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Theorem pi1xfrf 21421
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  pi1 
( F `  0
) )
pi1xfr.q  |-  Q  =  ( J  pi1 
( 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  pi1 
( F `  0
) )
3 pi1xfr.b . . . . 5  |-  B  =  ( Base `  P
)
4 pi1xfr.j . . . . . 6  |-  ( ph  ->  J  e.  (TopOn `  X ) )
54adantr 465 . . . . 5  |-  ( (
ph  /\  g  e.  U. B )  ->  J  e.  (TopOn `  X )
)
6 iitopon 21251 . . . . . . . . 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 19618 . . . . . . . 8  |-  ( ( II  e.  (TopOn `  ( 0 [,] 1
) )  /\  J  e.  (TopOn `  X )  /\  F  e.  (
II  Cn  J )
)  ->  F :
( 0 [,] 1
) --> X )
107, 4, 8, 9syl3anc 1228 . . . . . . 7  |-  ( ph  ->  F : ( 0 [,] 1 ) --> X )
11 0elunit 11650 . . . . . . 7  |-  0  e.  ( 0 [,] 1
)
12 ffvelrn 6030 . . . . . . 7  |-  ( ( F : ( 0 [,] 1 ) --> X  /\  0  e.  ( 0 [,] 1 ) )  ->  ( F `  0 )  e.  X )
1310, 11, 12sylancl 662 . . . . . 6  |-  ( ph  ->  ( F `  0
)  e.  X )
1413adantr 465 . . . . 5  |-  ( (
ph  /\  g  e.  U. B )  ->  ( F `  0 )  e.  X )
153a1i 11 . . . . . . . 8  |-  ( ph  ->  B  =  ( Base `  P ) )
162, 4, 13, 15pi1eluni 21410 . . . . . . 7  |-  ( ph  ->  ( g  e.  U. B 
<->  ( g  e.  ( II  Cn  J )  /\  ( g ` 
0 )  =  ( F `  0 )  /\  ( g ` 
1 )  =  ( F `  0 ) ) ) )
1716biimpa 484 . . . . . 6  |-  ( (
ph  /\  g  e.  U. B )  ->  (
g  e.  ( II 
Cn  J )  /\  ( g `  0
)  =  ( F `
 0 )  /\  ( g `  1
)  =  ( F `
 0 ) ) )
1817simp1d 1008 . . . . 5  |-  ( (
ph  /\  g  e.  U. B )  ->  g  e.  ( II  Cn  J
) )
1917simp2d 1009 . . . . 5  |-  ( (
ph  /\  g  e.  U. B )  ->  (
g `  0 )  =  ( F ` 
0 ) )
2017simp3d 1010 . . . . 5  |-  ( (
ph  /\  g  e.  U. B )  ->  (
g `  1 )  =  ( F ` 
0 ) )
212, 3, 5, 14, 18, 19, 20elpi1i 21414 . . . 4  |-  ( (
ph  /\  g  e.  U. B )  ->  [ g ] (  ~=ph  `  J
)  e.  B )
22 pi1xfr.q . . . . 5  |-  Q  =  ( J  pi1 
( F `  1
) )
23 eqid 2467 . . . . 5  |-  ( Base `  Q )  =  (
Base `  Q )
24 1elunit 11651 . . . . . . 7  |-  1  e.  ( 0 [,] 1
)
25 ffvelrn 6030 . . . . . . 7  |-  ( ( F : ( 0 [,] 1 ) --> X  /\  1  e.  ( 0 [,] 1 ) )  ->  ( F `  1 )  e.  X )
2610, 24, 25sylancl 662 . . . . . 6  |-  ( ph  ->  ( F `  1
)  e.  X )
2726adantr 465 . . . . 5  |-  ( (
ph  /\  g  e.  U. B )  ->  ( F `  1 )  e.  X )
28 pi1xfrval.i . . . . . . 7  |-  ( ph  ->  I  e.  ( II 
Cn  J ) )
2928adantr 465 . . . . . 6  |-  ( (
ph  /\  g  e.  U. B )  ->  I  e.  ( II  Cn  J
) )
308adantr 465 . . . . . . 7  |-  ( (
ph  /\  g  e.  U. B )  ->  F  e.  ( II  Cn  J
) )
3118, 30, 20pcocn 21385 . . . . . 6  |-  ( (
ph  /\  g  e.  U. B )  ->  (
g ( *p `  J ) F )  e.  ( II  Cn  J ) )
3218, 30pco0 21382 . . . . . . 7  |-  ( (
ph  /\  g  e.  U. B )  ->  (
