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Theorem pi1xfrf 18551
Description: Functionality of the loop transfer function on the equivalence class of a path. (Contributed by Mario Carneiro, 23-Dec-2016.)
Hypotheses
Ref Expression
pi1xfr.p  |-  P  =  ( J  pi 1 
( F `  0
) )
pi1xfr.q  |-  Q  =  ( J  pi 1 
( F `  1
) )
pi1xfr.b  |-  B  =  ( Base `  P
)
pi1xfr.g  |-  G  =  ran  ( g  e. 
U. B  |->  <. [ g ] (  ~=ph  `  J
) ,  [ ( I ( *p `  J ) ( g ( *p `  J
) F ) ) ] (  ~=ph  `  J
) >. )
pi1xfr.j  |-  ( ph  ->  J  e.  (TopOn `  X ) )
pi1xfr.f  |-  ( ph  ->  F  e.  ( II 
Cn  J ) )
pi1xfrval.i  |-  ( ph  ->  I  e.  ( II 
Cn  J ) )
pi1xfrval.1  |-  ( ph  ->  ( F `  1
)  =  ( I `
 0 ) )
pi1xfrval.2  |-  ( ph  ->  ( I `  1
)  =  ( F `
 0 ) )
Assertion
Ref Expression
pi1xfrf  |-  ( ph  ->  G : B --> ( Base `  Q ) )
Distinct variable groups:    B, g    g, F    g, I    ph, g    g, J    P, g    Q, g
Allowed substitution hints:    G( g)    X( g)

Proof of Theorem pi1xfrf
Dummy variables  h  s are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pi1xfr.g . . . 4  |-  G  =  ran  ( g  e. 
U. B  |->  <. [ g ] (  ~=ph  `  J
) ,  [ ( I ( *p `  J ) ( g ( *p `  J
) F ) ) ] (  ~=ph  `  J
) >. )
2 pi1xfr.p . . . . 5  |-  P  =  ( J  pi 1 
( F `  0
) )
3 pi1xfr.b . . . . 5  |-  B  =  ( Base `  P
)
4 pi1xfr.j . . . . . 6  |-  ( ph  ->  J  e.  (TopOn `  X ) )
54adantr 451 . . . . 5  |-  ( (
ph  /\  g  e.  U. B )  ->  J  e.  (TopOn `  X )
)
6 iitopon 18383 . . . . . . . . 9  |-  II  e.  (TopOn `  ( 0 [,] 1 ) )
76a1i 10 . . . . . . . 8  |-  ( ph  ->  II  e.  (TopOn `  ( 0 [,] 1
) ) )
8 pi1xfr.f . . . . . . . 8  |-  ( ph  ->  F  e.  ( II 
Cn  J ) )
9 cnf2 16979 . . . . . . . 8  |-  ( ( II  e.  (TopOn `  ( 0 [,] 1
) )  /\  J  e.  (TopOn `  X )  /\  F  e.  (
II  Cn  J )
)  ->  F :
( 0 [,] 1
) --> X )
107, 4, 8, 9syl3anc 1182 . . . . . . 7  |-  ( ph  ->  F : ( 0 [,] 1 ) --> X )
11 0elunit 10754 . . . . . . 7  |-  0  e.  ( 0 [,] 1
)
12 ffvelrn 5663 . . . . . . 7  |-  ( ( F : ( 0 [,] 1 ) --> X  /\  0  e.  ( 0 [,] 1 ) )  ->  ( F `  0 )  e.  X )
1310, 11, 12sylancl 643 . . . . . 6  |-  ( ph  ->  ( F `  0
)  e.  X )
1413adantr 451 . . . . 5  |-  ( (
ph  /\  g  e.  U. B )  ->  ( F `  0 )  e.  X )
153a1i 10 . . . . . . . 8  |-  ( ph  ->  B  =  ( Base `  P ) )
162, 4, 13, 15pi1eluni 18540 . . . . . . 7  |-  ( ph  ->  ( g  e.  U. B 
