Theorem prdsle 14878

Description: Structure product weak ordering. (Contributed by Mario Carneiro, 15-Aug-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.)

Hypotheses

Ref	Expression
prdsbas.p	|- P = ( S X_s R )
prdsbas.s	|- ( ph -> S e. V )
prdsbas.r	|- ( ph -> R e. W )
prdsbas.b	|- B = ( Base ` P )
prdsbas.i	|- ( ph -> dom R = I )
prdsle.l	|- .<_ = ( le ` P )

Assertion

Ref	Expression
prdsle	|- ( ph -> .<_ = { <. f , g >. | ( { f , g } C_ B /\ A. x e. I ( f ` x ) ( le ` ( R ` x ) ) ( g ` x ) ) } )

Distinct variable groups:   f, g, x, B   ph, f, g, x   f, I, g, x   P, f, g, x   R, f, g, x   S, f, g, x

Allowed substitution hints:   .<_ ( x, f, g)   V( x, f, g)   W( x, f, g)

Proof of Theorem prdsle

Dummy variables a c d e are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		prdsbas.p	. . 3 |- P = ( S X_s R )
2		eqid 2457	. . 3 |- ( Base ` S ) = ( Base ` S )
3		prdsbas.i	. . 3 |- ( ph -> dom R = I )
4		prdsbas.s	. . . 4 |- ( ph -> S e. V )
5		prdsbas.r	. . . 4 |- ( ph -> R e. W )
6		prdsbas.b	. . . 4 |- B = ( Base ` P )
7	1, 4, 5, 6, 3	prdsbas 14873	. . 3 |- ( ph -> B = X_ x e. I ( Base ` ( R ` x ) ) )
8		eqid 2457	. . . 4 |- ( +g ` P ) = ( +g ` P )
9	1, 4, 5, 6, 3, 8	prdsplusg 14874	. . 3 |- ( ph -> ( +g ` P ) = ( f e. B , g e. B |-> ( x e. I |-> ( ( f ` x ) ( +g ` ( R ` x ) ) ( g ` x ) ) ) ) )
10		eqid 2457	. . . 4 |- ( .r ` P ) = ( .r ` P )
11	1, 4, 5, 6, 3, 10	prdsmulr 14875	. . 3 |- ( ph -> ( .r ` P ) = ( f e. B , g e. B |-> ( x e. I |-> ( ( f ` x ) ( .r ` ( R ` x ) ) ( g ` x ) ) ) ) )
12		eqid 2457	. . . 4 |- ( .s ` P ) = ( .s ` P )
13	1, 4, 5, 6, 3, 2, 12	prdsvsca 14876	. . 3 |- ( ph -> ( .s ` P ) = ( f e. ( Base ` S ) , g e. B |-> ( x e. I |-> ( f ( .s ` ( R ` x ) ) ( g ` x ) ) ) ) )
14		eqidd 2458	. . 3 |- ( ph -> ( f e. B , g e. B |-> ( S gsumg ( x e. I |-> ( ( f ` x ) ( .i ` ( R ` x ) ) ( g ` x ) ) ) ) ) = ( f e. B , g e. B |-> ( S gsumg ( x e. I |-> ( ( f ` x ) ( .i ` ( R ` x ) ) ( g ` x ) ) ) ) ) )
15		eqidd 2458	. . 3 |- ( ph -> ( Xt_ ` ( TopOpen o. R ) ) = ( Xt_ ` ( TopOpen o. R ) ) )
16		eqidd 2458	. . 3 |- ( ph -> { <. f , g >. | ( { f , g } C_ B /\ A. x e. I ( f ` x ) ( le ` ( R ` x ) ) ( g ` x ) ) } = { <. f , g >. | ( { f , g } C_ B /\ A. x e. I ( f ` x ) ( le ` ( R ` x ) ) ( g ` x ) ) } )
17		eqidd 2458	. . 3 |- ( ph -> ( f e. B , g e. B |-> sup ( ( ran ( x e. I |-> ( ( f ` x ) ( dist ` ( R ` x ) ) ( g ` x ) ) ) u. { 0 } ) , RR* , < ) ) = ( f e. B , g e. B |-> sup ( ( ran ( x e. I |-> ( ( f ` x ) ( dist ` ( R ` x ) ) ( g ` x ) ) ) u. { 0 } ) , RR* , < ) ) )
18		eqidd 2458	. . 3 |- ( ph -> ( f e. B , g e. B |-> X_ x e. I ( ( f ` x ) ( Hom ` ( R ` x ) ) ( g ` x ) ) ) = ( f e. B , g e. B |-> X_ x e. I ( ( f ` x ) ( Hom ` ( R ` x ) ) ( g ` x ) ) ) )
