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Theorem pi1xfrf 18951
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 452 . . . . 5  |-  ( (
ph  /\  g  e.  U. B )  ->  J  e.  (TopOn `  X )
)
6 iitopon 18782 . . . . . . . . 9  |-  II  e.  (TopOn `  ( 0 [,] 1 ) )
76a1i 11 . . . . . . . 8  |-  ( ph  ->  II  e.  (TopOn `  ( 0 [,] 1
) ) )
8 pi1xfr.f . . . . . . . 8  |-  ( ph  ->  F  e.  ( II 
Cn  J ) )
9 cnf2 17237 . . . . . . . 8  |-  ( ( II  e.  (TopOn `  ( 0 [,] 1
) )  /\  J  e.  (TopOn `  X )  /\  F  e.  (
II  Cn  J )
)  ->  F :
( 0 [,] 1
) --> X )
107, 4, 8, 9syl3anc 1184 . . . . . . 7  |-  ( ph  ->  F : ( 0 [,] 1 ) --> X )
11 0elunit 10949 . . . . . . 7  |-  0  e.  ( 0 [,] 1
)
12 ffvelrn 5809 . . . . . . 7  |-  ( ( F : ( 0 [,] 1 ) --> X  /\  0  e.  ( 0 [,] 1 ) )  ->  ( F `  0 )  e.  X )
1310, 11, 12sylancl 644 . . . . . 6  |-  ( ph  ->  ( F `  0
)  e.  X )
1413adantr 452 . . . . 5  |-  ( (
ph  /\  g  e.  U. B )  ->  ( F `  0 )  e.  X )
153a1i 11 . . . . . . . 8  |-  ( ph  ->  B  =  ( Base `  P ) )
162, 4, 13, 15pi1eluni 18940 . . . . . . 7  |-  ( ph  ->  ( g  e.  U. B 
<->  ( g  e.  ( II  Cn  J )  /\  ( g ` 
0 )  =  ( F `  0 )  /\  ( g ` 
1 )  =  ( F `  0 ) ) ) )
1716biimpa 471 . . . . . 6  |-  ( (
ph  /\  g  e.  U. B )  ->  (
g  e.  ( II 
Cn  J )  /\  ( g `  0
)  =  ( F `
 0 )  /\  ( g `  1
)  =  ( F `
 0 ) ) )
1817simp1d 969 . . . . 5  |-  ( (
ph  /\  g  e.  U. B )  ->  g  e.  ( II  Cn  J
) )
1917simp2d 970 . . . . 5  |-  ( (
ph  /\  g  e.  U. B )  ->  (
g `  0 )  =  ( F ` 
0 ) )
2017simp3d 971 . . . . 5  |-  ( (
ph  /\  g  e.  U. B )  ->  (
g `  1 )  =  ( F ` 
0 ) )
212, 3, 5, 14, 18, 19, 20elpi1i 18944 . . . 4  |-  ( (
ph  /\  g  e.  U. B )  ->  [ g ] (  ~=ph  `  J
)  e.  B )
22 pi1xfr.q . . . . 5  |-  Q  =  ( J  pi 1 
( F `  1
) )
23 eqid 2389 . . . . 5  |-  ( Base `  Q )  =  (
Base `  Q )
24 1elunit 10950 . . . . . . 7  |-  1  e.  ( 0 [,] 1
)
25 ffvelrn 5809 . . . . . . 7  |-  ( ( F : ( 0 [,] 1 ) --> X  /\  1  e.  ( 0 [,] 1 ) )  ->  ( F `  1 )  e.  X )
2610, 24, 25sylancl 644 . . . . . 6  |-  ( ph  ->  ( F `  1
)  e.  X )
2726adantr 452 . . . . 5  |-  ( (
ph  /\  g  e.  U. B )  ->  ( F `  1 )  e.  X )
28 pi1xfrval.i . . . . . . 7  |-  ( ph  ->  I  e.  ( II 
Cn  J ) )
2928adantr 452 . . . . . 6  |-  ( (
ph  /\  g  e.  U. B )  ->  I  e.  ( II  Cn  J
) )
308adantr 452 . . . . . . 7  |-  ( (
ph  /\  g  e.  U. B )  ->  F  e.  ( II  Cn  J
) )
3118, 30, 20pcocn 18915 . . . . . 6  |-  ( (
ph  /\  g  e.  U. B )  ->  (
g ( *p `  J ) F )  e.  ( II  Cn  J ) )
3218, 30pco0 18912 . . . . . . 7  |-  ( (
ph  /\  g  e.  U. B )  ->  (
( g ( *p
`  J ) F ) `  0 )  =  ( g ` 
0 ) )
33 pi1xfrval.2 . . . . . . . 8  |-  ( ph  ->  ( I `  1
)  =  ( F `
 0 ) )
