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Theorem om1val 20602
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 20578 . . . 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 755 . . . . . . . 8  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  -> 
j  =  J )
54oveq2d 6107 . . . . . . 7  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  -> 
( II  Cn  j
)  =  ( II 
Cn  J ) )
6 simprr 756 . . . . . . . . 9  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  -> 
y  =  Y )
76eqeq2d 2454 . . . . . . . 8  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  -> 
( ( f ` 
0 )  =  y  <-> 
( f `  0
)  =  Y ) )
86eqeq2d 2454 . . . . . . . 8  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  -> 
( ( f ` 
1 )  =  y  <-> 
( f `  1
)  =  Y ) )
97, 8anbi12d 710 . . . . . . 7  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  -> 
( ( ( f `
 0 )  =  y  /\  ( f `
 1 )  =  y )  <->  ( (
f `  0 )  =  Y  /\  (
f `  1 )  =  Y ) ) )
105, 9rabeqbidv 2967 . . . . . 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 465 . . . . . 6  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  ->  B  =  { f  e.  ( II  Cn  J
)  |  ( ( f `  0 )  =  Y  /\  (
f `  1 )  =  Y ) } )
1310, 12eqtr4d 2478 . . . . 5  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  ->  { f  e.  ( II  Cn  j )  |  ( ( f `
 0 )  =  y  /\  ( f `
 1 )  =  y ) }  =  B )
1413opeq2d 4066 . . . 4  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  ->  <. ( Base `  ndx ) ,  { f  e.  ( II  Cn  j
)  |  ( ( f `  0 )  =  y  /\  (
f `  1 )  =  y ) }
>.  =  <. ( Base `  ndx ) ,  B >. )
154fveq2d 5695 . . . . . 6  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  -> 
( *p `  j
)  =  ( *p
`  J ) )
16 om1val.p . . . . . . 7  |-  ( ph  ->  .+  =  ( *p
`  J ) )
1716adantr 465 . . . . . 6  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  ->  .+  =  ( *p `  J ) )
1815, 17eqtr4d 2478 . . . . 5  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  -> 
( *p `  j
)  =  .+  )
1918opeq2d 4066 . . . 4  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  ->  <. ( +g  `  ndx ) ,  ( *p `  j ) >.  =  <. ( +g  `  ndx ) ,  .+  >. )
204oveq1d 6106 . . . . . 6  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  -> 
( j  ^ko  II )  =  ( J  ^ko  II ) )
21 om1val.k . . . . . . 7  |-  ( ph  ->  K  =  ( J  ^ko  II ) )
2221adantr 465 . . . . . 6  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  ->  K  =  ( J  ^ko  II ) )
2320, 22eqtr4d 2478 . . . . 5  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  -> 
( j  ^ko  II )  =  K )
2423opeq2d 4066 . . . 4  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  ->  <. (TopSet `  ndx ) ,  ( j  ^ko  II ) >.  =  <. (TopSet `  ndx ) ,  K >. )
2514, 19, 24tpeq123d 3969 . . 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 4099 . . . . 5  |-  ( j  =  J  ->  U. j  =  U. J )
2726adantl 466 . . . 4  |-  ( (
ph  /\  j  =  J )  ->  U. j  =  U. J )
28 om1val.j . . . . . 6  |-  ( ph  ->  J  e.  (TopOn `  X ) )
29 toponuni 18532 . . . . . 6  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
3028, 29syl 16 . . . . 5  |-  ( ph  ->  X  =  U. J
)
3130adantr 465 . . . 4  |-  ( (
ph  /\  j  =  J )  ->  X  =  U. J )
3227, 31eqtr4d 2478 . . 3  |-  ( (
ph  /\  j  =  J )  ->  U. j  =  X )
33 topontop 18531 . . . 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 6379 . . . 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 6217 . 2  |-  ( ph  ->  ( J  Om1  Y )  =  { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  .+  >. ,  <. (TopSet `  ndx ) ,  K >. } )
391, 38syl5eq 2487 1  |-  ( ph  ->  O  =  { <. (
Base `  ndx ) ,  B >. ,  <. ( +g  `  ndx ) , 
.+  >. ,  <. (TopSet ` 
ndx ) ,  K >. } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   {crab 2719   _Vcvv 2972   {ctp 3881   <.cop 3883   U.cuni 4091   ` cfv 5418  (class class class)co 6091    e. cmpt2 6093   0cc0 9282   1c1 9283   ndxcnx 14171   Basecbs 14174   +g cplusg 14238  TopSetcts 14244   Topctop 18498  TopOnctopon 18499    Cn ccn 18828    ^ko cxko 19134   IIcii 20451   *pcpco 20572    Om1 comi 20573
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-iota 5381  df-fun 5420  df-fv 5426  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-topon 18506  df-om1 20578
This theorem is referenced by:  om1bas  20603  om1plusg  20606  om1tset  20607
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