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Theorem tuslem 21280
Description: Lemma for tusbas 21281, tusunif 21282, and tustopn 21284. (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 15168 . . . 4  |-  Base  = Slot  ( Base `  ndx )
2 1re 9649 . . . . . 6  |-  1  e.  RR
3 1lt9 10818 . . . . . 6  |-  1  <  9
42, 3ltneii 9754 . . . . 5  |-  1  =/=  9
5 basendx 15172 . . . . . 6  |-  ( Base `  ndx )  =  1
6 tsetndx 15283 . . . . . 6  |-  (TopSet `  ndx )  =  9
75, 6neeq12i 2709 . . . . 5  |-  ( (
Base `  ndx )  =/=  (TopSet `  ndx )  <->  1  =/=  9 )
84, 7mpbir 212 . . . 4  |-  ( Base `  ndx )  =/=  (TopSet ` 
ndx )
91, 8setsnid 15164 . . 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 21238 . . . 4  |-  ( U  e.  (UnifOn `  X
)  ->  X  =  dom  U. U )
11 uniexg 6602 . . . . 5  |-  ( U  e.  (UnifOn `  X
)  ->  U. U  e. 
_V )
12 dmexg 6738 . . . . 5  |-  ( U. U  e.  _V  ->  dom  U. U  e.  _V )
13 eqid 2422 . . . . . 6  |-  { <. (
Base `  ndx ) ,  dom  U. U >. , 
<. ( UnifSet `  ndx ) ,  U >. }  =  { <. ( Base `  ndx ) ,  dom  U. U >. ,  <. ( UnifSet `  ndx ) ,  U >. }
14 df-unif 15212 . . . . . 6  |-  UnifSet  = Slot ; 1 3
15 1nn 10627 . . . . . . 7  |-  1  e.  NN
16 3nn0 10894 . . . . . . 7  |-  3  e.  NN0
17 1nn0 10892 . . . . . . 7  |-  1  e.  NN0
18 1lt10 10827 . . . . . . 7  |-  1  <  10
1915, 16, 17, 18declti 11083 . . . . . 6  |-  1  < ; 1
3
20 3nn 10775 . . . . . . 7  |-  3  e.  NN
2117, 20decnncl 11071 . . . . . 6  |- ; 1 3  e.  NN
2213, 14, 19, 212strbas 15230 . . . . 5  |-  ( dom  U. U  e.  _V  ->  dom  U. U  =  ( Base `  { <. ( Base `  ndx ) ,  dom  U. U >. ,  <. ( UnifSet `  ndx ) ,  U >. } ) )
2311, 12, 223syl 18 . . . 4  |-  ( U  e.  (UnifOn `  X
)  ->  dom  U. U  =  ( Base `  { <. ( Base `  ndx ) ,  dom  U. U >. ,  <. ( UnifSet `  ndx ) ,  U >. } ) )
2410, 23eqtrd 2463 . . 3  |-  ( U  e.  (UnifOn `  X
)  ->  X  =  ( Base `  { <. ( Base `  ndx ) ,  dom  U. U >. , 
<. ( UnifSet `  ndx ) ,  U >. } ) )
25 tuslem.k . . . . 5  |-  K  =  (toUnifSp `  U )
26 tusval 21279 . . . . 5  |-  ( U  e.  (UnifOn `  X
)  ->  (toUnifSp `  U
)  =  ( {
<. ( Base `  ndx ) ,  dom  U. U >. ,  <. ( UnifSet `  ndx ) ,  U >. } sSet  <. (TopSet `  ndx ) ,  (unifTop `  U ) >. ) )
2725, 26syl5eq 2475 . . . 4  |-  ( U  e.  (UnifOn `  X
)  ->  K  =  ( { <. ( Base `  ndx ) ,  dom  U. U >. ,  <. ( UnifSet `  ndx ) ,  U >. } sSet  <. (TopSet `  ndx ) ,  (unifTop `  U ) >. ) )
2827fveq2d 5885 . . 3  |-  ( U  e.  (UnifOn `  X
)  ->  ( Base `  K )  =  (
