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Theorem itg1addlem4 22706
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 22702 . . . 4  |-  ( ph  ->  ( F  oF  +  G )  e. 
dom  S.1 )
4 i1frn 22684 . . . . . . . 8  |-  ( F  e.  dom  S.1  ->  ran 
F  e.  Fin )
51, 4syl 17 . . . . . . 7  |-  ( ph  ->  ran  F  e.  Fin )
6 i1frn 22684 . . . . . . . 8  |-  ( G  e.  dom  S.1  ->  ran 
G  e.  Fin )
72, 6syl 17 . . . . . . 7  |-  ( ph  ->  ran  G  e.  Fin )
8 xpfi 7868 . . . . . . 7  |-  ( ( ran  F  e.  Fin  /\ 
ran  G  e.  Fin )  ->  ( ran  F  X.  ran  G )  e. 
Fin )
95, 7, 8syl2anc 671 . . . . . 6  |-  ( ph  ->  ( ran  F  X.  ran  G )  e.  Fin )
10 ax-addf 9644 . . . . . . . . . 10  |-  +  :
( CC  X.  CC )
--> CC
11 ffn 5751 . . . . . . . . . 10  |-  (  +  : ( CC  X.  CC ) --> CC  ->  +  Fn  ( CC  X.  CC ) )
1210, 11ax-mp 5 . . . . . . . . 9  |-  +  Fn  ( CC  X.  CC )
13 i1ff 22683 . . . . . . . . . . . . 13  |-  ( F  e.  dom  S.1  ->  F : RR --> RR )
141, 13syl 17 . . . . . . . . . . . 12  |-  ( ph  ->  F : RR --> RR )
15 frn 5758 . . . . . . . . . . . 12  |-  ( F : RR --> RR  ->  ran 
F  C_  RR )
1614, 15syl 17 . . . . . . . . . . 11  |-  ( ph  ->  ran  F  C_  RR )
17 ax-resscn 9622 . . . . . . . . . . 11  |-  RR  C_  CC
1816, 17syl6ss 3456 . . . . . . . . . 10  |-  ( ph  ->  ran  F  C_  CC )
19 i1ff 22683 . . . . . . . . . . . . 13  |-  ( G  e.  dom  S.1  ->  G : RR --> RR )
202, 19syl 17 . . . . . . . . . . . 12  |-  ( ph  ->  G : RR --> RR )
21 frn 5758 . . . . . . . . . . . 12  |-  ( G : RR --> RR  ->  ran 
G  C_  RR )
2220, 21syl 17 . . . . . . . . . . 11  |-  ( ph  ->  ran  G  C_  RR )
2322, 17syl6ss 3456 . . . . . . . . . 10  |-  ( ph  ->  ran  G  C_  CC )
24 xpss12 4959 . . . . . . . . . 10  |-  ( ( ran  F  C_  CC  /\ 
ran  G  C_  CC )  ->  ( ran  F  X.  ran  G )  C_  ( CC  X.  CC ) )
2518, 23, 24syl2anc 671 . . . . . . . . 9  |-  ( ph  ->  ( ran  F  X.  ran  G )  C_  ( CC  X.  CC ) )
26 fnssres 5711 . . . . . . . . 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 674 . . . . . . . 8  |-  ( ph  ->  (  +  |`  ( ran  F  X.  ran  G
) )  Fn  ( ran  F  X.  ran  G
) )
28 itg1add.4 . . . . . . . . 9  |-  P  =  (  +  |`  ( ran  F  X.  ran  G
) )
2928fneq1i 5692 . . . . . . . 8  |-  ( P  Fn  ( ran  F  X.  ran  G )  <->  (  +  |`  ( ran  F  X.  ran  G ) )  Fn  ( ran  F  X.  ran  G ) )
3027, 29sylibr 217 . . . . . . 7  |-  ( ph  ->  P  Fn  ( ran 
F  X.  ran  G
) )
31 dffn4 5822 . . . . . . 7  |-  ( P  Fn  ( ran  F  X.  ran  G )  <->  P :
( ran  F  X.  ran  G ) -onto-> ran  P
)
3230, 31sylib 201 . . . . . 6  |-  ( ph  ->  P : ( ran 
F  X.  ran  G
) -onto-> ran  P )
33 fofi 7886 . . . . . 6  |-  ( ( ( ran  F  X.  ran  G )  e.  Fin  /\  P : ( ran 
F  X.  ran  G
) -onto-> ran  P )  ->  ran  P  e.  Fin )
349, 32, 33syl2anc 671 . . . . 5  |-  ( ph  ->  ran  P  e.  Fin )
35 difss 3572 . . . . 5  |-  ( ran 
P  \  { 0 } )  C_  ran  P
36 ssfi 7818 . . . . 5  |-  ( ( ran  P  e.  Fin  /\  ( ran  P  \  { 0 } ) 
C_  ran  P )  ->  ( ran  P  \  { 0 } )  e.  Fin )
3734, 35, 36sylancl 673 . . . 4  |-  ( ph  ->  ( ran  P  \  { 0 } )  e.  Fin )
38 opelxpi 4885 . . . . . . . . . 10  |-  ( ( x  e.  ran  F  /\  y  e.  ran  G )  ->  <. x ,  y >.  e.  ( ran  F  X.  ran  G
) )
39 ffun 5754 . . . . . . . . . . . 12  |-  (  +  : ( CC  X.  CC ) --> CC  ->  Fun  +  )
4010, 39ax-mp 5 . . . . . . . . . . 11  |-  Fun  +
4110fdmi 5757 . . . . . . . . . . . 12  |-  dom  +  =  ( CC  X.  CC )
4225, 41syl6sseqr 3491 . . . . . . . . . . 11  |-  ( ph  ->  ( ran  F  X.  ran  G )  C_  dom  +  )
43 funfvima2 6166 . . . . . . . . . . 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 674 . . . . . . . . . 10  |-  ( ph  ->  ( <. x ,  y
>.  e.  ( ran  F  X.  ran  G )  -> 
(  +  `  <. x ,  y >. )  e.  (  +  " ( ran  F  X.  ran  G
) ) ) )
4538, 44syl5 33 . . . . . . . . 9  |-  ( ph  ->  ( ( x  e. 
