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Theorem pi1cof 21644
Description: Functionality of the loop transfer function on the equivalence class of a path. (Contributed by Mario Carneiro, 23-Dec-2016.)
Hypotheses
Ref Expression
pi1co.p  |-  P  =  ( J  pi1  A )
pi1co.q  |-  Q  =  ( K  pi1  B )
pi1co.v  |-  V  =  ( Base `  P
)
pi1co.g  |-  G  =  ran  ( g  e. 
U. V  |->  <. [ g ] (  ~=ph  `  J
) ,  [ ( F  o.  g ) ] (  ~=ph  `  K
) >. )
pi1co.j  |-  ( ph  ->  J  e.  (TopOn `  X ) )
pi1co.f  |-  ( ph  ->  F  e.  ( J  Cn  K ) )
pi1co.a  |-  ( ph  ->  A  e.  X )
pi1co.b  |-  ( ph  ->  ( F `  A
)  =  B )
Assertion
Ref Expression
pi1cof  |-  ( ph  ->  G : V --> ( Base `  Q ) )
Distinct variable groups:    A, g    g, F    g, J    ph, g    g, K    P, g    Q, g   
g, V
Allowed substitution hints:    B( g)    G( g)    X( g)

Proof of Theorem pi1cof
Dummy variables  s  h are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pi1co.g . . . 4  |-  G  =  ran  ( g  e. 
U. V  |->  <. [ g ] (  ~=ph  `  J
) ,  [ ( F  o.  g ) ] (  ~=ph  `  K
) >. )
2 fvex 5784 . . . . 5  |-  (  ~=ph  `  J )  e.  _V
3 ecexg 7233 . . . . 5  |-  ( ( 
~=ph  `  J )  e. 
_V  ->  [ g ] (  ~=ph  `  J )  e.  _V )
42, 3mp1i 12 . . . 4  |-  ( (
ph  /\  g  e.  U. V )  ->  [ g ] (  ~=ph  `  J
)  e.  _V )
5 pi1co.q . . . . 5  |-  Q  =  ( K  pi1  B )
6 eqid 2382 . . . . 5  |-  ( Base `  Q )  =  (
Base `  Q )
7 pi1co.f . . . . . . . 8  |-  ( ph  ->  F  e.  ( J  Cn  K ) )
8 cntop2 19828 . . . . . . . 8  |-  ( F  e.  ( J  Cn  K )  ->  K  e.  Top )
97, 8syl 16 . . . . . . 7  |-  ( ph  ->  K  e.  Top )
10 eqid 2382 . . . . . . . 8  |-  U. K  =  U. K
1110toptopon 19519 . . . . . . 7  |-  ( K  e.  Top  <->  K  e.  (TopOn `  U. K ) )
129, 11sylib 196 . . . . . 6  |-  ( ph  ->  K  e.  (TopOn `  U. K ) )
1312adantr 463 . . . . 5  |-  ( (
ph  /\  g  e.  U. V )  ->  K  e.  (TopOn `  U. K ) )
14 pi1co.b . . . . . . 7  |-  ( ph  ->  ( F `  A
)  =  B )
15 pi1co.j . . . . . . . . 9  |-  ( ph  ->  J  e.  (TopOn `  X ) )
16 cnf2 19836 . . . . . . . . 9  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  U. K )  /\  F  e.  ( J  Cn  K ) )  ->  F : X
--> U. K )
1715, 12, 7, 16syl3anc 1226 . . . . . . . 8  |-  ( ph  ->  F : X --> U. K