3433adantr 452 . . . . . . 7  |-  ( (
ph  /\  g  e.  U. B )  ->  (
I `  1 )  =  ( F ` 
0 ) )
3519, 32, 343eqtr4rd 2432 . . . . . 6  |-  ( (
ph  /\  g  e.  U. B )  ->  (
I `  1 )  =  ( ( g ( *p `  J
) F ) ` 
0 ) )
3629, 31, 35pcocn 18915 . . . . 5  |-  ( (
ph  /\  g  e.  U. B )  ->  (
I ( *p `  J ) ( g ( *p `  J
) F ) )  e.  ( II  Cn  J ) )
3729, 31pco0 18912 . . . . . 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 452 . . . . . 6  |-  ( (
ph  /\  g  e.  U. B )  ->  ( F `  1 )  =  ( I ` 
0 ) )
4037, 39eqtr4d 2424 . . . . 5  |-  ( (
ph  /\  g  e.  U. B )  ->  (
( I ( *p
`  J ) ( g ( *p `  J ) F ) ) `  0 )  =  ( F ` 
1 ) )
4129, 31pco1 18913 . . . . . 6  |-  ( (
ph  /\  g  e.  U. B )  ->  (
( I ( *p
`  J ) ( g ( *p `  J ) F ) ) `  1 )  =  ( ( g ( *p `  J
) F ) ` 
1 ) )
4218, 30pco1 18913 . . . . . 6  |-  ( (
ph  /\  g  e.  U. B )  ->  (
( g ( *p
`  J ) F ) `  1 )  =  ( F ` 
1 ) )
4341, 42eqtrd 2421 . . . . 5  |-  ( (
ph  /\  g  e.  U. B )  ->  (
( I ( *p
`  J ) ( g ( *p `  J ) F ) ) `  1 )  =  ( F ` 
1 ) )
4422, 23, 5, 27, 36, 40, 43elpi1i 18944 . . . 4  |-  ( (
ph  /\  g  e.  U. B )  ->  [ ( I ( *p `  J ) ( g ( *p `  J
) F ) ) ] (  ~=ph  `  J
)  e.  ( Base `  Q ) )
45 eceq1 6879 . . . 4  |-  ( g  =  h  ->  [ g ] (  ~=ph  `  J
)  =  [ h ] (  ~=ph  `  J
) )
46 oveq1 6029 . . . . . 6  |-  ( g  =  h  ->  (
g ( *p `  J ) F )  =  ( h ( *p `  J ) F ) )
4746oveq2d 6038 . . . . 5  |-  ( g  =  h  ->  (
I ( *p `  J ) ( g ( *p `  J
) F ) )  =  ( I ( *p `  J ) ( h ( *p
`  J ) F ) ) )
48 eceq1 6879 . . . . 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 16 . . . 4  |-  ( g  =  h  ->  [ ( I ( *p `  J ) ( g ( *p `  J
) F ) ) ] (  ~=ph  `  J
)  =  [ ( I ( *p `  J ) ( h ( *p `  J
) F ) ) ] (  ~=ph  `  J
) )
50 phtpcer 18893 . . . . . 6  |-  (  ~=ph  `  J )  Er  (
II  Cn  J )
5150a1i 11 . . . . 5  |-  ( (
ph  /\  ( g  e.  U. B  /\  h  e.  U. B  /\  [
g ] (  ~=ph  `  J )  =  [
h ] (  ~=ph  `  J ) ) )  ->  (  ~=ph  `  J
)  Er  ( II 
Cn  J ) )
52193ad2antr1 1122 . . . . . . 7  |-  ( (
ph  /\  ( g  e.  U. B  /\  h  e.  U. B  /\  [
g ] (  ~=ph  `  J )  =  [
h ] (  ~=ph  `  J ) ) )  ->  ( g ` 
0 )  =  ( F `  0 ) )
53183ad2antr1 1122 . . . . . . . 8  |-  ( (
ph  /\  ( g  e.  U. B  /\  h  e.  U. B  /\  [
g ] (  ~=ph  `  J )  =  [
h ] (  ~=ph  `  J ) ) )  ->  g  e.  ( II  Cn  J ) )
548adantr 452 . . . . . . . 8  |-  ( (
ph  /\  ( g  e.  U. B  /\  h  e.  U. B  /\  [
g ] (  ~=ph  `  J )  =  [
h ] (  ~=ph  `  J ) ) )  ->  F  e.  ( II  Cn  J ) )
5553, 54pco0 18912 . . . . . . 7  |-  ( (
ph  /\  ( g  e.  U. B  /\  h  e.  U. B  /\  [
g ] (  ~=ph  `  J )  =  [
h ] (  ~=ph  `  J ) ) )  ->  ( ( g ( *p `  J
) F ) ` 
0 )  =  ( g `  0 ) )
5633adantr 452 . . . . . . 7  |-  ( (