ran  F  /\  y  e.  ran  G )  -> 
(  +  `  <. x ,  y >. )  e.  (  +  " ( ran  F  X.  ran  G
) ) ) )
4645imp 435 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ran  F  /\  y  e.  ran  G ) )  ->  (  +  `  <. x ,  y >.
)  e.  (  + 
" ( ran  F  X.  ran  G ) ) )
47 df-ov 6318 . . . . . . . 8  |-  ( x  +  y )  =  (  +  `  <. x ,  y >. )
4828rneqi 5080 . . . . . . . . 9  |-  ran  P  =  ran  (  +  |`  ( ran  F  X.  ran  G
) )
49 df-ima 4866 . . . . . . . . 9  |-  (  + 
" ( ran  F  X.  ran  G ) )  =  ran  (  +  |`  ( ran  F  X.  ran  G ) )
5048, 49eqtr4i 2487 . . . . . . . 8  |-  ran  P  =  (  +  " ( ran  F  X.  ran  G
) )
5146, 47, 503eltr4g 2557 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ran  F  /\  y  e.  ran  G ) )  ->  ( x  +  y )  e.  ran  P )
52 ffn 5751 . . . . . . . . 9  |-  ( F : RR --> RR  ->  F  Fn  RR )
5314, 52syl 17 . . . . . . . 8  |-  ( ph  ->  F  Fn  RR )
54 dffn3 5759 . . . . . . . 8  |-  ( F  Fn  RR  <->  F : RR
--> ran  F )
5553, 54sylib 201 . . . . . . 7  |-  ( ph  ->  F : RR --> ran  F
)
56 ffn 5751 . . . . . . . . 9  |-  ( G : RR --> RR  ->  G  Fn  RR )
5720, 56syl 17 . . . . . . . 8  |-  ( ph  ->  G  Fn  RR )
58 dffn3 5759 . . . . . . . 8  |-  ( G  Fn  RR  <->  G : RR
--> ran  G )
5957, 58sylib 201 . . . . . . 7  |-  ( ph  ->  G : RR --> ran  G
)
60 reex 9656 . . . . . . . 8  |-  RR  e.  _V
6160a1i 11 . . . . . . 7  |-  ( ph  ->  RR  e.  _V )
62 inidm 3653 . . . . . . 7  |-  ( RR 
i^i  RR )  =  RR
6351, 55, 59, 61, 61, 62off 6573 . . . . . 6  |-  ( ph  ->  ( F  oF  +  G ) : RR --> ran  P )
64 frn 5758 . . . . . 6  |-  ( ( F  oF  +  G ) : RR --> ran  P  ->  ran  ( F  oF  +  G
)  C_  ran  P )
6563, 64syl 17 . . . . 5  |-  ( ph  ->  ran  ( F  oF  +  G )  C_ 
ran  P )
6665ssdifd 3581 . . . 4  |-  ( ph  ->  ( ran  ( F  oF  +  G
)  \  { 0 } )  C_  ( ran  P  \  { 0 } ) )
6716sselda 3444 . . . . . . . . . 10  |-  ( (
ph  /\  y  e.  ran  F )  ->  y  e.  RR )
6822sselda 3444 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  ran  G )  ->  z  e.  RR )
6967, 68anim12dan 853 . . . . . . . . 9  |-  ( (
ph  /\  ( y  e.  ran  F  /\  z  e.  ran  G ) )  ->  ( y  e.  RR  /\  z  e.  RR ) )
70 readdcl 9648 . . . . . . . . 9  |-  ( ( y  e.  RR  /\  z  e.  RR )  ->  ( y  +  z )  e.  RR )
7169, 70syl 17 . . . . . . . 8  |-  ( (
ph  /\  ( y  e.  ran  F  /\  z  e.  ran  G ) )  ->  ( y  +  z )  e.  RR )
7271ralrimivva 2821 . . . . . . 7  |-  ( ph  ->  A. y  e.  ran  F A. z  e.  ran  G ( y  +  z )  e.  RR )
73 funimassov 6473 . . . . . . . 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 674 . . . . . . 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 240 . . . . . 6  |-  ( ph  ->  (  +  " ( ran  F  X.  ran  G
) )  C_  RR )
7650, 75syl5eqss 3488 . . . . 5  |-  ( ph  ->  ran  P  C_  RR )
7776ssdifd 3581 . . . 4  |-  ( ph  ->  ( ran  P  \  { 0 } ) 
C_  ( RR  \  { 0 } ) )
78 itg1val2 22691 . . . 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 1278 . . 3  |-  ( ph  ->  ( S.1 `  ( F  oF  +  G
) )  =  sum_ w  e.  ( ran  P  \  { 0 } ) ( w  x.  ( vol `  ( `' ( F  oF  +  G ) " {
w } ) ) ) )
8020adantr 471 . . . . . . . 8  |-  ( (
ph  /\  w  e.  ( ran  P  \  {
0 } ) )  ->  G : RR --> RR )
817adantr 471 . . . . . . . 8  |-  ( (
ph  /\  w  e.  ( ran  P  \  {
0 } ) )  ->  ran  G  e.  Fin )