19		eqidd 2458	. . 3 |- ( ph -> ( a e. ( B X. B ) , c e. B |-> ( d e. ( c ( f e. B , g e. B |-> X_ x e. I ( ( f ` x ) ( Hom ` ( R ` x ) ) ( g ` x ) ) ) ( 2nd ` a ) ) , e e. ( ( f e. B , g e. B |-> X_ x e. I ( ( f ` x ) ( Hom ` ( R ` x ) ) ( g ` x ) ) ) ` a ) |-> ( x e. I |-> ( ( d ` x ) ( <. ( ( 1st ` a ) ` x ) , ( ( 2nd ` a ) ` x ) >. (comp ` ( R ` x ) ) ( c ` x ) ) ( e ` x ) ) ) ) ) = ( a e. ( B X. B ) , c e. B |-> ( d e. ( c ( f e. B , g e. B |-> X_ x e. I ( ( f ` x ) ( Hom ` ( R ` x ) ) ( g ` x ) ) ) ( 2nd ` a ) ) , e e. ( ( f e. B , g e. B |-> X_ x e. I ( ( f ` x ) ( Hom ` ( R ` x ) ) ( g ` x ) ) ) ` a ) |-> ( x e. I |-> ( ( d ` x ) ( <. ( ( 1st ` a ) ` x ) , ( ( 2nd ` a ) ` x ) >. (comp ` ( R ` x ) ) ( c ` x ) ) ( e ` x ) ) ) ) ) )
20	1, 2, 3, 7, 9, 11, 13, 14, 15, 16, 17, 18, 19, 4, 5	prdsval 14871	. 2 |- ( ph -> P = ( ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , ( +g ` P ) >. , <. ( .r ` ndx ) , ( .r ` P ) >. } u. { <. (Scalar ` ndx ) , S >. , <. ( .s ` ndx ) , ( .s ` P ) >. , <. ( .i ` ndx ) , ( f e. B , g e. B |-> ( S gsumg ( x e. I |-> ( ( f ` x ) ( .i ` ( R ` x ) ) ( g ` x ) ) ) ) ) >. } ) u. ( { <. (TopSet ` ndx ) , ( Xt_ ` ( TopOpen o. R ) ) >. , <. ( le ` ndx ) , { <. f , g >. | ( { f , g } C_ B /\ A. x e. I ( f ` x ) ( le ` ( R ` x ) ) ( g ` x ) ) } >. , <. ( dist ` ndx ) , ( f e. B , g e. B |-> sup ( ( ran ( x e. I |-> ( ( f ` x ) ( dist ` ( R ` x ) ) ( g ` x ) ) ) u. { 0 } ) , RR* , < ) ) >. } u. { <. ( Hom ` ndx ) , ( f e. B , g e. B |-> X_ x e. I ( ( f ` x ) ( Hom ` ( R ` x ) ) ( g ` x ) ) ) >. , <. (comp ` ndx ) , ( a e. ( B X. B ) , c e. B |-> ( d e. ( c ( f e. B , g e. B |-> X_ x e. I ( ( f ` x ) ( Hom ` ( R ` x ) ) ( g ` x ) ) ) ( 2nd ` a ) ) , e e. ( ( f e. B , g e. B |-> X_ x e. I ( ( f ` x ) ( Hom ` ( R ` x ) ) ( g ` x ) ) ) ` a ) |-> ( x e. I |-> ( ( d ` x ) ( <. ( ( 1st ` a ) ` x ) , ( ( 2nd ` a ) ` x ) >. (comp ` ( R ` x ) ) ( c ` x ) ) ( e ` x ) ) ) ) ) >. } ) ) )
21		prdsle.l	. 2 |- .<_ = ( le ` P )
22		pleid 14810	. 2 |- le = Slot ( le ` ndx )
23		fvex 5882	. . . . . 6 |- ( Base ` P ) e. _V
24	6, 23	eqeltri 2541	. . . . 5 |- B e. _V
25	24, 24	xpex 6603	. . . 4 |- ( B X. B ) e. _V
26		vex 3112	. . . . . . . 8 |- f e. _V
27		vex 3112	. . . . . . . 8 |- g e. _V
28	26, 27	prss 4186	. . . . . . 7 |- ( ( f e. B /\ g e. B ) <-> { f , g } C_ B )
29	28	anbi1i 695	. . . . . 6 |- ( ( ( f e. B /\ g e. B ) /\ A. x e. I ( f ` x ) ( le ` ( R ` x ) ) ( g ` x ) ) <-> ( { f , g } C_ B /\ A. x e. I ( f ` x ) ( le ` ( R ` x ) ) ( g ` x ) ) )
