Theorem pf1rcl 22234

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 4291	. 2 ⊢ (𝑋 ∈ 𝑄 → ¬ 𝑄 = ∅)
2		pf1rcl.q	. . . 4 ⊢ 𝑄 = ran (eval1‘𝑅)
3		eqid 2729	. . . . . 6 ⊢ (eval1‘𝑅) = (eval1‘𝑅)
4		eqid 2729	. . . . . 6 ⊢ (1o eval 𝑅) = (1o eval 𝑅)
5		eqid 2729	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
6	3, 4, 5	evl1fval 22213	. . . . 5 ⊢ (eval1‘𝑅) = ((𝑥 ∈ ((Base‘𝑅) ↑m ((Base‘𝑅) ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ (Base‘𝑅) ↦ (1o × {𝑦})))) ∘ (1o eval 𝑅))
7	6	rneqi 5879	. . . 4 ⊢ ran (eval1‘𝑅) = ran ((𝑥 ∈ ((Base‘𝑅) ↑m ((Base‘𝑅) ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ (Base‘𝑅) ↦ (1o × {𝑦})))) ∘ (1o eval 𝑅))
8		rnco2 6202	. . . 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 2756	. . 3 ⊢ 𝑄 = ((𝑥 ∈ ((Base‘𝑅) ↑m ((Base‘𝑅) ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ (Base‘𝑅) ↦ (1o × {𝑦})))) “ ran (1o eval 𝑅))
10		inss2 4189	. . . . 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 4303	. . . . . . 7 ⊢ (¬ ran (1o eval 𝑅) = ∅ ↔ ∃𝑥 𝑥 ∈ ran (1o eval 𝑅))
12	4, 5	evlval 22000	. . . . . . . . . . 11 ⊢ (1o eval 𝑅) = ((1o evalSub 𝑅)‘(Base‘𝑅))
13	12	rneqi 5879	. . . . . . . . . 10 ⊢ ran (1o eval 𝑅) = ran ((1o evalSub 𝑅)‘(Base‘𝑅))
14	13	mpfrcl 21990	. . . . . . . . 9 ⊢ (𝑥 ∈ ran (1o eval 𝑅) → (1o ∈ V ∧ 𝑅 ∈ CRing ∧ (Base‘𝑅) ∈ (SubRing‘𝑅)))
15	14	simp2d 1143	. . . . . . . 8 ⊢ (𝑥 ∈ ran (1o eval 𝑅) → 𝑅 ∈ CRing)
16	15	exlimiv 1930	. . . . . . 7 ⊢ (∃𝑥 𝑥 ∈ ran (1o eval 𝑅) → 𝑅 ∈ CRing)
17	11, 16	sylbi 217	. . . . . 6 ⊢ (¬ ran (1o eval 𝑅) = ∅ → 𝑅 ∈ CRing)
18	17	con1i 147	. . . . 5 ⊢ (¬ 𝑅 ∈ CRing → ran (1o eval 𝑅) = ∅)
19		sseq0 4354	. . . . 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 587	. . . 4 ⊢ (¬ 𝑅 ∈ CRing → (dom (𝑥 ∈ ((Base‘𝑅) ↑m ((Base‘𝑅) ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ (Base‘𝑅) ↦ (1o × {𝑦})))) ∩ ran (1o eval 𝑅)) = ∅)
21		imadisj 6031	. . . 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 234	. . 3 ⊢ (¬ 𝑅 ∈ CRing → ((𝑥 ∈ ((Base‘𝑅) ↑m ((Base‘𝑅) ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ (Base‘𝑅) ↦ (1o × {𝑦})))) “ ran (1o eval 𝑅)) = ∅)
23	9, 22	eqtrid 2776	. 2 ⊢ (¬ 𝑅 ∈ CRing → 𝑄 = ∅)
24	1, 23	nsyl2 141	1 ⊢ (𝑋 ∈ 𝑄 → 𝑅 ∈ CRing)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1540  ∃wex 1779   ∈ wcel 2109  Vcvv 3436   ∩ cin 3902   ⊆ wss 3903  ∅c0 4284  {csn 4577   ↦ cmpt 5173   × cxp 5617  dom cdm 5619  ran crn 5620   “ cima 5622   ∘ ccom 5623  ‘cfv 6482  (class class class)co 7349  1_oc1o 8381   ↑_m cmap 8753  Basecbs 17120  CRingccrg 20119  SubRingcsubrg 20454   evalSub ces 21977   eval cevl 21978  eval₁ce1 22199

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-evls 21979  df-evl 21980  df-evl1 22201

This theorem is referenced by:  pf1f  22235  pf1mpf  22237  pf1addcl  22238  pf1mulcl  22239  pf1ind  22240