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Theorem itg1addlem4 22272
Description: Lemma for itg1add . (Contributed by Mario Carneiro, 28-Jun-2014.)
Hypotheses
Ref Expression
i1fadd.1  |-  ( ph  ->  F  e.  dom  S.1 )
i1fadd.2  |-  ( ph  ->  G  e.  dom  S.1 )
itg1add.3  |-  I  =  ( i  e.  RR ,  j  e.  RR  |->  if ( ( i  =  0  /\  j  =  0 ) ,  0 ,  ( vol `  (
( `' F " { i } )  i^i  ( `' G " { j } ) ) ) ) )
itg1add.4  |-  P  =  (  +  |`  ( ran  F  X.  ran  G
) )
Assertion
Ref Expression
itg1addlem4  |-  ( ph  ->  ( S.1 `  ( F  oF  +  G
) )  =  sum_ y  e.  ran  F sum_ z  e.  ran  G ( ( y  +  z )  x.  ( y I z ) ) )
Distinct variable groups:    i, j,
y, z    y, I    y, P, z    i, F, j, y, z    i, G, j, y, z    ph, i,
j, y, z
Allowed substitution hints:    P( i, j)    I( z, i, j)

Proof of Theorem itg1addlem4
Dummy variables  w  v  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 i1fadd.1 . . . . 5  |-  ( ph  ->  F  e.  dom  S.1 )
2 i1fadd.2 . . . . 5  |-  ( ph  ->  G  e.  dom  S.1 )
31, 2i1fadd 22268 . . . 4  |-  ( ph  ->  ( F  oF  +  G )  e. 
dom  S.1 )
4 i1frn 22250 . . . . . . . 8  |-  ( F  e.  dom  S.1  ->  ran 
F  e.  Fin )
51, 4syl 16 . . . . . . 7  |-  ( ph  ->  ran  F  e.  Fin )
6 i1frn 22250 . . . . . . . 8  |-  ( G  e.  dom  S.1  ->  ran 
G  e.  Fin )
72, 6syl 16 . . . . . . 7  |-  ( ph  ->  ran  G  e.  Fin )
8 xpfi 7783 . . . . . . 7  |-  ( ( ran  F  e.  Fin  /\ 
ran  G  e.  Fin )  ->  ( ran  F  X.  ran  G )  e. 
Fin )
95, 7, 8syl2anc 659 . . . . . 6  |-  ( ph  ->  ( ran  F  X.  ran  G )  e.  Fin )
10 ax-addf 9560 . . . . . . . . . 10  |-  +  :
( CC  X.  CC )
--> CC
11 ffn 5713 . . . . . . . . . 10  |-  (  +  : ( CC  X.  CC ) --> CC  ->  +  Fn  ( CC  X.  CC ) )
1210, 11ax-mp 5 . . . . . . . . 9  |-  +  Fn  ( CC  X.  CC )
13 i1ff 22249 . . . . . . . . . . . . 13  |-  ( F  e.  dom  S.1  ->  F : RR --> RR )
141, 13syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  F : RR --> RR )
15 frn 5719 . . . . . . . . . . . 12  |-  ( F : RR --> RR  ->  ran 
F  C_  RR )
1614, 15syl 16 . . . . . . . . . . 11  |-  ( ph  ->  ran  F  C_  RR )
17 ax-resscn 9538 . . . . . . . . . . 11  |-  RR  C_  CC
1816, 17syl6ss 3501 . . . . . . . . . 10  |-  ( ph  ->  ran  F  C_  CC )
19 i1ff 22249 . . . . . . . . . . . . 13  |-  ( G  e.  dom  S.1  ->  G : RR --> RR )
202, 19syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  G : RR --> RR )
21 frn 5719 . . . . . . . . . . . 12  |-  ( G : RR --> RR  ->  ran 
G  C_  RR )
2220, 21syl 16 . . . . . . . . . . 11  |-  ( ph  ->  ran  G  C_  RR )
2322, 17syl6ss 3501 . . . . . . . . . 10  |-  ( ph  ->  ran  G  C_  CC )
24 xpss12 5096 . . . . . . . . . 10  |-  ( ( ran  F  C_  CC  /\ 
ran  G  C_  CC )  ->  ( ran  F  X.  ran  G )  C_  ( CC  X.  CC ) )
2518, 23, 24syl2anc 659 . . . . . . . . 9  |-  ( ph  ->  ( ran  F  X.  ran  G )  C_  ( CC  X.  CC ) )
26 fnssres 5676 . . . . . . . . 9  |-  ( (  +  Fn  ( CC 
X.  CC )  /\  ( ran  F  X.  ran  G )  C_  ( CC  X.  CC ) )  -> 
(  +  |`  ( ran  F  X.  ran  G
) )  Fn  ( ran  F  X.  ran  G
) )
2712, 25, 26sylancr 661 . . . . . . . 8  |-  ( ph  ->  (  +  |`  ( ran  F  X.  ran  G
) )  Fn  ( ran  F  X.  ran  G
) )
28 itg1add.4 . . . . . . . . 9  |-  P  =  (  +  |`  ( ran  F  X.  ran  G
) )
2928fneq1i 5657 . . . . . . . 8  |-  ( P  Fn  ( ran  F  X.  ran  G )  <->  (  +  |`  ( ran  F  X.  ran  G ) )  Fn  ( ran  F  X.  ran  G ) )
3027, 29sylibr 212 . . . . . . 7  |-  ( ph  ->  P  Fn  ( ran 
F  X.  ran  G
) )
31 dffn4 5783 . . . . . . 7  |-  ( P  Fn  ( ran  F  X.  ran  G )  <->  P :
( ran  F  X.  ran  G ) -onto-> ran  P
)
3230, 31sylib 196 . . . . . 6  |-  ( ph  ->  P : ( ran 
F  X.  ran  G
) -onto-> ran  P )
33 fofi 7798 . . . . . 6  |-  ( ( ( ran  F  X.  ran  G )  e.  Fin  /\  P : ( ran 
F  X.  ran  G
) -onto-> ran  P )  ->  ran  P  e.  Fin )
349, 32, 33syl2anc 659 . . . . 5  |-  ( ph  ->  ran  P  e.  Fin )
35 difss 3617 . . . . 5  |-  ( ran 
P  \  { 0 } )  C_  ran  P
36 ssfi 7733 . . . . 5  |-  ( ( ran  P  e.  Fin  /\  ( ran  P  \  { 0 } ) 
C_  ran  P )  ->  ( ran  P  \  { 0 } )  e.  Fin )
3734, 35, 36sylancl 660 . . . 4  |-  ( ph  ->  ( ran  P  \  { 0 } )  e.  Fin )
38 opelxpi 5020 . . . . . . . . . 10  |-  ( ( x  e.  ran  F  /\  y  e.  ran  G )  ->  <. x ,  y >.  e.  ( ran  F  X.  ran  G
) )
39 ffun 5715 . . . . . . . . . . . 12  |-  (  +  : ( CC  X.  CC ) --> CC  ->  Fun  +  )
4010, 39ax-mp 5 . . . . . . . . . . 11  |-  Fun  +
4110fdmi 5718 . . . . . . . . . . . 12  |-  dom  +  =  ( CC  X.  CC )
4225, 41syl6sseqr 3536 . . . . . . . . . . 11  |-  ( ph  ->  ( ran  F  X.  ran  G )  C_  dom  +  )
43 funfvima2 6123 . . . . . . . . . . 11  |-  ( ( Fun  +  /\  ( ran  F  X.  ran  G
)  C_  dom  +  )  ->  ( <. x ,  y >.  e.  ( ran  F  X.  ran  G )  ->  (  +  ` 
<. x ,  y >.