)
18 pi1co.a . . . . . . . 8  |-  ( ph  ->  A  e.  X )
1917, 18ffvelrnd 5934 . . . . . . 7  |-  ( ph  ->  ( F `  A
)  e.  U. K
)
2014, 19eqeltrrd 2471 . . . . . 6  |-  ( ph  ->  B  e.  U. K
)
2120adantr 463 . . . . 5  |-  ( (
ph  /\  g  e.  U. V )  ->  B  e.  U. K )
22 pi1co.p . . . . . . . . 9  |-  P  =  ( J  pi1  A )
23 pi1co.v . . . . . . . . . 10  |-  V  =  ( Base `  P
)
2423a1i 11 . . . . . . . . 9  |-  ( ph  ->  V  =  ( Base `  P ) )
2522, 15, 18, 24pi1eluni 21627 . . . . . . . 8  |-  ( ph  ->  ( g  e.  U. V 
<->  ( g  e.  ( II  Cn  J )  /\  ( g ` 
0 )  =  A  /\  ( g ` 
1 )  =  A ) ) )
2625biimpa 482 . . . . . . 7  |-  ( (
ph  /\  g  e.  U. V )  ->  (
g  e.  ( II 
Cn  J )  /\  ( g `  0
)  =  A  /\  ( g `  1
)  =  A ) )
2726simp1d 1006 . . . . . 6  |-  ( (
ph  /\  g  e.  U. V )  ->  g  e.  ( II  Cn  J
) )
287adantr 463 . . . . . 6  |-  ( (
ph  /\  g  e.  U. V )  ->  F  e.  ( J  Cn  K
) )
29 cnco 19853 . . . . . 6  |-  ( ( g  e.  ( II 
Cn  J )  /\  F  e.  ( J  Cn  K ) )  -> 
( F  o.  g
)  e.  ( II 
Cn  K ) )
3027, 28, 29syl2anc 659 . . . . 5  |-  ( (
ph  /\  g  e.  U. V )  ->  ( F  o.  g )  e.  ( II  Cn  K
) )
31 iitopon 21468 . . . . . . . . 9  |-  II  e.  (TopOn `  ( 0 [,] 1 ) )
3231a1i 11 . . . . . . . 8  |-  ( (
ph  /\  g  e.  U. V )  ->  II  e.  (TopOn `  ( 0 [,] 1 ) ) )
3315adantr 463 . . . . . . . 8  |-  ( (
ph  /\  g  e.  U. V )  ->  J  e.  (TopOn `  X )
)
34 cnf2 19836 . . . . . . . 8  |-  ( ( II  e.  (TopOn `  ( 0 [,] 1
) )  /\  J  e.  (TopOn `  X )  /\  g  e.  (
II  Cn  J )
)  ->  g :
( 0 [,] 1
) --> X )
3532, 33, 27, 34syl3anc 1226 . . . . . . 7  |-  ( (
ph  /\  g  e.  U. V )  ->  g : ( 0 [,] 1 ) --> X )
36 0elunit 11559 . . . . . . 7  |-  0  e.  ( 0 [,] 1
)
37 fvco3 5851 . . . . . . 7  |-  ( ( g : ( 0 [,] 1 ) --> X  /\  0  e.  ( 0 [,] 1 ) )  ->  ( ( F  o.  g ) `  0 )  =  ( F `  (
g `  0 )
) )
3835, 36, 37sylancl 660 . . . . . 6  |-  ( (
ph  /\  g  e.  U. V )  ->  (
( F  o.  g
) `  0 )  =  ( F `  ( g `  0
) ) )
3926simp2d 1007 . . . . . . 7  |-  ( (
ph  /\  g  e.  U. V )  ->  (
g `  0 )  =  A )
4039fveq2d 5778 . . . . . 6  |-  ( (
ph  /\  g  e.  U. V )  ->  ( F `  ( g `  0 ) )  =  ( F `  A ) )
