Theorem opsrle 18747

Description: An alternative expression for the set of polynomials, as the smallest subalgebra of the set of power series that contains all the variable generators. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)

Hypotheses

Ref	Expression
opsrle.s	|- S = ( I mPwSer R )
opsrle.o	|- O = ( ( I ordPwSer R ) ` T )
opsrle.b	|- B = ( Base ` S )
opsrle.q	|- .< = ( lt ` R )
opsrle.c	|- C = ( T <bag I )
opsrle.d	|- D = { h e. ( NN0 ^m I ) | ( `' h " NN ) e. Fin }
opsrle.l	|- .<_ = ( le ` O )
opsrle.t	|- ( ph -> T C_ ( I X. I ) )

Assertion

Ref	Expression
opsrle	|- ( ph -> .<_ = { <. x , y >. | ( { x , y } C_ B /\ ( E. z e. D ( ( x ` z ) .< ( y ` z ) /\ A. w e. D ( w C z -> ( x ` w ) = ( y ` w ) ) ) \/ x = y ) ) } )

Distinct variable groups:   x, y, B   z, w, D   w, h, x, y, z, I   w, R, x, y, z   ph, w, x, y, z   w, T, x, y, z

Allowed substitution hints:   ph( h)   B( z, w, h)   C( x, y, z, w, h)   D( x, y, h)   R( h)   S( x, y, z, w, h)   .< ( x, y, z, w, h)   T( h)   .<_ ( x, y, z, w, h)   O( x, y, z, w, h)

