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Theorem pi1xfrf 18567
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 18399 . . . . . . . . 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 16995 . . . . . . . 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 10770 . . . . . . 7  |-  0  e.  ( 0 [,] 1
)
12 ffvelrn 5679 . . . . . . 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 18556 . . . . . . 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 18560 . . . 4  |-  ( (
ph  /\  g  e.  U. B )  ->  [ g ] (  ~=ph  `  J
)  e.  B )
22 pi1xfr.q . . . . 5  |-  Q  =  ( J  pi 1 
( F `  1
) )
23 eqid 2296 . . . . 5  |-  ( Base `  Q )  =  (
Base `  Q )
24 1elunit 10771 . . . . . . 7  |-  1  e.  ( 0 [,] 1
)
25 ffvelrn 5679 . . . . . . 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 18531 . . . . . 6  |-  ( (
ph  /\  g  e.  U. B )  ->  (
g ( *p `  J ) F )  e.  ( II  Cn  J ) )
3218, 30pco0 18528 . . . . . . 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 2339 . . . . . 6  |-  ( (
ph  /\  g  e.  U. B )  ->  (
I `  1 )  =  ( ( g ( *p `  J
) F ) ` 
0 ) )
3629, 31, 35pcocn 18531 . . . . 5  |-  ( (
ph  /\  g  e.  U. B )  ->  (
I ( *p `  J ) ( g ( *p `  J
) F ) )  e.  ( II  Cn  J ) )
3729, 31pco0 18528 . . . . . 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 2331 . . . . 5  |-  ( (
ph  /\  g  e.  U. B )  ->  (
( I ( *p
`  J ) ( g ( *p `  J ) F ) ) `  0 )  =  ( F ` 
1 ) )
4129, 31pco1 18529 . . . . . 6  |-  ( (
ph  /\  g  e.  U. B )  ->  (
( I ( *p
`  J ) ( g ( *p `  J ) F ) ) `  1 )  =  ( ( g ( *p `  J
) F ) ` 
1 ) )
4218, 30pco1 18529 . . . . . 6  |-  ( (
ph  /\  g  e.  U. B )  ->  (
( g ( *p
`  J ) F ) `  1 )  =  ( F ` 
1 ) )
4341, 42eqtrd 2328 . . . . 5  |-  ( (
ph  /\  g  e.  U. B )  ->  (
( I ( *p
`  J ) ( g ( *p `  J ) F ) ) `  1 )  =  ( F ` 
1 ) )
4422, 23, 5, 27, 36, 40, 43elpi1i 18560 . . . 4  |-  ( (
ph  /\  g  e.  U. B )  ->  [ ( I ( *p `  J ) ( g ( *p `  J
) F ) ) ] (  ~=ph  `  J
)  e.  ( Base `  Q ) )
45 eceq1 6712 . . . 4  |-  ( g  =  h  ->  [ g ] (  ~=ph  `  J
)  =  [ h ] (  ~=ph  `  J
) )
46 oveq1 5881 . . . . . 6  |-  ( g  =  h  ->  (
g ( *p `  J ) F )  =  ( h ( *p `  J ) F ) )
4746oveq2d 5890 . . . . 5  |-  ( g  =  h  ->  (
I ( *p `  J ) ( g ( *p `  J
) F ) )  =  ( I ( *p `  J ) ( h ( *p
`  J ) F ) ) )
48 eceq1 6712 . . . . 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 18509 . . . . . 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 18528 . . . . . . 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 2339 . . . . . 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 6696 . . . . . 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 6720 . . . . . . . 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 6696 . . . . . . 7  |-  ( (
ph  /\  ( g  e.  U. B  /\  h  e.  U. B  /\  [
g ] (  ~=ph  `  J )  =  [
h ] (  ~=ph  `  J ) ) )  ->  F (  ~=ph  `  J ) F )
6560, 63, 64pcohtpy 18534 . . . . . 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 18534 . . . . 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 6722 . . . 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 5828 . . 3  |-  ( ph  ->  Fun  G )
691, 21, 44fliftf 5830 . . 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 18555 . . . 4  |-  ( ph  ->  B  =  ( U. B /. (  ~=ph  `  J
) ) )
72 df-qs 6682 . . . . 5  |-  ( U. B /. (  ~=ph  `  J
) )  =  {
s  |  E. g  e.  U. B s  =  [ g ] ( 
~=ph  `  J ) }
73 eqid 2296 . . . . . 6  |-  ( g  e.  U. B  |->  [ g ] (  ~=ph  `  J ) )  =  ( g  e.  U. B  |->  [ g ] (  ~=ph  `  J ) )
7473rnmpt 4941 . . . . 5  |-  ran  (
g  e.  U. B  |->  [ g ] ( 
~=ph  `  J ) )  =  { s  |  E. g  e.  U. B s  =  [
g ] (  ~=ph  `  J ) }
7572, 74eqtr4i 2319 . . . 4  |-  ( U. B /. (  ~=ph  `  J
) )  =  ran  ( g  e.  U. B  |->  [ g ] (  ~=ph  `  J ) )
7671, 75syl6eq 2344 . . 3  |-  ( ph  ->  B  =  ran  (
g  e.  U. B  |->  [ g ] ( 
~=ph  `  J ) ) )
7776feq2d 5396 . 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 1632    e. wcel 1696   {cab 2282   E.wrex 2557   <.cop 3656   U.cuni 3843   class class class wbr 4039    e. cmpt 4093   ran crn 4706   Fun wfun 5265   -->wf 5267   ` cfv 5271  (class class class)co 5874    Er wer 6673   [cec 6674   /.cqs 6675   0cc0 8753   1c1 8754   [,]cicc 10675   Basecbs 13164  TopOnctopon 16648    Cn ccn 16970   IIcii 18395    ~=ph cphtpc 18483   *pcpco 18514    pi 1 cpi1 18517
This theorem is referenced by:  pi1xfrval  18568  pi1xfr  18569
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-mulf 8833
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-ec 6678  df-qs 6682  df-map 6790  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-fi 7181  df-sup 7210  df-oi 7241  df-card 7588  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-ioo 10676  df-icc 10679  df-fz 10799  df-fzo 10887  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-starv 13239  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-hom 13248  df-cco 13249  df-rest 13343  df-topn 13344  df-topgen 13360  df-pt 13361  df-prds 13364  df-xrs 13419  df-0g 13420  df-gsum 13421  df-qtop 13426  df-imas 13427  df-divs 13428  df-xps 13429  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-submnd 14432  df-mulg 14508  df-cntz 14809  df-cmn 15107  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-cnfld 16394  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-cld 16772  df-cn 16973  df-cnp 16974  df-tx 17273  df-hmeo 17462  df-xms 17901  df-ms 17902  df-tms 17903  df-ii 18397  df-htpy 18484  df-phtpy 18485  df-phtpc 18506  df-pco 18519  df-om1 18520  df-pi1 18522
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