<->  ( g  e.  ( II  Cn  J )  /\  ( g ` 
0 )  =  ( F `  0 )  /\  ( g ` 
1 )  =  ( F `  0 ) ) ) )
1716biimpa 470 . . . . . 6  |-  ( (
ph  /\  g  e.  U. B )  ->  (
g  e.  ( II 
Cn  J )  /\  ( g `  0
)  =  ( F `
 0 )  /\  ( g `  1
)  =  ( F `
 0 ) ) )
1817simp1d 967 . . . . 5  |-  ( (
ph  /\  g  e.  U. B )  ->  g  e.  ( II  Cn  J
) )
1917simp2d 968 . . . . 5  |-  ( (
ph  /\  g  e.  U. B )  ->  (
g `  0 )  =  ( F ` 
0 ) )
2017simp3d 969 . . . . 5  |-  ( (
ph  /\  g  e.  U. B )  ->  (
g `  1 )  =  ( F ` 
0 ) )
212, 3, 5, 14, 18, 19, 20elpi1i 18544 . . . 4  |-  ( (
ph  /\  g  e.  U. B )  ->  [ g ] (  ~=ph  `  J
)  e.  B )
22 pi1xfr.q . . . . 5  |-  Q  =  ( J  pi 1 
( F `  1
) )
23 eqid 2283 . . . . 5  |-  ( Base `  Q )  =  (
Base `  Q )
24 1elunit 10755 . . . . . . 7  |-  1  e.  ( 0 [,] 1
)
25 ffvelrn 5663 . . . . . . 7  |-  ( ( F : ( 0 [,] 1 ) --> X  /\  1  e.  ( 0 [,] 1 ) )  ->  ( F `  1 )  e.  X )
2610, 24, 25sylancl 643 . . . . . 6  |-  ( ph  ->  ( F `  1
)  e.  X )
2726adantr 451 . . . . 5  |-  ( (
ph  /\  g  e.  U. B )  ->  ( F `  1 )  e.  X )
28 pi1xfrval.i . . . . . . 7  |-  ( ph  ->  I  e.  ( II 
Cn  J ) )
2928adantr 451 . . . . . 6  |-  ( (
ph  /\  g  e.  U. B )  ->  I  e.  ( II  Cn  J
) )
308adantr 451 . . . . . . 7  |-  ( (
ph  /\  g  e.  U. B )  ->  F  e.  ( II  Cn  J
) )
3118, 30, 20pcocn 18515 . . . . . 6  |-  ( (
ph  /\  g  e.  U. B )  ->  (
g ( *p `  J ) F )  e.  ( II  Cn  J ) )
3218, 30pco0 18512 . . . . . . 7  |-  ( (
ph  /\  g  e.  U. B )  ->  (
( g ( *p
`  J ) F ) `  0 )  =  ( g ` 
0 ) )
33 pi1xfrval.2 . . . . . . . 8  |-  ( ph  ->  ( I `  1
)  =  ( F `
 0 ) )
3433adantr 451 . . . . . . 7  |-  ( (
ph  /\  g  e.  U. B )  ->  (
I `  1 )  =  ( F ` 
0 ) )
3519, 32, 343eqtr4rd 2326 . . . . . 6  |-  ( (
ph  /\  g  e.  U. B )  ->  (
I `  1 )  =  ( ( g ( *p `  J
) F ) ` 
0 ) )
3629, 31, 35pcocn 18515 . . . . 5  |-  ( (
ph  /\  g  e.  U. B )  ->  (
I ( *p `  J ) ( g ( *p `  J
) F ) )  e.  ( II  Cn  J ) )
3729, 31pco0 18512 . . . . . 6  |-  ( (
ph  /\  g  e.  U. B )  ->  (
( I ( *p
`  J ) ( g ( *p `  J ) F ) ) `  0 )  =  ( I ` 
0 ) )
38 pi1xfrval.1 . . . . . . 7  |-  ( ph  ->  ( F `  1
)  =  ( I `
 0 ) )
3938adantr 451 . . . . . 6  |-  ( (
ph  /\  g  e.  U. B )  ->  ( F `  1 )  =  ( I ` 
0 ) )
4037, 39eqtr4d 2318 . . . . 5  |-  ( (
ph  /\  g  e.  U. B )  ->  (
( I ( *p
`  J ) ( g ( *p `  J ) F ) ) `  0 )  =  ( F ` 
1 ) )
