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Theorem ghgrplem2 21955
Description: Lemma for ghgrp 21956. (Contributed by Paul Chapman, 25-Apr-2008.) (Revised by Mario Carneiro, 12-May-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
ghgrp.1  |-  ( ph  ->  F : X -onto-> Y
)
ghgrp.2  |-  ( (
ph  /\  ( x  e.  X  /\  y  e.  X ) )  -> 
( F `  (
x G y ) )  =  ( ( F `  x ) O ( F `  y ) ) )
ghgrp.3  |-  H  =  ( O  |`  ( Y  X.  Y ) )
Assertion
Ref Expression
ghgrplem2  |-  ( (
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 ghgrplem2
Dummy variables  z  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ghgrp.2 . . . . 5  |-  ( (
ph  /\  ( x  e.  X  /\  y  e.  X ) )  -> 
( F `  (
x G y ) )  =  ( ( F `  x ) O ( F `  y ) ) )
21ralrimivva 2798 . . . 4  |-  ( ph  ->  A. x  e.  X  A. y  e.  X  ( F `  ( x G y ) )  =  ( ( F `
 x ) O ( F `  y
) ) )
3 oveq1 6088 . . . . . . 7  |-  ( x  =  z  ->  (
x G y )  =  ( z G y ) )
43fveq2d 5732 . . . . . 6  |-  ( x  =  z  ->  ( F `  ( x G y ) )  =  ( F `  ( z G y ) ) )
5 fveq2 5728 . . . . . . 7  |-  ( x  =  z  ->  ( F `  x )  =  ( F `  z ) )
65oveq1d 6096 . . . . . 6  |-  ( x  =  z  ->  (
( F `  x
) O ( F `
 y ) )  =  ( ( F `
 z ) O ( F `  y
) ) )
74, 6eqeq12d 2450 . . . . 5  |-  ( x  =  z  ->  (
( F `  (
x G y ) )  =  ( ( F `  x ) O ( F `  y ) )  <->  ( F `  ( z G y ) )  =  ( ( F `  z
) O ( F `
 y ) ) ) )
8 oveq2 6089 . . . . . . 7  |-  ( y  =  w  ->  (
z G y )  =  ( z G w ) )
98fveq2d 5732 . . . . . 6  |-  ( y  =  w  ->  ( F `  ( z G y ) )  =  ( F `  ( z G w ) ) )
10 fveq2 5728 . . . . . . 7  |-  ( y  =  w  ->  ( F `  y )  =  ( F `  w ) )
1110oveq2d 6097 . . . . . 6  |-  ( y  =  w  ->  (
( F `  z
) O ( F `
 y ) )  =  ( ( F `
 z ) O ( F `  w
) ) )
129, 11eqeq12d 2450 . . . . 5  |-  ( y  =  w  ->  (
( F `  (
z G y ) )  =  ( ( F `  z ) O ( F `  y ) )  <->  ( F `  ( z G w ) )  =  ( ( F `  z
) O ( F `
 w ) ) ) )
137, 12cbvral2v 2940 . . . 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 189 . . 3  |-  ( ph  ->  A. z  e.  X  A. w  e.  X  ( F `  ( z G w ) )  =  ( ( F `
 z ) O ( F `  w
) ) )
15 oveq1 6088 . . . . . 6  |-  ( z  =  C  ->  (
z G w )  =  ( C G w ) )
1615fveq2d 5732 . . . . 5  |-  ( z  =  C  ->  ( F `  ( z G w ) )  =  ( F `  ( C G w ) ) )
17 fveq2 5728 . . . . . 6  |-  ( z  =  C  ->  ( F `  z )  =  ( F `  C ) )
1817oveq1d 6096 . . . . 5  |-  ( z  =  C  ->  (
( F `  z
) O ( F `
 w ) )  =  ( ( F `
 C ) O ( F `  w
) ) )
1916, 18eqeq12d 2450 . . . 4  |-  ( z  =  C  ->  (
( F `  (
z G w ) )  =  ( ( F `  z ) O ( F `  w ) )  <->  ( F `  ( C G w ) )  =  ( ( F `  C
) O ( F `
 w ) ) ) )
20 oveq2 6089 . . . . . 6  |-  ( w  =  D  ->  ( C G w )  =  ( C G D ) )
2120fveq2d 5732 . . . . 5  |-  ( w  =  D  ->  ( F `  ( C G w ) )  =  ( F `  ( C G D ) ) )
22 fveq2 5728 . . . . . 6  |-  ( w  =  D  ->  ( F `  w )  =  ( F `  D ) )
2322oveq2d 6097 . . . . 5  |-  ( w  =  D  ->  (
( F `  C
) O ( F `
 w ) )  =  ( ( F `
 C ) O ( F `  D
) ) )
2421, 23eqeq12d 2450 . . . 4  |-  ( w  =  D  ->  (
( F `  ( C G w ) )  =  ( ( F `
 C ) O ( F `  w
) )  <->  ( F `  ( C G D ) )  =  ( ( F `  C
) O ( F `
 D ) ) ) )
2519, 24rspc2v 3058 . . 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 456 . 2  |-  ( (
ph  /\  ( C  e.  X  /\  D  e.  X ) )  -> 
( F `  ( C G D ) )  =  ( ( F `
 C ) O ( F `  D
) ) )
27 ghgrp.3 . . . 4  |-  H  =  ( O  |`  ( Y  X.  Y ) )
2827oveqi 6094 . . 3  |-  ( ( F `  C ) H ( F `  D ) )  =  ( ( F `  C ) ( O  |`  ( Y  X.  Y
) ) ( F `
 D ) )
29 ghgrp.1 . . . . . 6  |-  ( ph  ->  F : X -onto-> Y
)
30 fof 5653 . . . . . 6  |-  ( F : X -onto-> Y  ->  F : X --> Y )
3129, 30syl 16 . . . . 5  |-  ( ph  ->  F : X --> Y )
32 ffvelrn 5868 . . . . . 6  |-  ( ( F : X --> Y  /\  C  e.  X )  ->  ( F `  C
)  e.  Y )
33 ffvelrn 5868 . . . . . 6  |-  ( ( F : X --> Y  /\  D  e.  X )  ->  ( F `  D
)  e.  Y )
3432, 33anim12dan 811 . . . . 5  |-  ( ( F : X --> Y  /\  ( C  e.  X  /\  D  e.  X
) )  ->  (
( F `  C
)  e.  Y  /\  ( F `  D )  e.  Y ) )
3531, 34sylan 458 . . . 4  |-  ( (
ph  /\  ( C  e.  X  /\  D  e.  X ) )  -> 
( ( F `  C )  e.  Y  /\  ( F `  D
)  e.  Y ) )
36 ovres 6213 . . . 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 2480 . 2  |-  ( (
ph  /\  ( C  e.  X  /\  D  e.  X ) )  -> 
( ( F `  C ) H ( F `  D ) )  =  ( ( F `  C ) O ( F `  D ) ) )
3926, 38eqtr4d 2471 1  |-  ( (
ph  /\  ( C  e.  X  /\  D  e.  X ) )  -> 
( F `  ( C G D ) )  =  ( ( F `
 C ) H ( F `  D
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2705    X. cxp 4876    |` cres 4880   -->wf 5450   -onto->wfo 5452   ` cfv 5454  (class class class)co 6081
This theorem is referenced by:  ghgrp  21956  ghablo  21957
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-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fo 5460  df-fv 5462  df-ov 6084
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