Theorem pf1rcl 21515

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 4267	. 2 ⊢ (𝑋 ∈ 𝑄 → ¬ 𝑄 = ∅)
2		pf1rcl.q	. . . 4 ⊢ 𝑄 = ran (eval1‘𝑅)
3		eqid 2738	. . . . . 6 ⊢ (eval1‘𝑅) = (eval1‘𝑅)
4		eqid 2738	. . . . . 6 ⊢ (1o eval 𝑅) = (1o eval 𝑅)
5		eqid 2738	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
6	3, 4, 5	evl1fval 21494	. . . . 5 ⊢ (eval1‘𝑅) = ((𝑥 ∈ ((Base‘𝑅) ↑m ((Base‘𝑅) ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ (Base‘𝑅) ↦ (1o × {𝑦})))) ∘ (1o eval 𝑅))
7	6	rneqi 5846	. . . 4 ⊢ ran (eval1‘𝑅) = ran ((𝑥 ∈ ((Base‘𝑅) ↑m ((Base‘𝑅) ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ (Base‘𝑅) ↦ (1o × {𝑦})))) ∘ (1o eval 𝑅))
8		rnco2 6157	. . . 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 2770	. . 3 ⊢ 𝑄 = ((𝑥 ∈ ((Base‘𝑅) ↑m ((Base‘𝑅) ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ (Base‘𝑅) ↦ (1o × {𝑦})))) “ ran (1o eval 𝑅))
10		inss2 4163	. . . . 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 4279	. . . . . . 7 ⊢ (¬ ran (1o eval 𝑅) = ∅ ↔ ∃𝑥 𝑥 ∈ ran (1o eval 𝑅))
12	4, 5	evlval 21305	. . . . . . . . . . 11 ⊢ (1o eval 𝑅) = ((1o evalSub 𝑅)‘(Base‘𝑅))
13	12	rneqi 5846	. . . . . . . . . 10 ⊢ ran (1o eval 𝑅) = ran ((1o evalSub 𝑅)‘(Base‘𝑅))
14	13	mpfrcl 21295	. . . . . . . . 9 ⊢ (𝑥 ∈ ran (1o eval 𝑅) → (1o ∈ V ∧ 𝑅 ∈ CRing ∧ (Base‘𝑅) ∈ (SubRing‘𝑅)))
15	14	simp2d 1142	. . . . . . . 8 ⊢ (𝑥 ∈ ran (1o eval 𝑅) → 𝑅 ∈ CRing)
16	15	exlimiv 1933	. . . . . . 7 ⊢ (∃𝑥 𝑥 ∈ ran (1o eval 𝑅) → 𝑅 ∈ CRing)
17	11, 16	sylbi 216	. . . . . 6 ⊢ (¬ ran (1o eval 𝑅) = ∅ → 𝑅 ∈ CRing)
18	17	con1i 147	. . . . 5 ⊢ (¬ 𝑅 ∈ CRing → ran (1o eval 𝑅) = ∅)
19		sseq0 4333	. . . . 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 5988	. . . 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 233	. . 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 2790	. 2 ⊢ (¬ 𝑅 ∈ CRing → 𝑄 = ∅)
24	1, 23	nsyl2 141	1 ⊢ (𝑋 ∈ 𝑄 → 𝑅 ∈ CRing)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1539  ∃wex 1782   ∈ wcel 2106  Vcvv 3432   ∩ cin 3886   ⊆ wss 3887  ∅c0 4256  {csn 4561   ↦ cmpt 5157   × cxp 5587  dom cdm 5589  ran crn 5590   “ cima 5592   ∘ ccom 5593  ‘cfv 6433  (class class class)co 7275  1_oc1o 8290   ↑_m cmap 8615  Basecbs 16912  CRingccrg 19784  SubRingcsubrg 20020   evalSub ces 21280   eval cevl 21281  eval₁ce1 21480

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-evls 21282  df-evl 21283  df-evl1 21482

This theorem is referenced by:  pf1f  21516  pf1mpf  21518  pf1addcl  21519  pf1mulcl  21520  pf1ind  21521