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Theorem tuslem 19742
Description: Lemma for tusbas 19743, tusunif 19744, and tustopn 19746. (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 14216 . . . 4  |-  Base  = Slot  ( Base `  ndx )
2 1re 9381 . . . . . 6  |-  1  e.  RR
3 1lt9 10519 . . . . . 6  |-  1  <  9
42, 3ltneii 9483 . . . . 5  |-  1  =/=  9
5 basendx 14219 . . . . . 6  |-  ( Base `  ndx )  =  1
6 tsetndx 14321 . . . . . 6  |-  (TopSet `  ndx )  =  9
75, 6neeq12i 2618 . . . . 5  |-  ( (
Base `  ndx )  =/=  (TopSet `  ndx )  <->  1  =/=  9 )
84, 7mpbir 209 . . . 4  |-  ( Base `  ndx )  =/=  (TopSet ` 
ndx )
91, 8setsnid 14212 . . 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 19700 . . . 4  |-  ( U  e.  (UnifOn `  X
)  ->  X  =  dom  U. U )
11 uniexg 6376 . . . . 5  |-  ( U  e.  (UnifOn `  X
)  ->  U. U  e. 
_V )
12 dmexg 6508 . . . . 5  |-  ( U. U  e.  _V  ->  dom  U. U  e.  _V )
13 eqid 2441 . . . . . 6  |-  { <. (
Base `  ndx ) ,  dom  U. U >. , 
<. ( UnifSet `  ndx ) ,  U >. }  =  { <. ( Base `  ndx ) ,  dom  U. U >. ,  <. ( UnifSet `  ndx ) ,  U >. }
14 df-unif 14257 . . . . . 6  |-  UnifSet  = Slot ; 1 3
15 1nn 10329 . . . . . . 7  |-  1  e.  NN
16 3nn0 10593 . . . . . . 7  |-  3  e.  NN0
17 1nn0 10591 . . . . . . 7  |-  1  e.  NN0
18 1lt10 10528 . . . . . . 7  |-  1  <  10
1915, 16, 17, 18declti 10776 . . . . . 6  |-  1  < ; 1
3
20 3nn 10476 . . . . . . 7  |-  3  e.  NN
2117, 20decnncl 10764 . . . . . 6  |- ; 1 3  e.  NN
2213, 14, 19, 212strbas 14271 . . . . 5  |-  ( dom  U. U  e.  _V  ->  dom  U. U  =  ( Base `  { <. ( Base `  ndx ) ,  dom  U. U >. ,  <. ( UnifSet `  ndx ) ,  U >. } ) )
2311, 12, 223syl 20 . . . 4  |-  ( U  e.  (UnifOn `  X
)  ->  dom  U. U  =  ( Base `  { <. ( Base `  ndx ) ,  dom  U. U >. ,  <. ( UnifSet `  ndx ) ,  U >. } ) )
2410, 23eqtrd 2473 . . 3  |-  ( U  e.  (UnifOn `  X
)  ->  X  =  ( Base `  { <. ( Base `  ndx ) ,  dom  U. U >. , 
<. ( UnifSet `  ndx ) ,  U >. } ) )
25 tuslem.k . . . . 5  |-  K  =  (toUnifSp `  U )
26 tusval 19741 . . . . 5  |-  ( U  e.  (UnifOn `  X
)  ->  (toUnifSp `  U
)  =  ( {
<. ( Base `  ndx ) ,  dom  U. U >. ,  <. ( UnifSet `  ndx ) ,  U >. } sSet  <. (TopSet `  ndx ) ,  (unifTop `  U ) >. ) )
2725, 26syl5eq 2485 . . . 4  |-  ( U  e.  (UnifOn `  X
)  ->  K  =  ( { <. ( Base `  ndx ) ,  dom  U. U >. ,  <. ( UnifSet `  ndx ) ,  U >. } sSet  <. (TopSet `  ndx ) ,  (unifTop `  U ) >. ) )
2827fveq2d 5692 . . 3  |-  ( U  e.  (UnifOn `  X
)  ->  ( Base `  K )  =  (
Base `  ( { <. ( Base `  ndx ) ,  dom  U. U >. ,  <. ( UnifSet `  ndx ) ,  U >. } sSet  <. (TopSet `  ndx ) ,  (unifTop `  U ) >. ) ) )
299, 24, 283eqtr4a 2499 . 2  |-  ( U  e.  (UnifOn `  X
)  ->  X  =  ( Base `  K )
)
30 unifid 14340 . . . 4  |-  UnifSet  = Slot  ( UnifSet
`  ndx )
31 9re 10404 . . . . . 6  |-  9  e.  RR
32 9nn0 10599 . . . . . . 7  |-  9  e.  NN0
33 9lt10 10520 . . . . . . 7  |-  9  <  10
3415, 16, 32, 33declti 10776 . . . . . 6  |-  9  < ; 1
3
3531, 34gtneii 9482 . . . . 5  |- ; 1 3  =/=  9
36 unifndx 14339 . . . . . 6  |-  ( UnifSet ` 
ndx )  = ; 1 3
3736, 6neeq12i 2618 . . . . 5  |-  ( (
UnifSet `  ndx )  =/=  (TopSet `  ndx )  <-> ; 1 3  =/=  9
)
3835, 37mpbir 209 . . . 4  |-  ( UnifSet ` 
ndx )  =/=  (TopSet ` 
ndx )
