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Theorem pi1cof 20756
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 5802 . . . . 5  |-  (  ~=ph  `  J )  e.  _V
3 ecexg 7208 . . . . 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 2451 . . . . 5  |-  ( Base `  Q )  =  (
Base `  Q )
7 pi1co.f . . . . . . . 8  |-  ( ph  ->  F  e.  ( J  Cn  K ) )
8 cntop2 18970 . . . . . . . 8  |-  ( F  e.  ( J  Cn  K )  ->  K  e.  Top )
97, 8syl 16 . . . . . . 7  |-  ( ph  ->  K  e.  Top )
10 eqid 2451 . . . . . . . 8  |-  U. K  =  U. K
1110toptopon 18663 . . . . . . 7  |-  ( K  e.  Top  <->  K  e.  (TopOn `  U. K ) )
129, 11sylib 196 . . . . . 6  |-  ( ph  ->  K  e.  (TopOn `  U. K ) )
1312adantr 465 . . . . 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 18978 . . . . . . . . 9  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  U. K )  /\  F  e.  ( J  Cn  K ) )  ->  F : X
--> U. K )
1715, 12, 7, 16syl3anc 1219 . . . . . . . 8  |-  ( ph  ->  F : X --> U. K
)
18 pi1co.a . . . . . . . 8  |-  ( ph  ->  A  e.  X )
1917, 18ffvelrnd 5946 . . . . . . 7  |-  ( ph  ->  ( F `  A
)  e.  U. K
)
2014, 19eqeltrrd 2540 . . . . . 6  |-  ( ph  ->  B  e.  U. K
)
2120adantr 465 . . . . 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 20739 . . . . . . . 8  |-  ( ph  ->  ( g  e.  U. V 
<->  ( g  e.  ( II  Cn  J )  /\  ( g ` 
0 )  =  A  /\  ( g ` 
1 )  =  A ) ) )
2625biimpa 484 . . . . . . 7  |-  ( (
ph  /\  g  e.  U. V )  ->  (
g  e.  ( II 
Cn  J )  /\  ( g `  0
)  =  A  /\  ( g `  1
)  =  A ) )
2726simp1d 1000 . . . . . 6  |-  ( (
ph  /\  g  e.  U. V )  ->  g  e.  ( II  Cn  J
) )
287adantr 465 . . . . . 6  |-  ( (
ph  /\  g  e.  U. V )  ->  F  e.  ( J  Cn  K
) )
29 cnco 18995 . . . . . 6  |-  ( ( g  e.  ( II 
Cn  J )  /\  F  e.  ( J  Cn  K ) )  -> 
( F  o.  g
)  e.  ( II 
Cn  K ) )
3027, 28, 29syl2anc 661 . . . . 5  |-  ( (
ph  /\  g  e.  U. V )  ->  ( F  o.  g )  e.  ( II  Cn  K
) )
31 iitopon 20580 . . . . . . . . 9  |-  II  e.  (TopOn `  ( 0 [,] 1 ) )
3231a1i 11 . . . . . . . 8  |-  ( (
ph  /\  g  e.  U. V )  ->  II  e.  (TopOn `  ( 0 [,] 1 ) ) )
3315adantr 465 . . . . . . . 8  |-  ( (
ph  /\  g  e.  U. V )  ->  J  e.  (TopOn `  X )
)
34 cnf2 18978 . . . . . . . 8  |-  ( ( II  e.  (TopOn `  ( 0 [,] 1
) )  /\  J  e.  (TopOn `  X )  /\  g  e.  (
II  Cn  J )
)  ->  g :
( 0 [,] 1
) --> X )
3532, 33, 27, 34syl3anc 1219 . . . . . . 7  |-  ( (
ph  /\  g  e.  U. V )  ->  g : ( 0 [,] 1 ) --> X )
36 0elunit 11513 . . . . . . 7  |-  0  e.  ( 0 [,] 1
)
37 fvco3 5870 . . . . . . 7  |-  ( ( g : ( 0 [,] 1 ) --> X  /\  0  e.  ( 0 [,] 1 ) )  ->  ( ( F  o.  g ) `  0 )  =  ( F `  (
g `  0 )
) )
3835, 36, 37sylancl 662 . . . . . 6  |-  ( (
ph  /\  g  e.  U. V )  ->  (
( F  o.  g
) `  0 )  =  ( F `  ( g `  0
) ) )
3926simp2d 1001 . . . . . . 7  |-  ( (
ph  /\  g  e.  U. V )  ->  (
g `  0 )  =  A )
4039fveq2d 5796 . . . . . 6  |-  ( (
ph  /\  g  e.  U. V )  ->  ( F `  ( g `  0 ) )  =  ( F `  A ) )
4114adantr 465 . . . . . 6  |-  ( (
ph  /\  g  e.  U. V )  ->  ( F `  A )  =  B )
4238, 40, 413eqtrd 2496 . . . . 5  |-  ( (
ph  /\  g  e.  U. V )  ->  (
( F  o.  g
