MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pi1cof Structured version   Visualization version   Unicode version

Theorem pi1cof 22083
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 5873 . . . . 5  |-  (  ~=ph  `  J )  e.  _V
3 ecexg 7364 . . . . 5  |-  ( ( 
~=ph  `  J )  e. 
_V  ->  [ g ] (  ~=ph  `  J )  e.  _V )
42, 3mp1i 13 . . . 4  |-  ( (
ph  /\  g  e.  U. V )  ->  [ g ] (  ~=ph  `  J
)  e.  _V )
5 pi1co.q . . . . 5  |-  Q  =  ( K  pi1  B )
6 eqid 2450 . . . . 5  |-  ( Base `  Q )  =  (
Base `  Q )
7 pi1co.f . . . . . . . 8  |-  ( ph  ->  F  e.  ( J  Cn  K ) )
8 cntop2 20250 . . . . . . . 8  |-  ( F  e.  ( J  Cn  K )  ->  K  e.  Top )
97, 8syl 17 . . . . . . 7  |-  ( ph  ->  K  e.  Top )
10 eqid 2450 . . . . . . . 8  |-  U. K  =  U. K
1110toptopon 19941 . . . . . . 7  |-  ( K  e.  Top  <->  K  e.  (TopOn `  U. K ) )
129, 11sylib 200 . . . . . 6  |-  ( ph  ->  K  e.  (TopOn `  U. K ) )
1312adantr 467 . . . . 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 20258 . . . . . . . . 9  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  U. K )  /\  F  e.  ( J  Cn  K ) )  ->  F : X
--> U. K )
1715, 12, 7, 16syl3anc 1267 . . . . . . . 8  |-  ( ph  ->  F : X --> U. K
)
18 pi1co.a . . . . . . . 8  |-  ( ph  ->  A  e.  X )
1917, 18ffvelrnd 6021 . . . . . . 7  |-  ( ph  ->  ( F `  A
)  e.  U. K
)
2014, 19eqeltrrd 2529 . . . . . 6  |-  ( ph  ->  B  e.  U. K
)
2120adantr 467 . . . . 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 22066 . . . . . . . 8  |-  ( ph  ->  ( g  e.  U. V 
<->  ( g  e.  ( II  Cn  J )  /\  ( g ` 
0 )  =  A  /\  ( g ` 
1 )  =  A ) ) )
2625biimpa 487 . . . . . . 7  |-  ( (
ph  /\  g  e.  U. V )  ->  (
g  e.  ( II 
Cn  J )  /\  ( g `  0
)  =  A  /\  ( g `  1
)  =  A ) )
2726simp1d 1019 . . . . . 6  |-  ( (
ph  /\  g  e.  U. V )  ->  g  e.  ( II  Cn  J
) )
287adantr 467 . . . . . 6  |-  ( (
ph  /\  g  e.  U. V )  ->  F  e.  ( J  Cn  K
) )
29 cnco 20275 . . . . . 6  |-  ( ( g  e.  ( II 
Cn  J )  /\  F  e.  ( J  Cn  K ) )  -> 
( F  o.  g
)  e.  ( II 
Cn  K ) )
3027, 28, 29syl2anc 666 . . . . 5  |-  ( (
ph  /\  g  e.  U. V )  ->  ( F  o.  g )  e.  ( II  Cn  K
) )
31 iitopon 21904 . . . . . . . . 9  |-  II  e.  (TopOn `  ( 0 [,] 1 ) )
3231a1i 11 . . . . . . . 8  |-  ( (
ph  /\  g  e.  U. V )  ->  II  e.  (TopOn `  ( 0 [,] 1 ) ) )
3315adantr 467 . . . . . . . 8  |-  ( (
ph  /\  g  e.  U. V )  ->  J  e.  (TopOn `  X )
)
34 cnf2 20258 . . . . . . . 8  |-  ( ( II  e.  (TopOn `  ( 0 [,] 1
) )  /\  J  e.  (TopOn `  X )  /\  g  e.  (
II  Cn  J )
)  ->  g :
( 0 [,] 1
) --> X )
3532, 33, 27, 34syl3anc 1267 . . . . . . 7  |-  ( (
ph  /\  g  e.  U. V )  ->  g : ( 0 [,] 1 ) --> X )
36 0elunit 11747 . . . . . . 7  |-  0  e.  ( 0 [,] 1
)
37 fvco3 5940 . . . . . . 7  |-  ( ( g : ( 0 [,] 1 ) --> X  /\  0  e.  ( 0 [,] 1 ) )  ->  ( ( F  o.  g ) `  0 )  =  ( F `  (
g `  0 )
) )
3835, 36, 37sylancl 667 . . . . . 6  |-  ( (
ph  /\  g  e.  U. V )  ->  (
( F  o.  g
) `  0 )  =  ( F `  ( g `  0
) ) )
3926simp2d 1020 . . . . . . 7  |-  ( (
ph  /\  g  e.  U. V )  ->  (
g `  0 )  =  A )
4039fveq2d 5867 . . . . . 6  |-  ( (
ph  /\  g  e.  U. V )  ->  ( F `  ( g `  0 ) )  =  ( F `  A ) )
4114adantr 467 . . . . . 6  |-  ( (
ph  /\  g  e.  U. V )  ->  ( F `  A )  =  B )