4129, 31pco1 18513 . . . . . 6  |-  ( (
ph  /\  g  e.  U. B )  ->  (
( I ( *p
`  J ) ( g ( *p `  J ) F ) ) `  1 )  =  ( ( g ( *p `  J
) F ) ` 
1 ) )
4218, 30pco1 18513 . . . . . 6  |-  ( (
ph  /\  g  e.  U. B )  ->  (
( g ( *p
`  J ) F ) `  1 )  =  ( F ` 
1 ) )
4341, 42eqtrd 2315 . . . . 5  |-  ( (
ph  /\  g  e.  U. B )  ->  (
( I ( *p
`  J ) ( g ( *p `  J ) F ) ) `  1 )  =  ( F ` 
1 ) )
4422, 23, 5, 27, 36, 40, 43elpi1i 18544 . . . 4  |-  ( (
ph  /\  g  e.  U. B )  ->  [ ( I ( *p `  J ) ( g ( *p `  J
) F ) ) ] (  ~=ph  `  J
)  e.  ( Base `  Q ) )
45 eceq1 6696 . . . 4  |-  ( g  =  h  ->  [ g ] (  ~=ph  `  J
)  =  [ h ] (  ~=ph  `  J
) )
46 oveq1 5865 . . . . . 6  |-  ( g  =  h  ->  (
g ( *p `  J ) F )  =  ( h ( *p `  J ) F ) )
4746oveq2d 5874 . . . . 5  |-  ( g  =  h  ->  (
I ( *p `  J ) ( g ( *p `  J
) F ) )  =  ( I ( *p `  J ) ( h ( *p
`  J ) F ) ) )
48 eceq1 6696 . . . . 5  |-  ( ( I ( *p `  J ) ( g ( *p `  J
) F ) )  =  ( I ( *p `  J ) ( h ( *p
`  J ) F ) )  ->  [ ( I ( *p `  J ) ( g ( *p `  J
) F ) ) ] (  ~=ph  `  J
)  =  [ ( I ( *p `  J ) ( h ( *p `  J
) F ) ) ] (  ~=ph  `  J
) )
4947, 48syl 15 . . . 4  |-  ( g  =  h  ->  [ ( I ( *p `  J ) ( g ( *p `  J
) F ) ) ] (  ~=ph  `  J
)  =  [ ( I ( *p `  J ) ( h ( *p `  J
) F ) ) ] (  ~=ph  `  J
) )
50 phtpcer 18493 . . . . . 6  |-  (  ~=ph  `  J )  Er  (
II  Cn  J )
5150a1i 10 . . . . 5  |-  ( (
ph  /\  ( g  e.  U. B  /\  h  e.  U. B  /\  [
g ] (  ~=ph  `  J )  =  [
h ] (  ~=ph  `  J ) ) )  ->  (  ~=ph  `  J
)  Er  ( II 
Cn  J ) )
52193ad2antr1 1120 . . . . . . 7  |-  ( (
ph  /\  ( g  e.  U. B  /\  h  e.  U. B  /\  [
g ] (  ~=ph  `  J )  =  [
h ] (  ~=ph  `  J ) ) )  ->  ( g ` 
0 )  =  ( F `  0 ) )
53183ad2antr1 1120 . . . . . . . 8  |-  ( (
ph  /\  ( g  e.  U. B  /\  h  e.  U. B  /\  [
g ] (  ~=ph  `  J )  =  [
h ] (  ~=ph  `  J ) ) )  ->  g  e.  ( II  Cn  J ) )
548adantr 451 . . . . . . . 8  |-  ( (
ph  /\  ( g  e.  U. B  /\  h  e.  U. B  /\  [
g ] (  ~=ph  `  J )  =  [
h ] (  ~=ph  `  J ) ) )  ->  F  e.  ( II  Cn  J ) )
5553, 54pco0 18512 . . . . . . 7  |-  ( (
ph  /\  ( g  e.  U. B  /\  h  e.  U. B  /\  [
g ] (  ~=ph  `  J )  =  [
h ] (  ~=ph  `  J ) ) )  ->  ( ( g ( *p `  J
) F ) ` 
0 )  =  ( g `  0 ) )
5633adantr 451 . . . . . . 7  |-  ( (
ph  /\  ( g  e.  U. B  /\  h  e.  U. B  /\  [
g ] (  ~=ph  `  J )  =  [
h ] (  ~=ph  `  J ) ) )  ->  ( I ` 