( g ( *p
`  J ) F ) `  0 )  =  ( g ` 
0 ) )
33 pi1xfrval.2 . . . . . . . 8  |-  ( ph  ->  ( I `  1
)  =  ( F `
 0 ) )
3433adantr 465 . . . . . . 7  |-  ( (
ph  /\  g  e.  U. B )  ->  (
I `  1 )  =  ( F ` 
0 ) )
3519, 32, 343eqtr4rd 2519 . . . . . 6  |-  ( (
ph  /\  g  e.  U. B )  ->  (
I `  1 )  =  ( ( g ( *p `  J
) F ) ` 
0 ) )
3629, 31, 35pcocn 21385 . . . . 5  |-  ( (
ph  /\  g  e.  U. B )  ->  (
I ( *p `  J ) ( g ( *p `  J
) F ) )  e.  ( II  Cn  J ) )
3729, 31pco0 21382 . . . . . 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 465 . . . . . 6  |-  ( (
ph  /\  g  e.  U. B )  ->  ( F `  1 )  =  ( I ` 
0 ) )
4037, 39eqtr4d 2511 . . . . 5  |-  ( (
ph  /\  g  e.  U. B )  ->  (
( I ( *p
`  J ) ( g ( *p `  J ) F ) ) `  0 )  =  ( F ` 
1 ) )
4129, 31pco1 21383 . . . . . 6  |-  ( (
ph  /\  g  e.  U. B )  ->  (
( I ( *p
`  J ) ( g ( *p `  J ) F ) ) `  1 )  =  ( ( g ( *p `  J
) F ) ` 
1 ) )
4218, 30pco1 21383 . . . . . 6  |-  ( (
ph  /\  g  e.  U. B )  ->  (
( g ( *p
`  J ) F ) `  1 )  =  ( F ` 
1 ) )
4341, 42eqtrd 2508 . . . . 5  |-  ( (
ph  /\  g  e.  U. B )  ->  (
( I ( *p
`  J ) ( g ( *p `  J ) F ) ) `  1 )  =  ( F ` 
1 ) )
4422, 23, 5, 27, 36, 40, 43elpi1i 21414 . . . 4  |-  ( (
ph  /\  g  e.  U. B )  ->  [ ( I ( *p `  J ) ( g ( *p `  J
) F ) ) ] (  ~=ph  `  J
)  e.  ( Base `  Q ) )
45 eceq1 7359 . . . 4  |-  ( g  =  h  ->  [ g ] (  ~=ph  `  J
)  =  [ h ] (  ~=ph  `  J
) )
46 oveq1 6302 . . . . . 6  |-  ( g  =  h  ->  (
g ( *p `  J ) F )  =  ( h ( *p `  J ) F ) )
4746oveq2d 6311 . . . . 5  |-  ( g  =  h  ->  (
I ( *p `  J ) ( g ( *p `  J
) F ) )  =  ( I ( *p `  J ) ( h ( *p
`  J ) F ) ) )
4847eceq1d 7360 . . . 4  |-  ( g  =  h  ->  [ ( I ( *p `  J ) ( g ( *p `  J
) F ) ) ] (  ~=ph  `  J
)  =  [ ( I ( *p `  J ) ( h ( *p `  J
) F ) ) ] (  ~=ph  `  J
) )
49 phtpcer 21363 . . . . . 6  |-  (  ~=ph  `  J )  Er  (
II  Cn  J )
5049a1i 11 . . . . 5  |-  ( (
ph  /\  ( g  e.  U. B  /\  h  e.  U. B  /\  [
g ] (  ~=ph  `  J )  =  [
h ] (  ~=ph  `  J ) ) )  ->  (  ~=ph  `  J
)  Er  ( II 
Cn  J ) )
51193ad2antr1 1161 . . . . . . 7  |-  ( (
ph  /\  ( g  e.  U. B  /\  h  e.  U. B  /\  [
g ] (  ~=ph  `  J )  =  [
h ] (  ~=ph  `  J ) ) )  ->  ( g ` 
0 )  =  ( F `  0 ) )
52183ad2antr1 1161 . . . . . . . 8  |-  ( (
ph  /\  ( g  e.  U. B  /\  h  e.  U. B  /\  [
g ] (  ~=ph  `  J )  =  [
h ] (  ~=ph  `  J ) ) )  ->  g  e.  ( II  Cn  J ) )
538adantr 465 . . . . . . . 8  |-  ( (
ph  /\  ( g  e.  U. B  /\  h  e.  U. B  /\  [
g ] (  ~=ph  `  J )  =  [
h ] (  ~=ph  `  J ) ) )  ->  F  e.  ( II  Cn  J ) )
5452, 53pco0 21382 . . . . . . 7  |-  ( (
ph  /\  ( g  e.  U. B  /\  h  e.  U. B  /\  [
g ] (  ~=ph  `  J )  =  [
h ] (  ~=ph  `  J ) ) )  ->  ( ( g ( *p `  J