30	29	opabbii 4521	. . . . 5 |- { <. f , g >. | ( ( f e. B /\ g e. B ) /\ A. x e. I ( f ` x ) ( le ` ( R ` x ) ) ( g ` x ) ) } = { <. f , g >. | ( { f , g } C_ B /\ A. x e. I ( f ` x ) ( le ` ( R ` x ) ) ( g ` x ) ) }
31		opabssxp 5083	. . . . 5 |- { <. f , g >. | ( ( f e. B /\ g e. B ) /\ A. x e. I ( f ` x ) ( le ` ( R ` x ) ) ( g ` x ) ) } C_ ( B X. B )
32	30, 31	eqsstr3i 3530	. . . 4 |- { <. f , g >. | ( { f , g } C_ B /\ A. x e. I ( f ` x ) ( le ` ( R ` x ) ) ( g ` x ) ) } C_ ( B X. B )
33	25, 32	ssexi 4601	. . 3 |- { <. f , g >. | ( { f , g } C_ B /\ A. x e. I ( f ` x ) ( le ` ( R ` x ) ) ( g ` x ) ) } e. _V
34	33	a1i 11	. 2 |- ( ph -> { <. f , g >. | ( { f , g } C_ B /\ A. x e. I ( f ` x ) ( le ` ( R ` x ) ) ( g ` x ) ) } e. _V )
35		snsstp2 4184	. . . 4 |- { <. ( le ` ndx ) , { <. f , g >. | ( { f , g } C_ B /\ A. x e. I ( f ` x ) ( le ` ( R ` x ) ) ( g ` x ) ) } >. } C_ { <. (TopSet ` ndx ) , ( Xt_ ` ( TopOpen o. R ) ) >. , <. ( le ` ndx ) , { <. f , g >. | ( { f , g } C_ B /\ A. x e. I ( f ` x ) ( le ` ( R ` x ) ) ( g ` x ) ) } >. , <. ( dist ` ndx ) , ( f e. B , g e. B |-> sup ( ( ran ( x e. I |-> ( ( f ` x ) ( dist ` ( R ` x ) ) ( g ` x ) ) ) u. { 0 } ) , RR* , < ) ) >. }
36		ssun1 3663	. . . 4 |- { <. (TopSet ` ndx ) , ( Xt_ ` ( TopOpen o. R ) ) >. , <. ( le ` ndx ) , { <. f , g >. | ( { f , g } C_ B /\ A. x e. I ( f ` x ) ( le ` ( R ` x ) ) ( g ` x ) ) } >. , <. ( dist ` ndx ) , ( f e. B , g e. B |-> sup ( ( ran ( x e. I |-> ( ( f ` x ) ( dist ` ( R ` x ) ) ( g ` x ) ) ) u. { 0 } ) , RR* , < ) ) >. } C_ ( { <. (TopSet ` ndx ) , ( Xt_ ` ( TopOpen o. R ) ) >. , <. ( le ` ndx ) , { <. f , g >. | ( { f , g } C_ B /\ A. x e. I ( f ` x ) ( le ` ( R ` x ) ) ( g ` x ) ) } >. , <. ( dist ` ndx ) , ( f e. B , g e. B |-> sup ( ( ran ( x e. I |-> ( ( f ` x ) ( dist ` ( R ` x ) ) ( g ` x ) ) ) u. { 0 } ) , RR* , < ) ) >. } u. { <. ( Hom ` ndx ) , ( f e. B , g e. B |-> X_ x e. I ( ( f ` x ) ( Hom ` ( R ` x ) ) ( g ` x ) ) ) >. , <. (comp ` ndx ) , ( a e. ( B X. B ) , c e. B |-> ( d e. ( c ( f e. B , g e. B |-> X_ x e. I ( ( f ` x ) ( Hom ` ( R ` x ) ) ( g ` x ) ) ) ( 2nd ` a ) ) , e e. ( ( f e. B , g e. B |-> X_ x e. I ( ( f ` x ) ( Hom ` ( R ` x ) ) ( g ` x ) ) ) ` a ) |-> ( x e. I |-> ( ( d ` x ) ( <. ( ( 1st ` a ) ` x ) , ( ( 2nd ` a ) ` x ) >. (comp ` ( R ` x ) ) ( c ` x ) ) ( e ` x ) ) ) ) ) >. } )