82 inss2 3665 . . . . . . . . 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 22685 . . . . . . . . . . 11  |-  ( F  e.  dom  S.1  ->  ( `' F " { ( w  -  z ) } )  e.  dom  vol )
851, 84syl 17 . . . . . . . . . 10  |-  ( ph  ->  ( `' F " { ( w  -  z ) } )  e.  dom  vol )
86 i1fima 22685 . . . . . . . . . . 11  |-  ( G  e.  dom  S.1  ->  ( `' G " { z } )  e.  dom  vol )
872, 86syl 17 . . . . . . . . . 10  |-  ( ph  ->  ( `' G " { z } )  e.  dom  vol )
88 inmbl 22544 . . . . . . . . . 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 671 . . . . . . . . 9  |-  ( ph  ->  ( ( `' F " { ( w  -  z ) } )  i^i  ( `' G " { z } ) )  e.  dom  vol )
9089ad2antrr 737 . . . . . . . 8  |-  ( ( ( ph  /\  w  e.  ( ran  P  \  { 0 } ) )  /\  z  e. 
ran  G )  -> 
( ( `' F " { ( w  -  z ) } )  i^i  ( `' G " { z } ) )  e.  dom  vol )
9135, 76syl5ss 3455 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ran  P  \  { 0 } ) 
C_  RR )
9291sselda 3444 . . . . . . . . . . . 12  |-  ( (
ph  /\  w  e.  ( ran  P  \  {
0 } ) )  ->  w  e.  RR )
9392adantr 471 . . . . . . . . . . 11  |-  ( ( ( ph  /\  w  e.  ( ran  P  \  { 0 } ) )  /\  z  e. 
ran  G )  ->  w  e.  RR )
9468adantlr 726 . . . . . . . . . . 11  |-  ( ( ( ph  /\  w  e.  ( ran  P  \  { 0 } ) )  /\  z  e. 
ran  G )  -> 
z  e.  RR )
9593, 94resubcld 10075 . . . . . . . . . 10  |-  ( ( ( ph  /\  w  e.  ( ran  P  \  { 0 } ) )  /\  z  e. 
ran  G )  -> 
( w  -  z
)  e.  RR )
9693recnd 9695 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  w  e.  ( ran  P  \  { 0 } ) )  /\  z  e. 
ran  G )  ->  w  e.  CC )
9794recnd 9695 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  w  e.  ( ran  P  \  { 0 } ) )  /\  z  e. 
ran  G )  -> 
z  e.  CC )
9896, 97npcand 10016 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  w  e.  ( ran  P  \  { 0 } ) )  /\  z  e. 
ran  G )  -> 
( ( w  -  z )  +  z )  =  w )
99 eldifsni 4111 . . . . . . . . . . . . 13  |-  ( w  e.  ( ran  P  \  { 0 } )  ->  w  =/=  0
)
10099ad2antlr 738 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  w  e.  ( ran  P  \  { 0 } ) )  /\  z  e. 
ran  G )  ->  w  =/=  0 )
10198, 100eqnetrd 2703 . . . . . . . . . . 11  |-  ( ( ( ph  /\  w  e.  ( ran  P  \  { 0 } ) )  /\  z  e. 
ran  G )  -> 
( ( w  -  z )  +  z )  =/=  0 )
102 oveq12 6324 . . . . . . . . . . . . 13  |-  ( ( ( w  -  z
)  =  0  /\  z  =  0 )  ->  ( ( w  -  z )  +  z )  =  ( 0  +  0 ) )
103 00id 9834 . . . . . . . . . . . . 13  |-  ( 0  +  0 )  =  0
104102, 103syl6eq 2512 . . . . . . . . . . . 12  |-  ( ( ( w  -  z
)  =  0  /\  z  =  0 )  ->  ( ( w  -  z )  +  z )  =  0 )
105104necon3ai 2661 . . . . . . . . . . 11  |-  ( ( ( w  -  z
)  +  z )  =/=  0  ->  -.  ( ( w  -  z )  =  0  /\  z  =  0 ) )
106101, 105syl 17 . . . . . . . . . 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 22705 . . . . . . . . . 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 1275 . . . . . . . . 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 22704 . . . . . . . . . . 11  |-  ( ph  ->  I : ( RR 
X.  RR ) --> RR )
111110ad2antrr 737 . . . . . . . . . 10  |-  ( ( ( ph  /\  w  e.  ( ran  P  \  { 0 } ) )  /\  z  e. 
ran  G )  ->  I : ( RR  X.  RR ) --> RR )
112111, 95, 94fovrnd 6468 . . . . . . . . 9  |-  ( ( ( ph  /\  w  e.  ( ran  P  \  { 0 } ) )  /\  z  e. 