) F ) ` 
0 )  =  ( g `  0 ) )
5533adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( g  e.  U. B  /\  h  e.  U. B  /\  [
g ] (  ~=ph  `  J )  =  [
h ] (  ~=ph  `  J ) ) )  ->  ( I ` 
1 )  =  ( F `  0 ) )
5651, 54, 553eqtr4rd 2519 . . . . . 6  |-  ( (
ph  /\  ( g  e.  U. B  /\  h  e.  U. B  /\  [
g ] (  ~=ph  `  J )  =  [
h ] (  ~=ph  `  J ) ) )  ->  ( I ` 
1 )  =  ( ( g ( *p
`  J ) F ) `  0 ) )
5728adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( g  e.  U. B  /\  h  e.  U. B  /\  [
g ] (  ~=ph  `  J )  =  [
h ] (  ~=ph  `  J ) ) )  ->  I  e.  ( II  Cn  J ) )
5850, 57erref 7343 . . . . . 6  |-  ( (
ph  /\  ( g  e.  U. B  /\  h  e.  U. B  /\  [
g ] (  ~=ph  `  J )  =  [
h ] (  ~=ph  `  J ) ) )  ->  I (  ~=ph  `  J ) I )
59203ad2antr1 1161 . . . . . . 7  |-  ( (
ph  /\  ( g  e.  U. B  /\  h  e.  U. B  /\  [
g ] (  ~=ph  `  J )  =  [
h ] (  ~=ph  `  J ) ) )  ->  ( g ` 
1 )  =  ( F `  0 ) )
60 simpr3 1004 . . . . . . . 8  |-  ( (
ph  /\  ( g  e.  U. B  /\  h  e.  U. B  /\  [
g ] (  ~=ph  `  J )  =  [
h ] (  ~=ph  `  J ) ) )  ->  [ g ] (  ~=ph  `  J )  =  [ h ]
(  ~=ph  `  J )
)
6150, 52erth 7368 . . . . . . . 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 ) ) )
6260, 61mpbird 232 . . . . . . 7  |-  ( (
ph  /\  ( g  e.  U. B  /\  h  e.  U. B  /\  [
g ] (  ~=ph  `  J )  =  [
h ] (  ~=ph  `  J ) ) )  ->  g (  ~=ph  `  J ) h )
6350, 53erref 7343 . . . . . . 7  |-  ( (
ph  /\  ( g  e.  U. B  /\  h  e.  U. B  /\  [
g ] (  ~=ph  `  J )  =  [
h ] (  ~=ph  `  J ) ) )  ->  F (  ~=ph  `  J ) F )
6459, 62, 63pcohtpy 21388 . . . . . 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 ) )
6556, 58, 64pcohtpy 21388 . . . . 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 ) ) )
6650, 65erthi 7370 . . . 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 ) )
671, 21, 44, 45, 48, 66fliftfund 6210 . . 3  |-  ( ph  ->  Fun  G )
681, 21, 44fliftf 6212 . . 3  |-  ( ph  ->  ( Fun  G  <->  G : ran  ( g  e.  U. B  |->  [ g ] (  ~=ph  `  J ) ) --> ( Base `  Q
) ) )
6967, 68mpbid 210 . 2  |-  ( ph  ->  G : ran  (
g  e.  U. B  |->  [ g ] ( 
~=ph  `  J ) ) --> ( Base `  Q
) )
702, 4, 13, 15pi1bas2 21409 . . . 4  |-  ( ph  ->  B  =  ( U. B /. (  ~=ph  `  J
) ) )
71 df-qs 7329 . . . . 5  |-  ( U. B /. (  ~=ph  `  J
) )  =  {
s  |  E. g  e.  U. B s  =  [ g ] ( 
~=ph  `  J ) }
72 eqid 2467 . . . . . 6  |-  ( g  e.  U. B  |->  [ g ] (  ~=ph  `  J ) )  =  ( g  e.  U. B  |->  [ g ] (  ~=ph  `  J ) )
7372rnmpt 5254 . . . . 5  |-  ran  (
g  e.  U. B  |->  [ g ] ( 
~=ph  `  J ) )  =  { s  |  E. g  e.  U. B s  =  [
g ] (  ~=ph  `  J ) }
7471, 73eqtr4i 2499 . . . 4  |-  ( U. B /. (  ~=ph  `  J
) )  =  ran  ( g  e.  U. B  |->  [ g ] (  ~=ph  `  J ) )
