Theorem pf1rcl 19989

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 4086	. 2 ⊢ (𝑋 ∈ 𝑄 → ¬ 𝑄 = ∅)
2		pf1rcl.q	. . . 4 ⊢ 𝑄 = ran (eval1‘𝑅)
3		eqid 2765	. . . . . 6 ⊢ (eval1‘𝑅) = (eval1‘𝑅)
4		eqid 2765	. . . . . 6 ⊢ (1𝑜 eval 𝑅) = (1𝑜 eval 𝑅)
5		eqid 2765	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
6	3, 4, 5	evl1fval 19968	. . . . 5 ⊢ (eval1‘𝑅) = ((𝑥 ∈ ((Base‘𝑅) ↑𝑚 ((Base‘𝑅) ↑𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ (Base‘𝑅) ↦ (1𝑜 × {𝑦})))) ∘ (1𝑜 eval 𝑅))
7	6	rneqi 5522	. . . 4 ⊢ ran (eval1‘𝑅) = ran ((𝑥 ∈ ((Base‘𝑅) ↑𝑚 ((Base‘𝑅) ↑𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ (Base‘𝑅) ↦ (1𝑜 × {𝑦})))) ∘ (1𝑜 eval 𝑅))
8		rnco2 5830	. . . 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 2791	. . 3 ⊢ 𝑄 = ((𝑥 ∈ ((Base‘𝑅) ↑𝑚 ((Base‘𝑅) ↑𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ (Base‘𝑅) ↦ (1𝑜 × {𝑦})))) “ ran (1𝑜 eval 𝑅))
10		inss2 3995	. . . . 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 4096	. . . . . . 7 ⊢ (¬ ran (1𝑜 eval 𝑅) = ∅ ↔ ∃𝑥 𝑥 ∈ ran (1𝑜 eval 𝑅))
12	4, 5	evlval 19800	. . . . . . . . . . 11 ⊢ (1𝑜 eval 𝑅) = ((1𝑜 evalSub 𝑅)‘(Base‘𝑅))
13	12	rneqi 5522	. . . . . . . . . 10 ⊢ ran (1𝑜 eval 𝑅) = ran ((1𝑜 evalSub 𝑅)‘(Base‘𝑅))
14	13	mpfrcl 19794	. . . . . . . . 9 ⊢ (𝑥 ∈ ran (1𝑜 eval 𝑅) → (1𝑜 ∈ V ∧ 𝑅 ∈ CRing ∧ (Base‘𝑅) ∈ (SubRing‘𝑅)))
15	14	simp2d 1173	. . . . . . . 8 ⊢ (𝑥 ∈ ran (1𝑜 eval 𝑅) → 𝑅 ∈ CRing)
16	15	exlimiv 2025	. . . . . . 7 ⊢ (∃𝑥 𝑥 ∈ ran (1𝑜 eval 𝑅) → 𝑅 ∈ CRing)
17	11, 16	sylbi 208	. . . . . 6 ⊢ (¬ ran (1𝑜 eval 𝑅) = ∅ → 𝑅 ∈ CRing)
18	17	con1i 146	. . . . 5 ⊢ (¬ 𝑅 ∈ CRing → ran (1𝑜 eval 𝑅) = ∅)
19		sseq0 4139	. . . . 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 581	. . . 4 ⊢ (¬ 𝑅 ∈ CRing → (dom (𝑥 ∈ ((Base‘𝑅) ↑𝑚 ((Base‘𝑅) ↑𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ (Base‘𝑅) ↦ (1𝑜 × {𝑦})))) ∩ ran (1𝑜 eval 𝑅)) = ∅)
21		imadisj 5668	. . . 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 225	. . 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 2811	. 2 ⊢ (¬ 𝑅 ∈ CRing → 𝑄 = ∅)
24	1, 23	nsyl2 144	1 ⊢ (𝑋 ∈ 𝑄 → 𝑅 ∈ CRing)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1652  ∃wex 1874   ∈ wcel 2155  Vcvv 3350   ∩ cin 3733   ⊆ wss 3734  ∅c0 4081  {csn 4336   ↦ cmpt 4890   × cxp 5277  dom cdm 5279  ran crn 5280   “ cima 5282   ∘ ccom 5283  ‘cfv 6070  (class class class)co 6844  1_𝑜c1o 7759   ↑_𝑚 cmap 8062  Basecbs 16133  CRingccrg 18818  SubRingcsubrg 19048   evalSub ces 19780   eval cevl 19781  eval₁ce1 19955

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4932  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7149

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-op 4343  df-uni 4597  df-iun 4680  df-br 4812  df-opab 4874  df-mpt 4891  df-id 5187  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-riota 6805  df-ov 6847  df-oprab 6848  df-mpt2 6849  df-evls 19782  df-evl 19783  df-evl1 19957

This theorem is referenced by:  pf1f  19990  pf1mpf  19992  pf1addcl  19993  pf1mulcl  19994  pf1ind  19995