37	35, 36	sstri 3508	. . 3 |- { <. ( le ` ndx ) , { <. f , g >. | ( { f , g } C_ B /\ A. x e. I ( f ` x ) ( le ` ( R ` x ) ) ( g ` x ) ) } >. } C_ ( { <. (TopSet ` ndx ) , ( Xt_ ` ( TopOpen o. R ) ) >. , <. ( le ` ndx ) , { <. f , g >. | ( { f , g } C_ B /\ A. x e. I ( f ` x ) ( le ` ( R ` x ) ) ( g ` x ) ) } >. , <. ( dist ` ndx ) , ( f e. B , g e. B |-> sup ( ( ran ( x e. I |-> ( ( f ` x ) ( dist ` ( R ` x ) ) ( g ` x ) ) ) u. { 0 } ) , RR* , < ) ) >. } u. { <. ( Hom ` ndx ) , ( f e. B , g e. B |-> X_ x e. I ( ( f ` x ) ( Hom ` ( R ` x ) ) ( g ` x ) ) ) >. , <. (comp ` ndx ) , ( a e. ( B X. B ) , c e. B |-> ( d e. ( c ( f e. B , g e. B |-> X_ x e. I ( ( f ` x ) ( Hom ` ( R ` x ) ) ( g ` x ) ) ) ( 2nd ` a ) ) , e e. ( ( f e. B , g e. B |-> X_ x e. I ( ( f ` x ) ( Hom ` ( R ` x ) ) ( g ` x ) ) ) ` a ) |-> ( x e. I |-> ( ( d ` x ) ( <. ( ( 1st ` a ) ` x ) , ( ( 2nd ` a ) ` x ) >. (comp ` ( R ` x ) ) ( c ` x ) ) ( e ` x ) ) ) ) ) >. } )
38		ssun2 3664	. . 3 |- ( { <. (TopSet ` ndx ) , ( Xt_ ` ( TopOpen o. R ) ) >. , <. ( le ` ndx ) , { <. f , g >. | ( { f , g } C_ B /\ A. x e. I ( f ` x ) ( le ` ( R ` x ) ) ( g ` x ) ) } >. , <. ( dist ` ndx ) , ( f e. B , g e. B |-> sup ( ( ran ( x e. I |-> ( ( f ` x ) ( dist ` ( R ` x ) ) ( g ` x ) ) ) u. { 0 } ) , RR* , < ) ) >. } u. { <. ( Hom ` ndx ) , ( f e. B , g e. B |-> X_ x e. I ( ( f ` x ) ( Hom ` ( R ` x ) ) ( g ` x ) ) ) >. , <. (comp ` ndx ) , ( a e. ( B X. B ) , c e. B |-> ( d e. ( c ( f e. B , g e. B |-> X_ x e. I ( ( f ` x ) ( Hom ` ( R ` x ) ) ( g ` x ) ) ) ( 2nd ` a ) ) , e e. ( ( f e. B , g e. B |-> X_ x e. I ( ( f ` x ) ( Hom ` ( R ` x ) ) ( g ` x ) ) ) ` a ) |-> ( x e. I |-> ( ( d ` x ) ( <. ( ( 1st ` a ) ` x ) , ( ( 2nd ` a ) ` x ) >. (comp ` ( R ` x ) ) ( c ` x ) ) ( e ` x ) ) ) ) ) >. } ) C_ ( ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , ( +g ` P ) >. , <. ( .r ` ndx ) , ( .r ` P ) >. } u. { <. (Scalar ` ndx ) , S >. , <. ( .s ` ndx ) , ( .s ` P ) >. , <. ( .i ` ndx ) , ( f e. B , g e. B |-> ( S gsumg ( x e. I |-> ( ( f ` x ) ( .i ` ( R ` x ) ) ( g ` x ) ) ) ) ) >. } ) u. ( { <. (TopSet ` ndx ) , ( Xt_ ` ( TopOpen o. R ) ) >. , <. ( le ` ndx ) , { <. f , g >. | ( { f , g } C_ B /\ A. x e. I ( f ` x ) ( le ` ( R ` x ) ) ( g ` x ) ) } >. , <. ( dist ` ndx ) , ( f e. B , g e. B |-> sup ( ( ran ( x e. I |-> ( ( f ` x ) ( dist ` ( R ` x ) ) ( g ` x ) ) ) u. { 0 } ) , RR* , < ) ) >. } u. { <. ( Hom ` ndx ) , ( f e. B , g e. B |-> X_ x e. I ( ( f ` x ) ( Hom ` ( R ` x ) ) ( g ` x ) ) ) >. , <. (comp ` ndx ) , ( a e. ( B X. B ) , c e. B |-> ( d e. ( c ( f e. B , g e. B |-> X_ x e. I ( ( f ` x ) ( Hom ` ( R ` x ) ) ( g ` x ) ) ) ( 2nd ` a ) ) , e e. ( ( f e. B , g e. B |-> X_ x e. I ( ( f ` x ) ( Hom ` ( R ` x ) ) ( g ` x ) ) ) ` a ) |-> ( x e. I |-> ( ( d ` x ) ( <. ( ( 1st ` a ) ` x ) , ( ( 2nd ` a ) ` x ) >. (comp ` ( R ` x ) ) ( c ` x ) ) ( e ` x ) ) ) ) ) >. } ) )
