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Theorem ghgrplem2OLD 25583
Description: Obsolete as of 14-Mar-2020. Lemma for ghgrpOLD 25584. (Contributed by Paul Chapman, 25-Apr-2008.) (Revised by Mario Carneiro, 12-May-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
ghgrpOLD.1  |-  ( ph  ->  F : X -onto-> Y
)
ghgrpOLD.2  |-  ( (
ph  /\  ( x  e.  X  /\  y  e.  X ) )  -> 
( F `  (
x G y ) )  =  ( ( F `  x ) O ( F `  y ) ) )
ghgrpOLD.3  |-  H  =  ( O  |`  ( Y  X.  Y ) )
Assertion
Ref Expression
ghgrplem2OLD  |-  ( (
ph  /\  ( C  e.  X  /\  D  e.  X ) )  -> 
( F `  ( C G D ) )  =  ( ( F `
 C ) H ( F `  D
) ) )
Distinct variable groups:    x, y, F    x, G, y    ph, x, y    x, H, y    x, X, y    x, Y, y   
x, O, y
Allowed substitution hints:    C( x, y)    D( x, y)

Proof of Theorem ghgrplem2OLD
Dummy variables  z  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ghgrpOLD.2 . . . . 5  |-  ( (
ph  /\  ( x  e.  X  /\  y  e.  X ) )  -> 
( F `  (
x G y ) )  =  ( ( F `  x ) O ( F `  y ) ) )
21ralrimivva 2878 . . . 4  |-  ( ph  ->  A. x  e.  X  A. y  e.  X  ( F `  ( x G y ) )  =  ( ( F `
 x ) O ( F `  y
) ) )
3 oveq1 6303 . . . . . . 7  |-  ( x  =  z  ->  (
x G y )  =  ( z G y ) )
43fveq2d 5876 . . . . . 6  |-  ( x  =  z  ->  ( F `  ( x G y ) )  =  ( F `  ( z G y ) ) )
5 fveq2 5872 . . . . . . 7  |-  ( x  =  z  ->  ( F `  x )  =  ( F `  z ) )
65oveq1d 6311 . . . . . 6  |-  ( x  =  z  ->  (
( F `  x
) O ( F `
 y ) )  =  ( ( F `
 z ) O ( F `  y
) ) )
74, 6eqeq12d 2479 . . . . 5  |-  ( x  =  z  ->  (
( F `  (
x G y ) )  =  ( ( F `  x ) O ( F `  y ) )  <->  ( F `  ( z G y ) )  =  ( ( F `  z
) O ( F `
 y ) ) ) )
8 oveq2 6304 . . . . . . 7  |-  ( y  =  w  ->  (
z G y )  =  ( z G w ) )
98fveq2d 5876 . . . . . 6  |-  ( y  =  w  ->  ( F `  ( z G y ) )  =  ( F `  ( z G w ) ) )
10 fveq2 5872 . . . . . . 7  |-  ( y  =  w  ->  ( F `  y )  =  ( F `  w ) )
1110oveq2d 6312 . . . . . 6  |-  ( y  =  w  ->  (
( F `  z
) O ( F `
 y ) )  =  ( ( F `
 z ) O ( F `  w
) ) )
129, 11eqeq12d 2479 . . . . 5  |-  ( y  =  w  ->  (
( F `  (
z G y ) )  =  ( ( F `  z ) O ( F `  y ) )  <->  ( F `  ( z G w ) )  =  ( ( F `  z
) O ( F `
 w ) ) ) )
137, 12cbvral2v 3092 . . . 4  |-  ( A. x  e.  X  A. y  e.  X  ( F `  ( x G y ) )  =  ( ( F `
 x ) O ( F `  y
) )  <->  A. z  e.  X  A. w  e.  X  ( F `  ( z G w ) )  =  ( ( F `  z
) O ( F `
 w ) ) )
142, 13sylib 196 . . 3  |-  ( ph  ->  A. z  e.  X  A. w  e.  X  ( F `  ( z G w ) )  =  ( ( F `
 z ) O ( F `  w
) ) )
15 oveq1 6303 . . . . . 6  |-  ( z  =  C  ->  (
z G w )  =  ( C G w ) )
1615fveq2d 5876 . . . . 5  |-  ( z  =  C  ->  ( F `  ( z G w ) )  =  ( F `  ( C G w ) ) )
17 fveq2 5872 . . . . . 6  |-  ( z  =  C  ->  ( F `  z )  =  ( F `  C ) )
1817oveq1d 6311 . . . . 5  |-  ( z  =  C  ->  (
( F `  z
) O ( F `