1 )  =  ( F `  0 ) )
5752, 55, 563eqtr4rd 2326 . . . . . 6  |-  ( (
ph  /\  ( g  e.  U. B  /\  h  e.  U. B  /\  [
g ] (  ~=ph  `  J )  =  [
h ] (  ~=ph  `  J ) ) )  ->  ( I ` 
1 )  =  ( ( g ( *p
`  J ) F ) `  0 ) )
5828adantr 451 . . . . . . 7  |-  ( (
ph  /\  ( g  e.  U. B  /\  h  e.  U. B  /\  [
g ] (  ~=ph  `  J )  =  [
h ] (  ~=ph  `  J ) ) )  ->  I  e.  ( II  Cn  J ) )
5951, 58erref 6680 . . . . . 6  |-  ( (
ph  /\  ( g  e.  U. B  /\  h  e.  U. B  /\  [
g ] (  ~=ph  `  J )  =  [
h ] (  ~=ph  `  J ) ) )  ->  I (  ~=ph  `  J ) I )
60203ad2antr1 1120 . . . . . . 7  |-  ( (
ph  /\  ( g  e.  U. B  /\  h  e.  U. B  /\  [
g ] (  ~=ph  `  J )  =  [
h ] (  ~=ph  `  J ) ) )  ->  ( g ` 
1 )  =  ( F `  0 ) )
61 simpr3 963 . . . . . . . 8  |-  ( (
ph  /\  ( g  e.  U. B  /\  h  e.  U. B  /\  [
g ] (  ~=ph  `  J )  =  [
h ] (  ~=ph  `  J ) ) )  ->  [ g ] (  ~=ph  `  J )  =  [ h ]
(  ~=ph  `  J )
)
6251, 53erth 6704 . . . . . . . 8  |-  ( (
ph  /\  ( g  e.  U. B  /\  h  e.  U. B  /\  [
g ] (  ~=ph  `  J )  =  [
h ] (  ~=ph  `  J ) ) )  ->  ( g ( 
~=ph  `  J ) h  <->  [ g ] ( 
~=ph  `  J )  =  [ h ] ( 
~=ph  `  J ) ) )
6361, 62mpbird 223 . . . . . . 7  |-  ( (
ph  /\  ( g  e.  U. B  /\  h  e.  U. B  /\  [
g ] (  ~=ph  `  J )  =  [
h ] (  ~=ph  `  J ) ) )  ->  g (  ~=ph  `  J ) h )
6451, 54erref 6680 . . . . . . 7  |-  ( (
ph  /\  ( g  e.  U. B  /\  h  e.  U. B  /\  [
g ] (  ~=ph  `  J )  =  [
h ] (  ~=ph  `  J ) ) )  ->  F (  ~=ph  `  J ) F )
6560, 63, 64pcohtpy 18518 . . . . . 6  |-  ( (
ph  /\  ( g  e.  U. B  /\  h  e.  U. B  /\  [
g ] (  ~=ph  `  J )  =  [
h ] (  ~=ph  `  J ) ) )  ->  ( g ( *p `  J ) F ) (  ~=ph  `  J ) ( h ( *p `  J
) F ) )
6657, 59, 65pcohtpy 18518 . . . . 5  |-  ( (
ph  /\  ( g  e.  U. B  /\  h  e.  U. B  /\  [
g ] (  ~=ph  `  J )  =  [
h ] (  ~=ph  `  J ) ) )  ->  ( I ( *p `  J ) ( g ( *p
`  J ) F ) ) (  ~=ph  `  J ) ( I ( *p `  J
) ( h ( *p `  J ) F ) ) )
6751, 66erthi 6706 . . . 4  |-  ( (
ph  /\  ( g  e.  U. B  /\  h  e.  U. B  /\  [
g ] (  ~=ph  `  J )  =  [
h ] (  ~=ph  `  J ) ) )  ->  [ ( I ( *p `  J
) ( g ( *p `  J ) F ) ) ] (  ~=ph  `  J )  =  [ ( I ( *p `  J
) ( h ( *p `  J ) F ) ) ] (  ~=ph  `  J ) )
681, 21, 44, 45, 49, 67fliftfund 5812 . . 3  |-  ( ph  ->  Fun  G )
691, 21, 44fliftf 5814 . . 3  |-  ( ph  ->  ( Fun  G  <->  G : ran  ( g  e.  U. B  |->  [ g ] (  ~=ph  `  J ) ) --> ( Base `  Q
) ) )
