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Theorem pi1cof 19076
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  pi 1  A )
pi1co.q  |-  Q  =  ( K  pi 1  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 5734 . . . . 5  |-  (  ~=ph  `  J )  e.  _V
3 ecexg 6901 . . . . 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  pi 1  B )
6 eqid 2435 . . . . 5  |-  ( Base `  Q )  =  (
Base `  Q )
7 pi1co.f . . . . . . . 8  |-  ( ph  ->  F  e.  ( J  Cn  K ) )
8 cntop2 17297 . . . . . . . 8  |-  ( F  e.  ( J  Cn  K )  ->  K  e.  Top )
97, 8syl 16 . . . . . . 7  |-  ( ph  ->  K  e.  Top )
10 eqid 2435 . . . . . . . 8  |-  U. K  =  U. K
1110toptopon 16990 . . . . . . 7  |-  ( K  e.  Top  <->  K  e.  (TopOn `  U. K ) )
129, 11sylib 189 . . . . . 6  |-  ( ph  ->  K  e.  (TopOn `  U. K ) )
1312adantr 452 . . . . 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 17305 . . . . . . . . 9  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  U. K )  /\  F  e.  ( J  Cn  K ) )  ->  F : X
--> U. K )
1715, 12, 7, 16syl3anc 1184 . . . . . . . 8  |-  ( ph  ->  F : X --> U. K
)
18 pi1co.a . . . . . . . 8  |-  ( ph  ->  A  e.  X )
1917, 18ffvelrnd 5863 . . . . . . 7  |-  ( ph  ->  ( F `  A
)  e.  U. K
)
2014, 19eqeltrrd 2510 . . . . . 6  |-  ( ph  ->  B  e.  U. K
)
2120adantr 452 . . . . 5  |-  ( (
ph  /\  g  e.  U. V )  ->  B  e.  U. K )
22 pi1co.p . . . . . . . . 9  |-  P  =  ( J  pi 1  A )
23 pi1co.v . . . . . . . . . 10  |-  V  =  ( Base `  P
)
2423a1i 11 . . . . . . . . 9  |-  ( ph  ->  V  =  ( Base `  P ) )
2522, 15, 18, 24pi1eluni 19059 . . . . . . . 8  |-  ( ph  ->  ( g  e.  U. V 
<->  ( g  e.  ( II  Cn  J )  /\  ( g ` 
0 )  =  A  /\  ( g ` 
1 )  =  A ) ) )
2625biimpa 471 . . . . . . 7  |-  ( (
ph  /\  g  e.  U. V )  ->  (
g  e.  ( II 
Cn  J )  /\  ( g `  0
)  =  A  /\  ( g `  1
)  =  A ) )
2726simp1d 969 . . . . . 6  |-  ( (
ph  /\  g  e.  U. V )  ->  g  e.  ( II  Cn  J
) )
287adantr 452 . . . . . 6  |-  ( (
ph  /\  g  e.  U. V )  ->  F  e.  ( J  Cn  K
) )
29 cnco 17322 . . . . . 6  |-  ( ( g  e.  ( II 
Cn  J )  /\  F  e.  ( J  Cn  K ) )  -> 
( F  o.  g
)  e.  ( II 
Cn  K ) )
3027, 28, 29syl2anc 643 . . . . 5  |-  ( (
ph  /\  g  e.  U. V )  ->  ( F  o.  g )  e.  ( II  Cn  K
) )
31 iitopon 18901 . . . . . . . . 9  |-  II  e.  (TopOn `  ( 0 [,] 1 ) )
3231a1i 11 . . . . . . . 8  |-  ( (
ph  /\  g  e.  U. V )  ->  II  e.  (TopOn `  ( 0 [,] 1 ) ) )
3315adantr 452 . . . . . . . 8  |-  ( (