)  e.  (  + 
" ( ran  F  X.  ran  G ) ) ) )
4440, 42, 43sylancr 661 . . . . . . . . . 10  |-  ( ph  ->  ( <. x ,  y
>.  e.  ( ran  F  X.  ran  G )  -> 
(  +  `  <. x ,  y >. )  e.  (  +  " ( ran  F  X.  ran  G
) ) ) )
4538, 44syl5 32 . . . . . . . . 9  |-  ( ph  ->  ( ( x  e. 
ran  F  /\  y  e.  ran  G )  -> 
(  +  `  <. x ,  y >. )  e.  (  +  " ( ran  F  X.  ran  G
) ) ) )
4645imp 427 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ran  F  /\  y  e.  ran  G ) )  ->  (  +  `  <. x ,  y >.
)  e.  (  + 
" ( ran  F  X.  ran  G ) ) )
47 df-ov 6273 . . . . . . . 8  |-  ( x  +  y )  =  (  +  `  <. x ,  y >. )
4828rneqi 5218 . . . . . . . . 9  |-  ran  P  =  ran  (  +  |`  ( ran  F  X.  ran  G
) )
49 df-ima 5001 . . . . . . . . 9  |-  (  + 
" ( ran  F  X.  ran  G ) )  =  ran  (  +  |`  ( ran  F  X.  ran  G ) )
5048, 49eqtr4i 2486 . . . . . . . 8  |-  ran  P  =  (  +  " ( ran  F  X.  ran  G
) )
5146, 47, 503eltr4g 2560 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ran  F  /\  y  e.  ran  G ) )  ->  ( x  +  y )  e.  ran  P )
52 ffn 5713 . . . . . . . . 9  |-  ( F : RR --> RR  ->  F  Fn  RR )
5314, 52syl 16 . . . . . . . 8  |-  ( ph  ->  F  Fn  RR )
54 dffn3 5720 . . . . . . . 8  |-  ( F  Fn  RR  <->  F : RR
--> ran  F )
5553, 54sylib 196 . . . . . . 7  |-  ( ph  ->  F : RR --> ran  F
)
56 ffn 5713 . . . . . . . . 9  |-  ( G : RR --> RR  ->  G  Fn  RR )
5720, 56syl 16 . . . . . . . 8  |-  ( ph  ->  G  Fn  RR )
58 dffn3 5720 . . . . . . . 8  |-  ( G  Fn  RR  <->  G : RR
--> ran  G )
5957, 58sylib 196 . . . . . . 7  |-  ( ph  ->  G : RR --> ran  G
)
60 reex 9572 . . . . . . . 8  |-  RR  e.  _V
6160a1i 11 . . . . . . 7  |-  ( ph  ->  RR  e.  _V )
62 inidm 3693 . . . . . . 7  |-  ( RR 
i^i  RR )  =  RR
6351, 55, 59, 61, 61, 62off 6527 . . . . . 6  |-  ( ph  ->  ( F  oF  +  G ) : RR --> ran  P )
64 frn 5719 . . . . . 6  |-  ( ( F  oF  +  G ) : RR --> ran  P  ->  ran  ( F  oF  +  G
)  C_  ran  P )
6563, 64syl 16 . . . . 5  |-  ( ph  ->  ran  ( F  oF  +  G )  C_ 
ran  P )
6665ssdifd 3626 . . . 4  |-  ( ph  ->  ( ran  ( F  oF  +  G
)  \  { 0 } )  C_  ( ran  P  \  { 0 } ) )
6716sselda 3489 . . . . . . . . . 10  |-  ( (
ph  /\  y  e.  ran  F )  ->  y  e.  RR )
6822sselda 3489 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  ran  G )  ->  z  e.  RR )
6967, 68anim12dan 835 . . . . . . . . 9  |-  ( (
ph  /\  ( y  e.  ran  F  /\  z  e.  ran  G ) )  ->  ( y  e.  RR  /\  z  e.  RR ) )
70 readdcl 9564 . . . . . . . . 9  |-  ( ( y  e.  RR  /\  z  e.  RR )  ->  ( y  +  z )  e.  RR )
7169, 70syl 16 . . . . . . . 8  |-  ( (
ph  /\  ( y  e.  ran  F  /\  z  e.  ran  G ) )  ->  ( y  +  z )  e.  RR )
7271ralrimivva 2875 . . . . . . 7  |-  ( ph  ->  A. y  e.  ran  F A. z  e.  ran  G ( y  +  z )  e.  RR )
73 funimassov 6425 . . . . . . . 8  |-  ( ( Fun  +  /\  ( ran  F  X.  ran  G
)  C_  dom  +  )  ->  ( (  + 
" ( ran  F  X.  ran  G ) ) 
C_  RR  <->  A. y  e.  ran  F A. z  e.  ran  G ( y  +  z )  e.  RR ) )
7440, 42, 73sylancr 661 . . . . . . 7  |-  ( ph  ->  ( (  +  "
( ran  F  X.  ran  G ) )  C_  RR 
<-> 
A. y  e.  ran  F A. z  e.  ran  G ( y  +  z )  e.  RR ) )
7572, 74mpbird 232 . . . . . 6  |-  ( ph  ->  (  +  " ( ran  F  X.  ran  G
) )  C_  RR )
7650, 75syl5eqss 3533 . . . . 5  |-  ( ph  ->  ran  P  C_  RR )
7776ssdifd 3626 . . . 4  |-  ( ph  ->  ( ran  P  \  { 0 } ) 
C_  ( RR  \  { 0 } ) )
78 itg1val2 22257 . . . 4  |-  ( ( ( F  oF  +  G )  e. 
dom  S.1  /\  ( ( ran  P  \  {
0 } )  e. 