ran  G )  -> 
( ( w  -  z ) I z )  e.  RR )
113109, 112eqeltrrd 2541 . . . . . . . 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 22699 . . . . . . 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 9695 . . . . . . . . 9  |-  ( (
ph  /\  w  e.  ( ran  P  \  {
0 } ) )  ->  w  e.  CC )
1161, 2i1faddlem 22700 . . . . . . . . 9  |-  ( (
ph  /\  w  e.  CC )  ->  ( `' ( F  oF  +  G ) " { w } )  =  U_ z  e. 
ran  G ( ( `' F " { ( w  -  z ) } )  i^i  ( `' G " { z } ) ) )
117115, 116syldan 477 . . . . . . . 8  |-  ( (
ph  /\  w  e.  ( ran  P  \  {
0 } ) )  ->  ( `' ( F  oF  +  G ) " {
w } )  = 
U_ z  e.  ran  G ( ( `' F " { ( w  -  z ) } )  i^i  ( `' G " { z } ) ) )
118117fveq2d 5892 . . . . . . 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 13818 . . . . . . 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 2506 . . . . . 6  |-  ( (
ph  /\  w  e.  ( ran  P  \  {
0 } ) )  ->  ( vol `  ( `' ( F  oF  +  G ) " { w } ) )  =  sum_ z  e.  ran  G ( ( w  -  z ) I z ) )
121120oveq2d 6331 . . . . 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 9695 . . . . . 6  |-  ( ( ( ph  /\  w  e.  ( ran  P  \  { 0 } ) )  /\  z  e. 
ran  G )  -> 
( ( w  -  z ) I z )  e.  CC )
12381, 115, 122fsummulc2 13894 . . . . 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 2496 . . . 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 13818 . . 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 9689 . . . . 5  |-  ( ( ( ph  /\  w  e.  ( ran  P  \  { 0 } ) )  /\  z  e. 
ran  G )  -> 
( w  x.  (
( w  -  z
) I z ) )  e.  CC )
127126anasss 657 . . . 4  |-  ( (
ph  /\  ( w  e.  ( ran  P  \  { 0 } )  /\  z  e.  ran  G ) )  ->  (
w  x.  ( ( w  -  z ) I z ) )  e.  CC )
12837, 7, 127fsumcom 13885 . . 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 2500 . 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 6322 . . . . . . 7  |-  ( y  =  ( w  -  z )  ->  (
y  +  z )  =  ( ( w  -  z )  +  z ) )
131 oveq1 6322 . . . . . . 7  |-  ( y  =  ( w  -  z )  ->  (
y I z )  =  ( ( w  -  z ) I z ) )
132130, 131oveq12d 6333 . . . . . 6  |-  ( y  =  ( w  -  z )  ->  (
( y  +  z )  x.  ( y I z ) )  =  ( ( ( w  -  z )  +  z )  x.  ( ( w  -  z ) I z ) ) )
13334adantr 471 . . . . . 6  |-  ( (
ph  /\  z  e.  ran  G )  ->  ran  P  e.  Fin )
13476adantr 471 . . . . . . . . . . 11  |-  ( (
ph  /\  z  e.  ran  G )  ->  ran  P 
C_  RR )
135134sselda 3444 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  v  e.  ran  P )  ->  v  e.  RR )
13668adantr 471 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  v  e.  ran  P )  ->  z  e.  RR )
137135, 136resubcld 10075 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  v  e.  ran  P )  ->  ( v  -  z )  e.  RR )
138137ex 440 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ran  G )  ->  (
v  e.  ran  P  ->  ( v  -  z
)  e.  RR ) )
139135recnd 9695 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  v  e.  ran  P )  ->  v  e.  CC )
140139adantrr 728 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( v  e.  ran  P  /\  y  e.  ran  P ) )  ->  v  e.  CC )
14176sselda 3444 . . . . . . . . . . . 12  |-  ( (
ph  /\  y  e.  ran  P )  ->  y  e.  RR )
142141ad2ant2rl 760 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( v  e.  ran  P  /\  y  e.  ran  P ) )  ->  y  e.  RR )
143142recnd 9695 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( v  e.  ran  P  /\  y  e.  ran  P ) )  ->  y  e.  CC )
14468recnd 9695 . . . . . . . . . . 11  |-  ( (
ph  /\  z  e.  ran  G )  ->  z  e.  CC )
145144adantr 471 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( v  e.  ran  P  /\  y  e.  ran  P ) )  ->  z  e.  CC )
146140, 143, 145subcan2ad 10057 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( v  e.  ran  P  /\  y  e.  ran  P ) )  ->  (
( v  -  z
)  =  ( y  -  z )  <->  v  =  y ) )
147146ex 440 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ran  G )  ->  (
( v  e.  ran  P  /\  y  e.  ran  P )  ->  ( (
v  -  z )  =  ( y  -  z )  <->  v  =  y ) ) )
148138, 147dom2lem 7635 . . . . . . 7  |-  ( (
ph  /\  z  e.  ran  G )  ->  (
v  e.  ran  P  |->  ( v  -  z
) ) : ran  P
-1-1-> RR )
149 f1f1orn 5848 . . . . . . 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 17 . . . . . 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 6322 . . . . . . . 8  |-  ( v  =  w  ->  (
v  -  z )  =  ( w  -  z ) )
152 eqid 2462 . . . . . . . 8  |-  ( v  e.  ran  P  |->  ( v  -  z ) )  =  ( v  e.  ran  P  |->  ( v  -  z ) )
153 ovex 6343 . . . . . . . 8  |-  ( w  -  z )  e. 
