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Theorem tuslem 18335
Description: Lemma for tusbas 18336, tusunif 18337, and tustopn 18339. (Contributed by Thierry Arnoux, 5-Dec-2017.)
Hypothesis
Ref Expression
tuslem.k  |-  K  =  (toUnifSp `  U )
Assertion
Ref Expression
tuslem  |-  ( U  e.  (UnifOn `  X
)  ->  ( X  =  ( Base `  K
)  /\  U  =  ( UnifSet `  K )  /\  (unifTop `  U )  =  ( TopOpen `  K
) ) )

Proof of Theorem tuslem
StepHypRef Expression
1 baseid 13549 . . . 4  |-  Base  = Slot  ( Base `  ndx )
2 1re 9128 . . . . . 6  |-  1  e.  RR
3 1lt9 10215 . . . . . 6  |-  1  <  9
42, 3ltneii 9224 . . . . 5  |-  1  =/=  9
5 basendx 13552 . . . . . 6  |-  ( Base `  ndx )  =  1
6 tsetndx 13652 . . . . . 6  |-  (TopSet `  ndx )  =  9
75, 6neeq12i 2620 . . . . 5  |-  ( (
Base `  ndx )  =/=  (TopSet `  ndx )  <->  1  =/=  9 )
84, 7mpbir 202 . . . 4  |-  ( Base `  ndx )  =/=  (TopSet ` 
ndx )
91, 8setsnid 13547 . . 3  |-  ( Base `  { <. ( Base `  ndx ) ,  dom  U. U >. ,  <. ( UnifSet `  ndx ) ,  U >. } )  =  ( Base `  ( { <. ( Base `  ndx ) ,  dom  U. U >. , 
<. ( UnifSet `  ndx ) ,  U >. } sSet  <. (TopSet `  ndx ) ,  (unifTop `  U
) >. ) )
10 ustbas2 18293 . . . 4  |-  ( U  e.  (UnifOn `  X
)  ->  X  =  dom  U. U )
11 uniexg 4741 . . . . 5  |-  ( U  e.  (UnifOn `  X
)  ->  U. U  e. 
_V )
12 dmexg 5165 . . . . 5  |-  ( U. U  e.  _V  ->  dom  U. U  e.  _V )
13 eqid 2443 . . . . . 6  |-  { <. (
Base `  ndx ) ,  dom  U. U >. , 
<. ( UnifSet `  ndx ) ,  U >. }  =  { <. ( Base `  ndx ) ,  dom  U. U >. ,  <. ( UnifSet `  ndx ) ,  U >. }
14 df-unif 13590 . . . . . 6  |-  UnifSet  = Slot ; 1 3
15 1nn 10049 . . . . . . 7  |-  1  e.  NN
16 3nn0 10277 . . . . . . 7  |-  3  e.  NN0
17 1nn0 10275 . . . . . . 7  |-  1  e.  NN0
18 1lt10 10224 . . . . . . 7  |-  1  <  10
1915, 16, 17, 18declti 10445 . . . . . 6  |-  1  < ; 1
3
20 3nn 10172 . . . . . . 7  |-  3  e.  NN
2117, 20decnncl 10433 . . . . . 6  |- ; 1 3  e.  NN
2213, 14, 19, 212strbas 13604 . . . . 5  |-  ( dom  U. U  e.  _V  ->  dom  U. U  =  ( Base `  { <. ( Base `  ndx ) ,  dom  U. U >. ,  <. ( UnifSet `  ndx ) ,  U >. } ) )
2311, 12, 223syl 19 . . . 4  |-  ( U  e.  (UnifOn `  X
)  ->  dom  U. U  =  ( Base `  { <. ( Base `  ndx ) ,  dom  U. U >. ,  <. ( UnifSet `  ndx ) ,  U >. } ) )
2410, 23eqtrd 2475 . . 3  |-  ( U  e.  (UnifOn `  X
)  ->  X  =  ( Base `  { <. ( Base `  ndx ) ,  dom  U. U >. , 
<. ( UnifSet `  ndx ) ,  U >. } ) )
25 tuslem.k . . . . 5  |-  K  =  (toUnifSp `  U )
26 tusval 18334 . . . . 5  |-  ( U  e.  (UnifOn `  X
)  ->  (toUnifSp `  U
)  =  ( {
<. ( Base `  ndx ) ,  dom  U. U >. ,  <. ( UnifSet `  ndx ) ,  U >. } sSet  <. (TopSet `  ndx ) ,  (unifTop `  U ) >. ) )
2725, 26syl5eq 2487 . . . 4  |-  ( U  e.  (UnifOn `  X
)  ->  K  =  ( { <. ( Base `  ndx ) ,  dom  U. U >. ,  <. ( UnifSet `  ndx ) ,  U >. } sSet  <. (TopSet `  ndx ) ,  (unifTop `  U ) >. ) )
2827fveq2d 5767 . . 3  |-  ( U  e.  (UnifOn `  X
)  ->  ( Base `  K )  =  (
Base `  ( { <. ( Base `  ndx ) ,  dom  U. U >. ,  <. ( UnifSet `  ndx ) ,  U >. } sSet  <. (TopSet `  ndx ) ,  (unifTop `  U ) >. ) ) )
299, 24, 283eqtr4a 2501 . 2  |-  ( U  e.  (UnifOn `  X