Fin  /\  ( ran  ( F  oF  +  G )  \  {
0 } )  C_  ( ran  P  \  {
0 } )  /\  ( ran  P  \  {
0 } )  C_  ( RR  \  { 0 } ) ) )  ->  ( S.1 `  ( F  oF  +  G
) )  =  sum_ w  e.  ( ran  P  \  { 0 } ) ( w  x.  ( vol `  ( `' ( F  oF  +  G ) " {
w } ) ) ) )
793, 37, 66, 77, 78syl13anc 1228 . . 3  |-  ( ph  ->  ( S.1 `  ( F  oF  +  G
) )  =  sum_ w  e.  ( ran  P  \  { 0 } ) ( w  x.  ( vol `  ( `' ( F  oF  +  G ) " {
w } ) ) ) )
8020adantr 463 . . . . . . . 8  |-  ( (
ph  /\  w  e.  ( ran  P  \  {
0 } ) )  ->  G : RR --> RR )
817adantr 463 . . . . . . . 8  |-  ( (
ph  /\  w  e.  ( ran  P  \  {
0 } ) )  ->  ran  G  e.  Fin )
82 inss2 3705 . . . . . . . . 9  |-  ( ( `' F " { ( w  -  z ) } )  i^i  ( `' G " { z } ) )  C_  ( `' G " { z } )
8382a1i 11 . . . . . . . 8  |-  ( ( ( ph  /\  w  e.  ( ran  P  \  { 0 } ) )  /\  z  e. 
ran  G )  -> 
( ( `' F " { ( w  -  z ) } )  i^i  ( `' G " { z } ) )  C_  ( `' G " { z } ) )
84 i1fima 22251 . . . . . . . . . . 11  |-  ( F  e.  dom  S.1  ->  ( `' F " { ( w  -  z ) } )  e.  dom  vol )
851, 84syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( `' F " { ( w  -  z ) } )  e.  dom  vol )
86 i1fima 22251 . . . . . . . . . . 11  |-  ( G  e.  dom  S.1  ->  ( `' G " { z } )  e.  dom  vol )
872, 86syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( `' G " { z } )  e.  dom  vol )
88 inmbl 22118 . . . . . . . . . 10  |-  ( ( ( `' F " { ( w  -  z ) } )  e.  dom  vol  /\  ( `' G " { z } )  e.  dom  vol )  ->  ( ( `' F " { ( w  -  z ) } )  i^i  ( `' G " { z } ) )  e. 
dom  vol )
8985, 87, 88syl2anc 659 . . . . . . . . 9  |-  ( ph  ->  ( ( `' F " { ( w  -  z ) } )  i^i  ( `' G " { z } ) )  e.  dom  vol )
9089ad2antrr 723 . . . . . . . 8  |-  ( ( ( ph  /\  w  e.  ( ran  P  \  { 0 } ) )  /\  z  e. 
ran  G )  -> 
( ( `' F " { ( w  -  z ) } )  i^i  ( `' G " { z } ) )  e.  dom  vol )
9135, 76syl5ss 3500 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ran  P  \  { 0 } ) 
C_  RR )
9291sselda 3489 . . . . . . . . . . . 12  |-  ( (
ph  /\  w  e.  ( ran  P  \  {
0 } ) )  ->  w  e.  RR )
9392adantr 463 . . . . . . . . . . 11  |-  ( ( ( ph  /\  w  e.  ( ran  P  \  { 0 } ) )  /\  z  e. 
ran  G )  ->  w  e.  RR )
9468adantlr 712 . . . . . . . . . . 11  |-  ( ( ( ph  /\  w  e.  ( ran  P  \  { 0 } ) )  /\  z  e. 
ran  G )  -> 
z  e.  RR )
9593, 94resubcld 9983 . . . . . . . . . 10  |-  ( ( ( ph  /\  w  e.  ( ran  P  \  { 0 } ) )  /\  z  e. 
ran  G )  -> 
( w  -  z
)  e.  RR )
9693recnd 9611 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  w  e.  ( ran  P  \  { 0 } ) )  /\  z  e. 
ran  G )  ->  w  e.  CC )
9794recnd 9611 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  w  e.  ( ran  P  \  { 0 } ) )  /\  z  e. 
ran  G )  -> 
z  e.  CC )
9896, 97npcand 9926 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  w  e.  ( ran  P  \  { 0 } ) )  /\  z  e. 
ran  G )  -> 
( ( w  -  z )  +  z )  =  w )
99 eldifsni 4142 . . . . . . . . . . . . 13  |-  ( w  e.  ( ran  P  \  { 0 } )  ->  w  =/=  0
)
10099ad2antlr 724 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  w  e.  ( ran  P  \  { 0 } ) )  /\  z  e. 
ran  G )  ->  w  =/=  0 )
10198, 100eqnetrd 2747 . . . . . . . . . . 11  |-  ( ( ( ph  /\  w  e.  ( ran  P  \  { 0 } ) )  /\  z  e. 
ran  G )  -> 
( ( w  -  z )  +  z )  =/=  0 )
102 oveq12 6279 . . . . . . . . . . . . 13  |-  ( ( ( w  -  z
)  =  0  /\  z  =  0 )  ->  ( ( w  -  z )  +  z )  =  ( 0  +  0 ) )
103 00id 9744 . . . . . . . . . . . . 13  |-  ( 0  +  0 )  =  0
104102, 103syl6eq 2511 . . . . . . . . . . . 12  |-  ( ( ( w  -  z
)  =  0  /\  z  =  0 )  ->  ( ( w  -  z )  +  z )  =  0 )
105104necon3ai 2682 . . . . . . . . . . 11  |-  ( ( ( w  -  z
)  +  z )  =/=  0  ->  -.  ( ( w  -  z )  =  0  /\  z  =  0 ) )
106101, 105syl 16 . . . . . . . . . 10  |-  ( ( ( ph  /\  w  e.  ( ran  P  \  { 0 } ) )  /\  z  e. 
ran  G )  ->  -.  ( ( w  -  z )  =  0  /\  z  =  0 ) )
107 itg1add.3 . . . . . . . . . . 11  |-  I  =  ( i  e.  RR ,  j  e.  RR  |->  if ( ( i  =  0  /\  j  =  0 ) ,  0 ,  ( vol `  (
( `' F " { i } )  i^i  ( `' G " { j } ) ) ) ) )
1081, 2, 107itg1addlem3 22271 . . . . . . . . . 10  |-  ( ( ( ( w  -  z )  e.  RR  /\  z  e.  RR )  /\  -.  ( ( w  -  z )  =  0  /\  z  =  0 ) )  ->  ( ( w  -  z ) I z )  =  ( vol `  ( ( `' F " { ( w  -  z ) } )  i^i  ( `' G " { z } ) ) ) )
10995, 94, 106, 108syl21anc 1225 . . . . . . . . 9  |-  ( ( ( ph  /\  w  e.  ( ran  P  \  { 0 } ) )  /\  z  e. 
ran  G )  -> 
( ( w  -  z ) I z )  =  ( vol `  ( ( `' F " { ( w  -  z ) } )  i^i  ( `' G " { z } ) ) ) )
1101, 2, 107itg1addlem2 22270 . . . . . . . . . . 11  |-  ( ph  ->  I : ( RR 
X.  RR ) --> RR )
111110ad2antrr 723 . . . . . . . . . 10  |-  ( ( ( ph  /\  w  e.  ( ran  P  \  { 0 } ) )  /\  z  e. 
ran  G )  ->  I : ( RR  X.  RR ) --> RR )
112111, 95, 94fovrnd 6420 . . . . . . . . 9  |-  ( ( ( ph  /\  w  e.  ( ran  P  \  { 0 } ) )  /\  z  e. 
ran  G )  -> 
( ( w  -  z ) I z )  e.  RR )
113109, 112eqeltrrd 2543 . . . . . . . 8  |-  ( ( ( ph  /\  w  e.  ( ran  P  \  { 0 } ) )  /\  z  e. 
