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Theorem tuslem 20855
Description: Lemma for tusbas 20856, tusunif 20857, and tustopn 20859. (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 14682 . . . 4  |-  Base  = Slot  ( Base `  ndx )
2 1re 9506 . . . . . 6  |-  1  e.  RR
3 1lt9 10654 . . . . . 6  |-  1  <  9
42, 3ltneii 9608 . . . . 5  |-  1  =/=  9
5 basendx 14686 . . . . . 6  |-  ( Base `  ndx )  =  1
6 tsetndx 14793 . . . . . 6  |-  (TopSet `  ndx )  =  9
75, 6neeq12i 2671 . . . . 5  |-  ( (
Base `  ndx )  =/=  (TopSet `  ndx )  <->  1  =/=  9 )
84, 7mpbir 209 . . . 4  |-  ( Base `  ndx )  =/=  (TopSet ` 
ndx )
91, 8setsnid 14678 . . 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 20813 . . . 4  |-  ( U  e.  (UnifOn `  X
)  ->  X  =  dom  U. U )
11 uniexg 6496 . . . . 5  |-  ( U  e.  (UnifOn `  X
)  ->  U. U  e. 
_V )
12 dmexg 6630 . . . . 5  |-  ( U. U  e.  _V  ->  dom  U. U  e.  _V )
13 eqid 2382 . . . . . 6  |-  { <. (
Base `  ndx ) ,  dom  U. U >. , 
<. ( UnifSet `  ndx ) ,  U >. }  =  { <. ( Base `  ndx ) ,  dom  U. U >. ,  <. ( UnifSet `  ndx ) ,  U >. }
14 df-unif 14725 . . . . . 6  |-  UnifSet  = Slot ; 1 3
15 1nn 10463 . . . . . . 7  |-  1  e.  NN
16 3nn0 10730 . . . . . . 7  |-  3  e.  NN0
17 1nn0 10728 . . . . . . 7  |-  1  e.  NN0
18 1lt10 10663 . . . . . . 7  |-  1  <  10
1915, 16, 17, 18declti 10920 . . . . . 6  |-  1  < ; 1
3
20 3nn 10611 . . . . . . 7  |-  3  e.  NN
2117, 20decnncl 10908 . . . . . 6  |- ; 1 3  e.  NN
2213, 14, 19, 212strbas 14743 . . . . 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 2423 . . 3  |-  ( U  e.  (UnifOn `  X
)  ->  X  =  ( Base `  { <. ( Base `  ndx ) ,  dom  U. U >. , 
<. ( UnifSet `  ndx ) ,  U >. } ) )
25 tuslem.k . . . . 5  |-  K  =  (toUnifSp `  U )
26 tusval 20854 . . . . 5  |-  ( U  e.  (UnifOn `  X
)  ->  (toUnifSp `  U
)  =  ( {
<. ( Base `  ndx ) ,  dom  U. U >. ,  <. ( UnifSet `  ndx ) ,  U >. } sSet  <. (TopSet `  ndx ) ,  (unifTop `  U ) >. ) )
2725, 26syl5eq 2435 . . . 4  |-  ( U  e.  (UnifOn `  X
)  ->  K  =  ( { <. ( Base `  ndx ) ,  dom  U. U >. ,  <. ( UnifSet `  ndx ) ,  U >. } sSet  <. (TopSet `  ndx ) ,  (unifTop `  U ) >. ) )
2827fveq2d 5778 . . 3  |-  ( U  e.  (UnifOn `  X
)  ->  ( Base `  K )  =  (
Base `  ( { <. ( Base `  ndx ) ,  dom  U. U >. ,  <. ( UnifSet `  ndx ) ,  U >. } sSet  <. (TopSet `  ndx ) ,  (unifTop `  U ) >. ) ) )
299, 24, 283eqtr4a 2449 . 2  |-  ( U  e.  (UnifOn `  X
)  ->  X  =  ( Base `  K )
)
30 unifid 14812 . . . 4  |-  UnifSet  = Slot  ( UnifSet
`  ndx )
31 9re 10539 . . . . . 6  |-  9  e.  RR
32 9nn0 10736 . . . . . . 7  |-  9  e.  NN0
33 9lt10 10655 . . . . . . 7  |-  9  <  10
3415, 16, 32, 33declti 10920 . . . . . 6  |-  9  < ; 1
3
3531, 34gtneii 9607 . . . . 5  |- ; 1 3  =/=  9
36 unifndx 14811 . . . . . 6  |-  ( UnifSet ` 
ndx )  = ; 1 3
3736, 6neeq12i 2671 . . . . 5  |-  ( (
UnifSet `  ndx )  =/=  (TopSet `  ndx )  <-> ; 1 3  =/=  9
)
3835, 37mpbir 209 . . . 4  |-  ( UnifSet ` 
ndx )  =/=  (TopSet ` 
ndx )
3930, 38setsnid 14678 . . 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 14744 . . 3  |-  ( U  e.  (UnifOn `  X
)  ->  U  =  ( UnifSet `  { <. ( Base `  ndx ) ,  dom  U. U >. , 
<. ( UnifSet `  ndx ) ,  U >. } ) )
4127fveq2d 5778 . . 3  |-  ( U  e.  (UnifOn `  X
)  ->  ( UnifSet `  K )  =  (
UnifSet `  ( { <. (
Base `  ndx ) ,  dom  U. U >. , 