)  ->  X  =  ( Base `  K )
)
30 unifid 13671 . . . 4  |-  UnifSet  = Slot  ( UnifSet
`  ndx )
31 9re 10117 . . . . . 6  |-  9  e.  RR
32 9nn0 10283 . . . . . . 7  |-  9  e.  NN0
33 9lt10 10216 . . . . . . 7  |-  9  <  10
3415, 16, 32, 33declti 10445 . . . . . 6  |-  9  < ; 1
3
3531, 34gtneii 9223 . . . . 5  |- ; 1 3  =/=  9
36 unifndx 13670 . . . . . 6  |-  ( UnifSet ` 
ndx )  = ; 1 3
3736, 6neeq12i 2620 . . . . 5  |-  ( (
UnifSet `  ndx )  =/=  (TopSet `  ndx )  <-> ; 1 3  =/=  9
)
3835, 37mpbir 202 . . . 4  |-  ( UnifSet ` 
ndx )  =/=  (TopSet ` 
ndx )
3930, 38setsnid 13547 . . 3  |-  ( UnifSet `  { <. ( Base `  ndx ) ,  dom  U. U >. ,  <. ( UnifSet `  ndx ) ,  U >. } )  =  ( UnifSet `  ( { <. ( Base `  ndx ) ,  dom  U. U >. , 
<. ( UnifSet `  ndx ) ,  U >. } sSet  <. (TopSet `  ndx ) ,  (unifTop `  U
) >. ) )
4013, 14, 19, 212strop 13605 . . 3  |-  ( U  e.  (UnifOn `  X
)  ->  U  =  ( UnifSet `  { <. ( Base `  ndx ) ,  dom  U. U >. , 
<. ( UnifSet `  ndx ) ,  U >. } ) )
4127fveq2d 5767 . . 3  |-  ( U  e.  (UnifOn `  X
)  ->  ( UnifSet `  K )  =  (
UnifSet `  ( { <. (
Base `  ndx ) ,  dom  U. U >. , 
<. ( UnifSet `  ndx ) ,  U >. } sSet  <. (TopSet `  ndx ) ,  (unifTop `  U
) >. ) ) )
4239, 40, 413eqtr4a 2501 . 2  |-  ( U  e.  (UnifOn `  X
)  ->  U  =  ( UnifSet `  K )
)
4327fveq2d 5767 . . . 4  |-  ( U  e.  (UnifOn `  X
)  ->  (TopSet `  K
)  =  (TopSet `  ( { <. ( Base `  ndx ) ,  dom  U. U >. ,  <. ( UnifSet `  ndx ) ,  U >. } sSet  <. (TopSet `  ndx ) ,  (unifTop `  U ) >. ) ) )
44 prex 4441 . . . . 5  |-  { <. (
Base `  ndx ) ,  dom  U. U >. , 
<. ( UnifSet `  ndx ) ,  U >. }  e.  _V
45 fvex 5773 . . . . 5  |-  (unifTop `  U
)  e.  _V
46 tsetid 13653 . . . . . 6  |- TopSet  = Slot  (TopSet ` 
ndx )
4746setsid 13546 . . . . 5  |-  ( ( { <. ( Base `  ndx ) ,  dom  U. U >. ,  <. ( UnifSet `  ndx ) ,  U >. }  e.  _V  /\  (unifTop `  U )  e.  _V )  ->  (unifTop `  U )  =  (TopSet `  ( { <. ( Base `  ndx ) ,  dom  U. U >. ,  <. ( UnifSet `  ndx ) ,  U >. } sSet  <. (TopSet `  ndx ) ,  (unifTop `  U ) >. ) ) )
4844, 45, 47mp2an 655 . . . 4  |-  (unifTop `  U
)  =  (TopSet `  ( { <. ( Base `  ndx ) ,  dom  U. U >. ,  <. ( UnifSet `  ndx ) ,  U >. } sSet  <. (TopSet `  ndx ) ,  (unifTop `  U ) >. ) )
4943, 48syl6reqr 2494 . . 3  |-  ( U  e.  (UnifOn `  X
)  ->  (unifTop `  U
)  =  (TopSet `  K ) )
50 utopbas 18303 . . . . . 6  |-  ( U  e.  (UnifOn `  X
)  ->  X  =  U. (unifTop `  U )
)
5149unieqd 4055 . . . . . 6  |-  ( U  e.  (UnifOn `  X
)  ->  U. (unifTop `  U )  =  U. (TopSet `  K ) )
5250, 29, 513eqtr3rd 2484 . . . . 5  |-  ( U  e.  (UnifOn `  X
)  ->  U. (TopSet `  K )  =  (
Base `  K )
)
5352oveq2d 6133 . . . 4  |-  ( U  e.  (UnifOn `  X
)  ->  ( (TopSet `  K )t  U. (TopSet `  K
) )  =  ( (TopSet `  K )t  ( Base `  K ) ) )
54 fvex 5773 . . . . 5  |-  (TopSet `  K )  e.  _V
55 eqid 2443 . . . . . 6  |-  U. (TopSet `  K )  =  U. (TopSet `  K )
5655restid 13699 . . . . 5  |-  ( (TopSet `  K )  e.  _V  ->  ( (TopSet `  K
)t  U. (TopSet `  K )
)  =  (TopSet `  K ) )
5754, 56ax-mp 5 . . . 4  |-  ( (TopSet `  K )t  U. (TopSet `  K