ran  G )  -> 
( vol `  (
( `' F " { ( w  -  z ) } )  i^i  ( `' G " { z } ) ) )  e.  RR )
11480, 81, 83, 90, 113itg1addlem1 22265 . . . . . . 7  |-  ( (
ph  /\  w  e.  ( ran  P  \  {
0 } ) )  ->  ( vol `  U_ z  e.  ran  G ( ( `' F " { ( w  -  z ) } )  i^i  ( `' G " { z } ) ) )  =  sum_ z  e.  ran  G ( vol `  (
( `' F " { ( w  -  z ) } )  i^i  ( `' G " { z } ) ) ) )
11592recnd 9611 . . . . . . . . 9  |-  ( (
ph  /\  w  e.  ( ran  P  \  {
0 } ) )  ->  w  e.  CC )
1161, 2i1faddlem 22266 . . . . . . . . 9  |-  ( (
ph  /\  w  e.  CC )  ->  ( `' ( F  oF  +  G ) " { w } )  =  U_ z  e. 
ran  G ( ( `' F " { ( w  -  z ) } )  i^i  ( `' G " { z } ) ) )
117115, 116syldan 468 . . . . . . . 8  |-  ( (
ph  /\  w  e.  ( ran  P  \  {
0 } ) )  ->  ( `' ( F  oF  +  G ) " {
w } )  = 
U_ z  e.  ran  G ( ( `' F " { ( w  -  z ) } )  i^i  ( `' G " { z } ) ) )
118117fveq2d 5852 . . . . . . 7  |-  ( (
ph  /\  w  e.  ( ran  P  \  {
0 } ) )  ->  ( vol `  ( `' ( F  oF  +  G ) " { w } ) )  =  ( vol `  U_ z  e.  ran  G ( ( `' F " { ( w  -  z ) } )  i^i  ( `' G " { z } ) ) ) )
119109sumeq2dv 13607 . . . . . . 7  |-  ( (
ph  /\  w  e.  ( ran  P  \  {
0 } ) )  ->  sum_ z  e.  ran  G ( ( w  -  z ) I z )  =  sum_ z  e.  ran  G ( vol `  ( ( `' F " { ( w  -  z ) } )  i^i  ( `' G " { z } ) ) ) )
120114, 118, 1193eqtr4d 2505 . . . . . 6  |-  ( (
ph  /\  w  e.  ( ran  P  \  {
0 } ) )  ->  ( vol `  ( `' ( F  oF  +  G ) " { w } ) )  =  sum_ z  e.  ran  G ( ( w  -  z ) I z ) )
121120oveq2d 6286 . . . . 5  |-  ( (
ph  /\  w  e.  ( ran  P  \  {
0 } ) )  ->  ( w  x.  ( vol `  ( `' ( F  oF  +  G ) " { w } ) ) )  =  ( w  x.  sum_ z  e.  ran  G ( ( w  -  z ) I z ) ) )
122112recnd 9611 . . . . . 6  |-  ( ( ( ph  /\  w  e.  ( ran  P  \  { 0 } ) )  /\  z  e. 
ran  G )  -> 
( ( w  -  z ) I z )  e.  CC )
12381, 115, 122fsummulc2 13681 . . . . 5  |-  ( (
ph  /\  w  e.  ( ran  P  \  {
0 } ) )  ->  ( w  x. 
sum_ z  e.  ran  G ( ( w  -  z ) I z ) )  =  sum_ z  e.  ran  G ( w  x.  ( ( w  -  z ) I z ) ) )
124121, 123eqtrd 2495 . . . 4  |-  ( (
ph  /\  w  e.  ( ran  P  \  {
0 } ) )  ->  ( w  x.  ( vol `  ( `' ( F  oF  +  G ) " { w } ) ) )  =  sum_ z  e.  ran  G ( w  x.  ( ( w  -  z ) I z ) ) )
125124sumeq2dv 13607 . . 3  |-  ( ph  -> 
sum_ w  e.  ( ran  P  \  { 0 } ) ( w  x.  ( vol `  ( `' ( F  oF  +  G ) " { w } ) ) )  =  sum_ w  e.  ( ran  P  \  { 0 } )
sum_ z  e.  ran  G ( w  x.  (
( w  -  z
) I z ) ) )
12696, 122mulcld 9605 . . . . 5  |-  ( ( ( ph  /\  w  e.  ( ran  P  \  { 0 } ) )  /\  z  e. 
ran  G )  -> 
( w  x.  (
( w  -  z
) I z ) )  e.  CC )
127126anasss 645 . . . 4  |-  ( (
ph  /\  ( w  e.  ( ran  P  \  { 0 } )  /\  z  e.  ran  G ) )  ->  (
w  x.  ( ( w  -  z ) I z ) )  e.  CC )
12837, 7, 127fsumcom 13672 . . 3  |-  ( ph  -> 
sum_ w  e.  ( ran  P  \  { 0 } ) sum_ z  e.  ran  G ( w  x.  ( ( w  -  z ) I z ) )  = 
sum_ z  e.  ran  G
sum_ w  e.  ( ran  P  \  { 0 } ) ( w  x.  ( ( w  -  z ) I z ) ) )
12979, 125, 1283eqtrd 2499 . 2  |-  ( ph  ->  ( S.1 `  ( F  oF  +  G
) )  =  sum_ z  e.  ran  G sum_ w  e.  ( ran  P  \  { 0 } ) ( w  x.  (
( w  -  z
) I z ) ) )
130 oveq1 6277 . . . . . . 7  |-  ( y  =  ( w  -  z )  ->  (
y  +  z )  =  ( ( w  -  z )  +  z ) )
131 oveq1 6277 . . . . . . 7  |-  ( y  =  ( w  -  z )  ->  (
y I z )  =  ( ( w  -  z ) I z ) )
132130, 131oveq12d 6288 . . . . . 6  |-  ( y  =  ( w  -  z )  ->  (
( y  +  z )  x.  ( y I z ) )  =  ( ( ( w  -  z )  +  z )  x.  ( ( w  -  z ) I z ) ) )
13334adantr 463 . . . . . 6  |-  ( (
ph  /\  z  e.  ran  G )  ->  ran  P  e.  Fin )
13476adantr 463 . . . . . . . . . . 11  |-  ( (
ph  /\  z  e.  ran  G )  ->  ran  P 
C_  RR )
135134sselda 3489 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  v  e.  ran  P )  ->  v  e.  RR )
13668adantr 463 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  v  e.  ran  P )  ->  z  e.  RR )
137135, 136resubcld 9983 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  v  e.  ran  P )  ->  ( v  -  z )  e.  RR )
138137ex 432 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ran  G )  ->  (
v  e.  ran  P  ->  ( v  -  z
)  e.  RR ) )
139135recnd 9611 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  v  e.  ran  P )  ->  v  e.  CC )
140139adantrr 714 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( v  e.  ran  P  /\  y  e.  ran  P ) )  ->  v  e.  CC )
14176sselda 3489 . . . . . . . . . . . 12  |-  ( (
ph  /\  y  e.  ran  P )  ->  y  e.  RR )
142141ad2ant2rl 746 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( v  e.  ran  P  /\  y  e.  ran  P ) )  ->  y  e.  RR )
143142recnd 9611 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( v  e.  ran  P  /\  y  e.  ran  P ) )  ->  y  e.  CC )
14468recnd 9611 . . . . . . . . . . 11  |-  ( (
ph  /\  z  e.  ran  G )  ->  z  e.  CC )
145144adantr 463 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( v  e.  ran  P  /\  y  e.  ran  P ) )  ->  z  e.  CC )
