Theorem pf1rcl 19713

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 e. Q -> R e. CRing )

Proof of Theorem pf1rcl

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		n0i 3920	. 2 |- ( X e. Q -> -. Q = (/) )
2		pf1rcl.q	. . . 4 |- Q = ran (eval1 ` R )
3		eqid 2622	. . . . . 6 |- (eval1 ` R ) = (eval1 ` R )
4		eqid 2622	. . . . . 6 |- ( 1o eval R ) = ( 1o eval R )
5		eqid 2622	. . . . . 6 |- ( Base ` R ) = ( Base ` R )
6	3, 4, 5	evl1fval 19692	. . . . 5 |- (eval1 ` R ) = ( ( x e. ( ( Base ` R ) ^m ( ( Base ` R ) ^m 1o ) ) |-> ( x o. ( y e. ( Base ` R ) |-> ( 1o X. { y } ) ) ) ) o. ( 1o eval R ) )
7	6	rneqi 5352	. . . 4 |- ran (eval1 ` R ) = ran ( ( x e. ( ( Base ` R ) ^m ( ( Base ` R ) ^m 1o ) ) |-> ( x o. ( y e. ( Base ` R ) |-> ( 1o X. { y } ) ) ) ) o. ( 1o eval R ) )
8		rnco2 5642	. . . 4 |- ran ( ( x e. ( ( Base ` R ) ^m ( ( Base ` R ) ^m 1o ) ) |-> ( x o. ( y e. ( Base ` R ) |-> ( 1o X. { y } ) ) ) ) o. ( 1o eval R ) ) = ( ( x e. ( ( Base ` R ) ^m ( ( Base ` R ) ^m 1o ) ) |-> ( x o. ( y e. ( Base ` R ) |-> ( 1o X. { y } ) ) ) ) " ran ( 1o eval R ) )
9	2, 7, 8	3eqtri 2648	. . 3 |- Q = ( ( x e. ( ( Base ` R ) ^m ( ( Base ` R ) ^m 1o ) ) |-> ( x o. ( y e. ( Base ` R ) |-> ( 1o X. { y } ) ) ) ) " ran ( 1o eval R ) )
10		inss2 3834	. . . . 5 |- ( dom ( x e. ( ( Base ` R ) ^m ( ( Base ` R ) ^m 1o ) ) |-> ( x o. ( y e. ( Base ` R ) |-> ( 1o X. { y } ) ) ) ) i^i ran ( 1o eval R ) ) C_ ran ( 1o eval R )
11		neq0 3930	. . . . . . 7 |- ( -. ran ( 1o eval R ) = (/) <-> E. x x e. ran ( 1o eval R ) )
12	4, 5	evlval 19524	. . . . . . . . . . 11 |- ( 1o eval R ) = ( ( 1o evalSub R ) ` ( Base ` R ) )
13	12	rneqi 5352	. . . . . . . . . 10 |- ran ( 1o eval R ) = ran ( ( 1o evalSub R ) ` ( Base ` R ) )
14	13	mpfrcl 19518	. . . . . . . . 9 |- ( x e. ran ( 1o eval R ) -> ( 1o e. _V /\ R e. CRing /\ ( Base ` R ) e. (SubRing ` R ) ) )
15	14	simp2d 1074	. . . . . . . 8 |- ( x e. ran ( 1o eval R ) -> R e. CRing )
16	15	exlimiv 1858	. . . . . . 7 |- ( E. x x e. ran ( 1o eval R ) -> R e. CRing )
17	11, 16	sylbi 207	. . . . . 6 |- ( -. ran ( 1o eval R ) = (/) -> R e. CRing )
18	17	con1i 144	. . . . 5 |- ( -. R e. CRing -> ran ( 1o eval R ) = (/) )
19		sseq0 3975	. . . . 5 |- ( ( ( dom ( x e. ( ( Base ` R ) ^m ( ( Base ` R ) ^m 1o ) ) |-> ( x o. ( y e. ( Base ` R ) |-> ( 1o X. { y } ) ) ) ) i^i ran ( 1o eval R ) ) C_ ran ( 1o eval R ) /\ ran ( 1o eval R ) = (/) ) -> ( dom ( x e. ( ( Base ` R ) ^m ( ( Base ` R ) ^m 1o ) ) |-> ( x o. ( y e. ( Base ` R ) |-> ( 1o X. { y } ) ) ) ) i^i ran ( 1o eval R ) ) = (/) )
20	10, 18, 19	sylancr 695	. . . 4 |- ( -. R e. CRing -> ( dom ( x e. ( ( Base ` R ) ^m ( ( Base ` R ) ^m 1o ) ) |-> ( x o. ( y e. ( Base ` R ) |-> ( 1o X. { y } ) ) ) ) i^i ran ( 1o eval R ) ) = (/) )
21		imadisj 5484	. . . 4 |- ( ( ( x e. ( ( Base ` R ) ^m ( ( Base ` R ) ^m 1o ) ) |-> ( x o. ( y e. ( Base ` R ) |-> ( 1o X. { y } ) ) ) ) " ran ( 1o eval R ) ) = (/) <-> ( dom ( x e. ( ( Base ` R ) ^m ( ( Base ` R ) ^m 1o ) ) |-> ( x o. ( y e. ( Base ` R ) |-> ( 1o X. { y } ) ) ) ) i^i ran ( 1o eval R ) ) = (/) )
22	20, 21	sylibr 224	. . 3 |- ( -. R e. CRing -> ( ( x e. ( ( Base ` R ) ^m ( ( Base ` R ) ^m 1o ) ) |-> ( x o. ( y e. ( Base ` R ) |-> ( 1o X. { y } ) ) ) ) " ran ( 1o eval R ) ) = (/) )
23	9, 22	syl5eq 2668	. 2 |- ( -. R e. CRing -> Q = (/) )
24	1, 23	nsyl2 142	1 |- ( X e. Q -> R e. CRing )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1483   E.wex 1704   e. wcel 1990   _Vcvv 3200   i^i cin 3573   C_ wss 3574   (/)c0 3915   {csn 4177   |-> cmpt 4729   X. cxp 5112   dom cdm 5114   ran crn 5115   "cima 5117   o. ccom 5118   ` cfv 5888  (class class class)co 6650   1oc1o 7553   ^m cmap 7857   Basecbs 15857   CRingccrg 18548  SubRingcsubrg 18776   evalSub ces 19504   eval cevl 19505  eval₁ce1 19679

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-evls 19506  df-evl 19507  df-evl1 19681

This theorem is referenced by:  pf1f  19714  pf1mpf  19716  pf1addcl  19717  pf1mulcl  19718  pf1ind  19719