Proof of Theorem opsrle

Step	Hyp	Ref	Expression
1		opsrle.s	. . . . 5 |- S = ( I mPwSer R )
2		opsrle.o	. . . . 5 |- O = ( ( I ordPwSer R ) ` T )
3		opsrle.b	. . . . 5 |- B = ( Base ` S )
4		opsrle.q	. . . . 5 |- .< = ( lt ` R )
5		opsrle.c	. . . . 5 |- C = ( T <bag I )
6		opsrle.d	. . . . 5 |- D = { h e. ( NN0 ^m I ) | ( `' h " NN ) e. Fin }
7		eqid 2461	. . . . 5 |- { <. x , y >. | ( { x , y } C_ B /\ ( E. z e. D ( ( x ` z ) .< ( y ` z ) /\ A. w e. D ( w C z -> ( x ` w ) = ( y ` w ) ) ) \/ x = y ) ) } = { <. x , y >. | ( { x , y } C_ B /\ ( E. z e. D ( ( x ` z ) .< ( y ` z ) /\ A. w e. D ( w C z -> ( x ` w ) = ( y ` w ) ) ) \/ x = y ) ) }
8		simprl 769	. . . . 5 |- ( ( ph /\ ( I e. _V /\ R e. _V ) ) -> I e. _V )
9		simprr 771	. . . . 5 |- ( ( ph /\ ( I e. _V /\ R e. _V ) ) -> R e. _V )
10		opsrle.t	. . . . . 6 |- ( ph -> T C_ ( I X. I ) )
11	10	adantr 471	. . . . 5 |- ( ( ph /\ ( I e. _V /\ R e. _V ) ) -> T C_ ( I X. I ) )
12	1, 2, 3, 4, 5, 6, 7, 8, 9, 11	opsrval 18746	. . . 4 |- ( ( ph /\ ( I e. _V /\ R e. _V ) ) -> O = ( S sSet <. ( le ` ndx ) , { <. x , y >. | ( { x , y } C_ B /\ ( E. z e. D ( ( x ` z ) .< ( y ` z ) /\ A. w e. D ( w C z -> ( x ` w ) = ( y ` w ) ) ) \/ x = y ) ) } >. ) )
13	12	fveq2d 5891	. . 3 |- ( ( ph /\ ( I e. _V /\ R e. _V ) ) -> ( le ` O ) = ( le ` ( S sSet <. ( le ` ndx ) , { <. x , y >. | ( { x , y } C_ B /\ ( E. z e. D ( ( x ` z ) .< ( y ` z ) /\ A. w e. D ( w C z -> ( x ` w ) = ( y ` w ) ) ) \/ x = y ) ) } >. ) ) )
14		opsrle.l	. . 3 |- .<_ = ( le ` O )
15		ovex 6342	. . . . 5 |- ( I mPwSer R ) e. _V
16	1, 15	eqeltri 2535	. . . 4 |- S e. _V
17		fvex 5897	. . . . . . 7 |- ( Base ` S ) e. _V
18	3, 17	eqeltri 2535	. . . . . 6 |- B e. _V
19	18, 18	xpex 6621	. . . . 5 |- ( B X. B ) e. _V
20		vex 3059	. . . . . . . . 9 |- x e. _V
21		vex 3059	. . . . . . . . 9 |- y e. _V
22	20, 21	prss 4138	. . . . . . . 8 |- ( ( x e. B /\ y e. B ) <-> { x , y } C_ B )
23	22	anbi1i 706	. . . . . . 7 |- ( ( ( x e. B /\ y e. B ) /\ ( E. z e. D ( ( x ` z ) .< ( y ` z ) /\ A. w e. D ( w C z -> ( x ` w ) = ( y ` w ) ) ) \/ x = y ) ) <-> ( { x , y } C_ B /\ ( E. z e. D ( ( x ` z ) .< ( y ` z ) /\ A. w e. D ( w C z -> ( x ` w ) = ( y ` w ) ) ) \/ x = y ) ) )
24	23	opabbii 4480	. . . . . 6 |- { <. x , y >. | ( ( x e. B /\ y e. B ) /\ ( E. z e. D ( ( x ` z ) .< ( y ` z ) /\ A. w e. D ( w C z -> ( x ` w ) = ( y ` w ) ) ) \/ x = y ) ) } = { <. x , y >. | ( { x , y } C_ B /\ ( E. z e. D ( ( x ` z ) .< ( y ` z ) /\ A. w e. D ( w C z -> ( x ` w ) = ( y ` w ) ) ) \/ x = y ) ) }
25		opabssxp 4927	. . . . . 6 |- { <. x , y >. | ( ( x e. B /\ y e. B ) /\ ( E. z e. D ( ( x ` z ) .< ( y ` z ) /\ A. w e. D ( w C z -> ( x ` w ) = ( y ` w ) ) ) \/ x = y ) ) } C_ ( B X. B )
26	24, 25	eqsstr3i 3474	. . . . 5 |- { <. x , y >. | ( { x , y } C_ B /\ ( E. z e. D ( ( x ` z ) .< ( y ` z ) /\ A. w e. D ( w C z -> ( x ` w ) = ( y ` w ) ) ) \/ x = y ) ) } C_ ( B X. B )
27	19, 26	ssexi 4561	. . . 4 |- { <. x , y >. | ( { x , y } C_ B /\ ( E. z e. D ( ( x ` z ) .< ( y ` z ) /\ A. w e. D ( w C z -> ( x ` w ) = ( y ` w ) ) ) \/ x = y ) ) } e. _V
28		pleid 15340	. . . . 5 |- le = Slot ( le ` ndx )
29	28	setsid 15212	. . . 4 |- ( ( S e. _V /\ { <. x , y >. | ( { x , y } C_ B /\ ( E. z e. D ( ( x ` z ) .< ( y ` z ) /\ A. w e. D ( w C z -> ( x ` w ) = ( y ` w ) ) ) \/ x = y ) ) } e. _V ) -> { <. x , y >. | ( { x , y } C_ B /\ ( E. z e. D ( ( x ` z ) .< ( y ` z ) /\ A. w e. D ( w C z -> ( x ` w ) = ( y ` w ) ) ) \/ x = y ) ) } = ( le ` ( S sSet <. ( le ` ndx ) , { <. x , y >. | ( { x , y } C_ B /\ ( E. z e. D ( ( x ` z ) .< ( y ` z ) /\ A. w e. D ( w C z -> ( x ` w ) = ( y ` w ) ) ) \/ x = y ) ) } >. ) ) )