146140, 143, 145subcan2ad 9967 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( v  e.  ran  P  /\  y  e.  ran  P ) )  ->  (
( v  -  z
)  =  ( y  -  z )  <->  v  =  y ) )
147146ex 432 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ran  G )  ->  (
( v  e.  ran  P  /\  y  e.  ran  P )  ->  ( (
v  -  z )  =  ( y  -  z )  <->  v  =  y ) ) )
148138, 147dom2lem 7548 . . . . . . 7  |-  ( (
ph  /\  z  e.  ran  G )  ->  (
v  e.  ran  P  |->  ( v  -  z
) ) : ran  P
-1-1-> RR )
149 f1f1orn 5809 . . . . . . 7  |-  ( ( v  e.  ran  P  |->  ( v  -  z
) ) : ran  P
-1-1-> RR  ->  ( v  e.  ran  P  |->  ( v  -  z ) ) : ran  P -1-1-onto-> ran  (
v  e.  ran  P  |->  ( v  -  z
) ) )
150148, 149syl 16 . . . . . 6  |-  ( (
ph  /\  z  e.  ran  G )  ->  (
v  e.  ran  P  |->  ( v  -  z
) ) : ran  P -1-1-onto-> ran  ( v  e.  ran  P 
|->  ( v  -  z
) ) )
151 oveq1 6277 . . . . . . . 8  |-  ( v  =  w  ->  (
v  -  z )  =  ( w  -  z ) )
152 eqid 2454 . . . . . . . 8  |-  ( v  e.  ran  P  |->  ( v  -  z ) )  =  ( v  e.  ran  P  |->  ( v  -  z ) )
153 ovex 6298 . . . . . . . 8  |-  ( w  -  z )  e. 
_V
154151, 152, 153fvmpt 5931 . . . . . . 7  |-  ( w  e.  ran  P  -> 
( ( v  e. 
ran  P  |->  ( v  -  z ) ) `
 w )  =  ( w  -  z
) )
155154adantl 464 . . . . . 6  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  w  e.  ran  P )  ->  ( ( v  e.  ran  P  |->  ( v  -  z ) ) `  w )  =  ( w  -  z ) )
156 f1f 5763 . . . . . . . . . . 11  |-  ( ( v  e.  ran  P  |->  ( v  -  z
) ) : ran  P
-1-1-> RR  ->  ( v  e.  ran  P  |->  ( v  -  z ) ) : ran  P --> RR )
157 frn 5719 . . . . . . . . . . 11  |-  ( ( v  e.  ran  P  |->  ( v  -  z
) ) : ran  P --> RR  ->  ran  ( v  e.  ran  P  |->  ( v  -  z ) )  C_  RR )
158148, 156, 1573syl 20 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  ran  G )  ->  ran  ( v  e.  ran  P 
|->  ( v  -  z
) )  C_  RR )
159158sselda 3489 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  ( v  e.  ran  P  |->  ( v  -  z ) ) )  ->  y  e.  RR )
16068adantr 463 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  ( v  e.  ran  P  |->  ( v  -  z ) ) )  ->  z  e.  RR )
161159, 160readdcld 9612 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  ( v  e.  ran  P  |->  ( v  -  z ) ) )  ->  (
y  +  z )  e.  RR )
162110ad2antrr 723 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  ( v  e.  ran  P  |->  ( v  -  z ) ) )  ->  I : ( RR  X.  RR ) --> RR )
163162, 159, 160fovrnd 6420 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  ( v  e.  ran  P  |->  ( v  -  z ) ) )  ->  (
y I z )  e.  RR )
164161, 163remulcld 9613 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  ( v  e.  ran  P  |->  ( v  -  z ) ) )  ->  (
( y  +  z )  x.  ( y I z ) )  e.  RR )
165164recnd 9611 . . . . . 6  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  ( v  e.  ran  P  |->  ( v  -  z ) ) )  ->  (
( y  +  z )  x.  ( y I z ) )  e.  CC )
166132, 133, 150, 155, 165fsumf1o 13627 . . . . 5  |-  ( (
ph  /\  z  e.  ran  G )  ->  sum_ y  e.  ran  ( v  e. 
ran  P  |->  ( v  -  z ) ) ( ( y  +  z )  x.  (
y I z ) )  =  sum_ w  e.  ran  P ( ( ( w  -  z
)  +  z )  x.  ( ( w  -  z ) I z ) ) )
167134sselda 3489 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  w  e.  ran  P )  ->  w  e.  RR )
168167recnd 9611 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  w  e.  ran  P )  ->  w  e.  CC )
169144adantr 463 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  w  e.  ran  P )  ->  z  e.  CC )
170168, 169npcand 9926 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  w  e.  ran  P )  ->  ( ( w  -  z )  +  z )  =  w )
171170oveq1d 6285 . . . . . 6  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  w  e.  ran  P )  ->  ( ( ( w  -  z )  +  z )  x.  ( ( w  -  z ) I z ) )  =  ( w  x.  ( ( w  -  z ) I z ) ) )
172171sumeq2dv 13607 . . . . 5  |-  ( (
ph  /\  z  e.  ran  G )  ->  sum_ w  e.  ran  P ( ( ( w  -  z
)  +  z )  x.  ( ( w  -  z ) I z ) )  = 
sum_ w  e.  ran  P ( w  x.  (
( w  -  z
) I z ) ) )
173166, 172eqtrd 2495 . . . 4  |-  ( (
ph  /\  z  e.  ran  G )  ->  sum_ y  e.  ran  ( v  e. 
ran  P  |->  ( v  -  z ) ) ( ( y  +  z )  x.  (
y I z ) )  =  sum_ w  e.  ran  P ( w  x.  ( ( w  -  z ) I z ) ) )
17442ad2antrr 723 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  ( ran  F  X.  ran  G )  C_  dom  +  )
175 simpr 459 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  y  e.  ran  F )
176 simplr 753 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  z  e.  ran  G )
177 opelxpi 5020 . . . . . . . . . . . 12  |-  ( ( y  e.  ran  F  /\  z  e.  ran  G )  ->  <. y ,  z >.  e.  ( ran  F  X.  ran  G
) )
178175, 176, 177syl2anc 659 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  <. y ,  z
>.  e.  ( ran  F  X.  ran  G ) )
179 funfvima2 6123 . . . . . . . . . . . 12  |-  ( ( Fun  +  /\  ( ran  F  X.  ran  G
)  C_  dom  +  )  ->  ( <. y ,  z >.  e.  ( ran  F  X.  ran  G )  ->  (  +  ` 
<. y ,  z >.
