Theorem pf1rcl 20973

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 4249	. 2 ⊢ (𝑋 ∈ 𝑄 → ¬ 𝑄 = ∅)
2		pf1rcl.q	. . . 4 ⊢ 𝑄 = ran (eval1‘𝑅)
3		eqid 2798	. . . . . 6 ⊢ (eval1‘𝑅) = (eval1‘𝑅)
4		eqid 2798	. . . . . 6 ⊢ (1o eval 𝑅) = (1o eval 𝑅)
5		eqid 2798	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
6	3, 4, 5	evl1fval 20952	. . . . 5 ⊢ (eval1‘𝑅) = ((𝑥 ∈ ((Base‘𝑅) ↑m ((Base‘𝑅) ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ (Base‘𝑅) ↦ (1o × {𝑦})))) ∘ (1o eval 𝑅))
7	6	rneqi 5771	. . . 4 ⊢ ran (eval1‘𝑅) = ran ((𝑥 ∈ ((Base‘𝑅) ↑m ((Base‘𝑅) ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ (Base‘𝑅) ↦ (1o × {𝑦})))) ∘ (1o eval 𝑅))
8		rnco2 6073	. . . 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 2825	. . 3 ⊢ 𝑄 = ((𝑥 ∈ ((Base‘𝑅) ↑m ((Base‘𝑅) ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ (Base‘𝑅) ↦ (1o × {𝑦})))) “ ran (1o eval 𝑅))
10		inss2 4156	. . . . 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 4259	. . . . . . 7 ⊢ (¬ ran (1o eval 𝑅) = ∅ ↔ ∃𝑥 𝑥 ∈ ran (1o eval 𝑅))
12	4, 5	evlval 20767	. . . . . . . . . . 11 ⊢ (1o eval 𝑅) = ((1o evalSub 𝑅)‘(Base‘𝑅))
13	12	rneqi 5771	. . . . . . . . . 10 ⊢ ran (1o eval 𝑅) = ran ((1o evalSub 𝑅)‘(Base‘𝑅))
14	13	mpfrcl 20757	. . . . . . . . 9 ⊢ (𝑥 ∈ ran (1o eval 𝑅) → (1o ∈ V ∧ 𝑅 ∈ CRing ∧ (Base‘𝑅) ∈ (SubRing‘𝑅)))
15	14	simp2d 1140	. . . . . . . 8 ⊢ (𝑥 ∈ ran (1o eval 𝑅) → 𝑅 ∈ CRing)
16	15	exlimiv 1931	. . . . . . 7 ⊢ (∃𝑥 𝑥 ∈ ran (1o eval 𝑅) → 𝑅 ∈ CRing)
17	11, 16	sylbi 220	. . . . . 6 ⊢ (¬ ran (1o eval 𝑅) = ∅ → 𝑅 ∈ CRing)
18	17	con1i 149	. . . . 5 ⊢ (¬ 𝑅 ∈ CRing → ran (1o eval 𝑅) = ∅)
19		sseq0 4307	. . . . 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 590	. . . 4 ⊢ (¬ 𝑅 ∈ CRing → (dom (𝑥 ∈ ((Base‘𝑅) ↑m ((Base‘𝑅) ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ (Base‘𝑅) ↦ (1o × {𝑦})))) ∩ ran (1o eval 𝑅)) = ∅)
21		imadisj 5915	. . . 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 237	. . 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 2845	. 2 ⊢ (¬ 𝑅 ∈ CRing → 𝑄 = ∅)
24	1, 23	nsyl2 143	1 ⊢ (𝑋 ∈ 𝑄 → 𝑅 ∈ CRing)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1538  ∃wex 1781   ∈ wcel 2111  Vcvv 3441   ∩ cin 3880   ⊆ wss 3881  ∅c0 4243  {csn 4525   ↦ cmpt 5110   × cxp 5517  dom cdm 5519  ran crn 5520   “ cima 5522   ∘ ccom 5523  ‘cfv 6324  (class class class)co 7135  1_oc1o 8078   ↑_m cmap 8389  Basecbs 16475  CRingccrg 19291  SubRingcsubrg 19524   evalSub ces 20743   eval cevl 20744  eval₁ce1 20938

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-evls 20745  df-evl 20746  df-evl1 20940

This theorem is referenced by:  pf1f  20974  pf1mpf  20976  pf1addcl  20977  pf1mulcl  20978  pf1ind  20979