4114adantr 463 . . . . . 6  |-  ( (
ph  /\  g  e.  U. V )  ->  ( F `  A )  =  B )
4238, 40, 413eqtrd 2427 . . . . 5  |-  ( (
ph  /\  g  e.  U. V )  ->  (
( F  o.  g
) `  0 )  =  B )
43 1elunit 11560 . . . . . . 7  |-  1  e.  ( 0 [,] 1
)
44 fvco3 5851 . . . . . . 7  |-  ( ( g : ( 0 [,] 1 ) --> X  /\  1  e.  ( 0 [,] 1 ) )  ->  ( ( F  o.  g ) `  1 )  =  ( F `  (
g `  1 )
) )
4535, 43, 44sylancl 660 . . . . . 6  |-  ( (
ph  /\  g  e.  U. V )  ->  (
( F  o.  g
) `  1 )  =  ( F `  ( g `  1
) ) )
4626simp3d 1008 . . . . . . 7  |-  ( (
ph  /\  g  e.  U. V )  ->  (
g `  1 )  =  A )
4746fveq2d 5778 . . . . . 6  |-  ( (
ph  /\  g  e.  U. V )  ->  ( F `  ( g `  1 ) )  =  ( F `  A ) )
4845, 47, 413eqtrd 2427 . . . . 5  |-  ( (
ph  /\  g  e.  U. V )  ->  (
( F  o.  g
) `  1 )  =  B )
495, 6, 13, 21, 30, 42, 48elpi1i 21631 . . . 4  |-  ( (
ph  /\  g  e.  U. V )  ->  [ ( F  o.  g ) ] (  ~=ph  `  K
)  e.  ( Base `  Q ) )
50 eceq1 7265 . . . 4  |-  ( g  =  h  ->  [ g ] (  ~=ph  `  J
)  =  [ h ] (  ~=ph  `  J
) )
51 coeq2 5074 . . . . 5  |-  ( g  =  h  ->  ( F  o.  g )  =  ( F  o.  h ) )
5251eceq1d 7266 . . . 4  |-  ( g  =  h  ->  [ ( F  o.  g ) ] (  ~=ph  `  K
)  =  [ ( F  o.  h ) ] (  ~=ph  `  K
) )
53 phtpcer 21580 . . . . . 6  |-  (  ~=ph  `  K )  Er  (
II  Cn  K )
5453a1i 11 . . . . 5  |-  ( (
ph  /\  ( g  e.  U. V  /\  h  e.  U. V  /\  [
g ] (  ~=ph  `  J )  =  [
h ] (  ~=ph  `  J ) ) )  ->  (  ~=ph  `  K
)  Er  ( II 
Cn  K ) )
55 simpr3 1002 . . . . . . 7  |-  ( (
ph  /\  ( g  e.  U. V  /\  h  e.  U. V  /\  [
g ] (  ~=ph  `  J )  =  [
h ] (  ~=ph  `  J ) ) )  ->  [ g ] (  ~=ph  `  J )  =  [ h ]
(  ~=ph  `  J )
)
56 phtpcer 21580 . . . . . . . . 9  |-  (  ~=ph  `  J )  Er  (
II  Cn  J )
5756a1i 11 . . . . . . . 8  |-  ( (
ph  /\  ( g  e.  U. V  /\  h  e.  U. V  /\  [
g ] (  ~=ph  `  J )  =  [
h ] (  ~=ph  `  J ) ) )  ->  (  ~=ph  `  J
)  Er  ( II 
Cn  J ) )
58 simpr1 1000 . . . . . . . . . 10  |-  ( (
ph  /\  ( g  e.  U. V  /\  h  e.  U. V  /\  [
g ] (  ~=ph  `  J )  =  [
h ] (  ~=ph  `  J ) ) )  ->  g  e.  U. V )
5925adantr 463 . . . . . . . . . 10  |-  ( (
ph  /\  ( g  e.  U. V  /\  h  e.  U. V  /\  [
g ] (  ~=ph  `  J )  =  [
h ] (  ~=ph  `  J ) ) )  ->  ( g  e. 
