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Theorem om1val 22116
Description: The definition of the loop space. (Contributed by Mario Carneiro, 10-Jul-2015.)
Hypotheses
Ref Expression
om1val.o  |-  O  =  ( J  Om1  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  ^ko  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  Om1  Y )
2 df-om1 22092 . . . 4  |-  Om1 
=  ( 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  ^ko  II ) >. } )
32a1i 11 . . 3  |-  ( ph  ->  Om1  =  ( 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  ^ko  II ) >. } ) )
4 simprl 769 . . . . . . . 8  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  -> 
j  =  J )
54oveq2d 6336 . . . . . . 7  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  -> 
( II  Cn  j
)  =  ( II 
Cn  J ) )
6 simprr 771 . . . . . . . . 9  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  -> 
y  =  Y )
76eqeq2d 2472 . . . . . . . 8  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  -> 
( ( f ` 
0 )  =  y  <-> 
( f `  0
)  =  Y ) )
86eqeq2d 2472 . . . . . . . 8  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  -> 
( ( f ` 
1 )  =  y  <-> 
( f `  1
)  =  Y ) )
97, 8anbi12d 722 . . . . . . 7  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  -> 
( ( ( f `
 0 )  =  y  /\  ( f `
 1 )  =  y )  <->  ( (
f `  0 )  =  Y  /\  (
f `  1 )  =  Y ) ) )
105, 9rabeqbidv 3052 . . . . . 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 471 . . . . . 6  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  ->  B  =  { f  e.  ( II  Cn  J
)  |  ( ( f `  0 )  =  Y  /\  (
f `  1 )  =  Y ) } )
1310, 12eqtr4d 2499 . . . . 5  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  ->  { f  e.  ( II  Cn  j )  |  ( ( f `
 0 )  =  y  /\  ( f `
 1 )  =  y ) }  =  B )
1413opeq2d 4187 . . . 4  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  ->  <. ( Base `  ndx ) ,  { f  e.  ( II  Cn  j
)  |  ( ( f `  0 )  =  y  /\  (
f `  1 )  =  y ) }
>.  =  <. ( Base `  ndx ) ,  B >. )
154fveq2d 5896 . . . . . 6  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  -> 
( *p `  j
)  =  ( *p
`  J ) )
16 om1val.p . . . . . . 7  |-  ( ph  ->  .+  =  ( *p
`  J ) )
1716adantr 471 . . . . . 6  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  ->  .+  =  ( *p `  J ) )
1815, 17eqtr4d 2499 . . . . 5  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  -> 
( *p `  j
)  =  .+  )
1918opeq2d 4187 . . . 4  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  ->  <. ( +g  `  ndx ) ,  ( *p `  j ) >.  =  <. ( +g  `  ndx ) ,  .+  >. )
204oveq1d 6335 . . . . . 6  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  -> 
( j  ^ko  II )  =  ( J  ^ko  II ) )
21 om1val.k . . . . . . 7  |-  ( ph  ->  K  =  ( J  ^ko  II ) )
2221adantr 471 . . . . . 6  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  ->  K  =  ( J  ^ko  II ) )
2320, 22eqtr4d 2499 . . . . 5  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  -> 
( j  ^ko  II )  =  K )
2423opeq2d 4187 . . . 4  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  ->  <. (TopSet `  ndx ) ,  ( j  ^ko  II ) >.  =  <. (TopSet `  ndx ) ,  K >. )
2514, 19, 24tpeq123d 4079 . . 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  ^ko  II ) >. }  =  { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  .+  >. ,  <. (TopSet `  ndx ) ,  K >. } )
26 unieq 4220 . . . . 5  |-  ( j  =  J  ->  U. j  =  U. J )
2726adantl 472 . . . 4  |-  ( (
ph  /\  j  =  J )  ->  U. j  =  U. J )
28 om1val.j . . . . . 6  |-  ( ph  ->  J  e.  (TopOn `  X ) )
29 toponuni 19997 . . . . . 6  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
3028, 29syl 17 . . . . 5  |-  ( ph  ->  X  =  U. J
)
3130adantr 471 . . . 4  |-  ( (
ph  /\  j  =  J )  ->  X  =  U. J )
3227, 31eqtr4d 2499 . . 3  |-  ( (
ph  /\  j  =  J )  ->  U. j  =  X )
33 topontop 19996 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  Top )
3428, 33syl 17 . . 3  |-  ( ph  ->  J  e.  Top )
35 om1val.y . . 3  |-  ( ph  ->  Y  e.  X )
36 tpex 6622 . . . 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 6455 . 2  |-  ( ph  ->  ( J  Om1  Y )  =  { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  .+  >. ,  <. (TopSet `  ndx ) ,  K >. } )
391, 38syl5eq 2508 1  |-  ( ph  ->  O  =  { <. (
Base `  ndx ) ,  B >. ,  <. ( +g  `  ndx ) , 
.+  >. ,  <. (TopSet ` 
ndx ) ,  K >. } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 375    = wceq 1455    e. wcel 1898   {crab 2753   _Vcvv 3057   {ctp 3984   <.cop 3986   U.cuni 4212   ` cfv 5605  (class class class)co 6320    |-> cmpt2 6322   0cc0 9570   1c1 9571   ndxcnx 15173   Basecbs 15176   +g cplusg 15245  TopSetcts 15251   Topctop 19972  TopOnctopon 19973    Cn ccn 20295    ^ko cxko 20631   IIcii 21962   *pcpco 22086    Om1 comi 22087
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4541  ax-nul 4550  ax-pow 4598  ax-pr 4656  ax-un 6615
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-sbc 3280  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-br 4419  df-opab 4478  df-mpt 4479  df-id 4771  df-xp 4862  df-rel 4863  df-cnv 4864  df-co 4865  df-dm 4866  df-iota 5569  df-fun 5607  df-fv 5613  df-ov 6323  df-oprab 6324  df-mpt2 6325  df-topon 19978  df-om1 22092
This theorem is referenced by:  om1bas  22117  om1plusg  22120  om1tset  22121
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