Theorem opsrle 18011

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 2441	. . . . 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 755	. . . . 5 |- ( ( ph /\ ( I e. _V /\ R e. _V ) ) -> I e. _V )
9		simprr 756	. . . . 5 |- ( ( ph /\ ( I e. _V /\ R e. _V ) ) -> R e. _V )
10		opsrle.t	. . . . . 6 |- ( ph -> T C_ ( I X. I ) )
11	10	adantr 465	. . . . 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 18010	. . . 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 5857	. . 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 6306	. . . . 5 |- ( I mPwSer R ) e. _V
16	1, 15	eqeltri 2525	. . . 4 |- S e. _V
17		fvex 5863	. . . . . . 7 |- ( Base ` S ) e. _V
18	3, 17	eqeltri 2525	. . . . . 6 |- B e. _V
19	18, 18	xpex 6586	. . . . 5 |- ( B X. B ) e. _V
20		vex 3096	. . . . . . . . 9 |- x e. _V
21		vex 3096	. . . . . . . . 9 |- y e. _V
22	20, 21	prss 4166	. . . . . . . 8 |- ( ( x e. B /\ y e. B ) <-> { x , y } C_ B )
23	22	anbi1i 695	. . . . . . 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 4498	. . . . . 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 5061	. . . . . 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 3518	. . . . 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 4579	. . . 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 14666	. . . . 5 |- le = Slot ( le ` ndx )
29	28	setsid 14547	. . . 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 672	. . 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 2507	. 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 18009	. . . . . . . . . 10 |- Rel dom ordPwSer
33	32	ovprc 6308	. . . . . . . . 9 |- ( -. ( I e. _V /\ R e. _V ) -> ( I ordPwSer R ) = (/) )
34	33	adantl 466	. . . . . . . 8 |- ( ( ph /\ -. ( I e. _V /\ R e. _V ) ) -> ( I ordPwSer R ) = (/) )
35	34	fveq1d 5855	. . . . . . 7 |- ( ( ph /\ -. ( I e. _V /\ R e. _V ) ) -> ( ( I ordPwSer R ) ` T ) = ( (/) ` T ) )
36	2, 35	syl5eq 2494	. . . . . 6 |- ( ( ph /\ -. ( I e. _V /\ R e. _V ) ) -> O = ( (/) ` T ) )
37		0fv 5886	. . . . . 6 |- ( (/) ` T ) = (/)
38	36, 37	syl6eq 2498	. . . . 5 |- ( ( ph /\ -. ( I e. _V /\ R e. _V ) ) -> O = (/) )
39	38	fveq2d 5857	. . . 4 |- ( ( ph /\ -. ( I e. _V /\ R e. _V ) ) -> ( le ` O ) = ( le ` (/) ) )
40	28	str0 14544	. . . 4 |- (/) = ( le ` (/) )
41	39, 14, 40	3eqtr4g 2507	. . 3 |- ( ( ph /\ -. ( I e. _V /\ R e. _V ) ) -> .<_ = (/) )
42		reldmpsr 17881	. . . . . . . . . . 11 |- Rel dom mPwSer
43	42	ovprc 6308	. . . . . . . . . 10 |- ( -. ( I e. _V /\ R e. _V ) -> ( I mPwSer R ) = (/) )
44	43	adantl 466	. . . . . . . . 9 |- ( ( ph /\ -. ( I e. _V /\ R e. _V ) ) -> ( I mPwSer R ) = (/) )
45	1, 44	syl5eq 2494	. . . . . . . 8 |- ( ( ph /\ -. ( I e. _V /\ R e. _V ) ) -> S = (/) )
46	45	fveq2d 5857	. . . . . . 7 |- ( ( ph /\ -. ( I e. _V /\ R e. _V ) ) -> ( Base ` S ) = ( Base ` (/) ) )
47		base0 14545	. . . . . . 7 |- (/) = ( Base ` (/) )
48	46, 3, 47	3eqtr4g 2507	. . . . . 6 |- ( ( ph /\ -. ( I e. _V /\ R e. _V ) ) -> B = (/) )
49	48	xpeq2d 5010	. . . . 5 |- ( ( ph /\ -. ( I e. _V /\ R e. _V ) ) -> ( B X. B ) = ( B X. (/) ) )
50		xp0 5412	. . . . 5 |- ( B X. (/) ) = (/)
51	49, 50	syl6eq 2498	. . . 4 |- ( ( ph /\ -. ( I e. _V /\ R e. _V ) ) -> ( B X. B ) = (/) )
52		sseq0 3800	. . . 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 663	. . 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 2485	. 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 789	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 368   /\ wa 369   = wceq 1381   e. wcel 1802   A.wral 2791   E.wrex 2792   {crab 2795   _Vcvv 3093   C_ wss 3459   (/)c0 3768   {cpr 4013   <.cop 4017   class class class wbr 4434   {copab 4491   X. cxp 4984   `'ccnv 4985   "cima 4989   ` cfv 5575  (class class class)co 6278   ^m cmap 7419   Fincfn 7515   NNcn 10539   NN0cn0 10798   ndxcnx 14503   sSet csts 14504   Basecbs 14506   lecple 14578   ltcplt 15441   mPwSer cmps 17871   <_bag cltb 17874   ordPwSer copws 17875

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4545  ax-sep 4555  ax-nul 4563  ax-pow 4612  ax-pr 4673  ax-un 6574  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-i2m1 9560  ax-1ne0 9561  ax-rrecex 9564  ax-cnre 9565

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-reu 2798  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3419  df-dif 3462  df-un 3464  df-in 3466  df-ss 3473  df-pss 3475  df-nul 3769  df-if 3924  df-pw 3996  df-sn 4012  df-pr 4014  df-tp 4016  df-op 4018  df-uni 4232  df-iun 4314  df-br 4435  df-opab 4493  df-mpt 4494  df-tr 4528  df-eprel 4778  df-id 4782  df-po 4787  df-so 4788  df-fr 4825  df-we 4827  df-ord 4868  df-on 4869  df-lim 4870  df-suc 4871  df-xp 4992  df-rel 4993  df-cnv 4994  df-co 4995  df-dm 4996  df-rn 4997  df-res 4998  df-ima 4999  df-iota 5538  df-fun 5577  df-fn 5578  df-f 5579  df-f1 5580  df-fo 5581  df-f1o 5582  df-fv 5583  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6683  df-recs 7041  df-rdg 7075  df-nn 10540  df-2 10597  df-3 10598  df-4 10599  df-5 10600  df-6 10601  df-7 10602  df-8 10603  df-9 10604  df-10 10605  df-ndx 14509  df-slot 14510  df-base 14511  df-sets 14512  df-ple 14591  df-psr 17876  df-opsr 17880

This theorem is referenced by:  opsrval2  18012  opsrtoslem1  18019