Theorem prdsle 15408

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 2461	. . 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 15403	. . 3 |- ( ph -> B = X_ x e. I ( Base ` ( R ` x ) ) )
8		eqid 2461	. . . 4 |- ( +g ` P ) = ( +g ` P )
9	1, 4, 5, 6, 3, 8	prdsplusg 15404	. . 3 |- ( ph -> ( +g ` P ) = ( f e. B , g e. B |-> ( x e. I |-> ( ( f ` x ) ( +g ` ( R ` x ) ) ( g ` x ) ) ) ) )
10		eqid 2461	. . . 4 |- ( .r ` P ) = ( .r ` P )
11	1, 4, 5, 6, 3, 10	prdsmulr 15405	. . 3 |- ( ph -> ( .r ` P ) = ( f e. B , g e. B |-> ( x e. I |-> ( ( f ` x ) ( .r ` ( R ` x ) ) ( g ` x ) ) ) ) )
12		eqid 2461	. . . 4 |- ( .s ` P ) = ( .s ` P )
13	1, 4, 5, 6, 3, 2, 12	prdsvsca 15406	. . 3 |- ( ph -> ( .s ` P ) = ( f e. ( Base ` S ) , g e. B |-> ( x e. I |-> ( f ( .s ` ( R ` x ) ) ( g ` x ) ) ) ) )
14		eqidd 2462	. . 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 2462	. . 3 |- ( ph -> ( Xt_ ` ( TopOpen o. R ) ) = ( Xt_ ` ( TopOpen o. R ) ) )
16		eqidd 2462	. . 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 2462	. . 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 2462	. . 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 2462	. . 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 15401	. 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 15340	. 2 |- le = Slot ( le ` ndx )
23		fvex 5897	. . . . . 6 |- ( Base ` P ) e. _V
24	6, 23	eqeltri 2535	. . . . 5 |- B e. _V
25	24, 24	xpex 6621	. . . 4 |- ( B X. B ) e. _V
26		vex 3059	. . . . . . . 8 |- f e. _V
27		vex 3059	. . . . . . . 8 |- g e. _V
28	26, 27	prss 4138	. . . . . . 7 |- ( ( f e. B /\ g e. B ) <-> { f , g } C_ B )
29	28	anbi1i 706	. . . . . 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 4480	. . . . 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 4927	. . . . 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 3474	. . . 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 4561	. . 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 4136	. . . 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 3608	. . . 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 3452	. . 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 3609	. . 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 3452	. 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 15400	1 |- ( ph -> .<_ = { <. f , g >. | ( { f , g } C_ B /\ A. x e. I ( f ` x ) ( le ` ( R ` x ) ) ( g ` x ) ) } )

Syntax hints:   -> wi 4   /\ wa 375   = wceq 1454   e. wcel 1897   A.wral 2748   _Vcvv 3056   u. cun 3413   C_ wss 3415   {csn 3979   {cpr 3981   {ctp 3983   <.cop 3985   class class class wbr 4415   {copab 4473   |-> cmpt 4474   X. cxp 4850   dom cdm 4852   ran crn 4853   o. ccom 4856   ` cfv 5600  (class class class)co 6314   |-> cmpt2 6316   1stc1st 6817   2ndc2nd 6818   X_cixp 7547   supcsup 7979   0cc0 9564   RR*cxr 9699   < clt 9700   ndxcnx 15166   Basecbs 15169   +g cplusg 15238   .rcmulr 15239  Scalarcsca 15241   .scvsca 15242   .icip 15243  TopSetcts 15244   lecple 15245   distcds 15247   Hom chom 15249  compcco 15250   TopOpenctopn 15368   Xt_cpt 15385   gsum_g cgsu 15387   X__scprds 15392

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-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-mulcom 9628  ax-addass 9629  ax-mulass 9630  ax-distr 9631  ax-i2m1 9632  ax-1ne0 9633  ax-1rid 9634  ax-rnegex 9635  ax-rrecex 9636  ax-cnre 9637  ax-pre-lttri 9638  ax-pre-lttrn 9639  ax-pre-ltadd 9640  ax-pre-mulgt0 9641

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-nel 2635  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-int 4248  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-riota 6276  df-ov 6317  df-oprab 6318  df-mpt2 6319  df-om 6719  df-1st 6819  df-2nd 6820  df-wrecs 7053  df-recs 7115  df-rdg 7153  df-1o 7207  df-oadd 7211  df-er 7388  df-map 7499  df-ixp 7548  df-en 7595  df-dom 7596  df-sdom 7597  df-fin 7598  df-sup 7981  df-pnf 9702  df-mnf 9703  df-xr 9704  df-ltxr 9705  df-le 9706  df-sub 9887  df-neg 9888  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-n0 10898  df-z 10966  df-dec 11080  df-uz 11188  df-fz 11813  df-struct 15171  df-ndx 15172  df-slot 15173  df-base 15174  df-plusg 15251  df-mulr 15252  df-sca 15254  df-vsca 15255  df-ip 15256  df-tset 15257  df-ple 15258  df-ds 15260  df-hom 15262  df-cco 15263  df-prds 15394

This theorem is referenced by:  prdsless  15409  prdsds  15410  prdstset  15412  prdshom  15413  prdsco  15414  prdsleval  15423  pwsle  15438