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Theorem tuslem 19984
Description: Lemma for tusbas 19985, tusunif 19986, and tustopn 19988. (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 14342 . . . 4  |-  Base  = Slot  ( Base `  ndx )
2 1re 9500 . . . . . 6  |-  1  e.  RR
3 1lt9 10638 . . . . . 6  |-  1  <  9
42, 3ltneii 9602 . . . . 5  |-  1  =/=  9
5 basendx 14346 . . . . . 6  |-  ( Base `  ndx )  =  1
6 tsetndx 14448 . . . . . 6  |-  (TopSet `  ndx )  =  9
75, 6neeq12i 2741 . . . . 5  |-  ( (
Base `  ndx )  =/=  (TopSet `  ndx )  <->  1  =/=  9 )
84, 7mpbir 209 . . . 4  |-  ( Base `  ndx )  =/=  (TopSet ` 
ndx )
91, 8setsnid 14338 . . 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 19942 . . . 4  |-  ( U  e.  (UnifOn `  X
)  ->  X  =  dom  U. U )
11 uniexg 6490 . . . . 5  |-  ( U  e.  (UnifOn `  X
)  ->  U. U  e. 
_V )
12 dmexg 6622 . . . . 5  |-  ( U. U  e.  _V  ->  dom  U. U  e.  _V )
13 eqid 2454 . . . . . 6  |-  { <. (
Base `  ndx ) ,  dom  U. U >. , 
<. ( UnifSet `  ndx ) ,  U >. }  =  { <. ( Base `  ndx ) ,  dom  U. U >. ,  <. ( UnifSet `  ndx ) ,  U >. }
14 df-unif 14384 . . . . . 6  |-  UnifSet  = Slot ; 1 3
15 1nn 10448 . . . . . . 7  |-  1  e.  NN
16 3nn0 10712 . . . . . . 7  |-  3  e.  NN0
17 1nn0 10710 . . . . . . 7  |-  1  e.  NN0
18 1lt10 10647 . . . . . . 7  |-  1  <  10
1915, 16, 17, 18declti 10895 . . . . . 6  |-  1  < ; 1
3
20 3nn 10595 . . . . . . 7  |-  3  e.  NN
2117, 20decnncl 10883 . . . . . 6  |- ; 1 3  e.  NN
2213, 14, 19, 212strbas 14398 . . . . 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 2495 . . 3  |-  ( U  e.  (UnifOn `  X
)  ->  X  =  ( Base `  { <. ( Base `  ndx ) ,  dom  U. U >. , 
<. ( UnifSet `  ndx ) ,  U >. } ) )
25 tuslem.k . . . . 5  |-  K  =  (toUnifSp `  U )
26 tusval 19983 . . . . 5  |-  ( U  e.  (UnifOn `  X
)  ->  (toUnifSp `  U
)  =  ( {
<. ( Base `  ndx ) ,  dom  U. U >. ,  <. ( UnifSet `  ndx ) ,  U >. } sSet  <. (TopSet `  ndx ) ,  (unifTop `  U ) >. ) )
2725, 26syl5eq 2507 . . . 4  |-  ( U  e.  (UnifOn `  X
)  ->  K  =  ( { <. ( Base `  ndx ) ,  dom  U. U >. ,  <. ( UnifSet `  ndx ) ,  U >. } sSet  <. (TopSet `  ndx ) ,  (unifTop `  U ) >. ) )
2827fveq2d 5806 . . 3  |-  ( U  e.  (UnifOn `  X
)  ->  ( Base `  K )  =  (
Base `  ( { <. ( Base `  ndx ) ,  dom  U. U >. ,  <. ( UnifSet `  ndx ) ,  U >. } sSet  <. (TopSet `  ndx ) ,  (unifTop `  U ) >. ) ) )
299, 24, 283eqtr4a 2521 . 2  |-  ( U  e.  (UnifOn `  X
)  ->  X  =  ( Base `  K )
)
30 unifid 14467 . . . 4  |-  UnifSet  = Slot  ( UnifSet
`  ndx )
31 9re 10523 . . . . . 6  |-  9  e.  RR
32 9nn0 10718 . . . . . . 7  |-  9  e.  NN0
33 9lt10 10639 . . . . . . 7  |-  9  <  10
3415, 16, 32, 33declti 10895 . . . . . 6  |-  9  < ; 1
3
3531, 34gtneii 9601 . . . . 5  |- ; 1 3  =/=  9
36 unifndx 14466 . . . . . 6  |-  ( UnifSet ` 
ndx )  = ; 1 3
3736, 6neeq12i 2741 . . . . 5  |-  ( (
UnifSet `  ndx )  =/=  (TopSet `  ndx )  <-> ; 1 3  =/=  9
)
3835, 37mpbir 209 . . . 4  |-  ( UnifSet ` 
ndx )  =/=  (TopSet ` 
ndx )
3930, 38setsnid 14338 . . 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 14399 . . 3  |-  ( U  e.  (UnifOn `  X
)  ->  U  =  ( UnifSet `  { <. ( Base `  ndx ) ,  dom  U. U >. , 
<. ( UnifSet `  ndx ) ,  U >. } ) )
4127fveq2d 5806 . . 3  |-  ( U  e.  (UnifOn `  X
)  ->  ( UnifSet `  K )  =  (
UnifSet `  ( { <. (
Base `  ndx ) ,  dom  U. U >. , 
<. ( UnifSet `  ndx ) ,  U >. } sSet  <. (TopSet `  ndx ) ,  (unifTop `  U
) >. ) ) )
4239, 40, 413eqtr4a 2521 . 2  |-  ( U  e.  (UnifOn `  X