Base `  ( { <. ( Base `  ndx ) ,  dom  U. U >. ,  <. ( UnifSet `  ndx ) ,  U >. } sSet  <. (TopSet `  ndx ) ,  (unifTop `  U ) >. ) ) )
299, 24, 283eqtr4a 2489 . 2  |-  ( U  e.  (UnifOn `  X
)  ->  X  =  ( Base `  K )
)
30 unifid 15302 . . . 4  |-  UnifSet  = Slot  ( UnifSet
`  ndx )
31 9re 10703 . . . . . 6  |-  9  e.  RR
32 9nn0 10900 . . . . . . 7  |-  9  e.  NN0
33 9lt10 10819 . . . . . . 7  |-  9  <  10
3415, 16, 32, 33declti 11083 . . . . . 6  |-  9  < ; 1
3
3531, 34gtneii 9753 . . . . 5  |- ; 1 3  =/=  9
36 unifndx 15301 . . . . . 6  |-  ( UnifSet ` 
ndx )  = ; 1 3
3736, 6neeq12i 2709 . . . . 5  |-  ( (
UnifSet `  ndx )  =/=  (TopSet `  ndx )  <-> ; 1 3  =/=  9
)
3835, 37mpbir 212 . . . 4  |-  ( UnifSet ` 
ndx )  =/=  (TopSet ` 
ndx )
3930, 38setsnid 15164 . . 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 15231 . . 3  |-  ( U  e.  (UnifOn `  X
)  ->  U  =  ( UnifSet `  { <. ( Base `  ndx ) ,  dom  U. U >. , 
<. ( UnifSet `  ndx ) ,  U >. } ) )
4127fveq2d 5885 . . 3  |-  ( U  e.  (UnifOn `  X
)  ->  ( UnifSet `  K )  =  (
UnifSet `  ( { <. (
Base `  ndx ) ,  dom  U. U >. , 
<. ( UnifSet `  ndx ) ,  U >. } sSet  <. (TopSet `  ndx ) ,  (unifTop `  U
) >. ) ) )
4239, 40, 413eqtr4a 2489 . 2  |-  ( U  e.  (UnifOn `  X
)  ->  U  =  ( UnifSet `  K )
)
4327fveq2d 5885 . . . 4  |-  ( U  e.  (UnifOn `  X
)  ->  (TopSet `  K
)  =  (TopSet `  ( { <. ( Base `  ndx ) ,  dom  U. U >. ,  <. ( UnifSet `  ndx ) ,  U >. } sSet  <. (TopSet `  ndx ) ,  (unifTop `  U ) >. ) ) )
44 prex 4663 . . . . 5  |-  { <. (
Base `  ndx ) ,  dom  U. U >. , 
<. ( UnifSet `  ndx ) ,  U >. }  e.  _V
45 fvex 5891 . . . . 5  |-  (unifTop `  U
)  e.  _V
46 tsetid 15284 . . . . . 6  |- TopSet  = Slot  (TopSet ` 
ndx )
4746setsid 15163 . . . . 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 676 . . . 4  |-  (unifTop `  U
)  =  (TopSet `  ( { <. ( Base `  ndx ) ,  dom  U. U >. ,  <. ( UnifSet `  ndx ) ,  U >. } sSet  <. (TopSet `  ndx ) ,  (unifTop `  U ) >. ) )
4943, 48syl6reqr 2482 . . 3  |-  ( U  e.  (UnifOn `  X
)  ->  (unifTop `  U
)  =  (TopSet `  K ) )
50 utopbas 21248 . . . . . 6  |-  ( U  e.  (UnifOn `  X
)  ->  X  =  U. (unifTop `  U )
)
5149unieqd 4229 . . . . . 6  |-  ( U  e.  (UnifOn `  X
)  ->  U. (unifTop `  U )  =  U. (TopSet `  K ) )
5250, 29, 513eqtr3rd 2472 . . . . 5  |-  ( U  e.  (UnifOn `  X
)  ->  U. (TopSet `  K )  =  (
Base `  K )
)
5352oveq2d 6321 . . . 4  |-  ( U  e.  (UnifOn `  X
)  ->  ( (TopSet `  K )t  U. (TopSet `  K