) `  0 )  =  B )
43 1elunit 11514 . . . . . . 7  |-  1  e.  ( 0 [,] 1
)
44 fvco3 5870 . . . . . . 7  |-  ( ( g : ( 0 [,] 1 ) --> X  /\  1  e.  ( 0 [,] 1 ) )  ->  ( ( F  o.  g ) `  1 )  =  ( F `  (
g `  1 )
) )
4535, 43, 44sylancl 662 . . . . . 6  |-  ( (
ph  /\  g  e.  U. V )  ->  (
( F  o.  g
) `  1 )  =  ( F `  ( g `  1
) ) )
4626simp3d 1002 . . . . . . 7  |-  ( (
ph  /\  g  e.  U. V )  ->  (
g `  1 )  =  A )
4746fveq2d 5796 . . . . . 6  |-  ( (
ph  /\  g  e.  U. V )  ->  ( F `  ( g `  1 ) )  =  ( F `  A ) )
4845, 47, 413eqtrd 2496 . . . . 5  |-  ( (
ph  /\  g  e.  U. V )  ->  (
( F  o.  g
) `  1 )  =  B )
495, 6, 13, 21, 30, 42, 48elpi1i 20743 . . . 4  |-  ( (
ph  /\  g  e.  U. V )  ->  [ ( F  o.  g ) ] (  ~=ph  `  K
)  e.  ( Base `  Q ) )
50 eceq1 7240 . . . 4  |-  ( g  =  h  ->  [ g ] (  ~=ph  `  J
)  =  [ h ] (  ~=ph  `  J
) )
51 coeq2 5099 . . . . 5  |-  ( g  =  h  ->  ( F  o.  g )  =  ( F  o.  h ) )
52 eceq1 7240 . . . . 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 20692 . . . . . 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 996 . . . . . . 7  |-  ( (
ph  /\  ( g  e.  U. V  /\  h  e.  U. V  /\  [
g ] (  ~=ph  `  J )  =  [
h ] (  ~=ph  `  J ) ) )  ->  [ g ] (  ~=ph  `  J )  =  [ h ]
(  ~=ph  `  J )
)
57 phtpcer 20692 . . . . . . . . 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 994 . . . . . . . . . 10  |-  ( (
ph  /\  ( g  e.  U. V  /\  h  e.  U. V  /\  [
g ] (  ~=ph  `  J )  =  [
h ] (  ~=ph  `  J ) ) )  ->  g  e.  U. V )
6025adantr 465 . . . . . . . . . 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 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 ) )
6261simp1d 1000 . . . . . . . 8  |-  ( (
ph  /\  ( g  e.  U. V  /\  h  e.  U. V  /\  [
g ] (  ~=ph  `  J )  =  [
h ] (  ~=ph  `  J ) ) )  ->  g  e.  ( II  Cn  J ) )
6358, 62erth 7248 . . . . . . 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 232 . . . . . 6  |-  ( (
ph  /\  ( g  e.  U. V  /\  h  e.  U. V  /\  [
g ] (  ~=ph  `  J )  =  [
h ] (  ~=ph  `  J ) ) )  ->  g (  ~=ph  `  J ) h )
657adantr 465 . . . . . 6  |-  ( (
ph  /\  ( g  e.  U. V  /\  h  e.  U. V  /\  [
g ] (  ~=ph  `  J )  =  [
h ] (  ~=ph  `  J ) ) )  ->  F  e.  ( J  Cn  K ) )
6664, 65phtpcco2 20696 . . . . 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 7250 . . . 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 6108 . . 3  |-  ( ph  ->  Fun  G )
691, 4, 49fliftf 6110 . . 3  |-  ( ph  ->  ( Fun  G  <->  G : ran  ( g  e.  U. V  |->  [ g ] (  ~=ph  `  J ) ) --> ( Base `  Q
) ) )
7068, 69mpbid 210 . 2  |-  ( ph  ->  G : ran  (
g  e.  U. V  |->  [ g ] ( 
~=ph  `  J ) ) --> ( Base `  Q
) )
7122, 15, 18, 24pi1bas2 20738 . . . 4  |-  ( ph  ->  V  =  ( U. V /. (  ~=ph  `  J
) ) )
72 df-qs 7210 . . . . 5  |-  ( U. V /. (  ~=ph  `  J
) )  =  {
s  |  E. g  e.  U. V s  =  [ g ] ( 
~=ph  `  J ) }
73 eqid 2451 . . . . . 6  |-  ( g  e.  U. V  |->  [ g ] (  ~=ph  `  J ) )  =  ( g  e.  U. V  |->  [ g ] (  ~=ph  `  J ) )
7473rnmpt 5186 . . . . 5  |-  ran  (
g  e.  U. V  |->  [ g ] ( 
~=ph  `  J ) )  =  { s  |  E. g  e.  U. V s  =  [
g ] (  ~=ph  `  J ) }
