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Theorem ressuss 20494
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 2460 . . . . 5  |-  ( Base `  W )  =  (
Base `  W )
2 eqid 2460 . . . . 5  |-  ( UnifSet `  W )  =  (
UnifSet `  W )
31, 2ussval 20490 . . . 4  |-  ( (
UnifSet `  W )t  ( (
Base `  W )  X.  ( Base `  W
) ) )  =  (UnifSt `  W )
43oveq1i 6285 . . 3  |-  ( ( ( UnifSet `  W )t  (
( Base `  W )  X.  ( Base `  W
) ) )t  ( A  X.  A ) )  =  ( (UnifSt `  W )t  ( A  X.  A ) )
5 fvex 5867 . . . . 5  |-  ( UnifSet `  W )  e.  _V
65a1i 11 . . . 4  |-  ( A  e.  V  ->  ( UnifSet
`  W )  e. 
_V )
7 fvex 5867 . . . . . 6  |-  ( Base `  W )  e.  _V
87, 7xpex 6704 . . . . 5  |-  ( (
Base `  W )  X.  ( Base `  W
) )  e.  _V
98a1i 11 . . . 4  |-  ( A  e.  V  ->  (
( Base `  W )  X.  ( Base `  W
) )  e.  _V )
10 xpexg 6702 . . . . 5  |-  ( ( A  e.  V  /\  A  e.  V )  ->  ( A  X.  A
)  e.  _V )
1110anidms 645 . . . 4  |-  ( A  e.  V  ->  ( A  X.  A )  e. 
_V )
12 restco 19424 . . . 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
) ) ) )
136, 9, 11, 12syl3anc 1223 . . 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
) ) ) )
144, 13syl5eqr 2515 . 2  |-  ( A  e.  V  ->  (
(UnifSt `  W )t  ( A  X.  A ) )  =  ( ( UnifSet `  W )t  ( ( (
Base `  W )  X.  ( Base `  W
) )  i^i  ( A  X.  A ) ) ) )
15 inxp 5126 . . . . 5  |-  ( ( ( Base `  W
)  X.  ( Base `  W ) )  i^i  ( A  X.  A
) )  =  ( ( ( Base `  W
)  i^i  A )  X.  ( ( Base `  W
)  i^i  A )
)
16 incom 3684 . . . . . . 7  |-  ( A  i^i  ( Base `  W
) )  =  ( ( Base `  W
)  i^i  A )
17 eqid 2460 . . . . . . . 8  |-  ( Ws  A )  =  ( Ws  A )
1817, 1ressbas 14534 . . . . . . 7  |-  ( A  e.  V  ->  ( A  i^i  ( Base `  W
) )  =  (
Base `  ( Ws  A
) ) )
1916, 18syl5eqr 2515 . . . . . 6  |-  ( A  e.  V  ->  (
( Base `  W )  i^i  A )  =  (
Base `  ( Ws  A
) ) )
2019, 19xpeq12d 5017 . . . . 5  |-  ( A  e.  V  ->  (
( ( Base `  W
)  i^i  A )  X.  ( ( Base `  W
)  i^i  A )
)  =  ( (
Base `  ( Ws  A
) )  X.  ( Base `  ( Ws  A ) ) ) )
2115, 20syl5eq 2513 . . . 4  |-  ( A  e.  V  ->  (
( ( Base `  W
)  X.  ( Base `  W ) )  i^i  ( A  X.  A
) )  =  ( ( Base `  ( Ws  A ) )  X.  ( Base `  ( Ws  A ) ) ) )
2221oveq2d 6291 . . 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 ) ) ) ) )
2317, 2ressunif 20493 . . . 4  |-  ( A  e.  V  ->  ( UnifSet
`  W )  =  ( UnifSet `  ( Ws  A
) ) )
2423oveq1d 6290 . . 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 ) ) ) ) )
25 eqid 2460 . . . . 5  |-  ( Base `  ( Ws  A ) )  =  ( Base `  ( Ws  A ) )
26 eqid 2460 . . . . 5  |-  ( UnifSet `  ( Ws  A ) )  =  ( UnifSet `  ( Ws  A
) )
2725, 26ussval 20490 . . . 4  |-  ( (
UnifSet `  ( Ws  A ) )t  ( ( Base `  ( Ws  A ) )  X.  ( Base `  ( Ws  A ) ) ) )  =  (UnifSt `  ( Ws  A ) )
2827a1i 11 . . 3  |-  ( A  e.  V  ->  (
( UnifSet `  ( Ws  A
) )t  ( ( Base `  ( Ws  A ) )  X.  ( Base `  ( Ws  A ) ) ) )  =  (UnifSt `  ( Ws  A ) ) )
2922, 24, 283eqtrd 2505 . 2  |-  ( A  e.  V  ->  (
( UnifSet `  W )t  (
( ( Base `  W
)  X.  ( Base `  W ) )  i^i  ( A  X.  A
) ) )  =  (UnifSt `  ( Ws  A
) ) )
3014, 29eqtr2d 2502 1  |-  ( A  e.  V  ->  (UnifSt `  ( Ws  A ) )  =  ( (UnifSt `  W
)t  ( A  X.  A
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1374    e. wcel 1762   _Vcvv 3106    i^i cin 3468    X. cxp 4990   ` cfv 5579  (class class class)co 6275   Basecbs 14479   ↾s cress 14480   UnifSetcunif 14554   ↾t crest 14665  UnifStcuss 20484
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-2 10583  df-3 10584  df-4 10585  df-5 10586  df-6 10587  df-7 10588  df-8 10589  df-9 10590  df-10 10591  df-n0 10785  df-z 10854  df-dec 10966  df-ndx 14482  df-slot 14483  df-base 14484  df-sets 14485  df-ress 14486  df-unif 14567  df-rest 14667  df-uss 20487
This theorem is referenced by:  ressust  20495  ressusp  20496  ucnextcn  20535  reust  21541  qqhucn  27595
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