_V
154151, 152, 153fvmpt 5971 . . . . . . 7  |-  ( w  e.  ran  P  -> 
( ( v  e. 
ran  P  |->  ( v  -  z ) ) `
 w )  =  ( w  -  z
) )
155154adantl 472 . . . . . 6  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  w  e.  ran  P )  ->  ( ( v  e.  ran  P  |->  ( v  -  z ) ) `  w )  =  ( w  -  z ) )
156 f1f 5802 . . . . . . . . . . 11  |-  ( ( v  e.  ran  P  |->  ( v  -  z
) ) : ran  P
-1-1-> RR  ->  ( v  e.  ran  P  |->  ( v  -  z ) ) : ran  P --> RR )
157 frn 5758 . . . . . . . . . . 11  |-  ( ( v  e.  ran  P  |->  ( v  -  z
) ) : ran  P --> RR  ->  ran  ( v  e.  ran  P  |->  ( v  -  z ) )  C_  RR )
158148, 156, 1573syl 18 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  ran  G )  ->  ran  ( v  e.  ran  P 
|->  ( v  -  z
) )  C_  RR )
159158sselda 3444 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  ( v  e.  ran  P  |->  ( v  -  z ) ) )  ->  y  e.  RR )
16068adantr 471 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  ( v  e.  ran  P  |->  ( v  -  z ) ) )  ->  z  e.  RR )
161159, 160readdcld 9696 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  ( v  e.  ran  P  |->  ( v  -  z ) ) )  ->  (
y  +  z )  e.  RR )
162110ad2antrr 737 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  ( v  e.  ran  P  |->  ( v  -  z ) ) )  ->  I : ( RR  X.  RR ) --> RR )
163162, 159, 160fovrnd 6468 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  ( v  e.  ran  P  |->  ( v  -  z ) ) )  ->  (
y I z )  e.  RR )
164161, 163remulcld 9697 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  ( v  e.  ran  P  |->  ( v  -  z ) ) )  ->  (
( y  +  z )  x.  ( y I z ) )  e.  RR )
165164recnd 9695 . . . . . 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 13838 . . . . 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 3444 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  w  e.  ran  P )  ->  w  e.  RR )
168167recnd 9695 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  w  e.  ran  P )  ->  w  e.  CC )
169144adantr 471 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  w  e.  ran  P )  ->  z  e.  CC )
170168, 169npcand 10016 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  w  e.  ran  P )  ->  ( ( w  -  z )  +  z )  =  w )
171170oveq1d 6330 . . . . . 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 13818 . . . . 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 2496 . . . 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 737 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  ( ran  F  X.  ran  G )  C_  dom  +  )
175 simpr 467 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  y  e.  ran  F )
176 simplr 767 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  z  e.  ran  G )
177 opelxpi 4885 . . . . . . . . . . . 12  |-  ( ( y  e.  ran  F  /\  z  e.  ran  G )  ->  <. y ,  z >.  e.  ( ran  F  X.  ran  G
) )
178175, 176, 177syl2anc 671 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  <. y ,  z
>.  e.  ( ran  F  X.  ran  G ) )
179 funfvima2 6166 . . . . . . . . . . . 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 681 . . . . . . . . . . 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 62 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  (  +  `  <. y ,  z >.
)  e.  (  + 
" ( ran  F  X.  ran  G ) ) )
182 df-ov 6318 . . . . . . . . . 10  |-  ( y  +  z )  =  (  +  `  <. y ,  z >. )
183181, 182, 503eltr4g 2557 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  ( y  +  z )  e.  ran  P )
18467adantlr 726 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  y  e.  RR )
185184recnd 9695 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  y  e.  CC )
186144adantr 471 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  z  e.  CC )
187185, 186pncand 10013 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  ( ( y  +  z )  -  z )  =  y )
188187eqcomd 2468 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  y  =  ( ( y  +  z )  -  z ) )
189 oveq1 6322 . . . . . . . . . . 11  |-  ( v  =  ( y  +  z )  ->  (