7572, 74eqtr4i 2483 . . . 4  |-  ( U. V /. (  ~=ph  `  J
) )  =  ran  ( g  e.  U. V  |->  [ g ] (  ~=ph  `  J ) )
7671, 75syl6eq 2508 . . 3  |-  ( ph  ->  V  =  ran  (
g  e.  U. V  |->  [ g ] ( 
~=ph  `  J ) ) )
7776feq2d 5648 . 2  |-  ( ph  ->  ( G : V --> ( Base `  Q )  <->  G : ran  ( g  e.  U. V  |->  [ g ] (  ~=ph  `  J ) ) --> (
Base `  Q )
) )
7870, 77mpbird 232 1  |-  ( ph  ->  G : V --> ( Base `  Q ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   {cab 2436   E.wrex 2796   _Vcvv 3071   <.cop 3984   U.cuni 4192   class class class wbr 4393    |-> cmpt 4451   ran crn 4942    o. ccom 4945   Fun wfun 5513   -->wf 5515   ` cfv 5519  (class class class)co 6193    Er wer 7201   [cec 7202   /.cqs 7203   0cc0 9386   1c1 9387   [,]cicc 11407   Basecbs 14285   Topctop 18623  TopOnctopon 18624    Cn ccn 18953   IIcii 20576    ~=ph cphtpc 20666    pi1 cpi1 20700
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-inf2 7951  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463  ax-pre-sup 9464  ax-mulf 9466
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-int 4230  df-iun 4274  df-iin 4275  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-se 4781  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-isom 5528  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-of 6423  df-om 6580  df-1st 6680  df-2nd 6681  df-supp 6794  df-recs 6935  df-rdg 6969  df-1o 7023  df-2o 7024  df-oadd 7027  df-er 7204  df-ec 7206  df-qs 7210  df-map 7319  df-ixp 7367  df-en 7414  df-dom 7415  df-sdom 7416  df-fin 7417  df-fsupp 7725  df-fi 7765  df-sup 7795  df-oi 7828  df-card 8213  df-cda 8441  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-div 10098  df-nn 10427  df-2 10484  df-3 10485  df-4 10486  df-5 10487  df-6 10488  df-7 10489  df-8 10490  df-9 10491  df-10 10492  df-n0 10684  df-z 10751  df-dec 10860  df-uz 10966  df-q 11058  df-rp 11096  df-xneg 11193  df-xadd 11194  df-xmul 11195  df-ioo 11408  df-icc 11411  df-fz 11548  df-fzo 11659  df-seq 11917  df-exp 11976  df-hash 12214  df-cj 12699  df-re 12700  df-im 12701  df-sqr 12835  df-abs 12836  df-struct 14287  df-ndx 14288  df-slot 14289  df-base 14290  df-sets 14291  df-ress 14292  df-plusg 14362  df-mulr 14363  df-starv 14364  df-sca 14365  df-vsca 14366  df-ip 14367  df-tset 14368  df-ple 14369  df-ds 14371  df-unif 14372  df-hom 14373  df-cco 14374  df-rest 14472  df-topn 14473  df-0g 14491  df-gsum 14492  df-topgen 14493  df-pt 14494  df-prds 14497  df-xrs 14551  df-qtop 14556  df-imas 14557  df-divs 14558  df-xps 14559  df-mre 14635  df-mrc 14636  df-acs 14638  df-mnd 15526  df-submnd 15576  df-mulg 15659  df-cntz 15946  df-cmn 16392  df-psmet 17927  df-xmet 17928  df-met 17929  df-bl 17930  df-mopn 17931  df-cnfld 17937  df-top 18628  df-bases 18630  df-topon 18631  df-topsp 18632  df-cld 18748  df-cn 18956  df-cnp 18957  df-tx 19260  df-hmeo 19453  df-xms 20020  df-ms 20021  df-tms 20022  df-ii 20578  df-htpy 20667  df-phtpy 20668  df-phtpc 20689  df-om1 20703  df-pi1 20705
This theorem is referenced by:  pi1coval  20757  pi1coghm  20758
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