) )  =  ( (TopSet `  K )t  ( Base `  K ) ) )
54 fvex 5891 . . . . 5  |-  (TopSet `  K )  e.  _V
55 eqid 2422 . . . . . 6  |-  U. (TopSet `  K )  =  U. (TopSet `  K )
5655restid 15331 . . . . 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 2422 . . . . 5  |-  ( Base `  K )  =  (
Base `  K )
59 eqid 2422 . . . . 5  |-  (TopSet `  K )  =  (TopSet `  K )
6058, 59topnval 15332 . . . 4  |-  ( (TopSet `  K )t  ( Base `  K
) )  =  (
TopOpen `  K )
6153, 57, 603eqtr3g 2486 . . 3  |-  ( U  e.  (UnifOn `  X
)  ->  (TopSet `  K
)  =  ( TopOpen `  K ) )
6249, 61eqtrd 2463 . 2  |-  ( U  e.  (UnifOn `  X
)  ->  (unifTop `  U
)  =  ( TopOpen `  K ) )
6329, 42, 623jca 1185 1  |-  ( U  e.  (UnifOn `  X
)  ->  ( X  =  ( Base `  K
)  /\  U  =  ( UnifSet `  K )  /\  (unifTop `  U )  =  ( TopOpen `  K
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 982    = wceq 1437    e. wcel 1872    =/= wne 2614   _Vcvv 3080   {cpr 4000   <.cop 4004   U.cuni 4219   dom cdm 4853   ` cfv 5601  (class class class)co 6305   1c1 9547   3c3 10667   9c9 10673  ;cdc 11058   ndxcnx 15117   sSet csts 15118   Basecbs 15120  TopSetcts 15195   UnifSetcunif 15199   ↾t crest 15318   TopOpenctopn 15319  UnifOncust 21212  unifTopcutop 21243  toUnifSpctus 21268
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597  ax-cnex 9602  ax-resscn 9603  ax-1cn 9604  ax-icn 9605  ax-addcl 9606  ax-addrcl 9607  ax-mulcl 9608  ax-mulrcl 9609  ax-mulcom 9610  ax-addass 9611  ax-mulass 9612  ax-distr 9613  ax-i2m1 9614  ax-1ne0 9615  ax-1rid 9616  ax-rnegex 9617  ax-rrecex 9618  ax-cnre 9619  ax-pre-lttri 9620  ax-pre-lttrn 9621  ax-pre-ltadd 9622  ax-pre-mulgt0 9623
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-int 4256  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7039  df-recs 7101  df-rdg 7139  df-1o 7193  df-oadd 7197  df-er 7374  df-en 7581  df-dom 7582  df-sdom 7583  df-fin 7584  df-pnf 9684  df-mnf 9685  df-xr 9686  df-ltxr 9687  df-le 9688  df-sub 9869  df-neg 9870  df-nn 10617  df-2 10675  df-3 10676  df-4 10677  df-5 10678  df-6 10679  df-7 10680  df-8 10681  df-9 10682  df-10 10683  df-n0 10877  df-z 10945  df-dec 11059  df-uz 11167  df-fz 11792  df-struct 15122  df-ndx 15123  df-slot 15124  df-base 15125  df-sets 15126  df-tset 15208  df-unif 15212  df-rest 15320  df-topn 15321  df-ust 21213  df-utop 21244  df-tus 21271
This theorem is referenced by:  tusbas  21281  tusunif  21282  tustopn  21284  tususp  21285
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