MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pi1val Structured version   Unicode version

Theorem pi1val 19054
Description: The definition of the fundamental group. (Contributed by Mario Carneiro, 11-Feb-2015.) (Revised by Mario Carneiro, 10-Jul-2015.)
Hypotheses
Ref Expression
pi1val.g  |-  G  =  ( J  pi 1  Y )
pi1val.1  |-  ( ph  ->  J  e.  (TopOn `  X ) )
pi1val.2  |-  ( ph  ->  Y  e.  X )
pi1val.o  |-  O  =  ( J  Om 1  Y )
Assertion
Ref Expression
pi1val  |-  ( ph  ->  G  =  ( O 
/.s  (  ~=ph  `  J ) ) )

Proof of Theorem pi1val
Dummy variables  j 
y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pi1val.g . 2  |-  G  =  ( J  pi 1  Y )
2 df-pi1 19025 . . . 4  |-  pi 1  =  ( j  e. 
Top ,  y  e.  U. j  |->  ( ( j 
Om 1  y ) 
/.s  (  ~=ph  `  j ) ) )
32a1i 11 . . 3  |-  ( ph  ->  pi 1  =  ( j  e.  Top , 
y  e.  U. j  |->  ( ( j  Om 1  y )  /.s  (  ~=ph  `  j ) ) ) )
4 simprl 733 . . . . . 6  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  -> 
j  =  J )
5 simprr 734 . . . . . 6  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  -> 
y  =  Y )
64, 5oveq12d 6091 . . . . 5  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  -> 
( j  Om 1 
y )  =  ( J  Om 1  Y
) )
7 pi1val.o . . . . 5  |-  O  =  ( J  Om 1  Y )
86, 7syl6eqr 2485 . . . 4  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  -> 
( j  Om 1 
y )  =  O )
94fveq2d 5724 . . . 4  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  -> 
(  ~=ph  `  j )  =  (  ~=ph  `  J
) )
108, 9oveq12d 6091 . . 3  |-  ( (
ph  /\  ( j  =  J  /\  y  =  Y ) )  -> 
( ( j  Om 1  y )  /.s  (  ~=ph  `  j ) )  =  ( O  /.s  (  ~=ph  `  J ) ) )
11 unieq 4016 . . . . 5  |-  ( j  =  J  ->  U. j  =  U. J )
1211adantl 453 . . . 4  |-  ( (
ph  /\  j  =  J )  ->  U. j  =  U. J )
13 pi1val.1 . . . . . 6  |-  ( ph  ->  J  e.  (TopOn `  X ) )
14 toponuni 16984 . . . . . 6  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
1513, 14syl 16 . . . . 5  |-  ( ph  ->  X  =  U. J
)
1615adantr 452 . . . 4  |-  ( (
ph  /\  j  =  J )  ->  X  =  U. J )
1712, 16eqtr4d 2470 . . 3  |-  ( (
ph  /\  j  =  J )  ->  U. j  =  X )
18 topontop 16983 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  Top )
1913, 18syl 16 . . 3  |-  ( ph  ->  J  e.  Top )
20 pi1val.2 . . 3  |-  ( ph  ->  Y  e.  X )
21 ovex 6098 . . . 4  |-  ( O 
/.s  (  ~=ph  `  J ) )  e.  _V
2221a1i 11 . . 3  |-  ( ph  ->  ( O  /.s  (  ~=ph  `  J ) )  e. 
_V )
233, 10, 17, 19, 20, 22ovmpt2dx 6192 . 2  |-  ( ph  ->  ( J  pi 1  Y )  =  ( O  /.s  (  ~=ph  `  J ) ) )
241, 23syl5eq 2479 1  |-  ( ph  ->  G  =  ( O 
/.s  (  ~=ph  `  J ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2948   U.cuni 4007   ` cfv 5446  (class class class)co 6073    e. cmpt2 6075    /.s cqus 13723   Topctop 16950  TopOnctopon 16951    ~=ph cphtpc 18986    Om 1 comi 19018    pi 1 cpi1 19020
This theorem is referenced by:  pi1bas  19055  pi1addf  19064  pi1addval  19065  pi1grplem  19066
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-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  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-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-topon 16958  df-pi1 19025
  Copyright terms: Public domain W3C validator