)  e.  (  + 
" ( ran  F  X.  ran  G ) ) ) )
18040, 179mpan 668 . . . . . . . . . . 11  |-  ( ( ran  F  X.  ran  G )  C_  dom  +  ->  (
<. y ,  z >.  e.  ( ran  F  X.  ran  G )  ->  (  +  `  <. y ,  z
>. )  e.  (  +  " ( ran  F  X.  ran  G ) ) ) )
181174, 178, 180sylc 60 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  (  +  `  <. y ,  z >.
)  e.  (  + 
" ( ran  F  X.  ran  G ) ) )
182 df-ov 6273 . . . . . . . . . 10  |-  ( y  +  z )  =  (  +  `  <. y ,  z >. )
183181, 182, 503eltr4g 2560 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  ( y  +  z )  e.  ran  P )
18467adantlr 712 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  y  e.  RR )
185184recnd 9611 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  y  e.  CC )
186144adantr 463 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  z  e.  CC )
187185, 186pncand 9923 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  ( ( y  +  z )  -  z )  =  y )
188187eqcomd 2462 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  y  =  ( ( y  +  z )  -  z ) )
189 oveq1 6277 . . . . . . . . . . 11  |-  ( v  =  ( y  +  z )  ->  (
v  -  z )  =  ( ( y  +  z )  -  z ) )
190189eqeq2d 2468 . . . . . . . . . 10  |-  ( v  =  ( y  +  z )  ->  (
y  =  ( v  -  z )  <->  y  =  ( ( y  +  z )  -  z
) ) )
191190rspcev 3207 . . . . . . . . 9  |-  ( ( ( y  +  z )  e.  ran  P  /\  y  =  (
( y  +  z )  -  z ) )  ->  E. v  e.  ran  P  y  =  ( v  -  z
) )
192183, 188, 191syl2anc 659 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  E. v  e.  ran  P  y  =  ( v  -  z ) )
193192ralrimiva 2868 . . . . . . 7  |-  ( (
ph  /\  z  e.  ran  G )  ->  A. y  e.  ran  F E. v  e.  ran  P  y  =  ( v  -  z
) )
194 ssabral 3557 . . . . . . 7  |-  ( ran 
F  C_  { y  |  E. v  e.  ran  P  y  =  ( v  -  z ) }  <->  A. y  e.  ran  F E. v  e.  ran  P  y  =  ( v  -  z ) )
195193, 194sylibr 212 . . . . . 6  |-  ( (
ph  /\  z  e.  ran  G )  ->  ran  F 
C_  { y  |  E. v  e.  ran  P  y  =  ( v  -  z ) } )
196152rnmpt 5237 . . . . . 6  |-  ran  (
v  e.  ran  P  |->  ( v  -  z
) )  =  {
y  |  E. v  e.  ran  P  y  =  ( v  -  z
) }
197195, 196syl6sseqr 3536 . . . . 5  |-  ( (
ph  /\  z  e.  ran  G )  ->  ran  F 
C_  ran  ( v  e.  ran  P  |->  ( v  -  z ) ) )
19868adantr 463 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  z  e.  RR )
199184, 198readdcld 9612 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  ( y  +  z )  e.  RR )
200110ad2antrr 723 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  I : ( RR  X.  RR ) --> RR )
201200, 184, 198fovrnd 6420 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  ( y I z )  e.  RR )
202199, 201remulcld 9613 . . . . . 6  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  ( ( y  +  z )  x.  ( y I z ) )  e.  RR )
203202recnd 9611 . . . . 5  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  ( ( y  +  z )  x.  ( y I z ) )  e.  CC )
204158ssdifd 3626 . . . . . . 7  |-  ( (
ph  /\  z  e.  ran  G )  ->  ( ran  ( v  e.  ran  P 
|->  ( v  -  z
) )  \  ran  F )  C_  ( RR  \  ran  F ) )
205204sselda 3489 . . . . . 6  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ( ran  ( v  e.  ran  P 
|->  ( v  -  z
) )  \  ran  F ) )  ->  y  e.  ( RR  \  ran  F ) )
206 eldifi 3612 . . . . . . . . . . . . 13  |-  ( y  e.  ( RR  \  ran  F )  ->  y  e.  RR )
207206ad2antrl 725 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( y  e.  ( RR  \  ran  F
)  /\  -.  (
y  =  0  /\  z  =  0 ) ) )  ->  y  e.  RR )
20868adantr 463 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( y  e.  ( RR  \  ran  F
)  /\  -.  (
y  =  0  /\  z  =  0 ) ) )  ->  z  e.  RR )
209 simprr 755 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( y  e.  ( RR  \  ran  F
)  /\  -.  (
y  =  0  /\  z  =  0 ) ) )  ->  -.  ( y  =  0  /\  z  =  0 ) )
2101, 2, 107itg1addlem3 22271 . . . . . . . . . . . 12  |-  ( ( ( y  e.  RR  /\  z  e.  RR )  /\  -.  ( y  =  0  /\  z  =  0 ) )  ->  ( y I z )  =  ( vol `  ( ( `' F " { y } )  i^i  ( `' G " { z } ) ) ) )
211207, 208, 209, 210syl21anc 1225 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( y  e.  ( RR  \  ran  F
)  /\  -.  (
y  =  0  /\  z  =  0 ) ) )  ->  (
y I z )  =  ( vol `  (
( `' F " { y } )  i^i  ( `' G " { z } ) ) ) )
212 inss1 3704 . . . . . . . . . . . . . . 15  |-  ( ( `' F " { y } )  i^i  ( `' G " { z } ) )  C_  ( `' F " { y } )
213 eldifn 3613 . . . . . . . . . . . . . . . . . . 19  |-  ( y  e.  ( RR  \  ran  F )  ->  -.  y  e.  ran  F )
214213ad2antrl 725 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( y  e.  ( RR  \  ran  F
)  /\  -.  (
y  =  0  /\  z  =  0 ) ) )  ->  -.  y  e.  ran  F )
215 vex 3109 . . . . . . . . . . . . . . . . . . . 20  |-  y  e. 
_V
216 vex 3109 . . . . . . . . . . . . . . . . . . . . 21  |-  v  e. 