39	37, 38	sstri 3508	. 2 |- { <. ( le ` ndx ) , { <. f , g >. | ( { f , g } C_ B /\ A. x e. I ( f ` x ) ( le ` ( R ` x ) ) ( g ` x ) ) } >. } C_ ( ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , ( +g ` P ) >. , <. ( .r ` ndx ) , ( .r ` P ) >. } u. { <. (Scalar ` ndx ) , S >. , <. ( .s ` ndx ) , ( .s ` P ) >. , <. ( .i ` ndx ) , ( f e. B , g e. B |-> ( S gsumg ( x e. I |-> ( ( f ` x ) ( .i ` ( R ` x ) ) ( g ` x ) ) ) ) ) >. } ) u. ( { <. (TopSet ` ndx ) , ( Xt_ ` ( TopOpen o. R ) ) >. , <. ( le ` ndx ) , { <. f , g >. | ( { f , g } C_ B /\ A. x e. I ( f ` x ) ( le ` ( R ` x ) ) ( g ` x ) ) } >. , <. ( dist ` ndx ) , ( f e. B , g e. B |-> sup ( ( ran ( x e. I |-> ( ( f ` x ) ( dist ` ( R ` x ) ) ( g ` x ) ) ) u. { 0 } ) , RR* , < ) ) >. } u. { <. ( Hom ` ndx ) , ( f e. B , g e. B |-> X_ x e. I ( ( f ` x ) ( Hom ` ( R ` x ) ) ( g ` x ) ) ) >. , <. (comp ` ndx ) , ( a e. ( B X. B ) , c e. B |-> ( d e. ( c ( f e. B , g e. B |-> X_ x e. I ( ( f ` x ) ( Hom ` ( R ` x ) ) ( g ` x ) ) ) ( 2nd ` a ) ) , e e. ( ( f e. B , g e. B |-> X_ x e. I ( ( f ` x ) ( Hom ` ( R ` x ) ) ( g ` x ) ) ) ` a ) |-> ( x e. I |-> ( ( d ` x ) ( <. ( ( 1st ` a ) ` x ) , ( ( 2nd ` a ) ` x ) >. (comp ` ( R ` x ) ) ( c ` x ) ) ( e ` x ) ) ) ) ) >. } ) )
40	20, 21, 22, 34, 39	prdsvallem 14870	1 |- ( ph -> .<_ = { <. f , g >. | ( { f , g } C_ B /\ A. x e. I ( f ` x ) ( le ` ( R ` x ) ) ( g ` x ) ) } )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1395   e. wcel 1819   A.wral 2807   _Vcvv 3109   u. cun 3469   C_ wss 3471   {csn 4032   {cpr 4034   {ctp 4036   <.cop 4038   class class class wbr 4456   {copab 4514   |-> cmpt 4515   X. cxp 5006   dom cdm 5008   ran crn 5009   o. ccom 5012   ` cfv 5594  (class class class)co 6296   |-> cmpt2 6298   1stc1st 6797   2ndc2nd 6798   X_cixp 7488   supcsup 7918   0cc0 9509   RR*cxr 9644   < clt 9645   ndxcnx 14640   Basecbs 14643   +g cplusg 14711   .rcmulr 14712  Scalarcsca 14714   .scvsca 14715   .icip 14716  TopSetcts 14717   lecple 14718   distcds 14720   Hom chom 14722  compcco 14723   TopOpenctopn 14838   Xt_cpt 14855   gsum_g cgsu 14857   X__scprds 14862

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-ixp 7489  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-sup 7919  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-10 10623  df-n0 10817  df-z 10886  df-dec 11001  df-uz 11107  df-fz 11698  df-struct 14645  df-ndx 14646  df-slot 14647  df-base 14648  df-plusg 14724  df-mulr 14725  df-sca 14727  df-vsca 14728  df-ip 14729  df-tset 14730  df-ple 14731  df-ds 14733  df-hom 14735  df-cco 14736  df-prds 14864

This theorem is referenced by:  prdsless  14879  prdsds  14880  prdstset  14882  prdshom  14883  prdsco  14884  prdsleval  14893  pwsle  14908