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Theorem om1val 19057
Description: The definition of the loop space. (Contributed by Mario Carneiro, 10-Jul-2015.)
Hypotheses
Ref Expression
om1val.o  |-  O  =  ( J  Om 1  Y )
om1val.b  |-  ( ph  ->  B  =  { f  e.  ( II  Cn  J )  |  ( ( f `  0
)  =  Y  /\  ( f `  1
)  =  Y ) } )
om1val.p  |-  ( ph  ->  .+  =  ( *p
`  J ) )
om1val.k  |-  ( ph  ->  K  =  ( J  ^ k o  II ) )
om1val.j  |-  ( ph  ->  J  e.  (TopOn `  X ) )
om1val.y  |-  ( ph  ->  Y  e.  X )
Assertion
Ref Expression
om1val  |-  ( ph  ->  O  =  { <. (
Base `  ndx ) ,  B >. ,  <. ( +g  `  ndx ) , 
.+  >. ,  <. (TopSet ` 
ndx ) ,  K >. } )
Distinct variable groups:    f, J    ph, f    f, Y
Allowed substitution hints:    B( f)    .+ ( f)    K( f)    O( f)    X( f)

Proof of Theorem om1val
Dummy variables  j 
y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 om1val.o . 2  |-  O  =  ( J  Om 1  Y )
2 df-om1 19033 . . . 4  |-  Om 1  =  ( j  e. 
Top ,  y  e.  U. j  |->  { <. ( Base `  ndx ) ,  { f  e.  ( II  Cn  j )  |  ( ( f `
 0 )  =  y  /\  ( f `
 1 )  =  y ) } >. , 
<. ( +g  `  ndx ) ,  ( *p `  j ) >. ,  <. (TopSet `  ndx ) ,  ( j  ^ k o  II ) >. } )
32a1i 11 . . 3  |-  ( ph  ->  Om 1  =  ( j  e.  Top , 
y  e.  U. j  |->  { <. ( Base `  ndx ) ,  { f  e.  ( II  Cn  j
)  |  ( ( f `  0 )  =  y  /\  (
f `  1 )  =  y ) }
>. ,  <. ( +g  ` 
ndx ) ,  ( *p `  j )
>. ,  <. (TopSet `  ndx ) ,  ( j  ^ k o  II ) >. } ) )
4 simprl 734 . . . . . . . 8  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  -> 
j  =  J )
54oveq2d 6099 . . . . . . 7  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  -> 
( II  Cn  j
)  =  ( II 
Cn  J ) )
6 simprr 735 . . . . . . . . 9  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  -> 
y  =  Y )
76eqeq2d 2449 . . . . . . . 8  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  -> 
( ( f ` 
0 )  =  y  <-> 
( f `  0
)  =  Y ) )
86eqeq2d 2449 . . . . . . . 8  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  -> 
( ( f ` 
1 )  =  y  <-> 
( f `  1
)  =  Y ) )
97, 8anbi12d 693 . . . . . . 7  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  -> 
( ( ( f `
 0 )  =  y  /\  ( f `
 1 )  =  y )  <->  ( (
f `  0 )  =  Y  /\  (
f `  1 )  =  Y ) ) )
105, 9rabeqbidv 2953 . . . . . 6  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  ->  { f  e.  ( II  Cn  j )  |  ( ( f `
 0 )  =  y  /\  ( f `
 1 )  =  y ) }  =  { f  e.  ( II  Cn  J )  |  ( ( f `
 0 )  =  Y  /\  ( f `
 1 )  =  Y ) } )
11 om1val.b . . . . . . 7  |-  ( ph  ->  B  =  { f  e.  ( II  Cn  J )  |  ( ( f `  0
)  =  Y  /\  ( f `  1
)  =  Y ) } )
1211adantr 453 . . . . . 6  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  ->  B  =  { f  e.  ( II  Cn  J
)  |  ( ( f `  0 )  =  Y  /\  (
f `  1 )  =  Y ) } )
1310, 12eqtr4d 2473 . . . . 5  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  ->  { f  e.  ( II  Cn  j )  |  ( ( f `
 0 )  =  y  /\  ( f `
 1 )  =  y ) }  =  B )
1413opeq2d 3993 . . . 4  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  ->  <. ( Base `  ndx ) ,  { f  e.  ( II  Cn  j
)  |  ( ( f `  0 )  =  y  /\  (
f `  1 )  =  y ) }
>.  =  <. ( Base `  ndx ) ,  B >. )
154fveq2d 5734 . . . . . 6  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  -> 
( *p `  j
)  =  ( *p
`  J ) )