v  -  z )  =  ( ( y  +  z )  -  z ) )
190189eqeq2d 2472 . . . . . . . . . 10  |-  ( v  =  ( y  +  z )  ->  (
y  =  ( v  -  z )  <->  y  =  ( ( y  +  z )  -  z
) ) )
191190rspcev 3162 . . . . . . . . 9  |-  ( ( ( y  +  z )  e.  ran  P  /\  y  =  (
( y  +  z )  -  z ) )  ->  E. v  e.  ran  P  y  =  ( v  -  z
) )
192183, 188, 191syl2anc 671 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  E. v  e.  ran  P  y  =  ( v  -  z ) )
193192ralrimiva 2814 . . . . . . 7  |-  ( (
ph  /\  z  e.  ran  G )  ->  A. y  e.  ran  F E. v  e.  ran  P  y  =  ( v  -  z
) )
194 ssabral 3512 . . . . . . 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 217 . . . . . 6  |-  ( (
ph  /\  z  e.  ran  G )  ->  ran  F 
C_  { y  |  E. v  e.  ran  P  y  =  ( v  -  z ) } )
196152rnmpt 5099 . . . . . 6  |-  ran  (
v  e.  ran  P  |->  ( v  -  z
) )  =  {
y  |  E. v  e.  ran  P  y  =  ( v  -  z
) }
197195, 196syl6sseqr 3491 . . . . 5  |-  ( (
ph  /\  z  e.  ran  G )  ->  ran  F 
C_  ran  ( v  e.  ran  P  |->  ( v  -  z ) ) )
19868adantr 471 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  z  e.  RR )
199184, 198readdcld 9696 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  ( y  +  z )  e.  RR )
200110ad2antrr 737 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  I : ( RR  X.  RR ) --> RR )
201200, 184, 198fovrnd 6468 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  ( y I z )  e.  RR )
202199, 201remulcld 9697 . . . . . 6  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  ( ( y  +  z )  x.  ( y I z ) )  e.  RR )
203202recnd 9695 . . . . 5  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  ( ( y  +  z )  x.  ( y I z ) )  e.  CC )
204158ssdifd 3581 . . . . . . 7  |-  ( (
ph  /\  z  e.  ran  G )  ->  ( ran  ( v  e.  ran  P 
|->  ( v  -  z
) )  \  ran  F )  C_  ( RR  \  ran  F ) )
205204sselda 3444 . . . . . 6  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ( ran  ( v  e.  ran  P 
|->  ( v  -  z
) )  \  ran  F ) )  ->  y  e.  ( RR  \  ran  F ) )
206 eldifi 3567 . . . . . . . . . . . . 13  |-  ( y  e.  ( RR  \  ran  F )  ->  y  e.  RR )
207206ad2antrl 739 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( y  e.  ( RR  \  ran  F
)  /\  -.  (
y  =  0  /\  z  =  0 ) ) )  ->  y  e.  RR )
20868adantr 471 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( y  e.  ( RR  \  ran  F
)  /\  -.  (
y  =  0  /\  z  =  0 ) ) )  ->  z  e.  RR )
209 simprr 771 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( y  e.  ( RR  \  ran  F
)  /\  -.  (
y  =  0  /\  z  =  0 ) ) )  ->  -.  ( y  =  0  /\  z  =  0 ) )
2101, 2, 107itg1addlem3 22705 . . . . . . . . . . . 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 1275 . . . . . . . . . . 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 3664 . . . . . . . . . . . . . . 15  |-  ( ( `' F " { y } )  i^i  ( `' G " { z } ) )  C_  ( `' F " { y } )
213 eldifn 3568 . . . . . . . . . . . . . . . . . . 19  |-  ( y  e.  ( RR  \  ran  F )  ->  -.  y  e.  ran  F )
214213ad2antrl 739 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( y  e.  ( RR  \  ran  F
)  /\  -.  (
y  =  0  /\  z  =  0 ) ) )  ->  -.  y  e.  ran  F )
215 vex 3060 . . . . . . . . . . . . . . . . . . . 20  |-  y  e. 
_V
216 vex 3060 . . . . . . . . . . . . . . . . . . . . 21  |-  v  e. 
_V
217216eliniseg 5216 . . . . . . . . . . . . . . . . . . . 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 5084 . . . . . . . . . . . . . . . . . . 19  |-  ( v F y  ->  y  e.  ran  F )
220218, 219sylbi 200 . . . . . . . . . . . . . . . . . 18  |-  ( v  e.  ( `' F " { y } )  ->  y  e.  ran  F )
221214, 220nsyl 126 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( y  e.  ( RR  \  ran  F
)  /\  -.  (
y  =  0  /\  z  =  0 ) ) )  ->  -.  v  e.  ( `' F " { y } ) )
222221pm2.21d 110 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( y  e.  ( RR  \  ran  F
)  /\  -.  (
y  =  0  /\  z  =  0 ) ) )  ->  (
v  e.  ( `' F " { y } )  ->  v  e.  (/) ) )
223222ssrdv 3450 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( y  e.  ( RR  \  ran  F
