Theorem pf1rcl 19534

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 3879	. 2 ⊢ (𝑋 ∈ 𝑄 → ¬ 𝑄 = ∅)
2		pf1rcl.q	. . . 4 ⊢ 𝑄 = ran (eval1‘𝑅)
3		eqid 2610	. . . . . 6 ⊢ (eval1‘𝑅) = (eval1‘𝑅)
4		eqid 2610	. . . . . 6 ⊢ (1𝑜 eval 𝑅) = (1𝑜 eval 𝑅)
5		eqid 2610	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
6	3, 4, 5	evl1fval 19513	. . . . 5 ⊢ (eval1‘𝑅) = ((𝑥 ∈ ((Base‘𝑅) ↑𝑚 ((Base‘𝑅) ↑𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ (Base‘𝑅) ↦ (1𝑜 × {𝑦})))) ∘ (1𝑜 eval 𝑅))
7	6	rneqi 5273	. . . 4 ⊢ ran (eval1‘𝑅) = ran ((𝑥 ∈ ((Base‘𝑅) ↑𝑚 ((Base‘𝑅) ↑𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ (Base‘𝑅) ↦ (1𝑜 × {𝑦})))) ∘ (1𝑜 eval 𝑅))
8		rnco2 5559	. . . 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 2636	. . 3 ⊢ 𝑄 = ((𝑥 ∈ ((Base‘𝑅) ↑𝑚 ((Base‘𝑅) ↑𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ (Base‘𝑅) ↦ (1𝑜 × {𝑦})))) “ ran (1𝑜 eval 𝑅))
10		inss2 3796	. . . . 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 3889	. . . . . . 7 ⊢ (¬ ran (1𝑜 eval 𝑅) = ∅ ↔ ∃𝑥 𝑥 ∈ ran (1𝑜 eval 𝑅))
12	4, 5	evlval 19345	. . . . . . . . . . 11 ⊢ (1𝑜 eval 𝑅) = ((1𝑜 evalSub 𝑅)‘(Base‘𝑅))
13	12	rneqi 5273	. . . . . . . . . 10 ⊢ ran (1𝑜 eval 𝑅) = ran ((1𝑜 evalSub 𝑅)‘(Base‘𝑅))
14	13	mpfrcl 19339	. . . . . . . . 9 ⊢ (𝑥 ∈ ran (1𝑜 eval 𝑅) → (1𝑜 ∈ V ∧ 𝑅 ∈ CRing ∧ (Base‘𝑅) ∈ (SubRing‘𝑅)))
15	14	simp2d 1067	. . . . . . . 8 ⊢ (𝑥 ∈ ran (1𝑜 eval 𝑅) → 𝑅 ∈ CRing)
16	15	exlimiv 1845	. . . . . . 7 ⊢ (∃𝑥 𝑥 ∈ ran (1𝑜 eval 𝑅) → 𝑅 ∈ CRing)
17	11, 16	sylbi 206	. . . . . 6 ⊢ (¬ ran (1𝑜 eval 𝑅) = ∅ → 𝑅 ∈ CRing)
18	17	con1i 143	. . . . 5 ⊢ (¬ 𝑅 ∈ CRing → ran (1𝑜 eval 𝑅) = ∅)
19		sseq0 3927	. . . . 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 694	. . . 4 ⊢ (¬ 𝑅 ∈ CRing → (dom (𝑥 ∈ ((Base‘𝑅) ↑𝑚 ((Base‘𝑅) ↑𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ (Base‘𝑅) ↦ (1𝑜 × {𝑦})))) ∩ ran (1𝑜 eval 𝑅)) = ∅)
21		imadisj 5403	. . . 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 223	. . 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 2656	. 2 ⊢ (¬ 𝑅 ∈ CRing → 𝑄 = ∅)
24	1, 23	nsyl2 141	1 ⊢ (𝑋 ∈ 𝑄 → 𝑅 ∈ CRing)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1475  ∃wex 1695   ∈ wcel 1977  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  {csn 4125   ↦ cmpt 4643   × cxp 5036  dom cdm 5038  ran crn 5039   “ cima 5041   ∘ ccom 5042  ‘cfv 5804  (class class class)co 6549  1_𝑜c1o 7440   ↑_𝑚 cmap 7744  Basecbs 15695  CRingccrg 18371  SubRingcsubrg 18599   evalSub ces 19325   eval cevl 19326  eval₁ce1 19500

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-evls 19327  df-evl 19328  df-evl1 19502

This theorem is referenced by:  pf1f  19535  pf1mpf  19537  pf1addcl  19538  pf1mulcl  19539  pf1ind  19540