7570, 74syl6eq 2524 . . 3  |-  ( ph  ->  B  =  ran  (
g  e.  U. B  |->  [ g ] ( 
~=ph  `  J ) ) )
7675feq2d 5724 . 2  |-  ( ph  ->  ( G : B --> ( Base `  Q )  <->  G : ran  ( g  e.  U. B  |->  [ g ] (  ~=ph  `  J ) ) --> (
Base `  Q )
) )
7769, 76mpbird 232 1  |-  ( ph  ->  G : B --> ( Base `  Q ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   {cab 2452   E.wrex 2818   <.cop 4039   U.cuni 4251   class class class wbr 4453    |-> cmpt 4511   ran crn 5006   Fun wfun 5588   -->wf 5590   ` cfv 5594  (class class class)co 6295    Er wer 7320   [cec 7321   /.cqs 7322   0cc0 9504   1c1 9505   [,]cicc 11544   Basecbs 14507  TopOnctopon 19264    Cn ccn 19593   IIcii 21247    ~=ph cphtpc 21337   *pcpco 21368    pi1 cpi1 21371
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582  ax-mulf 9584
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-iin 4334  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-of 6535  df-om 6696  df-1st 6795  df-2nd 6796  df-supp 6914  df-recs 7054  df-rdg 7088  df-1o 7142  df-2o 7143  df-oadd 7146  df-er 7323  df-ec 7325  df-qs 7329  df-map 7434  df-ixp 7482  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-fsupp 7842  df-fi 7883  df-sup 7913  df-oi 7947  df-card 8332  df-cda 8560  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-9 10613  df-10 10614  df-n0 10808  df-z 10877  df-dec 10989  df-uz 11095  df-q 11195  df-rp 11233  df-xneg 11330  df-xadd 11331  df-xmul 11332  df-ioo 11545  df-icc 11548  df-fz 11685  df-fzo 11805  df-seq 12088  df-exp 12147  df-hash 12386  df-cj 12912  df-re 12913  df-im 12914  df-sqrt 13048  df-abs 13049  df-struct 14509  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-ress 14514  df-plusg 14585  df-mulr 14586  df-starv 14587  df-sca 14588  df-vsca 14589  df-ip 14590  df-tset 14591  df-ple 14592  df-ds 14594  df-unif 14595  df-hom 14596  df-cco 14597  df-rest 14695  df-topn 14696  df-0g 14714  df-gsum 14715  df-topgen 14716  df-pt 14717  df-prds 14720  df-xrs 14774  df-qtop 14779  df-imas 14780  df-qus 14781  df-xps 14782  df-mre 14858  df-mrc 14859  df-acs 14861  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-submnd 15840  df-mulg 15932  df-cntz 16227  df-cmn 16673  df-psmet 18281  df-xmet 18282  df-met 18283  df-bl 18284  df-mopn 18285  df-cnfld 18291  df-top 19268  df-bases 19270  df-topon 19271  df-topsp 19272  df-cld 19388  df-cn 19596  df-cnp 19597  df-tx 19931  df-hmeo 20124  df-xms 20691  df-ms 20692  df-tms 20693  df-ii 21249  df-htpy 21338  df-phtpy 21339  df-phtpc 21360  df-pco 21373  df-om1 21374  df-pi1 21376
This theorem is referenced by:  pi1xfrval  21422  pi1xfr  21423
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