U. V  <->  ( g  e.  ( II  Cn  J
)  /\  ( g `  0 )  =  A  /\  ( g `
 1 )  =  A ) ) )
6058, 59mpbid 210 . . . . . . . . 9  |-  ( (
ph  /\  ( g  e.  U. V  /\  h  e.  U. V  /\  [
g ] (  ~=ph  `  J )  =  [
h ] (  ~=ph  `  J ) ) )  ->  ( g  e.  ( II  Cn  J
)  /\  ( g `  0 )  =  A  /\  ( g `
 1 )  =  A ) )
6160simp1d 1006 . . . . . . . 8  |-  ( (
ph  /\  ( g  e.  U. V  /\  h  e.  U. V  /\  [
g ] (  ~=ph  `  J )  =  [
h ] (  ~=ph  `  J ) ) )  ->  g  e.  ( II  Cn  J ) )
6257, 61erth 7274 . . . . . . 7  |-  ( (
ph  /\  ( g  e.  U. V  /\  h  e.  U. V  /\  [
g ] (  ~=ph  `  J )  =  [
h ] (  ~=ph  `  J ) ) )  ->  ( g ( 
~=ph  `  J ) h  <->  [ g ] ( 
~=ph  `  J )  =  [ h ] ( 
~=ph  `  J ) ) )
6355, 62mpbird 232 . . . . . 6  |-  ( (
ph  /\  ( g  e.  U. V  /\  h  e.  U. V  /\  [
g ] (  ~=ph  `  J )  =  [
h ] (  ~=ph  `  J ) ) )  ->  g (  ~=ph  `  J ) h )
647adantr 463 . . . . . 6  |-  ( (
ph  /\  ( g  e.  U. V  /\  h  e.  U. V  /\  [
g ] (  ~=ph  `  J )  =  [
h ] (  ~=ph  `  J ) ) )  ->  F  e.  ( J  Cn  K ) )
6563, 64phtpcco2 21584 . . . . 5  |-  ( (
ph  /\  ( g  e.  U. V  /\  h  e.  U. V  /\  [
g ] (  ~=ph  `  J )  =  [
h ] (  ~=ph  `  J ) ) )  ->  ( F  o.  g ) (  ~=ph  `  K ) ( F  o.  h ) )
6654, 65erthi 7276 . . . 4  |-  ( (
ph  /\  ( g  e.  U. V  /\  h  e.  U. V  /\  [
g ] (  ~=ph  `  J )  =  [
h ] (  ~=ph  `  J ) ) )  ->  [ ( F  o.  g ) ] (  ~=ph  `  K )  =  [ ( F  o.  h ) ] (  ~=ph  `  K ) )
671, 4, 49, 50, 52, 66fliftfund 6112 . . 3  |-  ( ph  ->  Fun  G )
681, 4, 49fliftf 6114 . . 3  |-  ( ph  ->  ( Fun  G  <->  G : ran  ( g  e.  U. V  |->  [ g ] (  ~=ph  `  J ) ) --> ( Base `  Q
) ) )
6967, 68mpbid 210 . 2  |-  ( ph  ->  G : ran  (
g  e.  U. V  |->  [ g ] ( 
~=ph  `  J ) ) --> ( Base `  Q
) )
7022, 15, 18, 24pi1bas2 21626 . . . 4  |-  ( ph  ->  V  =  ( U. V /. (  ~=ph  `  J
) ) )
71 df-qs 7235 . . . . 5  |-  ( U. V /. (  ~=ph  `  J
) )  =  {
s  |  E. g  e.  U. V s  =  [ g ] ( 
~=ph  `  J ) }
72 eqid 2382 . . . . . 6  |-  ( g  e.  U. V  |->  [ g ] (  ~=ph  `  J ) )  =  ( g  e.  U. V  |->  [ g ] (  ~=ph  `  J ) )
7372rnmpt 5161 . . . . 5  |-  ran  (
g  e.  U. V  |->  [ g ] ( 
~=ph  `  J ) )  =  { s  |  E. g  e.  U. V s  =  [
g ] (  ~=ph  `  J ) }
7471, 73eqtr4i 2414 . . . 4  |-  ( U. V /. (  ~=ph  `  J
) )  =  ran  ( g  e.  U. V  |->  [ g ] (  ~=ph  `  J ) )
7570, 74syl6eq 2439 . . 3  |-  ( ph  ->  V  =  ran  (
g  e.  U. V  |->  [ g ] ( 
~=ph  `  J ) ) )
7675feq2d 5626 . 2  |-  ( ph  ->  ( G : V --> ( Base `  Q )  <->  G : ran  ( g  e.  U. V  |->  [ g ] (  ~=ph  `  J ) ) --> (
Base `  Q )
) )