7068, 69mpbid 201 . 2  |-  ( ph  ->  G : ran  (
g  e.  U. B  |->  [ g ] ( 
~=ph  `  J ) ) --> ( Base `  Q
) )
712, 4, 13, 15pi1bas2 18539 . . . 4  |-  ( ph  ->  B  =  ( U. B /. (  ~=ph  `  J
) ) )
72 df-qs 6666 . . . . 5  |-  ( U. B /. (  ~=ph  `  J
) )  =  {
s  |  E. g  e.  U. B s  =  [ g ] ( 
~=ph  `  J ) }
73 eqid 2283 . . . . . 6  |-  ( g  e.  U. B  |->  [ g ] (  ~=ph  `  J ) )  =  ( g  e.  U. B  |->  [ g ] (  ~=ph  `  J ) )
7473rnmpt 4925 . . . . 5  |-  ran  (
g  e.  U. B  |->  [ g ] ( 
~=ph  `  J ) )  =  { s  |  E. g  e.  U. B s  =  [
g ] (  ~=ph  `  J ) }
7572, 74eqtr4i 2306 . . . 4  |-  ( U. B /. (  ~=ph  `  J
) )  =  ran  ( g  e.  U. B  |->  [ g ] (  ~=ph  `  J ) )
7671, 75syl6eq 2331 . . 3  |-  ( ph  ->  B  =  ran  (
g  e.  U. B  |->  [ g ] ( 
~=ph  `  J ) ) )
7776feq2d 5380 . 2  |-  ( ph  ->  ( G : B --> ( Base `  Q )  <->  G : ran  ( g  e.  U. B  |->  [ g ] (  ~=ph  `  J ) ) --> (
Base `  Q )
) )
7870, 77mpbird 223 1  |-  ( ph  ->  G : B --> ( Base `  Q ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   {cab 2269   E.wrex 2544   <.cop 3643   U.cuni 3827   class class class wbr 4023    e. cmpt 4077   ran crn 4690   Fun wfun 5249   -->wf 5251   ` cfv 5255  (class class class)co 5858    Er wer 6657   [cec 6658   /.cqs 6659   0cc0 8737   1c1 8738   [,]cicc 10659   Basecbs 13148  TopOnctopon 16632    Cn ccn 16954   IIcii 18379    ~=ph cphtpc 18467   *pcpco 18498    pi 1 cpi1 18501
This theorem is referenced by:  pi1xfrval  18552  pi1xfr  18553
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-mulf 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-ec 6662  df-qs 6666  df-map 6774  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-icc 10663  df-fz 10783  df-fzo 10871  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-rest 13327  df-topn 13328  df-topgen 13344  df-pt 13345  df-prds 13348  df-xrs 13403  df-0g 13404  df-gsum 13405  df-qtop 13410  df-imas 13411  df-divs 13412  df-xps 13413  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-mulg 14492  df-cntz 14793  df-cmn 15091  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cld 16756  df-cn 16957  df-cnp 16958  df-tx 17257  df-hmeo 17446  df-xms 17885  df-ms 17886  df-tms 17887  df-ii 18381  df-htpy 18468  df-phtpy 18469  df-phtpc 18490  df-pco 18503  df-om1 18504  df-pi1 18506
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