ph  /\  g  e.  U. V )  ->  J  e.  (TopOn `  X )
)
34 cnf2 17305 . . . . . . . 8  |-  ( ( II  e.  (TopOn `  ( 0 [,] 1
) )  /\  J  e.  (TopOn `  X )  /\  g  e.  (
II  Cn  J )
)  ->  g :
( 0 [,] 1
) --> X )
3532, 33, 27, 34syl3anc 1184 . . . . . . 7  |-  ( (
ph  /\  g  e.  U. V )  ->  g : ( 0 [,] 1 ) --> X )
36 0elunit 11007 . . . . . . 7  |-  0  e.  ( 0 [,] 1
)
37 fvco3 5792 . . . . . . 7  |-  ( ( g : ( 0 [,] 1 ) --> X  /\  0  e.  ( 0 [,] 1 ) )  ->  ( ( F  o.  g ) `  0 )  =  ( F `  (
g `  0 )
) )
3835, 36, 37sylancl 644 . . . . . 6  |-  ( (
ph  /\  g  e.  U. V )  ->  (
( F  o.  g
) `  0 )  =  ( F `  ( g `  0
) ) )
3926simp2d 970 . . . . . . 7  |-  ( (
ph  /\  g  e.  U. V )  ->  (
g `  0 )  =  A )
4039fveq2d 5724 . . . . . 6  |-  ( (
ph  /\  g  e.  U. V )  ->  ( F `  ( g `  0 ) )  =  ( F `  A ) )
4114adantr 452 . . . . . 6  |-  ( (
ph  /\  g  e.  U. V )  ->  ( F `  A )  =  B )
4238, 40, 413eqtrd 2471 . . . . 5  |-  ( (
ph  /\  g  e.  U. V )  ->  (
( F  o.  g
) `  0 )  =  B )
43 1elunit 11008 . . . . . . 7  |-  1  e.  ( 0 [,] 1
)
44 fvco3 5792 . . . . . . 7  |-  ( ( g : ( 0 [,] 1 ) --> X  /\  1  e.  ( 0 [,] 1 ) )  ->  ( ( F  o.  g ) `  1 )  =  ( F `  (
g `  1 )
) )
4535, 43, 44sylancl 644 . . . . . 6  |-  ( (
ph  /\  g  e.  U. V )  ->  (
( F  o.  g
) `  1 )  =  ( F `  ( g `  1
) ) )
4626simp3d 971 . . . . . . 7  |-  ( (
ph  /\  g  e.  U. V )  ->  (
g `  1 )  =  A )
4746fveq2d 5724 . . . . . 6  |-  ( (
ph  /\  g  e.  U. V )  ->  ( F `  ( g `  1 ) )  =  ( F `  A ) )
4845, 47, 413eqtrd 2471 . . . . 5  |-  ( (
ph  /\  g  e.  U. V )  ->  (
( F  o.  g
) `  1 )  =  B )
495, 6, 13, 21, 30, 42, 48elpi1i 19063 . . . 4  |-  ( (
ph  /\  g  e.  U. V )  ->  [ ( F  o.  g ) ] (  ~=ph  `  K
)  e.  ( Base `  Q ) )
50 eceq1 6933 . . . 4  |-  ( g  =  h  ->  [ g ] (  ~=ph  `  J
)  =  [ h ] (  ~=ph  `  J
) )
51 coeq2 5023 . . . . 5  |-  ( g  =  h  ->  ( F  o.  g )  =  ( F  o.  h ) )
52 eceq1 6933 . . . . 5  |-  ( ( F  o.  g )  =  ( F  o.  h )  ->  [ ( F  o.  g ) ] (  ~=ph  `  K
)  =  [ ( F  o.  h ) ] (  ~=ph  `  K
) )
5351, 52syl 16 . . . 4  |-  ( g  =  h  ->  [ ( F  o.  g ) ] (  ~=ph  `  K
)  =  [ ( F  o.  h ) ] (  ~=ph  `  K
) )
54 phtpcer 19012 . . . . . 6  |-  (  ~=ph  `  K )  Er  (
II  Cn  K )
5554a1i 11 . . . . 5  |-  ( (
ph  /\  ( g  e.  U. V  /\  h  e.  U. V  /\  [
g ] (  ~=ph  `  J )  =  [
h ] (  ~=ph  `  J ) ) )  ->  (  ~=ph  `  K
)  Er  ( II 
Cn  K ) )
56 simpr3 965 . . . . . . 7  |-  ( (