_V
217216eliniseg 5354 . . . . . . . . . . . . . . . . . . . 20  |-  ( y  e.  _V  ->  (
v  e.  ( `' F " { y } )  <->  v F
y ) )
218215, 217ax-mp 5 . . . . . . . . . . . . . . . . . . 19  |-  ( v  e.  ( `' F " { y } )  <-> 
v F y )
219216, 215brelrn 5222 . . . . . . . . . . . . . . . . . . 19  |-  ( v F y  ->  y  e.  ran  F )
220218, 219sylbi 195 . . . . . . . . . . . . . . . . . 18  |-  ( v  e.  ( `' F " { y } )  ->  y  e.  ran  F )
221214, 220nsyl 121 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( y  e.  ( RR  \  ran  F
)  /\  -.  (
y  =  0  /\  z  =  0 ) ) )  ->  -.  v  e.  ( `' F " { y } ) )
222221pm2.21d 106 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( y  e.  ( RR  \  ran  F
)  /\  -.  (
y  =  0  /\  z  =  0 ) ) )  ->  (
v  e.  ( `' F " { y } )  ->  v  e.  (/) ) )
223222ssrdv 3495 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( y  e.  ( RR  \  ran  F
)  /\  -.  (
y  =  0  /\  z  =  0 ) ) )  ->  ( `' F " { y } )  C_  (/) )
224212, 223syl5ss 3500 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( y  e.  ( RR  \  ran  F
)  /\  -.  (
y  =  0  /\  z  =  0 ) ) )  ->  (
( `' F " { y } )  i^i  ( `' G " { z } ) )  C_  (/) )
225 ss0 3815 . . . . . . . . . . . . . 14  |-  ( ( ( `' F " { y } )  i^i  ( `' G " { z } ) )  C_  (/)  ->  (
( `' F " { y } )  i^i  ( `' G " { z } ) )  =  (/) )
226224, 225syl 16 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( y  e.  ( RR  \  ran  F
)  /\  -.  (
y  =  0  /\  z  =  0 ) ) )  ->  (
( `' F " { y } )  i^i  ( `' G " { z } ) )  =  (/) )
227226fveq2d 5852 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( y  e.  ( RR  \  ran  F
)  /\  -.  (
y  =  0  /\  z  =  0 ) ) )  ->  ( vol `  ( ( `' F " { y } )  i^i  ( `' G " { z } ) ) )  =  ( vol `  (/) ) )
228 0mbl 22116 . . . . . . . . . . . . . 14  |-  (/)  e.  dom  vol
229 mblvol 22107 . . . . . . . . . . . . . 14  |-  ( (/)  e.  dom  vol  ->  ( vol `  (/) )  =  ( vol* `  (/) ) )
230228, 229ax-mp 5 . . . . . . . . . . . . 13  |-  ( vol `  (/) )  =  ( vol* `  (/) )
231 ovol0 22070 . . . . . . . . . . . . 13  |-  ( vol* `  (/) )  =  0
232230, 231eqtri 2483 . . . . . . . . . . . 12  |-  ( vol `  (/) )  =  0
233227, 232syl6eq 2511 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( y  e.  ( RR  \  ran  F
)  /\  -.  (
y  =  0  /\  z  =  0 ) ) )  ->  ( vol `  ( ( `' F " { y } )  i^i  ( `' G " { z } ) ) )  =  0 )
234211, 233eqtrd 2495 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( y  e.  ( RR  \  ran  F
)  /\  -.  (
y  =  0  /\  z  =  0 ) ) )  ->  (
y I z )  =  0 )
235234oveq2d 6286 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( y  e.  ( RR  \  ran  F
)  /\  -.  (
y  =  0  /\  z  =  0 ) ) )  ->  (
( y  +  z )  x.  ( y I z ) )  =  ( ( y  +  z )  x.  0 ) )
236207, 208readdcld 9612 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( y  e.  ( RR  \  ran  F
)  /\  -.  (
y  =  0  /\  z  =  0 ) ) )  ->  (
y  +  z )  e.  RR )
237236recnd 9611 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( y  e.  ( RR  \  ran  F
)  /\  -.  (
y  =  0  /\  z  =  0 ) ) )  ->  (
y  +  z )  e.  CC )
238237mul01d 9768 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( y  e.  ( RR  \  ran  F
)  /\  -.  (
y  =  0  /\  z  =  0 ) ) )  ->  (
( y  +  z )  x.  0 )  =  0 )
239235, 238eqtrd 2495 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( y  e.  ( RR  \  ran  F
)  /\  -.  (
y  =  0  /\  z  =  0 ) ) )  ->  (
( y  +  z )  x.  ( y I z ) )  =  0 )
240239expr 613 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ( RR  \  ran  F ) )  ->  ( -.  (
y  =  0  /\  z  =  0 )  ->  ( ( y  +  z )  x.  ( y I z ) )  =  0 ) )
241 oveq12 6279 . . . . . . . . . 10  |-  ( ( y  =  0  /\  z  =  0 )  ->  ( y  +  z )  =  ( 0  +  0 ) )
242241, 103syl6eq 2511 . . . . . . . . 9  |-  ( ( y  =  0  /\  z  =  0 )  ->  ( y  +  z )  =  0 )
243 oveq12 6279 . . . . . . . . . 10  |-  ( ( y  =  0  /\  z  =  0 )  ->  ( y I z )  =  ( 0 I 0 ) )
244 0re 9585 . . . . . . . . . . 11  |-  0  e.  RR
245 iftrue 3935 . . . . . . . . . . . 12  |-  ( ( i  =  0  /\  j  =  0 )  ->  if ( ( i  =  0  /\  j  =  0 ) ,  0 ,  ( vol `  ( ( `' F " { i } )  i^i  ( `' G " { j } ) ) ) )  =  0 )
246 c0ex 9579 . . . . . . . . . . . 12  |-  0  e.  _V
247245, 107, 246ovmpt2a 6406 . . . . . . . . . . 11  |-  ( ( 0  e.  RR  /\  0  e.  RR )  ->  ( 0 I 0 )  =  0 )
248244, 244, 247mp2an 670 . . . . . . . . . 10  |-  ( 0 I 0 )  =  0
249243, 248syl6eq 2511 . . . . . . . . 9  |-  ( ( y  =  0  /\  z  =  0 )  ->  ( y I z )  =  0 )
250242, 249oveq12d 6288 . . . . . . . 8  |-  ( ( y  =  0  /\  z  =  0 )  ->  ( ( y  +  z )  x.  ( y I z ) )  =  ( 0  x.  0 ) )
251 0cn 9577 . . . . . . . . 9  |-  0  e.  CC
252251mul01i 9759 . . . . . . . 8  |-  ( 0  x.  0 )  =  0
253250, 252syl6eq 2511 . . . . . . 7  |-  ( ( y  =  0  /\  z  =  0 )  ->  ( ( y  +  z )  x.  ( y I z ) )  =  0 )
254240, 253pm2.61d2 160 . . . . . 6  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ( RR  \  ran  F ) )  ->  ( ( y  +  z )  x.  ( y I z ) )  =  0 )
255205, 254syldan 468 . . . . 5  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ( ran  ( v  e.  ran  P 
|->  ( v  -  z
) )  \  ran  F ) )  ->  (
( y  +  z )  x.  ( y I z ) )  =  0 )
256 f1ofo 5805 . . . . . . 7  |-  ( ( v  e.  ran  P  |->  ( v  -  z
) ) : ran  P -1-1-onto-> ran  ( v  e.  ran  P 
|->  ( v  -  z
) )  ->  (
v  e.  ran  P  |->  ( v  -  z
) ) : ran  P
-onto->
ran  ( v  e. 
ran  P  |->  ( v  -  z ) ) )
257150, 256syl 16 . . . . . 6  |-  ( (
ph  /\  z  e.  ran  G )  ->  (
v  e.  ran  P  |->  ( v  -  z
) ) : ran  P
-onto->
ran  ( v  e. 
