Theorem prdstset 16126

Description: Structure product topology. (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 )
prdstset.l	|- O = (TopSet ` P )

Assertion

Ref	Expression
prdstset	|- ( ph -> O = ( Xt_ ` ( TopOpen o. R ) ) )

Proof of Theorem prdstset

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

Step	Hyp	Ref	Expression
1		prdsbas.p	. . 3 |- P = ( S X_s R )
2		eqid 2622	. . 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 16117	. . 3 |- ( ph -> B = X_ x e. I ( Base ` ( R ` x ) ) )
8		eqid 2622	. . . 4 |- ( +g ` P ) = ( +g ` P )
9	1, 4, 5, 6, 3, 8	prdsplusg 16118	. . 3 |- ( ph -> ( +g ` P ) = ( f e. B , g e. B |-> ( x e. I |-> ( ( f ` x ) ( +g ` ( R ` x ) ) ( g ` x ) ) ) ) )
10		eqid 2622	. . . 4 |- ( .r ` P ) = ( .r ` P )
11	1, 4, 5, 6, 3, 10	prdsmulr 16119	. . 3 |- ( ph -> ( .r ` P ) = ( f e. B , g e. B |-> ( x e. I |-> ( ( f ` x ) ( .r ` ( R ` x ) ) ( g ` x ) ) ) ) )
12		eqid 2622	. . . 4 |- ( .s ` P ) = ( .s ` P )
13	1, 4, 5, 6, 3, 2, 12	prdsvsca 16120	. . 3 |- ( ph -> ( .s ` P ) = ( f e. ( Base ` S ) , g e. B |-> ( x e. I |-> ( f ( .s ` ( R ` x ) ) ( g ` x ) ) ) ) )
14		eqidd 2623	. . 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 2623	. . 3 |- ( ph -> ( Xt_ ` ( TopOpen o. R ) ) = ( Xt_ ` ( TopOpen o. R ) ) )
16		eqid 2622	. . . 4 |- ( le ` P ) = ( le ` P )
17	1, 4, 5, 6, 3, 16	prdsle 16122	. . 3 |- ( ph -> ( le ` P ) = { <. f , g >. | ( { f , g } C_ B /\ A. x e. I ( f ` x ) ( le ` ( R ` x ) ) ( g ` x ) ) } )
18		eqid 2622	. . . 4 |- ( dist ` P ) = ( dist ` P )
19	1, 4, 5, 6, 3, 18	prdsds 16124	. . 3 |- ( ph -> ( dist ` P ) = ( f e. B , g e. B |-> sup ( ( ran ( x e. I |-> ( ( f ` x ) ( dist ` ( R ` x ) ) ( g ` x ) ) ) u. { 0 } ) , RR* , < ) ) )
20		eqidd 2623	. . 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 ) ) ) )
21		eqidd 2623	. . 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 ) ) ) ) ) )
22	1, 2, 3, 7, 9, 11, 13, 14, 15, 17, 19, 20, 21, 4, 5	prdsval 16115	. 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 ) , ( le ` P ) >. , <. ( dist ` ndx ) , ( dist ` P ) >. } 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 ) ) ) ) ) >. } ) ) )
23		prdstset.l	. 2 |- O = (TopSet ` P )
24		tsetid 16041	. 2 |- TopSet = Slot (TopSet ` ndx )
25		fvexd 6203	. 2 |- ( ph -> ( Xt_ ` ( TopOpen o. R ) ) e. _V )
26		snsstp1 4347	. . . 4 |- { <. (TopSet ` ndx ) , ( Xt_ ` ( TopOpen o. R ) ) >. } C_ { <. (TopSet ` ndx ) , ( Xt_ ` ( TopOpen o. R ) ) >. , <. ( le ` ndx ) , ( le ` P ) >. , <. ( dist ` ndx ) , ( dist ` P ) >. }
27		ssun1 3776	. . . 4 |- { <. (TopSet ` ndx ) , ( Xt_ ` ( TopOpen o. R ) ) >. , <. ( le ` ndx ) , ( le ` P ) >. , <. ( dist ` ndx ) , ( dist ` P ) >. } C_ ( { <. (TopSet ` ndx ) , ( Xt_ ` ( TopOpen o. R ) ) >. , <. ( le ` ndx ) , ( le ` P ) >. , <. ( dist ` ndx ) , ( dist ` P ) >. } 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 ) ) ) ) ) >. } )
28	26, 27	sstri 3612	. . 3 |- { <. (TopSet ` ndx ) , ( Xt_ ` ( TopOpen o. R ) ) >. } C_ ( { <. (TopSet ` ndx ) , ( Xt_ ` ( TopOpen o. R ) ) >. , <. ( le ` ndx ) , ( le ` P ) >. , <. ( dist ` ndx ) , ( dist ` P ) >. } 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 ) ) ) ) ) >. } )
29		ssun2 3777	. . 3 |- ( { <. (TopSet ` ndx ) , ( Xt_ ` ( TopOpen o. R ) ) >. , <. ( le ` ndx ) , ( le ` P ) >. , <. ( dist ` ndx ) , ( dist ` P ) >. } 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 ) , ( le ` P ) >. , <. ( dist ` ndx ) , ( dist ` P ) >. } 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 ) ) ) ) ) >. } ) )
30	28, 29	sstri 3612	. 2 |- { <. (TopSet ` ndx ) , ( Xt_ ` ( TopOpen o. R ) ) >. } 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 ) , ( le ` P ) >. , <. ( dist ` ndx ) , ( dist ` P ) >. } 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 ) ) ) ) ) >. } ) )
31	22, 23, 24, 25, 30	prdsvallem 16114	1 |- ( ph -> O = ( Xt_ ` ( TopOpen o. R ) ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   u. cun 3572   {csn 4177   {cpr 4179   {ctp 4181   <.cop 4183   |-> cmpt 4729   X. cxp 5112   dom cdm 5114   o. ccom 5118   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   1stc1st 7166   2ndc2nd 7167   X_cixp 7908   ndxcnx 15854   Basecbs 15857   +g cplusg 15941   .rcmulr 15942  Scalarcsca 15944   .scvsca 15945   .icip 15946  TopSetcts 15947   lecple 15948   distcds 15950   Hom chom 15952  compcco 15953   TopOpenctopn 16082   Xt_cpt 16099   gsum_g cgsu 16101   X__scprds 16106

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-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  ax-pre-mulgt0 10013

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-int 4476  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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  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-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-plusg 15954  df-mulr 15955  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-hom 15966  df-cco 15967  df-prds 16108

This theorem is referenced by:  prdshom  16127  prdsco  16128  prdstopn  21431  prdstps  21432