ph  /\  ( g  e.  U. B  /\  h  e.  U. B  /\  [
g ] (  ~=ph  `  J )  =  [
h ] (  ~=ph  `  J ) ) )  ->  ( I ` 
1 )  =  ( F `  0 ) )
5752, 55, 563eqtr4rd 2432 . . . . . 6  |-  ( (
ph  /\  ( g  e.  U. B  /\  h  e.  U. B  /\  [
g ] (  ~=ph  `  J )  =  [
h ] (  ~=ph  `  J ) ) )  ->  ( I ` 
1 )  =  ( ( g ( *p
`  J ) F ) `  0 ) )
5828adantr 452 . . . . . . 7  |-  ( (
ph  /\  ( g  e.  U. B  /\  h  e.  U. B  /\  [
g ] (  ~=ph  `  J )  =  [
h ] (  ~=ph  `  J ) ) )  ->  I  e.  ( II  Cn  J ) )
5951, 58erref 6863 . . . . . 6  |-  ( (
ph  /\  ( g  e.  U. B  /\  h  e.  U. B  /\  [
g ] (  ~=ph  `  J )  =  [
h ] (  ~=ph  `  J ) ) )  ->  I (  ~=ph  `  J ) I )
60203ad2antr1 1122 . . . . . . 7  |-  ( (
ph  /\  ( g  e.  U. B  /\  h  e.  U. B  /\  [
g ] (  ~=ph  `  J )  =  [
h ] (  ~=ph  `  J ) ) )  ->  ( g ` 
1 )  =  ( F `  0 ) )
61 simpr3 965 . . . . . . . 8  |-  ( (
ph  /\  ( g  e.  U. B  /\  h  e.  U. B  /\  [
g ] (  ~=ph  `  J )  =  [
h ] (  ~=ph  `  J ) ) )  ->  [ g ] (  ~=ph  `  J )  =  [ h ]
(  ~=ph  `  J )
)
6251, 53erth 6887 . . . . . . . 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 224 . . . . . . 7  |-  ( (
ph  /\  ( g  e.  U. B  /\  h  e.  U. B  /\  [
g ] (  ~=ph  `  J )  =  [
h ] (  ~=ph  `  J ) ) )  ->  g (  ~=ph  `  J ) h )
6451, 54erref 6863 . . . . . . 7  |-  ( (
ph  /\  ( g  e.  U. B  /\  h  e.  U. B  /\  [
g ] (  ~=ph  `  J )  =  [
h ] (  ~=ph  `  J ) ) )  ->  F (  ~=ph  `  J ) F )
6560, 63, 64pcohtpy 18918 . . . . . 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 18918 . . . . 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 6889 . . . 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 5976 . . 3  |-  ( ph  ->  Fun  G )
691, 21, 44fliftf 5978 . . 3  |-  ( ph  ->  ( Fun  G  <->  G : ran  ( g  e.  U. B  |->  [ g ] (  ~=ph  `  J ) ) --> ( Base `  Q
) ) )
7068, 69mpbid 202 . 2  |-  ( ph  ->  G : ran  (
g  e.  U. B  |->  [ g ] ( 
~=ph  `  J ) ) --> ( Base `  Q
) )
712, 4, 13, 15pi1bas2 18939 . . . 4  |-  ( ph  ->  B  =  ( U. B /. (  ~=ph  `  J
) ) )
72 df-qs 6849 . . . . 5  |-  ( U. B /. (  ~=ph  `  J
) )  =  {
s  |  E. g  e.  U. B s  =  [ g ] ( 
~=ph  `  J ) }
73 eqid 2389 . . . . . 6  |-  ( g  e.  U. B  |->  [ g ] (  ~=ph  `  J ) )  =  ( g  e.  U. B  |->  [ g ] (  ~=ph  `  J ) )
7473rnmpt 5058 . . . . 5  |-  ran  (
g  e.  U. B  |->  [ g ] ( 
~=ph  `  J ) )  =  { s  |  E. g  e.  U. B s  =  [
g ] (  ~=ph  `  J ) }
7572, 74eqtr4i 2412 . . . 4  |-  ( U. B /. (  ~=ph  `  J
) )  =  ran  ( g  e.  U. B  |->  [ g ] (  ~=ph  `  J ) )
7671, 75syl6eq 2437 . . 3  |-  ( ph  ->  B  =  ran  (
g  e.  U. B  |->  [ g ] ( 
~=ph  `  J ) ) )
7776feq2d 5523 . 2  |-  ( ph  ->  ( G : B --> ( Base `  Q )  <->  G : ran  ( g  e.  U. B  |->  [ g ] (  ~=ph  `  J ) ) --> (
Base `  Q )