)  /\  -.  (
y  =  0  /\  z  =  0 ) ) )  ->  ( `' F " { y } )  C_  (/) )
224212, 223syl5ss 3455 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( y  e.  ( RR  \  ran  F
)  /\  -.  (
y  =  0  /\  z  =  0 ) ) )  ->  (
( `' F " { y } )  i^i  ( `' G " { z } ) )  C_  (/) )
225 ss0 3777 . . . . . . . . . . . . . 14  |-  ( ( ( `' F " { y } )  i^i  ( `' G " { z } ) )  C_  (/)  ->  (
( `' F " { y } )  i^i  ( `' G " { z } ) )  =  (/) )
226224, 225syl 17 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( y  e.  ( RR  \  ran  F
)  /\  -.  (
y  =  0  /\  z  =  0 ) ) )  ->  (
( `' F " { y } )  i^i  ( `' G " { z } ) )  =  (/) )
227226fveq2d 5892 . . . . . . . . . . . 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 22542 . . . . . . . . . . . . . 14  |-  (/)  e.  dom  vol
229 mblvol 22533 . . . . . . . . . . . . . 14  |-  ( (/)  e.  dom  vol  ->  ( vol `  (/) )  =  ( vol* `  (/) ) )
230228, 229ax-mp 5 . . . . . . . . . . . . 13  |-  ( vol `  (/) )  =  ( vol* `  (/) )
231 ovol0 22495 . . . . . . . . . . . . 13  |-  ( vol* `  (/) )  =  0
232230, 231eqtri 2484 . . . . . . . . . . . 12  |-  ( vol `  (/) )  =  0
233227, 232syl6eq 2512 . . . . . . . . . . 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 2496 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( y  e.  ( RR  \  ran  F
)  /\  -.  (
y  =  0  /\  z  =  0 ) ) )  ->  (
y I z )  =  0 )
235234oveq2d 6331 . . . . . . . . 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 9696 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( y  e.  ( RR  \  ran  F
)  /\  -.  (
y  =  0  /\  z  =  0 ) ) )  ->  (
y  +  z )  e.  RR )
237236recnd 9695 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( y  e.  ( RR  \  ran  F
)  /\  -.  (
y  =  0  /\  z  =  0 ) ) )  ->  (
y  +  z )  e.  CC )
238237mul01d 9858 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( y  e.  ( RR  \  ran  F
)  /\  -.  (
y  =  0  /\  z  =  0 ) ) )  ->  (
( y  +  z )  x.  0 )  =  0 )
239235, 238eqtrd 2496 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( y  e.  ( RR  \  ran  F
)  /\  -.  (
y  =  0  /\  z  =  0 ) ) )  ->  (
( y  +  z )  x.  ( y I z ) )  =  0 )
240239expr 624 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ( RR  \  ran  F ) )  ->  ( -.  (
y  =  0  /\  z  =  0 )  ->  ( ( y  +  z )  x.  ( y I z ) )  =  0 ) )
241 oveq12 6324 . . . . . . . . . 10  |-  ( ( y  =  0  /\  z  =  0 )  ->  ( y  +  z )  =  ( 0  +  0 ) )
242241, 103syl6eq 2512 . . . . . . . . 9  |-  ( ( y  =  0  /\  z  =  0 )  ->  ( y  +  z )  =  0 )
243 oveq12 6324 . . . . . . . . . 10  |-  ( ( y  =  0  /\  z  =  0 )  ->  ( y I z )  =  ( 0 I 0 ) )
244 0re 9669 . . . . . . . . . . 11  |-  0  e.  RR
245 iftrue 3899 . . . . . . . . . . . 12  |-  ( ( i  =  0  /\  j  =  0 )  ->  if ( ( i  =  0  /\  j  =  0 ) ,  0 ,  ( vol `  ( ( `' F " { i } )  i^i  ( `' G " { j } ) ) ) )  =  0 )
246 c0ex 9663 . . . . . . . . . . . 12  |-  0  e.  _V
247245, 107, 246ovmpt2a 6454 . . . . . . . . . . 11  |-  ( ( 0  e.  RR  /\  0  e.  RR )  ->  ( 0 I 0 )  =  0 )
248244, 244, 247mp2an 683 . . . . . . . . . 10  |-  ( 0 I 0 )  =  0
249243, 248syl6eq 2512 . . . . . . . . 9  |-  ( ( y  =  0  /\  z  =  0 )  ->  ( y I z )  =  0 )
250242, 249oveq12d 6333 . . . . . . . 8  |-  ( ( y  =  0  /\  z  =  0 )  ->  ( ( y  +  z )  x.  ( y I z ) )  =  ( 0  x.  0 ) )
251 0cn 9661 . . . . . . . . 9  |-  0  e.  CC
252251mul01i 9849 . . . . . . . 8  |-  ( 0  x.  0 )  =  0
253250, 252syl6eq 2512 . . . . . . 7  |-  ( ( y  =  0  /\  z  =  0 )  ->  ( ( y  +  z )  x.  ( y I z ) )  =  0 )
254240, 253pm2.61d2 165 . . . . . 6  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ( RR  \  ran  F ) )  ->  ( ( y  +  z )  x.  ( y I z ) )  =  0 )
255205, 254syldan 477 . . . . 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 5844 . . . . . . 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 17 . . . . . 6  |-  ( (
ph  /\  z  e.  ran  G )  ->  (
v  e.  ran  P  |->  ( v  -  z
) ) : ran  P
-onto->
ran  ( v  e. 