)  ->  U  =  ( UnifSet `  K )
)
4327fveq2d 5806 . . . 4  |-  ( U  e.  (UnifOn `  X
)  ->  (TopSet `  K
)  =  (TopSet `  ( { <. ( Base `  ndx ) ,  dom  U. U >. ,  <. ( UnifSet `  ndx ) ,  U >. } sSet  <. (TopSet `  ndx ) ,  (unifTop `  U ) >. ) ) )
44 prex 4645 . . . . 5  |-  { <. (
Base `  ndx ) ,  dom  U. U >. , 
<. ( UnifSet `  ndx ) ,  U >. }  e.  _V
45 fvex 5812 . . . . 5  |-  (unifTop `  U
)  e.  _V
46 tsetid 14449 . . . . . 6  |- TopSet  = Slot  (TopSet ` 
ndx )
4746setsid 14337 . . . . 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 672 . . . 4  |-  (unifTop `  U
)  =  (TopSet `  ( { <. ( Base `  ndx ) ,  dom  U. U >. ,  <. ( UnifSet `  ndx ) ,  U >. } sSet  <. (TopSet `  ndx ) ,  (unifTop `  U ) >. ) )
4943, 48syl6reqr 2514 . . 3  |-  ( U  e.  (UnifOn `  X
)  ->  (unifTop `  U
)  =  (TopSet `  K ) )
50 utopbas 19952 . . . . . 6  |-  ( U  e.  (UnifOn `  X
)  ->  X  =  U. (unifTop `  U )
)
5149unieqd 4212 . . . . . 6  |-  ( U  e.  (UnifOn `  X
)  ->  U. (unifTop `  U )  =  U. (TopSet `  K ) )
5250, 29, 513eqtr3rd 2504 . . . . 5  |-  ( U  e.  (UnifOn `  X
)  ->  U. (TopSet `  K )  =  (
Base `  K )
)
5352oveq2d 6219 . . . 4  |-  ( U  e.  (UnifOn `  X
)  ->  ( (TopSet `  K )t  U. (TopSet `  K
) )  =  ( (TopSet `  K )t  ( Base `  K ) ) )
54 fvex 5812 . . . . 5  |-  (TopSet `  K )  e.  _V
55 eqid 2454 . . . . . 6  |-  U. (TopSet `  K )  =  U. (TopSet `  K )
5655restid 14495 . . . . 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 2454 . . . . 5  |-  ( Base `  K )  =  (
Base `  K )
59 eqid 2454 . . . . 5  |-  (TopSet `  K )  =  (TopSet `  K )
6058, 59topnval 14496 . . . 4  |-  ( (TopSet `  K )t  ( Base `  K
) )  =  (
TopOpen `  K )
6153, 57, 603eqtr3g 2518 . . 3  |-  ( U  e.  (UnifOn `  X
)  ->  (TopSet `  K
)  =  ( TopOpen `  K ) )
6249, 61eqtrd 2495 . 2  |-  ( U  e.  (UnifOn `  X
)  ->  (unifTop `  U
)  =  ( TopOpen `  K ) )
6329, 42, 623jca 1168 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 965    = wceq 1370    e. wcel 1758    =/= wne 2648   _Vcvv 3078   {cpr 3990   <.cop 3994   U.cuni 4202   dom cdm 4951   ` cfv 5529  (class class class)co 6203   1c1 9398   3c3 10487   9c9 10493  ;cdc 10870   ndxcnx 14293   sSet csts 14294   Basecbs 14296  TopSetcts 14367   UnifSetcunif 14371   ↾t crest 14482   TopOpenctopn 14483  UnifOncust 19916  unifTopcutop 19947  toUnifSpctus 19972
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9453  ax-resscn 9454  ax-1cn 9455  ax-icn 9456  ax-addcl 9457  ax-addrcl 9458  ax-mulcl 9459  ax-mulrcl 9460  ax-mulcom 9461  ax-addass 9462  ax-mulass 9463  ax-distr 9464  ax-i2m1 9465  ax-1ne0 9466  ax-1rid 9467  ax-rnegex 9468  ax-rrecex 9469  ax-cnre 9470  ax-pre-lttri 9471  ax-pre-lttrn 9472  ax-pre-ltadd 9473  ax-pre-mulgt0 9474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-1o 7033  df-oadd 7037  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-pnf 9535  df-mnf 9536  df-xr 9537  df-ltxr 9538  df-le 9539  df-sub 9712  df-neg 9713  df-nn 10438  df-2 10495  df-3 10496  df-4 10497  df-5 10498  df-6 10499  df-7 10500  df-8 10501  df-9 10502  df-10 10503  df-n0 10695  df-z 10762  df-dec 10871  df-uz 10977  df-fz 11559  df-struct 14298  df-ndx 14299  df-slot 14300  df-base 14301  df-sets 14302  df-tset 14380  df-unif 14384  df-rest 14484  df-topn 14485  df-ust 19917  df-utop 19948  df-tus 19975
This theorem is referenced by:  tusbas  19985  tusunif  19986  tustopn  19988  tususp  19989
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