<. ( UnifSet `  ndx ) ,  U >. } sSet  <. (TopSet `  ndx ) ,  (unifTop `  U
) >. ) ) )
4239, 40, 413eqtr4a 2449 . 2  |-  ( U  e.  (UnifOn `  X
)  ->  U  =  ( UnifSet `  K )
)
4327fveq2d 5778 . . . 4  |-  ( U  e.  (UnifOn `  X
)  ->  (TopSet `  K
)  =  (TopSet `  ( { <. ( Base `  ndx ) ,  dom  U. U >. ,  <. ( UnifSet `  ndx ) ,  U >. } sSet  <. (TopSet `  ndx ) ,  (unifTop `  U ) >. ) ) )
44 prex 4604 . . . . 5  |-  { <. (
Base `  ndx ) ,  dom  U. U >. , 
<. ( UnifSet `  ndx ) ,  U >. }  e.  _V
45 fvex 5784 . . . . 5  |-  (unifTop `  U
)  e.  _V
46 tsetid 14794 . . . . . 6  |- TopSet  = Slot  (TopSet ` 
ndx )
4746setsid 14677 . . . . 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 670 . . . 4  |-  (unifTop `  U
)  =  (TopSet `  ( { <. ( Base `  ndx ) ,  dom  U. U >. ,  <. ( UnifSet `  ndx ) ,  U >. } sSet  <. (TopSet `  ndx ) ,  (unifTop `  U ) >. ) )
4943, 48syl6reqr 2442 . . 3  |-  ( U  e.  (UnifOn `  X
)  ->  (unifTop `  U
)  =  (TopSet `  K ) )
50 utopbas 20823 . . . . . 6  |-  ( U  e.  (UnifOn `  X
)  ->  X  =  U. (unifTop `  U )
)
5149unieqd 4173 . . . . . 6  |-  ( U  e.  (UnifOn `  X
)  ->  U. (unifTop `  U )  =  U. (TopSet `  K ) )
5250, 29, 513eqtr3rd 2432 . . . . 5  |-  ( U  e.  (UnifOn `  X
)  ->  U. (TopSet `  K )  =  (
Base `  K )
)
5352oveq2d 6212 . . . 4  |-  ( U  e.  (UnifOn `  X
)  ->  ( (TopSet `  K )t  U. (TopSet `  K
) )  =  ( (TopSet `  K )t  ( Base `  K ) ) )
54 fvex 5784 . . . . 5  |-  (TopSet `  K )  e.  _V
55 eqid 2382 . . . . . 6  |-  U. (TopSet `  K )  =  U. (TopSet `  K )
5655restid 14841 . . . . 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 2382 . . . . 5  |-  ( Base `  K )  =  (
Base `  K )
59 eqid 2382 . . . . 5  |-  (TopSet `  K )  =  (TopSet `  K )
6058, 59topnval 14842 . . . 4  |-  ( (TopSet `  K )t  ( Base `  K
) )  =  (
TopOpen `  K )
6153, 57, 603eqtr3g 2446 . . 3  |-  ( U  e.  (UnifOn `  X
)  ->  (TopSet `  K
)  =  ( TopOpen `  K ) )
6249, 61eqtrd 2423 . 2  |-  ( U  e.  (UnifOn `  X
)  ->  (unifTop `  U
)  =  ( TopOpen `  K ) )
6329, 42, 623jca 1174 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 971    = wceq 1399    e. wcel 1826    =/= wne 2577   _Vcvv 3034   {cpr 3946   <.cop 3950   U.cuni 4163   dom cdm 4913   ` cfv 5496  (class class class)co 6196   1c1 9404   3c3 10503   9c9 10509  ;cdc 10895   ndxcnx 14631   sSet csts 14632   Basecbs 14634  TopSetcts 14708   UnifSetcunif 14712   ↾t crest 14828   TopOpenctopn 14829  UnifOncust 20787  unifTopcutop 20818  toUnifSpctus 20843
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-int 4200  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  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 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-1st 6699  df-2nd 6700  df-recs 6960  df-rdg 6994  df-1o 7048  df-oadd 7052  df-er 7229  df-en 7436  df-dom 7437  df-sdom 7438  df-fin 7439  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-nn 10453  df-2 10511  df-3 10512  df-4 10513  df-5 10514  df-6 10515  df-7 10516  df-8 10517  df-9 10518  df-10 10519  df-n0 10713  df-z 10782  df-dec 10896  df-uz 11002  df-fz 11594  df-struct 14636  df-ndx 14637  df-slot 14638  df-base 14639  df-sets 14640  df-tset 14721  df-unif 14725  df-rest 14830  df-topn 14831  df-ust 20788  df-utop 20819  df-tus 20846
This theorem is referenced by:  tusbas  20856  tusunif  20857  tustopn  20859  tususp  20860
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