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Theorem om1val 18544
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 18520 . . . 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 10 . . 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 732 . . . . . . . 8  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  -> 
j  =  J )
54oveq2d 5890 . . . . . . 7  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  -> 
( II  Cn  j
)  =  ( II 
Cn  J ) )
6 simprr 733 . . . . . . . . 9  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  -> 
y  =  Y )
76eqeq2d 2307 . . . . . . . 8  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  -> 
( ( f ` 
0 )  =  y  <-> 
( f `  0
)  =  Y ) )
86eqeq2d 2307 . . . . . . . 8  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  -> 
( ( f ` 
1 )  =  y  <-> 
( f `  1
)  =  Y ) )
97, 8anbi12d 691 . . . . . . 7  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  -> 
( ( ( f `
 0 )  =  y  /\  ( f `
 1 )  =  y )  <->  ( (
f `  0 )  =  Y  /\  (
f `  1 )  =  Y ) ) )
105, 9rabeqbidv 2796 . . . . . 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 451 . . . . . 6  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  ->  B  =  { f  e.  ( II  Cn  J
)  |  ( ( f `  0 )  =  Y  /\  (
f `  1 )  =  Y ) } )
1310, 12eqtr4d 2331 . . . . 5  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  ->  { f  e.  ( II  Cn  j )  |  ( ( f `
 0 )  =  y  /\  ( f `
 1 )  =  y ) }  =  B )
1413opeq2d 3819 . . . 4  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  ->  <. ( Base `  ndx ) ,  { f  e.  ( II  Cn  j
)  |  ( ( f `  0 )  =  y  /\  (
f `  1 )  =  y ) }
>.  =  <. ( Base `  ndx ) ,  B >. )
154fveq2d 5545 . . . . . 6  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  -> 
( *p `  j
)  =  ( *p
`  J ) )
16 om1val.p . . . . . . 7  |-  ( ph  ->  .+  =  ( *p
`  J ) )
1716adantr 451 . . . . . 6  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  ->  .+  =  ( *p `  J ) )
1815, 17eqtr4d 2331 . . . . 5  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  -> 
( *p `  j
)  =  .+  )
1918opeq2d 3819 . . . 4  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  ->  <. ( +g  `  ndx ) ,  ( *p `  j ) >.  =  <. ( +g  `  ndx ) ,  .+  >. )
204oveq1d 5889 . . . . . 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 451 . . . . . 6  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  ->  K  =  ( J  ^ k o  II ) )
2320, 22eqtr4d 2331 . . . . 5  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  -> 
( j  ^ k o  II )  =  K )
2423opeq2d 3819 . . . 4  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  ->  <. (TopSet `  ndx ) ,  ( j  ^ k o  II ) >.  =  <. (TopSet `  ndx ) ,  K >. )
2514, 19, 24tpeq123d 3734 . . 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 3852 . . . . 5  |-  ( j  =  J  ->  U. j  =  U. J )
2726adantl 452 . . . 4  |-  ( (
ph  /\  j  =  J )  ->  U. j  =  U. J )
28 om1val.j . . . . . 6  |-  ( ph  ->  J  e.  (TopOn `  X ) )
29 toponuni 16681 . . . . . 6  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
3028, 29syl 15 . . . . 5  |-  ( ph  ->  X  =  U. J
)
3130adantr 451 . . . 4  |-  ( (
ph  /\  j  =  J )  ->  X  =  U. J )
3227, 31eqtr4d 2331 . . 3  |-  ( (
ph  /\  j  =  J )  ->  U. j  =  X )
33 topontop 16680 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  Top )
3428, 33syl 15 . . 3  |-  ( ph  ->  J  e.  Top )
35 om1val.y . . 3  |-  ( ph  ->  Y  e.  X )
36 tpex 4535 . . . 4  |-  { <. (
Base `  ndx ) ,  B >. ,  <. ( +g  `  ndx ) , 
.+  >. ,  <. (TopSet ` 
ndx ) ,  K >. }  e.  _V
3736a1i 10 . . 3  |-  ( ph  ->  { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  .+  >. ,  <. (TopSet `  ndx ) ,  K >. }  e.  _V )
383, 25, 32, 34, 35, 37ovmpt2dx 5990 . 2  |-  ( ph  ->  ( J  Om 1  Y )  =  { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  .+  >. ,  <. (TopSet `  ndx ) ,  K >. } )
391, 38syl5eq 2340 1  |-  ( ph  ->  O  =  { <. (
Base `  ndx ) ,  B >. ,  <. ( +g  `  ndx ) , 
.+  >. ,  <. (TopSet ` 
ndx ) ,  K >. } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   {crab 2560   _Vcvv 2801   {ctp 3655   <.cop 3656   U.cuni 3843   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   0cc0 8753   1c1 8754   ndxcnx 13161   Basecbs 13164   +g cplusg 13224  TopSetcts 13230   Topctop 16647  TopOnctopon 16648    Cn ccn 16970    ^ k o cxko 17272   IIcii 18395   *pcpco 18514    Om 1 comi 18515
This theorem is referenced by:  om1bas  18545  om1plusg  18548  om1tset  18549
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-topon 16655  df-om1 18520
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