅)
19		sseq0 3975	. . . . 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)) = ∅)
20	10, 18, 19	sylancr 695	. . . 4 ⊢ (¬ 𝑅 ∈ CRing → (dom (𝑥 ∈ ((Base‘𝑅) ↑𝑚 ((Base‘𝑅) ↑𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ (Base‘𝑅) ↦ (1𝑜 × {𝑦})))) ∩ ran (1𝑜 eval 𝑅)) = ∅)
21		imadisj 5484	. . . 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)) = ∅)
22	20, 21	sylibr 224	. . 3 ⊢ (¬ 𝑅 ∈ CRing → ((𝑥 ∈ ((Base‘𝑅) ↑𝑚 ((Base‘𝑅) ↑𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ (Base‘𝑅) ↦ (1𝑜 × {𝑦})))) “ ran (1𝑜 eval 𝑅)) = ∅)
23	9, 22	syl5eq 2668	. 2 ⊢ (¬ 𝑅 ∈ CRing → 𝑄 = ∅)
24	1, 23	nsyl2 142	1 ⊢ (𝑋 ∈ 𝑄 → 𝑅 ∈ CRing)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1483  ∃wex 1704   ∈ wcel 1990  Vcvv 3200   ∩ cin 3573   ⊆ wss 3574  ∅c0 3915  {csn 4177   ↦ cmpt 4729   × cxp 5112  dom cdm 5114  ran crn 5115   “ cima 5117   ∘ ccom 5118  ‘cfv 5888  (class class class)co 6650  1_𝑜c1o 7553   ↑_𝑚 cmap 7857  Basecbs 15857  CRingccrg 18548  SubRingcsubrg 18776   evalSub ces 19504   eval cevl 19505  eval₁ce1 19679

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-evls 19506  df-evl 19507  df-evl1 19681

This theorem is referenced by:  pf1f  19714  pf1mpf  19716  pf1addcl  19717  pf1mulcl  19718  pf1ind  19719