3930, 38setsnid 14212 . . 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 14272 . . 3  |-  ( U  e.  (UnifOn `  X
)  ->  U  =  ( UnifSet `  { <. ( Base `  ndx ) ,  dom  U. U >. , 
<. ( UnifSet `  ndx ) ,  U >. } ) )
4127fveq2d 5692 . . 3  |-  ( U  e.  (UnifOn `  X
)  ->  ( UnifSet `  K )  =  (
UnifSet `  ( { <. (
Base `  ndx ) ,  dom  U. U >. , 
<. ( UnifSet `  ndx ) ,  U >. } sSet  <. (TopSet `  ndx ) ,  (unifTop `  U
) >. ) ) )
4239, 40, 413eqtr4a 2499 . 2  |-  ( U  e.  (UnifOn `  X
)  ->  U  =  ( UnifSet `  K )
)
4327fveq2d 5692 . . . 4  |-  ( U  e.  (UnifOn `  X
)  ->  (TopSet `  K
)  =  (TopSet `  ( { <. ( Base `  ndx ) ,  dom  U. U >. ,  <. ( UnifSet `  ndx ) ,  U >. } sSet  <. (TopSet `  ndx ) ,  (unifTop `  U ) >. ) ) )
44 prex 4531 . . . . 5  |-  { <. (
Base `  ndx ) ,  dom  U. U >. , 
<. ( UnifSet `  ndx ) ,  U >. }  e.  _V
45 fvex 5698 . . . . 5  |-  (unifTop `  U
)  e.  _V
46 tsetid 14322 . . . . . 6  |- TopSet  = Slot  (TopSet ` 
ndx )
4746setsid 14211 . . . . 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 667 . . . 4  |-  (unifTop `  U
)  =  (TopSet `  ( { <. ( Base `  ndx ) ,  dom  U. U >. ,  <. ( UnifSet `  ndx ) ,  U >. } sSet  <. (TopSet `  ndx ) ,  (unifTop `  U ) >. ) )
4943, 48syl6reqr 2492 . . 3  |-  ( U  e.  (UnifOn `  X
)  ->  (unifTop `  U
)  =  (TopSet `  K ) )
50 utopbas 19710 . . . . . 6  |-  ( U  e.  (UnifOn `  X
)  ->  X  =  U. (unifTop `  U )
)
5149unieqd 4098 . . . . . 6  |-  ( U  e.  (UnifOn `  X
)  ->  U. (unifTop `  U )  =  U. (TopSet `  K ) )
5250, 29, 513eqtr3rd 2482 . . . . 5  |-  ( U  e.  (UnifOn `  X
)  ->  U. (TopSet `  K )  =  (
Base `  K )
)
5352oveq2d 6106 . . . 4  |-  ( U  e.  (UnifOn `  X
)  ->  ( (TopSet `  K )t  U. (TopSet `  K
) )  =  ( (TopSet `  K )t  ( Base `  K ) ) )
54 fvex 5698 . . . . 5  |-  (TopSet `  K )  e.  _V
55 eqid 2441 . . . . . 6  |-  U. (TopSet `  K )  =  U. (TopSet `  K )
5655restid 14368 . . . . 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 2441 . . . . 5  |-  ( Base `  K )  =  (
Base `  K )
59 eqid 2441 . . . . 5  |-  (TopSet `  K )  =  (TopSet `  K )
6058, 59topnval 14369 . . . 4  |-  ( (TopSet `  K )t  ( Base `  K
) )  =  (
TopOpen `  K )
6153, 57, 603eqtr3g 2496 . . 3  |-  ( U  e.  (UnifOn `  X
)  ->  (TopSet `  K
)  =  ( TopOpen `  K ) )
6249, 61eqtrd 2473 . 2  |-  ( U  e.  (UnifOn `  X
)  ->  (unifTop `  U
)  =  ( TopOpen `  K ) )
6329, 42, 623jca 1163 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 960    = wceq 1364    e. wcel 1761    =/= wne 2604   _Vcvv 2970   {cpr 3876   <.cop 3880   U.cuni 4088   dom cdm 4836   ` cfv 5415  (class class class)co 6090   1c1 9279   3c3 10368   9c9 10374  ;cdc 10751   ndxcnx 14167   sSet csts 14168   Basecbs 14170  TopSetcts 14240   UnifSetcunif 14244   ↾t crest 14355   TopOpenctopn 14356  UnifOncust 19674  unifTopcutop 19705  toUnifSpctus 19730
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 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-uz 10858  df-fz 11434  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-tset 14253  df-unif 14257  df-rest 14357  df-topn 14358  df-ust 19675  df-utop 19706  df-tus 19733
This theorem is referenced by:  tusbas  19743  tusunif  19744  tustopn  19746  tususp  19747
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