ran  P  |->  ( v  -  z ) ) )
258 fofi 7798 . . . . . 6  |-  ( ( ran  P  e.  Fin  /\  ( v  e.  ran  P 
|->  ( v  -  z
) ) : ran  P
-onto->
ran  ( v  e. 
ran  P  |->  ( v  -  z ) ) )  ->  ran  ( v  e.  ran  P  |->  ( v  -  z ) )  e.  Fin )
259133, 257, 258syl2anc 659 . . . . 5  |-  ( (
ph  /\  z  e.  ran  G )  ->  ran  ( v  e.  ran  P 
|->  ( v  -  z
) )  e.  Fin )
260197, 203, 255, 259fsumss 13629 . . . 4  |-  ( (
ph  /\  z  e.  ran  G )  ->  sum_ y  e.  ran  F ( ( y  +  z )  x.  ( y I z ) )  = 
sum_ y  e.  ran  ( v  e.  ran  P 
|->  ( v  -  z
) ) ( ( y  +  z )  x.  ( y I z ) ) )
26135a1i 11 . . . . 5  |-  ( (
ph  /\  z  e.  ran  G )  ->  ( ran  P  \  { 0 } )  C_  ran  P )
262126an32s 802 . . . . 5  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  w  e.  ( ran  P 
\  { 0 } ) )  ->  (
w  x.  ( ( w  -  z ) I z ) )  e.  CC )
263 dfin4 3735 . . . . . . . 8  |-  ( ran 
P  i^i  { 0 } )  =  ( ran  P  \  ( ran  P  \  { 0 } ) )
264 inss2 3705 . . . . . . . 8  |-  ( ran 
P  i^i  { 0 } )  C_  { 0 }
265263, 264eqsstr3i 3520 . . . . . . 7  |-  ( ran 
P  \  ( ran  P 
\  { 0 } ) )  C_  { 0 }
266265sseli 3485 . . . . . 6  |-  ( w  e.  ( ran  P  \  ( ran  P  \  { 0 } ) )  ->  w  e.  { 0 } )
267 elsni 4041 . . . . . . . . 9  |-  ( w  e.  { 0 }  ->  w  =  0 )
268267adantl 464 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  w  e.  { 0 } )  ->  w  =  0 )
269268oveq1d 6285 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  w  e.  { 0 } )  ->  (
w  x.  ( ( w  -  z ) I z ) )  =  ( 0  x.  ( ( w  -  z ) I z ) ) )
270110ad2antrr 723 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  w  e.  { 0 } )  ->  I : ( RR  X.  RR ) --> RR )
271268, 244syl6eqel 2550 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  w  e.  { 0 } )  ->  w  e.  RR )
27268adantr 463 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  w  e.  { 0 } )  ->  z  e.  RR )
273271, 272resubcld 9983 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  w  e.  { 0 } )  ->  (
w  -  z )  e.  RR )
274270, 273, 272fovrnd 6420 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  w  e.  { 0 } )  ->  (
( w  -  z
) I z )  e.  RR )
275274recnd 9611 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  w  e.  { 0 } )  ->  (
( w  -  z
) I z )  e.  CC )
276275mul02d 9767 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  w  e.  { 0 } )  ->  (
0  x.  ( ( w  -  z ) I z ) )  =  0 )
277269, 276eqtrd 2495 . . . . . 6  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  w  e.  { 0 } )  ->  (
w  x.  ( ( w  -  z ) I z ) )  =  0 )
278266, 277sylan2 472 . . . . 5  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  w  e.  ( ran  P 
\  ( ran  P  \  { 0 } ) ) )  ->  (
w  x.  ( ( w  -  z ) I z ) )  =  0 )
279261, 262, 278, 133fsumss 13629 . . . 4  |-  ( (
ph  /\  z  e.  ran  G )  ->  sum_ w  e.  ( ran  P  \  { 0 } ) ( w  x.  (
( w  -  z
) I z ) )  =  sum_ w  e.  ran  P ( w  x.  ( ( w  -  z ) I z ) ) )
280173, 260, 2793eqtr4d 2505 . . 3  |-  ( (
ph  /\  z  e.  ran  G )  ->  sum_ y  e.  ran  F ( ( y  +  z )  x.  ( y I z ) )  = 
sum_ w  e.  ( ran  P  \  { 0 } ) ( w  x.  ( ( w  -  z ) I z ) ) )
281280sumeq2dv 13607 . 2  |-  ( ph  -> 
sum_ z  e.  ran  G
sum_ y  e.  ran  F ( ( y  +  z )  x.  (
y I z ) )  =  sum_ z  e.  ran  G sum_ w  e.  ( ran  P  \  { 0 } ) ( w  x.  (
( w  -  z
) I z ) ) )
282203anasss 645 . . 3  |-  ( (
ph  /\  ( z  e.  ran  G  /\  y  e.  ran  F ) )  ->  ( ( y  +  z )  x.  ( y I z ) )  e.  CC )
2837, 5, 282fsumcom 13672 . 2  |-  ( ph  -> 
sum_ z  e.  ran  G
sum_ y  e.  ran  F ( ( y  +  z )  x.  (
y I z ) )  =  sum_ y  e.  ran  F sum_ z  e.  ran  G ( ( y  +  z )  x.  ( y I z ) ) )
284129, 281, 2833eqtr2d 2501 1  |-  ( ph  ->  ( S.1 `  ( F  oF  +  G
) )  =  sum_ y  e.  ran  F sum_ z  e.  ran  G ( ( y  +  z )  x.  ( y I z ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823   {cab 2439    =/= wne 2649   A.wral 2804   E.wrex 2805   _Vcvv 3106    \ cdif 3458    i^i cin 3460    C_ wss 3461   (/)c0 3783   ifcif 3929   {csn 4016   <.cop 4022   U_ciun 4315   class class class wbr 4439    |-> cmpt 4497    X. cxp 4986   `'ccnv 4987   dom cdm 4988   ran crn 4989    |` cres 4990   "cima 4991   Fun wfun 5564    Fn wfn 5565   -->wf 5566   -1-1->wf1 5567   -onto->wfo 5568   -1-1-onto->wf1o 5569   ` cfv 5570  (class class class)co 6270    |-> cmpt2 6272    oFcof 6511   Fincfn 7509   CCcc 9479   RRcr 9480   0cc0 9481    + caddc 9484    x. cmul 9486    - cmin 9796   sum_csu 13590   vol*covol 22040   volcvol 22041   S.1citg1 22190
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559  ax-addf 9560
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-disj 4411  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-of 6513  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-2o 7123  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-sup 7893  df-oi 7927  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11083  df-q 11184  df-rp 11222  df-xadd 11322  df-ioo 11536  df-ico 11538  df-icc 11539  df-fz 11676  df-fzo 11800  df-fl 11910  df-seq 12090  df-exp 12149  df-hash 12388  df-cj 13014  df-re 13015  df-im 13016  df-sqrt 13150  df-abs 13151  df-clim 13393  df-sum 13591  df-xmet 18607  df-met 18608  df-ovol 22042  df-vol 22043  df-mbf 22194  df-itg1 22195
This theorem is referenced by:  itg1addlem5  22273
  Copyright terms: Public domain W3C validator