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Theorem ressuss 20743
Description: Value of the uniform structure of a restricted space. (Contributed by Thierry Arnoux, 12-Dec-2017.)
Assertion
Ref Expression
ressuss  |-  ( A  e.  V  ->  (UnifSt `  ( Ws  A ) )  =  ( (UnifSt `  W
)t  ( A  X.  A
) ) )

Proof of Theorem ressuss
StepHypRef Expression
1 eqid 2443 . . . . 5  |-  ( Base `  W )  =  (
Base `  W )
2 eqid 2443 . . . . 5  |-  ( UnifSet `  W )  =  (
UnifSet `  W )
31, 2ussval 20739 . . . 4  |-  ( (
UnifSet `  W )t  ( (
Base `  W )  X.  ( Base `  W
) ) )  =  (UnifSt `  W )
43oveq1i 6291 . . 3  |-  ( ( ( UnifSet `  W )t  (
( Base `  W )  X.  ( Base `  W
) ) )t  ( A  X.  A ) )  =  ( (UnifSt `  W )t  ( A  X.  A ) )
5 fvex 5866 . . . . 5  |-  ( UnifSet `  W )  e.  _V
65a1i 11 . . . 4  |-  ( A  e.  V  ->  ( UnifSet
`  W )  e. 
_V )
7 fvex 5866 . . . . . 6  |-  ( Base `  W )  e.  _V
87, 7xpex 6589 . . . . 5  |-  ( (
Base `  W )  X.  ( Base `  W
) )  e.  _V
98a1i 11 . . . 4  |-  ( A  e.  V  ->  (
( Base `  W )  X.  ( Base `  W
) )  e.  _V )
10 sqxpexg 6590 . . . 4  |-  ( A  e.  V  ->  ( A  X.  A )  e. 
_V )
11 restco 19642 . . . 4  |-  ( ( ( UnifSet `  W )  e.  _V  /\  ( (
Base `  W )  X.  ( Base `  W
) )  e.  _V  /\  ( A  X.  A
)  e.  _V )  ->  ( ( ( UnifSet `  W )t  ( ( Base `  W )  X.  ( Base `  W ) ) )t  ( A  X.  A
) )  =  ( ( UnifSet `  W )t  (
( ( Base `  W
)  X.  ( Base `  W ) )  i^i  ( A  X.  A
) ) ) )
126, 9, 10, 11syl3anc 1229 . . 3  |-  ( A  e.  V  ->  (
( ( UnifSet `  W
)t  ( ( Base `  W
)  X.  ( Base `  W ) ) )t  ( A  X.  A ) )  =  ( (
UnifSet `  W )t  ( ( ( Base `  W
)  X.  ( Base `  W ) )  i^i  ( A  X.  A
) ) ) )
134, 12syl5eqr 2498 . 2  |-  ( A  e.  V  ->  (
(UnifSt `  W )t  ( A  X.  A ) )  =  ( ( UnifSet `  W )t  ( ( (
Base `  W )  X.  ( Base `  W
) )  i^i  ( A  X.  A ) ) ) )
14 inxp 5125 . . . . 5  |-  ( ( ( Base `  W
)  X.  ( Base `  W ) )  i^i  ( A  X.  A
) )  =  ( ( ( Base `  W
)  i^i  A )  X.  ( ( Base `  W
)  i^i  A )
)
15 incom 3676 . . . . . . 7  |-  ( A  i^i  ( Base `  W
) )  =  ( ( Base `  W
)  i^i  A )
16 eqid 2443 . . . . . . . 8  |-  ( Ws  A )  =  ( Ws  A )
1716, 1ressbas 14668 . . . . . . 7  |-  ( A  e.  V  ->  ( A  i^i  ( Base `  W
) )  =  (
Base `  ( Ws  A
) ) )
1815, 17syl5eqr 2498 . . . . . 6  |-  ( A  e.  V  ->  (
( Base `  W )  i^i  A )  =  (
Base `  ( Ws  A
) ) )
1918sqxpeqd 5015 . . . . 5  |-  ( A  e.  V  ->  (
( ( Base `  W
)  i^i  A )  X.  ( ( Base `  W
)  i^i  A )
)  =  ( (
Base `  ( Ws  A
) )  X.  ( Base `  ( Ws  A ) ) ) )
2014, 19syl5eq 2496 . . . 4  |-  ( A  e.  V  ->  (
( ( Base `  W
)  X.  ( Base `  W ) )  i^i  ( A  X.  A
) )  =  ( ( Base `  ( Ws  A ) )  X.  ( Base `  ( Ws  A ) ) ) )
2120oveq2d 6297 . . 3  |-  ( A  e.  V  ->  (
( UnifSet `  W )t  (
( ( Base `  W
)  X.  ( Base `  W ) )  i^i  ( A  X.  A
) ) )  =  ( ( UnifSet `  W
)t  ( ( Base `  ( Ws  A ) )  X.  ( Base `  ( Ws  A ) ) ) ) )
2216, 2ressunif 20742 . . . 4  |-  ( A  e.  V  ->  ( UnifSet
`  W )  =  ( UnifSet `  ( Ws  A
) ) )
2322oveq1d 6296 . . 3  |-  ( A  e.  V  ->  (
( UnifSet `  W )t  (
( Base `  ( Ws  A
) )  X.  ( Base `  ( Ws  A ) ) ) )  =  ( ( UnifSet `  ( Ws  A ) )t  ( (
Base `  ( Ws  A
) )  X.  ( Base `  ( Ws  A ) ) ) ) )
24 eqid 2443 . . . . 5  |-  ( Base `  ( Ws  A ) )  =  ( Base `  ( Ws  A ) )
25 eqid 2443 . . . . 5  |-  ( UnifSet `  ( Ws  A ) )  =  ( UnifSet `  ( Ws  A
) )
2624, 25ussval 20739 . . . 4  |-  ( (
UnifSet `  ( Ws  A ) )t  ( ( Base `  ( Ws  A ) )  X.  ( Base `  ( Ws  A ) ) ) )  =  (UnifSt `  ( Ws  A ) )
2726a1i 11 . . 3  |-  ( A  e.  V  ->  (
( UnifSet `  ( Ws  A
) )t  ( ( Base `  ( Ws  A ) )  X.  ( Base `  ( Ws  A ) ) ) )  =  (UnifSt `  ( Ws  A ) ) )
2821, 23, 273eqtrd 2488 . 2  |-  ( A  e.  V  ->  (
( UnifSet `  W )t  (
( ( Base `  W
)  X.  ( Base `  W ) )  i^i  ( A  X.  A
) ) )  =  (UnifSt `  ( Ws  A
) ) )
2913, 28eqtr2d 2485 1  |-  ( A  e.  V  ->  (UnifSt `  ( Ws  A ) )  =  ( (UnifSt `  W
)t  ( A  X.  A
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1383    e. wcel 1804   _Vcvv 3095    i^i cin 3460    X. cxp 4987   ` cfv 5578  (class class class)co 6281   Basecbs 14613   ↾s cress 14614   UnifSetcunif 14688   ↾t crest 14799  UnifStcuss 20733
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10544  df-2 10601  df-3 10602  df-4 10603  df-5 10604  df-6 10605  df-7 10606  df-8 10607  df-9 10608  df-10 10609  df-n0 10803  df-z 10872  df-dec 10986  df-ndx 14616  df-slot 14617  df-base 14618  df-sets 14619  df-ress 14620  df-unif 14701  df-rest 14801  df-uss 20736
This theorem is referenced by:  ressust  20744  ressusp  20745  ucnextcn  20784  reust  21790  qqhucn  27950
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