30	16, 27, 29	mp2an 683	. . 3 |- { <. x , y >. | ( { x , y } C_ B /\ ( E. z e. D ( ( x ` z ) .< ( y ` z ) /\ A. w e. D ( w C z -> ( x ` w ) = ( y ` w ) ) ) \/ x = y ) ) } = ( le ` ( S sSet <. ( le ` ndx ) , { <. x , y >. | ( { x , y } C_ B /\ ( E. z e. D ( ( x ` z ) .< ( y ` z ) /\ A. w e. D ( w C z -> ( x ` w ) = ( y ` w ) ) ) \/ x = y ) ) } >. ) )
31	13, 14, 30	3eqtr4g 2520	. 2 |- ( ( ph /\ ( I e. _V /\ R e. _V ) ) -> .<_ = { <. x , y >. | ( { x , y } C_ B /\ ( E. z e. D ( ( x ` z ) .< ( y ` z ) /\ A. w e. D ( w C z -> ( x ` w ) = ( y ` w ) ) ) \/ x = y ) ) } )
32		reldmopsr 18745	. . . . . . . . . 10 |- Rel dom ordPwSer
33	32	ovprc 6344	. . . . . . . . 9 |- ( -. ( I e. _V /\ R e. _V ) -> ( I ordPwSer R ) = (/) )
34	33	adantl 472	. . . . . . . 8 |- ( ( ph /\ -. ( I e. _V /\ R e. _V ) ) -> ( I ordPwSer R ) = (/) )
35	34	fveq1d 5889	. . . . . . 7 |- ( ( ph /\ -. ( I e. _V /\ R e. _V ) ) -> ( ( I ordPwSer R ) ` T ) = ( (/) ` T ) )
36	2, 35	syl5eq 2507	. . . . . 6 |- ( ( ph /\ -. ( I e. _V /\ R e. _V ) ) -> O = ( (/) ` T ) )
37		0fv 5920	. . . . . 6 |- ( (/) ` T ) = (/)
38	36, 37	syl6eq 2511	. . . . 5 |- ( ( ph /\ -. ( I e. _V /\ R e. _V ) ) -> O = (/) )
39	38	fveq2d 5891	. . . 4 |- ( ( ph /\ -. ( I e. _V /\ R e. _V ) ) -> ( le ` O ) = ( le ` (/) ) )
40	28	str0 15209	. . . 4 |- (/) = ( le ` (/) )
41	39, 14, 40	3eqtr4g 2520	. . 3 |- ( ( ph /\ -. ( I e. _V /\ R e. _V ) ) -> .<_ = (/) )
42		reldmpsr 18633	. . . . . . . . . . 11 |- Rel dom mPwSer
43	42	ovprc 6344	. . . . . . . . . 10 |- ( -. ( I e. _V /\ R e. _V ) -> ( I mPwSer R ) = (/) )
44	43	adantl 472	. . . . . . . . 9 |- ( ( ph /\ -. ( I e. _V /\ R e. _V ) ) -> ( I mPwSer R ) = (/) )
45	1, 44	syl5eq 2507	. . . . . . . 8 |- ( ( ph /\ -. ( I e. _V /\ R e. _V ) ) -> S = (/) )
46	45	fveq2d 5891	. . . . . . 7 |- ( ( ph /\ -. ( I e. _V /\ R e. _V ) ) -> ( Base ` S ) = ( Base ` (/) ) )
47		base0 15210	. . . . . . 7 |- (/) = ( Base ` (/) )
48	46, 3, 47	3eqtr4g 2520	. . . . . 6 |- ( ( ph /\ -. ( I e. _V /\ R e. _V ) ) -> B = (/) )
49	48	xpeq2d 4876	. . . . 5 |- ( ( ph /\ -. ( I e. _V /\ R e. _V ) ) -> ( B X. B ) = ( B X. (/) ) )
50		xp0 5273	. . . . 5 |- ( B X. (/) ) = (/)
51	49, 50	syl6eq 2511	. . . 4 |- ( ( ph /\ -. ( I e. _V /\ R e. _V ) ) -> ( B X. B ) = (/) )
52		sseq0 3777	. . . 4 |- ( ( { <. x , y >. | ( { x , y } C_ B /\ ( E. z e. D ( ( x ` z ) .< ( y ` z ) /\ A. w e. D ( w C z -> ( x ` w ) = ( y ` w ) ) ) \/ x = y ) ) } C_ ( B X. B ) /\ ( B X. B ) = (/) ) -> { <. x , y >. | ( { x , y } C_ B /\ ( E. z e. D ( ( x ` z ) .< ( y ` z ) /\ A. w e. D ( w C z -> ( x ` w ) = ( y ` w ) ) ) \/ x = y ) ) } = (/) )
53	26, 51, 52	sylancr 674	. . 3 |- ( ( ph /\ -. ( I e. _V /\ R e. _V ) ) -> { <. x , y >. | ( { x , y } C_ B /\ ( E. z e. D ( ( x ` z ) .< ( y ` z ) /\ A. w e. D ( w C z -> ( x ` w ) = ( y ` w ) ) ) \/ x = y ) ) } = (/) )
54	41, 53	eqtr4d 2498	. 2 |- ( ( ph /\ -. ( I e. _V /\ R e. _V ) ) -> .<_ = { <. x , y >. | ( { x , y } C_ B /\ ( E. z e. D ( ( x ` z ) .< ( y ` z ) /\ A. w e. D ( w C z -> ( x ` w ) = ( y ` w ) ) ) \/ x = y ) ) } )
55	31, 54	pm2.61dan 805	1 |- ( ph -> .<_ = { <. x , y >. | ( { x , y } C_ B /\ ( E. z e. D ( ( x ` z ) .< ( y ` z ) /\ A. w e. D ( w C z -> ( x ` w ) = ( y ` w ) ) ) \/ x = y ) ) } )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 374   /\ wa 375   = wceq 1454   e. wcel 1897   A.wral 2748   E.wrex 2749   {crab 2752   _Vcvv 3056   C_ wss 3415   (/)c0 3742   {cpr 3981   <.cop 3985   class class class wbr 4415   {copab 4473   X. cxp 4850   `'ccnv 4851   "cima 4855   ` cfv 5600  (class class class)co 6314   ^m cmap 7497   Fincfn 7594   NNcn 10636   NN0cn0 10897   ndxcnx 15166   sSet csts 15167   Basecbs 15169   lecple 15245   ltcplt 16234   mPwSer cmps 18623   <_bag cltb 18626   ordPwSer copws 18627