) )
7870, 77mpbird 224 1  |-  ( ph  ->  G : B --> ( Base `  Q ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   {cab 2375   E.wrex 2652   <.cop 3762   U.cuni 3959   class class class wbr 4155    e. cmpt 4209   ran crn 4821   Fun wfun 5390   -->wf 5392   ` cfv 5396  (class class class)co 6022    Er wer 6840   [cec 6841   /.cqs 6842   0cc0 8925   1c1 8926   [,]cicc 10853   Basecbs 13398  TopOnctopon 16884    Cn ccn 17212   IIcii 18778    ~=ph cphtpc 18867   *pcpco 18898    pi 1 cpi1 18901
This theorem is referenced by:  pi1xfrval  18952  pi1xfr  18953
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-inf2 7531  ax-cnex 8981  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002  ax-pre-sup 9003  ax-mulf 9005
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-int 3995  df-iun 4039  df-iin 4040  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-se 4485  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-isom 5405  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-of 6246  df-1st 6290  df-2nd 6291  df-riota 6487  df-recs 6571  df-rdg 6606  df-1o 6662  df-2o 6663  df-oadd 6666  df-er 6843  df-ec 6845  df-qs 6849  df-map 6958  df-ixp 7002  df-en 7048  df-dom 7049  df-sdom 7050  df-fin 7051  df-fi 7353  df-sup 7383  df-oi 7414  df-card 7761  df-cda 7983  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-div 9612  df-nn 9935  df-2 9992  df-3 9993  df-4 9994  df-5 9995  df-6 9996  df-7 9997  df-8 9998  df-9 9999  df-10 10000  df-n0 10156  df-z 10217  df-dec 10317  df-uz 10423  df-q 10509  df-rp 10547  df-xneg 10644  df-xadd 10645  df-xmul 10646  df-ioo 10854  df-icc 10857  df-fz 10978  df-fzo 11068  df-seq 11253  df-exp 11312  df-hash 11548  df-cj 11833  df-re 11834  df-im 11835  df-sqr 11969  df-abs 11970  df-struct 13400  df-ndx 13401  df-slot 13402  df-base 13403  df-sets 13404  df-ress 13405  df-plusg 13471  df-mulr 13472  df-starv 13473  df-sca 13474  df-vsca 13475  df-tset 13477  df-ple 13478  df-ds 13480  df-unif 13481  df-hom 13482  df-cco 13483  df-rest 13579  df-topn 13580  df-topgen 13596  df-pt 13597  df-prds 13600  df-xrs 13655  df-0g 13656  df-gsum 13657  df-qtop 13662  df-imas 13663  df-divs 13664  df-xps 13665  df-mre 13740  df-mrc 13741  df-acs 13743  df-mnd 14619  df-submnd 14668  df-mulg 14744  df-cntz 15045  df-cmn 15343  df-xmet 16621  df-met 16622  df-bl 16623  df-mopn 16624  df-cnfld 16629  df-top 16888  df-bases 16890  df-topon 16891  df-topsp 16892  df-cld 17008  df-cn 17215  df-cnp 17216  df-tx 17517  df-hmeo 17710  df-xms 18261  df-ms 18262  df-tms 18263  df-ii 18780  df-htpy 18868  df-phtpy 18869  df-phtpc 18890  df-pco 18903  df-om1 18904  df-pi1 18906
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