ph  /\  ( g  e.  U. V  /\  h  e.  U. V  /\  [
g ] (  ~=ph  `  J )  =  [
h ] (  ~=ph  `  J ) ) )  ->  [ g ] (  ~=ph  `  J )  =  [ h ]
(  ~=ph  `  J )
)
57 phtpcer 19012 . . . . . . . . 9  |-  (  ~=ph  `  J )  Er  (
II  Cn  J )
5857a1i 11 . . . . . . . 8  |-  ( (
ph  /\  ( g  e.  U. V  /\  h  e.  U. V  /\  [
g ] (  ~=ph  `  J )  =  [
h ] (  ~=ph  `  J ) ) )  ->  (  ~=ph  `  J
)  Er  ( II 
Cn  J ) )
59 simpr1 963 . . . . . . . . . 10  |-  ( (
ph  /\  ( g  e.  U. V  /\  h  e.  U. V  /\  [
g ] (  ~=ph  `  J )  =  [
h ] (  ~=ph  `  J ) ) )  ->  g  e.  U. V )
6025adantr 452 . . . . . . . . . 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 ) ) )
6159, 60mpbid 202 . . . . . . . . 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 ) )
6261simp1d 969 . . . . . . . 8  |-  ( (
ph  /\  ( g  e.  U. V  /\  h  e.  U. V  /\  [
g ] (  ~=ph  `  J )  =  [
h ] (  ~=ph  `  J ) ) )  ->  g  e.  ( II  Cn  J ) )
6358, 62erth 6941 . . . . . . 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 ) ) )
6456, 63mpbird 224 . . . . . 6  |-  ( (
ph  /\  ( g  e.  U. V  /\  h  e.  U. V  /\  [
g ] (  ~=ph  `  J )  =  [
h ] (  ~=ph  `  J ) ) )  ->  g (  ~=ph  `  J ) h )
657adantr 452 . . . . . 6  |-  ( (
ph  /\  ( g  e.  U. V  /\  h  e.  U. V  /\  [
g ] (  ~=ph  `  J )  =  [
h ] (  ~=ph  `  J ) ) )  ->  F  e.  ( J  Cn  K ) )
6664, 65phtpcco2 19016 . . . . 5  |-  ( (
ph  /\  ( g  e.  U. V  /\  h  e.  U. V  /\  [
g ] (  ~=ph  `  J )  =  [
h ] (  ~=ph  `  J ) ) )  ->  ( F  o.  g ) (  ~=ph  `  K ) ( F  o.  h ) )
6755, 66erthi 6943 . . . 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 ) )
681, 4, 49, 50, 53, 67fliftfund 6027 . . 3  |-  ( ph  ->  Fun  G )
691, 4, 49fliftf 6029 . . 3  |-  ( ph  ->  ( Fun  G  <->  G : ran  ( g  e.  U. V  |->  [ g ] (  ~=ph  `  J ) ) --> ( Base `  Q
) ) )
7068, 69mpbid 202 . 2  |-  ( ph  ->  G : ran  (
g  e.  U. V  |->  [ g ] ( 
~=ph  `  J ) ) --> ( Base `  Q
) )
7122, 15, 18, 24pi1bas2 19058 . . . 4  |-  ( ph  ->  V  =  ( U. V /. (  ~=ph  `  J
) ) )
72 df-qs 6903 . . . . 5  |-  ( U. V /. (  ~=ph  `  J
) )  =  {
s  |  E. g  e.  U. V s  =  [ g ] ( 
~=ph  `  J ) }
73 eqid 2435 . . . . . 6  |-  ( g  e.  U. V  |->  [ g ] (  ~=ph  `  J ) )  =  ( g  e.  U. V  |->  [ g ] (  ~=ph  `  J ) )
7473rnmpt 5108 . . . . 5  |-  ran  (
g  e.  U. V  |->  [ g ] ( 
~=ph  `  J ) )  =  { s  |  E. g  e.  U. V s  =  [
g ] (  ~=ph  `  J ) }
7572, 74eqtr4i 2458 . . . 4  |-  ( U. V /. (  ~=ph  `  J
) )  =  ran  ( g  e.  U. V  |->  [ g ] (  ~=ph  `  J ) )