7769, 76mpbird 232 1  |-  ( ph  ->  G : V --> ( Base `  Q ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1399    e. wcel 1826   {cab 2367   E.wrex 2733   _Vcvv 3034   <.cop 3950   U.cuni 4163   class class class wbr 4367    |-> cmpt 4425   ran crn 4914    o. ccom 4917   Fun wfun 5490   -->wf 5492   ` cfv 5496  (class class class)co 6196    Er wer 7226   [cec 7227   /.cqs 7228   0cc0 9403   1c1 9404   [,]cicc 11453   Basecbs 14634   Topctop 19479  TopOnctopon 19480    Cn ccn 19811   IIcii 21464    ~=ph cphtpc 21554    pi1 cpi1 21588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-inf2 7972  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480  ax-pre-sup 9481  ax-mulf 9483
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-int 4200  df-iun 4245  df-iin 4246  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-se 4753  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-isom 5505  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-of 6439  df-om 6600  df-1st 6699  df-2nd 6700  df-supp 6818  df-recs 6960  df-rdg 6994  df-1o 7048  df-2o 7049  df-oadd 7052  df-er 7229  df-ec 7231  df-qs 7235  df-map 7340  df-ixp 7389  df-en 7436  df-dom 7437  df-sdom 7438  df-fin 7439  df-fsupp 7745  df-fi 7786  df-sup 7816  df-oi 7850  df-card 8233  df-cda 8461  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-div 10124  df-nn 10453  df-2 10511  df-3 10512  df-4 10513  df-5 10514  df-6 10515  df-7 10516  df-8 10517  df-9 10518  df-10 10519  df-n0 10713  df-z 10782  df-dec 10896  df-uz 11002  df-q 11102  df-rp 11140  df-xneg 11239  df-xadd 11240  df-xmul 11241  df-ioo 11454  df-icc 11457  df-fz 11594  df-fzo 11718  df-seq 12011  df-exp 12070  df-hash 12308  df-cj 12934  df-re 12935  df-im 12936  df-sqrt 13070  df-abs 13071  df-struct 14636  df-ndx 14637  df-slot 14638  df-base 14639  df-sets 14640  df-ress 14641  df-plusg 14715  df-mulr 14716  df-starv 14717  df-sca 14718  df-vsca 14719  df-ip 14720  df-tset 14721  df-ple 14722  df-ds 14724  df-unif 14725  df-hom 14726  df-cco 14727  df-rest 14830  df-topn 14831  df-0g 14849  df-gsum 14850  df-topgen 14851  df-pt 14852  df-prds 14855  df-xrs 14909  df-qtop 14914  df-imas 14915  df-qus 14916  df-xps 14917  df-mre 14993  df-mrc 14994  df-acs 14996  df-mgm 15989  df-sgrp 16028  df-mnd 16038  df-submnd 16084  df-mulg 16177  df-cntz 16472  df-cmn 16917  df-psmet 18524  df-xmet 18525  df-met 18526  df-bl 18527  df-mopn 18528  df-cnfld 18534  df-top 19484  df-bases 19486  df-topon 19487  df-topsp 19488  df-cld 19605  df-cn 19814  df-cnp 19815  df-tx 20148  df-hmeo 20341  df-xms 20908  df-ms 20909  df-tms 20910  df-ii 21466  df-htpy 21555  df-phtpy 21556  df-phtpc 21577  df-om1 21591  df-pi1 21593
This theorem is referenced by:  pi1coval  21645  pi1coghm  21646
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