ran  P  |->  ( v  -  z ) ) )
258 fofi 7886 . . . . . 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 671 . . . . 5  |-  ( (
ph  /\  z  e.  ran  G )  ->  ran  ( v  e.  ran  P 
|->  ( v  -  z
) )  e.  Fin )
260197, 203, 255, 259fsumss 13840 . . . 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 818 . . . . 5  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  w  e.  ( ran  P 
\  { 0 } ) )  ->  (
w  x.  ( ( w  -  z ) I z ) )  e.  CC )
263 dfin4 3695 . . . . . . . 8  |-  ( ran 
P  i^i  { 0 } )  =  ( ran  P  \  ( ran  P  \  { 0 } ) )
264 inss2 3665 . . . . . . . 8  |-  ( ran 
P  i^i  { 0 } )  C_  { 0 }
265263, 264eqsstr3i 3475 . . . . . . 7  |-  ( ran 
P  \  ( ran  P 
\  { 0 } ) )  C_  { 0 }
266265sseli 3440 . . . . . 6  |-  ( w  e.  ( ran  P  \  ( ran  P  \  { 0 } ) )  ->  w  e.  { 0 } )
267 elsni 4005 . . . . . . . . 9  |-  ( w  e.  { 0 }  ->  w  =  0 )
268267adantl 472 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  w  e.  { 0 } )  ->  w  =  0 )
269268oveq1d 6330 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  w  e.  { 0 } )  ->  (
w  x.  ( ( w  -  z ) I z ) )  =  ( 0  x.  ( ( w  -  z ) I z ) ) )
270110ad2antrr 737 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  w  e.  { 0 } )  ->  I : ( RR  X.  RR ) --> RR )
271268, 244syl6eqel 2548 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  w  e.  { 0 } )  ->  w  e.  RR )
27268adantr 471 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  w  e.  { 0 } )  ->  z  e.  RR )
273271, 272resubcld 10075 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  w  e.  { 0 } )  ->  (
w  -  z )  e.  RR )
274270, 273, 272fovrnd 6468 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  w  e.  { 0 } )  ->  (
( w  -  z
) I z )  e.  RR )
275274recnd 9695 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  w  e.  { 0 } )  ->  (
( w  -  z
) I z )  e.  CC )
276275mul02d 9857 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  w  e.  { 0 } )  ->  (
0  x.  ( ( w  -  z ) I z ) )  =  0 )
277269, 276eqtrd 2496 . . . . . 6  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  w  e.  { 0 } )  ->  (
w  x.  ( ( w  -  z ) I z ) )  =  0 )
278266, 277sylan2 481 . . . . 5  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  w  e.  ( ran  P 
\  ( ran  P  \  { 0 } ) ) )  ->  (
w  x.  ( ( w  -  z ) I z ) )  =  0 )
279261, 262, 278, 133fsumss 13840 . . . 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 2506 . . 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 13818 . 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 657 . . 3  |-  ( (
ph  /\  ( z  e.  ran  G  /\  y  e.  ran  F ) )  ->  ( ( y  +  z )  x.  ( y I z ) )  e.  CC )
2837, 5, 282fsumcom 13885 . 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 2502 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 189    /\ wa 375    = wceq 1455    e. wcel 1898   {cab 2448    =/= wne 2633   A.wral 2749   E.wrex 2750   _Vcvv 3057    \ cdif 3413    i^i cin 3415    C_ wss 3416   (/)c0 3743   ifcif 3893   {csn 3980   <.cop 3986   U_ciun 4292   class class class wbr 4416    |-> cmpt 4475    X. cxp 4851   `'ccnv 4852   dom cdm 4853   ran crn 4854    |` cres 4855   "cima 4856   Fun wfun 5595    Fn wfn 5596   -->wf 5597   -1-1->wf1 5598   -onto->wfo 5599   -1-1-onto->wf1o 5600   ` cfv 5601  (class class class)co 6315    |-> cmpt2 6317    oFcof 6556   Fincfn 7595   CCcc 9563   RRcr 9564   0cc0 9565    + caddc 9568    x. cmul 9570    - cmin 9886   sum_csu 13801   vol*covol 22462   volcvol 22464   S.1citg1 22622
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4529  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6610  ax-inf2 8172  ax-cnex 9621  ax-resscn 9622  ax-1cn 9623  ax-icn 9624  ax-addcl 9625  ax-addrcl 9626  ax-mulcl 9627  ax-mulrcl 9628  ax-mulcom 9629  ax-addass 9630  ax-mulass 9631  ax-distr 9632  ax-i2m1 9633  ax-1ne0 9634  ax-1rid 9635  ax-rnegex 9636  ax-rrecex 9637  ax-cnre 9638  ax-pre-lttri 9639  ax-pre-lttrn 9640  ax-pre-ltadd 9641  ax-pre-mulgt0 9642  ax-pre-sup 9643  ax-addf 9644
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-fal 1461  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-ral 2754  df-rex 2755  df-reu 2756  df-rmo 2757  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-int 4249  df-iun 4294  df-disj 4388  df-br 4417  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-se 4813  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6277  df-ov 6318  df-oprab 6319  df-mpt2 6320  df-of 6558  df-om 6720  df-1st 6820  df-2nd 6821  df-wrecs 7054  df-recs 7116  df-rdg 7154  df-1o 7208  df-2o 7209  df-oadd 7212  df-er 7389  df-map 7500  df-pm 7501  df-en 7596  df-dom 7597  df-sdom 7598  df-fin 7599  df-sup 7982  df-inf 7983  df-oi 8051  df-card 8399  df-cda 8624  df-pnf 9703  df-mnf 9704  df-xr 9705  df-ltxr 9706  df-le 9707  df-sub 9888  df-neg 9889  df-div 10298  df-nn 10638  df-2 10696  df-3 10697  df-n0 10899  df-z 10967  df-uz 11189  df-q 11294  df-rp 11332  df-xadd 11439  df-ioo 11668  df-ico 11670  df-icc 11671  df-fz 11814  df-fzo 11947  df-fl 12060  df-seq 12246  df-exp 12305  df-hash 12548  df-cj 13211  df-re 13212  df-im 13213  df-sqrt 13347  df-abs 13348  df-clim 13601  df-sum 13802  df-xmet 19012  df-met 19013  df-ovol 22465  df-vol 22467  df-mbf 22626  df-itg1 22627
This theorem is referenced by:  itg1addlem5  22707
  Copyright terms: Public domain W3C validator