Theorem opsrle 19475

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 2622	. . . . 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 794	. . . . 5 |- ( ( ph /\ ( I e. _V /\ R e. _V ) ) -> I e. _V )
9		simprr 796	. . . . 5 |- ( ( ph /\ ( I e. _V /\ R e. _V ) ) -> R e. _V )
10		opsrle.t	. . . . . 6 |- ( ph -> T C_ ( I X. I ) )
11	10	adantr 481	. . . . 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 19474	. . . 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 6195	. . 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 6678	. . . . 5 |- ( I mPwSer R ) e. _V
16	1, 15	eqeltri 2697	. . . 4 |- S e. _V
17		fvex 6201	. . . . . . 7 |- ( Base ` S ) e. _V
18	3, 17	eqeltri 2697	. . . . . 6 |- B e. _V
19	18, 18	xpex 6962	. . . . 5 |- ( B X. B ) e. _V
20		vex 3203	. . . . . . . . 9 |- x e. _V
21		vex 3203	. . . . . . . . 9 |- y e. _V
22	20, 21	prss 4351	. . . . . . . 8 |- ( ( x e. B /\ y e. B ) <-> { x , y } C_ B )
23	22	anbi1i 731	. . . . . . 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 4717	. . . . . 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 5193	. . . . . 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 3636	. . . . 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 4803	. . . 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 16049	. . . . 5 |- le = Slot ( le ` ndx )
29	28	setsid 15914	. . . 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 708	. . 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 2681	. 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 19473	. . . . . . . . . 10 |- Rel dom ordPwSer
33	32	ovprc 6683	. . . . . . . . 9 |- ( -. ( I e. _V /\ R e. _V ) -> ( I ordPwSer R ) = (/) )
34	33	adantl 482	. . . . . . . 8 |- ( ( ph /\ -. ( I e. _V /\ R e. _V ) ) -> ( I ordPwSer R ) = (/) )
35	34	fveq1d 6193	. . . . . . 7 |- ( ( ph /\ -. ( I e. _V /\ R e. _V ) ) -> ( ( I ordPwSer R ) ` T ) = ( (/) ` T ) )
36	2, 35	syl5eq 2668	. . . . . 6 |- ( ( ph /\ -. ( I e. _V /\ R e. _V ) ) -> O = ( (/) ` T ) )
37		0fv 6227	. . . . . 6 |- ( (/) ` T ) = (/)
38	36, 37	syl6eq 2672	. . . . 5 |- ( ( ph /\ -. ( I e. _V /\ R e. _V ) ) -> O = (/) )
39	38	fveq2d 6195	. . . 4 |- ( ( ph /\ -. ( I e. _V /\ R e. _V ) ) -> ( le ` O ) = ( le ` (/) ) )
40	28	str0 15911	. . . 4 |- (/) = ( le ` (/) )
41	39, 14, 40	3eqtr4g 2681	. . 3 |- ( ( ph /\ -. ( I e. _V /\ R e. _V ) ) -> .<_ = (/) )
42		reldmpsr 19361	. . . . . . . . . . 11 |- Rel dom mPwSer
43	42	ovprc 6683	. . . . . . . . . 10 |- ( -. ( I e. _V /\ R e. _V ) -> ( I mPwSer R ) = (/) )
44	43	adantl 482	. . . . . . . . 9 |- ( ( ph /\ -. ( I e. _V /\ R e. _V ) ) -> ( I mPwSer R ) = (/) )
45	1, 44	syl5eq 2668	. . . . . . . 8 |- ( ( ph /\ -. ( I e. _V /\ R e. _V ) ) -> S = (/) )
46	45	fveq2d 6195	. . . . . . 7 |- ( ( ph /\ -. ( I e. _V /\ R e. _V ) ) -> ( Base ` S ) = ( Base ` (/) ) )
47		base0 15912	. . . . . . 7 |- (/) = ( Base ` (/) )
48	46, 3, 47	3eqtr4g 2681	. . . . . 6 |- ( ( ph /\ -. ( I e. _V /\ R e. _V ) ) -> B = (/) )
49	48	xpeq2d 5139	. . . . 5 |- ( ( ph /\ -. ( I e. _V /\ R e. _V ) ) -> ( B X. B ) = ( B X. (/) ) )
50		xp0 5552	. . . . 5 |- ( B X. (/) ) = (/)
51	49, 50	syl6eq 2672	. . . 4 |- ( ( ph /\ -. ( I e. _V /\ R e. _V ) ) -> ( B X. B ) = (/) )
52		sseq0 3975	. . . 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 695	. . 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 2659	. 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 832	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 383   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   {crab 2916   _Vcvv 3200   C_ wss 3574   (/)c0 3915   {cpr 4179   <.cop 4183   class class class wbr 4653   {copab 4712   X. cxp 5112   `'ccnv 5113   "cima 5117   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   Fincfn 7955   NNcn 11020   NN0cn0 11292   ndxcnx 15854   sSet csts 15855   Basecbs 15857   lecple 15948   ltcplt 16941   mPwSer cmps 19351   <_bag cltb 19354   ordPwSer copws 19355

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  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-nel 2898  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-ltxr 10079  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-dec 11494  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ple 15961  df-psr 19356  df-opsr 19360

This theorem is referenced by:  opsrval2  19476  opsrtoslem1  19484