4238, 40, 413eqtrd 2488 . . . . 5  |-  ( (
ph  /\  g  e.  U. V )  ->  (
( F  o.  g
) `  0 )  =  B )
43 1elunit 11748 . . . . . . 7  |-  1  e.  ( 0 [,] 1
)
44 fvco3 5940 . . . . . . 7  |-  ( ( g : ( 0 [,] 1 ) --> X  /\  1  e.  ( 0 [,] 1 ) )  ->  ( ( F  o.  g ) `  1 )  =  ( F `  (
g `  1 )
) )
4535, 43, 44sylancl 667 . . . . . 6  |-  ( (
ph  /\  g  e.  U. V )  ->  (
( F  o.  g
) `  1 )  =  ( F `  ( g `  1
) ) )
4626simp3d 1021 . . . . . . 7  |-  ( (
ph  /\  g  e.  U. V )  ->  (
g `  1 )  =  A )
4746fveq2d 5867 . . . . . 6  |-  ( (
ph  /\  g  e.  U. V )  ->  ( F `  ( g `  1 ) )  =  ( F `  A ) )
4845, 47, 413eqtrd 2488 . . . . 5  |-  ( (
ph  /\  g  e.  U. V )  ->  (
( F  o.  g
) `  1 )  =  B )
495, 6, 13, 21, 30, 42, 48elpi1i 22070 . . . 4  |-  ( (
ph  /\  g  e.  U. V )  ->  [ ( F  o.  g ) ] (  ~=ph  `  K
)  e.  ( Base `  Q ) )
50 eceq1 7396 . . . 4  |-  ( g  =  h  ->  [ g ] (  ~=ph  `  J
)  =  [ h ] (  ~=ph  `  J
) )
51 coeq2 4992 . . . . 5  |-  ( g  =  h  ->  ( F  o.  g )  =  ( F  o.  h ) )
5251eceq1d 7397 . . . 4  |-  ( g  =  h  ->  [ ( F  o.  g ) ] (  ~=ph  `  K
)  =  [ ( F  o.  h ) ] (  ~=ph  `  K
) )
53 phtpcer 22019 . . . . . 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 1015 . . . . . . 7  |-  ( (
ph  /\  ( g  e.  U. V  /\  h  e.  U. V  /\  [
g ] (  ~=ph  `  J )  =  [
h ] (  ~=ph  `  J ) ) )  ->  [ g ] (  ~=ph  `  J )  =  [ h ]
(  ~=ph  `  J )
)
56 phtpcer 22019 . . . . . . . . 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 1013 . . . . . . . . . 10  |-  ( (
ph  /\  ( g  e.  U. V  /\  h  e.  U. V  /\  [
g ] (  ~=ph  `  J )  =  [
h ] (  ~=ph  `  J ) ) )  ->  g  e.  U. V )
5925adantr 467 . . . . . . . . . 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 214 . . . . . . . . 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 1019 . . . . . . . 8  |-  ( (
ph  /\  ( g  e.  U. V  /\  h  e.  U. V  /\  [
g ] (  ~=ph  `  J )  =  [
h ] (  ~=ph  `  J ) ) )  ->  g  e.  ( II  Cn  J ) )
6257, 61erth 7405 . . . . . . 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 236 . . . . . 6  |-  ( (
ph  /\  ( g  e.  U. V  /\  h  e.  U. V  /\  [
g ] (  ~=ph  `  J )  =  [
h ] (  ~=ph  `  J ) ) )  ->  g (  ~=ph  `  J ) h )
647adantr 467 . . . . . 6  |-  ( (
ph  /\  ( g  e.  U. V  /\  h  e.  U. V  /\  [
g ] (  ~=ph  `  J )  =  [
h ] (  ~=ph  `  J ) ) )  ->  F  e.  ( J  Cn  K ) )
6563, 64phtpcco2 22023 . . . . 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 7407 . . . 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 6204 . . 3  |-  ( ph  ->  Fun  G )
681, 4, 49fliftf 6206 . . 3  |-  ( ph  ->  ( Fun  G  <->  G : ran  ( g  e.  U. V  |->  [ g ] (  ~=ph  `  J ) ) --> ( Base `  Q
) ) )
6967, 68mpbid 214 . 2  |-  ( ph  ->  G : ran  (
g  e.  U. V  |->  [ g ] ( 
~=ph  `  J ) ) --> ( Base `  Q
) )
7022, 15, 18, 24pi1bas2 22065 . . . 4  |-  ( ph  ->  V  =  ( U. V /. (  ~=ph  `  J
) ) )
71 df-qs 7366 . . . . 5  |-  ( U. V /. (  ~=ph  `  J
) )  =  {
s  |  E. g  e.  U. V s  =  [ g ] ( 
~=ph  `  J ) }
72 eqid 2450 . . . . . 6  |-  ( g  e.  U. V  |->  [ g ] (  ~=ph  `  J ) )  =  ( g  e.  U. V  |->  [ g ] (  ~=ph  `  J ) )
7372rnmpt 5079 . . . . 5  |-  ran  (
g  e.  U. V  |->  [ g ] ( 
~=ph  `  J ) )  =  { s  |  E. g  e.  U. V s  =  [
g ] (  ~=ph  `  J ) }