 w ) )  =  ( ( F `
 C ) O ( F `  w
) ) )
1916, 18eqeq12d 2479 . . . 4  |-  ( z  =  C  ->  (
( F `  (
z G w ) )  =  ( ( F `  z ) O ( F `  w ) )  <->  ( F `  ( C G w ) )  =  ( ( F `  C
) O ( F `
 w ) ) ) )
20 oveq2 6304 . . . . . 6  |-  ( w  =  D  ->  ( C G w )  =  ( C G D ) )
2120fveq2d 5876 . . . . 5  |-  ( w  =  D  ->  ( F `  ( C G w ) )  =  ( F `  ( C G D ) ) )
22 fveq2 5872 . . . . . 6  |-  ( w  =  D  ->  ( F `  w )  =  ( F `  D ) )
2322oveq2d 6312 . . . . 5  |-  ( w  =  D  ->  (
( F `  C
) O ( F `
 w ) )  =  ( ( F `
 C ) O ( F `  D
) ) )
2421, 23eqeq12d 2479 . . . 4  |-  ( w  =  D  ->  (
( F `  ( C G w ) )  =  ( ( F `
 C ) O ( F `  w
) )  <->  ( F `  ( C G D ) )  =  ( ( F `  C
) O ( F `
 D ) ) ) )
2519, 24rspc2v 3219 . . 3  |-  ( ( C  e.  X  /\  D  e.  X )  ->  ( A. z  e.  X  A. w  e.  X  ( F `  ( z G w ) )  =  ( ( F `  z
) O ( F `
 w ) )  ->  ( F `  ( C G D ) )  =  ( ( F `  C ) O ( F `  D ) ) ) )
2614, 25mpan9 469 . 2  |-  ( (
ph  /\  ( C  e.  X  /\  D  e.  X ) )  -> 
( F `  ( C G D ) )  =  ( ( F `
 C ) O ( F `  D
) ) )
27 ghgrpOLD.3 . . . 4  |-  H  =  ( O  |`  ( Y  X.  Y ) )
2827oveqi 6309 . . 3  |-  ( ( F `  C ) H ( F `  D ) )  =  ( ( F `  C ) ( O  |`  ( Y  X.  Y
) ) ( F `
 D ) )
29 ghgrpOLD.1 . . . . . 6  |-  ( ph  ->  F : X -onto-> Y
)
30 fof 5801 . . . . . 6  |-  ( F : X -onto-> Y  ->  F : X --> Y )
3129, 30syl 16 . . . . 5  |-  ( ph  ->  F : X --> Y )
32 ffvelrn 6030 . . . . . 6  |-  ( ( F : X --> Y  /\  C  e.  X )  ->  ( F `  C
)  e.  Y )
33 ffvelrn 6030 . . . . . 6  |-  ( ( F : X --> Y  /\  D  e.  X )  ->  ( F `  D
)  e.  Y )
3432, 33anim12dan 837 . . . . 5  |-  ( ( F : X --> Y  /\  ( C  e.  X  /\  D  e.  X
) )  ->  (
( F `  C
)  e.  Y  /\  ( F `  D )  e.  Y ) )
3531, 34sylan 471 . . . 4  |-  ( (
ph  /\  ( C  e.  X  /\  D  e.  X ) )  -> 
( ( F `  C )  e.  Y  /\  ( F `  D
)  e.  Y ) )
36 ovres 6441 . . . 4  |-  ( ( ( F `  C
)  e.  Y  /\  ( F `  D )  e.  Y )  -> 
( ( F `  C ) ( O  |`  ( Y  X.  Y
) ) ( F `
 D ) )  =  ( ( F `
 C ) O ( F `  D
) ) )
3735, 36syl 16 . . 3  |-  ( (
ph  /\  ( C  e.  X  /\  D  e.  X ) )  -> 
( ( F `  C ) ( O  |`  ( Y  X.  Y
) ) ( F `
 D ) )  =  ( ( F `
 C ) O ( F `  D
) ) )
3828, 37syl5eq 2510 . 2  |-  ( (
ph  /\  ( C  e.  X  /\  D  e.  X ) )  -> 
( ( F `  C ) H ( F `  D ) )  =  ( ( F `  C ) O ( F `  D ) ) )
3926, 38eqtr4d 2501 1  |-  ( (
ph  /\  ( C  e.  X  /\  D  e.  X ) )  -> 
( F `  ( C G D ) )  =  ( ( F `
 C ) H ( F `  D
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819   A.wral 2807    X. cxp 5006    |` cres 5010   -->wf 5590   -onto->wfo 5592   ` cfv 5594  (class class class)co 6296
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fo 5600  df-fv 5602  df-ov 6299
This theorem is referenced by:  ghgrpOLD  25584  ghabloOLD  25585
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