16 om1val.p . . . . . . 7  |-  ( ph  ->  .+  =  ( *p
`  J ) )
1716adantr 453 . . . . . 6  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  ->  .+  =  ( *p `  J ) )
1815, 17eqtr4d 2473 . . . . 5  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  -> 
( *p `  j
)  =  .+  )
1918opeq2d 3993 . . . 4  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  ->  <. ( +g  `  ndx ) ,  ( *p `  j ) >.  =  <. ( +g  `  ndx ) ,  .+  >. )
204oveq1d 6098 . . . . . 6  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  -> 
( j  ^ k o  II )  =  ( J  ^ k o  II ) )
21 om1val.k . . . . . . 7  |-  ( ph  ->  K  =  ( J  ^ k o  II ) )
2221adantr 453 . . . . . 6  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  ->  K  =  ( J  ^ k o  II ) )
2320, 22eqtr4d 2473 . . . . 5  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  -> 
( j  ^ k o  II )  =  K )
2423opeq2d 3993 . . . 4  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  ->  <. (TopSet `  ndx ) ,  ( j  ^ k o  II ) >.  =  <. (TopSet `  ndx ) ,  K >. )
2514, 19, 24tpeq123d 3900 . . 3  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  ->  { <. ( Base `  ndx ) ,  { f  e.  ( II  Cn  j
)  |  ( ( f `  0 )  =  y  /\  (
f `  1 )  =  y ) }
>. ,  <. ( +g  ` 
ndx ) ,  ( *p `  j )
>. ,  <. (TopSet `  ndx ) ,  ( j  ^ k o  II ) >. }  =  { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  .+  >. ,  <. (TopSet `  ndx ) ,  K >. } )
26 unieq 4026 . . . . 5  |-  ( j  =  J  ->  U. j  =  U. J )
2726adantl 454 . . . 4  |-  ( (
ph  /\  j  =  J )  ->  U. j  =  U. J )
28 om1val.j . . . . . 6  |-  ( ph  ->  J  e.  (TopOn `  X ) )
29 toponuni 16994 . . . . . 6  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
3028, 29syl 16 . . . . 5  |-  ( ph  ->  X  =  U. J
)
3130adantr 453 . . . 4  |-  ( (
ph  /\  j  =  J )  ->  X  =  U. J )
3227, 31eqtr4d 2473 . . 3  |-  ( (
ph  /\  j  =  J )  ->  U. j  =  X )
33 topontop 16993 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  Top )
3428, 33syl 16 . . 3  |-  ( ph  ->  J  e.  Top )
35 om1val.y . . 3  |-  ( ph  ->  Y  e.  X )
36 tpex 4710 . . . 4  |-  { <. (
Base `  ndx ) ,  B >. ,  <. ( +g  `  ndx ) , 
.+  >. ,  <. (TopSet ` 
ndx ) ,  K >. }  e.  _V
3736a1i 11 . . 3  |-  ( ph  ->  { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  .+  >. ,  <. (TopSet `  ndx ) ,  K >. }  e.  _V )
383, 25, 32, 34, 35, 37ovmpt2dx 6202 . 2  |-  ( ph  ->  ( J  Om 1  Y )  =  { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  .+  >. ,  <. (TopSet `  ndx ) ,  K >. } )
391, 38syl5eq 2482 1  |-  ( ph  ->  O  =  { <. (
Base `  ndx ) ,  B >. ,  <. ( +g  `  ndx ) , 
.+  >. ,  <. (TopSet ` 
ndx ) ,  K >. } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   {crab 2711   _Vcvv 2958   {ctp 3818   <.cop 3819   U.cuni 4017   ` cfv 5456  (class class class)co 6083    e. cmpt2 6085   0cc0 8992   1c1 8993   ndxcnx 13468   Basecbs 13471   +g cplusg 13531  TopSetcts 13537   Topctop 16960  TopOnctopon 16961    Cn ccn 17290    ^ k o cxko 17595   IIcii 18907   *pcpco 19027    Om 1 comi 19028
This theorem is referenced by:  om1bas  19058  om1plusg  19061  om1tset  19062
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-iota 5420  df-fun 5458  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-topon 16968  df-om1 19033
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