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Theorem om1val 22060
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 22036 . . . 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 762 . . . . . . . 8  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  -> 
j  =  J )
54oveq2d 6322 . . . . . . 7  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  -> 
( II  Cn  j
)  =  ( II 
Cn  J ) )
6 simprr 764 . . . . . . . . 9  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  -> 
y  =  Y )
76eqeq2d 2436 . . . . . . . 8  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  -> 
( ( f ` 
0 )  =  y  <-> 
( f `  0
)  =  Y ) )
86eqeq2d 2436 . . . . . . . 8  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  -> 
( ( f ` 
1 )  =  y  <-> 
( f `  1
)  =  Y ) )
97, 8anbi12d 715 . . . . . . 7  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  -> 
( ( ( f `
 0 )  =  y  /\  ( f `
 1 )  =  y )  <->  ( (
f `  0 )  =  Y  /\  (
f `  1 )  =  Y ) ) )
105, 9rabeqbidv 3075 . . . . . 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 466 . . . . . 6  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  ->  B  =  { f  e.  ( II  Cn  J
)  |  ( ( f `  0 )  =  Y  /\  (
f `  1 )  =  Y ) } )
1310, 12eqtr4d 2466 . . . . 5  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  ->  { f  e.  ( II  Cn  j )  |  ( ( f `
 0 )  =  y  /\  ( f `
 1 )  =  y ) }  =  B )
1413opeq2d 4194 . . . 4  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  ->  <. ( Base `  ndx ) ,  { f  e.  ( II  Cn  j
)  |  ( ( f `  0 )  =  y  /\  (
f `  1 )  =  y ) }
>.  =  <. ( Base `  ndx ) ,  B >. )
154fveq2d 5886 . . . . . 6  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  -> 
( *p `  j
)  =  ( *p
`  J ) )
16 om1val.p . . . . . . 7  |-  ( ph  ->  .+  =  ( *p
`  J ) )
1716adantr 466 . . . . . 6  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  ->  .+  =  ( *p `  J ) )
1815, 17eqtr4d 2466 . . . . 5  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  -> 
( *p `  j
)  =  .+  )
1918opeq2d 4194 . . . 4  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  ->  <. ( +g  `  ndx ) ,  ( *p `  j ) >.  =  <. ( +g  `  ndx ) ,  .+  >. )
204oveq1d 6321 . . . . . 6  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  -> 
( j  ^ko  II )  =  ( J  ^ko  II ) )
21 om1val.k . . . . . . 7  |-  ( ph  ->  K  =  ( J  ^ko  II ) )
2221adantr 466 . . . . . 6  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  ->  K  =  ( J  ^ko  II ) )
2320, 22eqtr4d 2466 . . . . 5  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  -> 
( j  ^ko  II )  =  K )
2423opeq2d 4194 . . . 4  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  ->  <. (TopSet `  ndx ) ,  ( j  ^ko  II ) >.  =  <. (TopSet `  ndx ) ,  K >. )
2514, 19, 24tpeq123d 4094 . . 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 4227 . . . . 5  |-  ( j  =  J  ->  U. j  =  U. J )
2726adantl 467 . . . 4  |-  ( (
ph  /\  j  =  J )  ->  U. j  =  U. J )
28 om1val.j . . . . . 6  |-  ( ph  ->  J  e.  (TopOn `  X ) )
29 toponuni 19941 . . . . . 6  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
3028, 29syl 17 . . . . 5  |-  ( ph  ->  X  =  U. J
)
3130adantr 466 . . . 4  |-  ( (
ph  /\  j  =  J )  ->  X  =  U. J )
3227, 31eqtr4d 2466 . . 3  |-  ( (
ph  /\  j  =  J )  ->  U. j  =  X )
33 topontop 19940 . . . 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 6605 . . . 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 6438 . 2  |-  ( ph  ->  ( J  Om1  Y )  =  { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  .+  >. ,  <. (TopSet `  ndx ) ,  K >. } )
391, 38syl5eq 2475 1  |-  ( ph  ->  O  =  { <. (
Base `  ndx ) ,  B >. ,  <. ( +g  `  ndx ) , 
.+  >. ,  <. (TopSet ` 
ndx ) ,  K >. } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1872   {crab 2775   _Vcvv 3080   {ctp 4002   <.cop 4004   U.cuni 4219   ` cfv 5601  (class class class)co 6306    |-> cmpt2 6308   0cc0 9547   1c1 9548   ndxcnx 15118   Basecbs 15121   +g cplusg 15190  TopSetcts 15196   Topctop 19916  TopOnctopon 19917    Cn ccn 20239    ^ko cxko 20575   IIcii 21906   *pcpco 22030    Om1 comi 22031
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6598
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-br 4424  df-opab 4483  df-mpt 4484  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-iota 5565  df-fun 5603  df-fv 5609  df-ov 6309  df-oprab 6310  df-mpt2 6311  df-topon 19922  df-om1 22036
This theorem is referenced by:  om1bas  22061  om1plusg  22064  om1tset  22065
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