7471, 73eqtr4i 2475 . . . 4  |-  ( U. V /. (  ~=ph  `  J
) )  =  ran  ( g  e.  U. V  |->  [ g ] (  ~=ph  `  J ) )
7570, 74syl6eq 2500 . . 3  |-  ( ph  ->  V  =  ran  (
g  e.  U. V  |->  [ g ] ( 
~=ph  `  J ) ) )
7675feq2d 5713 . 2  |-  ( ph  ->  ( G : V --> ( Base `  Q )  <->  G : ran  ( g  e.  U. V  |->  [ g ] (  ~=ph  `  J ) ) --> (
Base `  Q )
) )
7769, 76mpbird 236 1  |-  ( ph  ->  G : V --> ( Base `  Q ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 984    = wceq 1443    e. wcel 1886   {cab 2436   E.wrex 2737   _Vcvv 3044   <.cop 3973   U.cuni 4197   class class class wbr 4401    |-> cmpt 4460   ran crn 4834    o. ccom 4837   Fun wfun 5575   -->wf 5577   ` cfv 5581  (class class class)co 6288    Er wer 7357   [cec 7358   /.cqs 7359   0cc0 9536   1c1 9537   [,]cicc 11635   Basecbs 15114   Topctop 19910  TopOnctopon 19911    Cn ccn 20233   IIcii 21900    ~=ph cphtpc 21993    pi1 cpi1 22027
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-inf2 8143  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613  ax-pre-sup 9614  ax-mulf 9616
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-int 4234  df-iun 4279  df-iin 4280  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-se 4793  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-isom 5590  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-of 6528  df-om 6690  df-1st 6790  df-2nd 6791  df-supp 6912  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-1o 7179  df-2o 7180  df-oadd 7183  df-er 7360  df-ec 7362  df-qs 7366  df-map 7471  df-ixp 7520  df-en 7567  df-dom 7568  df-sdom 7569  df-fin 7570  df-fsupp 7881  df-fi 7922  df-sup 7953  df-inf 7954  df-oi 8022  df-card 8370  df-cda 8595  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-div 10267  df-nn 10607  df-2 10665  df-3 10666  df-4 10667  df-5 10668  df-6 10669  df-7 10670  df-8 10671  df-9 10672  df-10 10673  df-n0 10867  df-z 10935  df-dec 11049  df-uz 11157  df-q 11262  df-rp 11300  df-xneg 11406  df-xadd 11407  df-xmul 11408  df-ioo 11636  df-icc 11639  df-fz 11782  df-fzo 11913  df-seq 12211  df-exp 12270  df-hash 12513  df-cj 13155  df-re 13156  df-im 13157  df-sqrt 13291  df-abs 13292  df-struct 15116  df-ndx 15117  df-slot 15118  df-base 15119  df-sets 15120  df-ress 15121  df-plusg 15196  df-mulr 15197  df-starv 15198  df-sca 15199  df-vsca 15200  df-ip 15201  df-tset 15202  df-ple 15203  df-ds 15205  df-unif 15206  df-hom 15207  df-cco 15208  df-rest 15314  df-topn 15315  df-0g 15333  df-gsum 15334  df-topgen 15335  df-pt 15336  df-prds 15339  df-xrs 15393  df-qtop 15399  df-imas 15400  df-qus 15402  df-xps 15403  df-mre 15485  df-mrc 15486  df-acs 15488  df-mgm 16481  df-sgrp 16520  df-mnd 16530  df-submnd 16576  df-mulg 16669  df-cntz 16964  df-cmn 17425  df-psmet 18955  df-xmet 18956  df-met 18957  df-bl 18958  df-mopn 18959  df-cnfld 18964  df-top 19914  df-bases 19915  df-topon 19916  df-topsp 19917  df-cld 20027  df-cn 20236  df-cnp 20237  df-tx 20570  df-hmeo 20763  df-xms 21328  df-ms 21329  df-tms 21330  df-ii 21902  df-htpy 21994  df-phtpy 21995  df-phtpc 22016  df-om1 22030  df-pi1 22032
This theorem is referenced by:  pi1coval  22084  pi1coghm  22085
  Copyright terms: Public domain W3C validator