Theorem opsrle 17911

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 2467	. . . . 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 17910	. . . 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 5868	. . 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 6307	. . . . 5 |- ( I mPwSer R ) e. _V
16	1, 15	eqeltri 2551	. . . 4 |- S e. _V
17		fvex 5874	. . . . . . 7 |- ( Base ` S ) e. _V
18	3, 17	eqeltri 2551	. . . . . 6 |- B e. _V
19	18, 18	xpex 6711	. . . . 5 |- ( B X. B ) e. _V
20		vex 3116	. . . . . . . . 9 |- x e. _V
21		vex 3116	. . . . . . . . 9 |- y e. _V
22	20, 21	prss 4181	. . . . . . . 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 4511	. . . . . 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 5072	. . . . . 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 3535	. . . . 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 4592	. . . 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 14646	. . . . 5 |- le = Slot ( le ` ndx )
29	28	setsid 14527	. . . 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 2533	. 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 17909	. . . . . . . . . 10 |- Rel dom ordPwSer
33	32	ovprc 6309	. . . . . . . . 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 5866	. . . . . . 7 |- ( ( ph /\ -. ( I e. _V /\ R e. _V ) ) -> ( ( I ordPwSer R ) ` T ) = ( (/) ` T ) )
36	2, 35	syl5eq 2520	. . . . . 6 |- ( ( ph /\ -. ( I e. _V /\ R e. _V ) ) -> O = ( (/) ` T ) )
37		0fv 5897	. . . . . 6 |- ( (/) ` T ) = (/)
38	36, 37	syl6eq 2524	. . . . 5 |- ( ( ph /\ -. ( I e. _V /\ R e. _V ) ) -> O = (/) )
39	38	fveq2d 5868	. . . 4 |- ( ( ph /\ -. ( I e. _V /\ R e. _V ) ) -> ( le ` O ) = ( le ` (/) ) )
40	28	str0 14524	. . . 4 |- (/) = ( le ` (/) )
41	39, 14, 40	3eqtr4g 2533	. . 3 |- ( ( ph /\ -. ( I e. _V /\ R e. _V ) ) -> .<_ = (/) )
42		reldmpsr 17781	. . . . . . . . . . 11 |- Rel dom mPwSer
43	42	ovprc 6309	. . . . . . . . . 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 2520	. . . . . . . 8 |- ( ( ph /\ -. ( I e. _V /\ R e. _V ) ) -> S = (/) )
46	45	fveq2d 5868	. . . . . . 7 |- ( ( ph /\ -. ( I e. _V /\ R e. _V ) ) -> ( Base ` S ) = ( Base ` (/) ) )
47		base0 14525	. . . . . . 7 |- (/) = ( Base ` (/) )
48	46, 3, 47	3eqtr4g 2533	. . . . . 6 |- ( ( ph /\ -. ( I e. _V /\ R e. _V ) ) -> B = (/) )
49	48	xpeq2d 5023	. . . . 5 |- ( ( ph /\ -. ( I e. _V /\ R e. _V ) ) -> ( B X. B ) = ( B X. (/) ) )
50		xp0 5423	. . . . 5 |- ( B X. (/) ) = (/)
51	49, 50	syl6eq 2524	. . . 4 |- ( ( ph /\ -. ( I e. _V /\ R e. _V ) ) -> ( B X. B ) = (/) )
52		sseq0 3817	. . . 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 2511	. 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 1379   e. wcel 1767   A.wral 2814   E.wrex 2815   {crab 2818   _Vcvv 3113   C_ wss 3476   (/)c0 3785   {cpr 4029   <.cop 4033   class class class wbr 4447   {copab 4504   X. cxp 4997   `'ccnv 4998   "cima 5002   ` cfv 5586  (class class class)co 6282   ^m cmap 7417   Fincfn 7513   NNcn 10532   NN0cn0 10791   ndxcnx 14483   sSet csts 14484   Basecbs 14486   lecple 14558   ltcplt 15424   mPwSer cmps 17771   <_bag cltb 17774   ordPwSer copws 17775

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-i2m1 9556  ax-1ne0 9557  ax-rrecex 9560  ax-cnre 9561

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-recs 7039  df-rdg 7073  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-ndx 14489  df-slot 14490  df-base 14491  df-sets 14492  df-ple 14571  df-psr 17776  df-opsr 17780

This theorem is referenced by:  opsrval2  17912  opsrtoslem1  17919