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-rep 4528  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6609  ax-cnex 9620  ax-resscn 9621  ax-1cn 9622  ax-icn 9623  ax-addcl 9624  ax-addrcl 9625  ax-mulcl 9626  ax-mulrcl 9627  ax-i2m1 9632  ax-1ne0 9633  ax-rrecex 9636  ax-cnre 9637

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-ral 2753  df-rex 2754  df-reu 2755  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-pss 3431  df-nul 3743  df-if 3893  df-pw 3964  df-sn 3980  df-pr 3982  df-tp 3984  df-op 3986  df-uni 4212  df-iun 4293  df-br 4416  df-opab 4475  df-mpt 4476  df-tr 4511  df-eprel 4763  df-id 4767  df-po 4773  df-so 4774  df-fr 4811  df-we 4813  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-pred 5398  df-ord 5444  df-on 5445  df-lim 5446  df-suc 5447  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608  df-ov 6317  df-oprab 6318  df-mpt2 6319  df-om 6719  df-wrecs 7053  df-recs 7115  df-rdg 7153  df-nn 10637  df-2 10695  df-3 10696  df-4 10697  df-5 10698  df-6 10699  df-7 10700  df-8 10701  df-9 10702  df-10 10703  df-ndx 15172  df-slot 15173  df-base 15174  df-sets 15175  df-ple 15258  df-psr 18628  df-opsr 18632

This theorem is referenced by:  opsrval2  18748  opsrtoslem1  18755