) )  =  (TopSet `  K )
58 eqid 2443 . . . . 5  |-  ( Base `  K )  =  (
Base `  K )
59 eqid 2443 . . . . 5  |-  (TopSet `  K )  =  (TopSet `  K )
6058, 59topnval 13700 . . . 4  |-  ( (TopSet `  K )t  ( Base `  K
) )  =  (
TopOpen `  K )
6153, 57, 603eqtr3g 2498 . . 3  |-  ( U  e.  (UnifOn `  X
)  ->  (TopSet `  K
)  =  ( TopOpen `  K ) )
6249, 61eqtrd 2475 . 2  |-  ( U  e.  (UnifOn `  X
)  ->  (unifTop `  U
)  =  ( TopOpen `  K ) )
6329, 42, 623jca 1135 1  |-  ( U  e.  (UnifOn `  X
)  ->  ( X  =  ( Base `  K
)  /\  U  =  ( UnifSet `  K )  /\  (unifTop `  U )  =  ( TopOpen `  K
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 937    = wceq 1654    e. wcel 1728    =/= wne 2606   _Vcvv 2965   {cpr 3844   <.cop 3846   U.cuni 4044   dom cdm 4913   ` cfv 5489  (class class class)co 6117   1c1 9029   3c3 10088   9c9 10094  ;cdc 10420   ndxcnx 13504   sSet csts 13505   Basecbs 13507  TopSetcts 13573   UnifSetcunif 13577   ↾t crest 13686   TopOpenctopn 13687  UnifOncust 18267  unifTopcutop 18298  toUnifSpctus 18323
This theorem is referenced by:  tusbas  18336  tusunif  18337  tustopn  18339  tususp  18340
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-13 1730  ax-14 1732  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424  ax-rep 4351  ax-sep 4361  ax-nul 4369  ax-pow 4412  ax-pr 4438  ax-un 4736  ax-cnex 9084  ax-resscn 9085  ax-1cn 9086  ax-icn 9087  ax-addcl 9088  ax-addrcl 9089  ax-mulcl 9090  ax-mulrcl 9091  ax-mulcom 9092  ax-addass 9093  ax-mulass 9094  ax-distr 9095  ax-i2m1 9096  ax-1ne0 9097  ax-1rid 9098  ax-rnegex 9099  ax-rrecex 9100  ax-cnre 9101  ax-pre-lttri 9102  ax-pre-lttrn 9103  ax-pre-ltadd 9104  ax-pre-mulgt0 9105
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1661  df-eu 2292  df-mo 2293  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2717  df-rex 2718  df-reu 2719  df-rab 2721  df-v 2967  df-sbc 3171  df-csb 3271  df-dif 3312  df-un 3314  df-in 3316  df-ss 3323  df-pss 3325  df-nul 3617  df-if 3768  df-pw 3830  df-sn 3849  df-pr 3850  df-tp 3851  df-op 3852  df-uni 4045  df-int 4080  df-iun 4124  df-br 4244  df-opab 4298  df-mpt 4299  df-tr 4334  df-eprel 4529  df-id 4533  df-po 4538  df-so 4539  df-fr 4576  df-we 4578  df-ord 4619  df-on 4620  df-lim 4621  df-suc 4622  df-om 4881  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5453  df-fun 5491  df-fn 5492  df-f 5493  df-f1 5494  df-fo 5495  df-f1o 5496  df-fv 5497  df-ov 6120  df-oprab 6121  df-mpt2 6122  df-1st 6385  df-2nd 6386  df-riota 6585  df-recs 6669  df-rdg 6704  df-1o 6760  df-oadd 6764  df-er 6941  df-en 7146  df-dom 7147  df-sdom 7148  df-fin 7149  df-pnf 9160  df-mnf 9161  df-xr 9162  df-ltxr 9163  df-le 9164  df-sub 9331  df-neg 9332  df-nn 10039  df-2 10096  df-3 10097  df-4 10098  df-5 10099  df-6 10100  df-7 10101  df-8 10102  df-9 10103  df-10 10104  df-n0 10260  df-z 10321  df-dec 10421  df-uz 10527  df-fz 11082  df-struct 13509  df-ndx 13510  df-slot 13511  df-base 13512  df-sets 13513  df-tset 13586  df-unif 13590  df-rest 13688  df-topn 13689  df-ust 18268  df-utop 18299  df-tus 18326
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