7671, 75syl6eq 2483 . . 3  |-  ( ph  ->  V  =  ran  (
g  e.  U. V  |->  [ g ] ( 
~=ph  `  J ) ) )
7776feq2d 5573 . 2  |-  ( ph  ->  ( G : V --> ( Base `  Q )  <->  G : ran  ( g  e.  U. V  |->  [ g ] (  ~=ph  `  J ) ) --> (
Base `  Q )
) )
7870, 77mpbird 224 1  |-  ( ph  ->  G : V --> ( Base `  Q ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   {cab 2421   E.wrex 2698   _Vcvv 2948   <.cop 3809   U.cuni 4007   class class class wbr 4204    e. cmpt 4258   ran crn 4871    o. ccom 4874   Fun wfun 5440   -->wf 5442   ` cfv 5446  (class class class)co 6073    Er wer 6894   [cec 6895   /.cqs 6896   0cc0 8982   1c1 8983   [,]cicc 10911   Basecbs 13461   Topctop 16950  TopOnctopon 16951    Cn ccn 17280   IIcii 18897    ~=ph cphtpc 18986    pi 1 cpi1 19020
This theorem is referenced by:  pi1coval  19077  pi1coghm  19078
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060  ax-mulf 9062
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-oadd 6720  df-er 6897  df-ec 6899  df-qs 6903  df-map 7012  df-ixp 7056  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-fi 7408  df-sup 7438  df-oi 7471  df-card 7818  df-cda 8040  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-7 10055  df-8 10056  df-9 10057  df-10 10058  df-n0 10214  df-z 10275  df-dec 10375  df-uz 10481  df-q 10567  df-rp 10605  df-xneg 10702  df-xadd 10703  df-xmul 10704  df-ioo 10912  df-icc 10915  df-fz 11036  df-fzo 11128  df-seq 11316  df-exp 11375  df-hash 11611  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-struct 13463  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-mulr 13535  df-starv 13536  df-sca 13537  df-vsca 13538  df-tset 13540  df-ple 13541  df-ds 13543  df-unif 13544  df-hom 13545  df-cco 13546  df-rest 13642  df-topn 13643  df-topgen 13659  df-pt 13660  df-prds 13663  df-xrs 13718  df-0g 13719  df-gsum 13720  df-qtop 13725  df-imas 13726  df-divs 13727  df-xps 13728  df-mre 13803  df-mrc 13804  df-acs 13806  df-mnd 14682  df-submnd 14731  df-mulg 14807  df-cntz 15108  df-cmn 15406  df-psmet 16686  df-xmet 16687  df-met 16688  df-bl 16689  df-mopn 16690  df-cnfld 16696  df-top 16955  df-bases 16957  df-topon 16958  df-topsp 16959  df-cld 17075  df-cn 17283  df-cnp 17284  df-tx 17586  df-hmeo 17779  df-xms 18342  df-ms 18343  df-tms 18344  df-ii 18899  df-htpy 18987  df-phtpy 18988  df-phtpc 19009  df-om1 19023  df-pi1 19025
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