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Theorem itg1addlem4 21182
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 21178 . . . 4  |-  ( ph  ->  ( F  oF  +  G )  e. 
dom  S.1 )
4 i1frn 21160 . . . . . . . 8  |-  ( F  e.  dom  S.1  ->  ran 
F  e.  Fin )
51, 4syl 16 . . . . . . 7  |-  ( ph  ->  ran  F  e.  Fin )
6 i1frn 21160 . . . . . . . 8  |-  ( G  e.  dom  S.1  ->  ran 
G  e.  Fin )
72, 6syl 16 . . . . . . 7  |-  ( ph  ->  ran  G  e.  Fin )
8 xpfi 7588 . . . . . . 7  |-  ( ( ran  F  e.  Fin  /\ 
ran  G  e.  Fin )  ->  ( ran  F  X.  ran  G )  e. 
Fin )
95, 7, 8syl2anc 661 . . . . . 6  |-  ( ph  ->  ( ran  F  X.  ran  G )  e.  Fin )
10 ax-addf 9366 . . . . . . . . . 10  |-  +  :
( CC  X.  CC )
--> CC
11 ffn 5564 . . . . . . . . . 10  |-  (  +  : ( CC  X.  CC ) --> CC  ->  +  Fn  ( CC  X.  CC ) )
1210, 11ax-mp 5 . . . . . . . . 9  |-  +  Fn  ( CC  X.  CC )
13 i1ff 21159 . . . . . . . . . . . . 13  |-  ( F  e.  dom  S.1  ->  F : RR --> RR )
141, 13syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  F : RR --> RR )
15 frn 5570 . . . . . . . . . . . 12  |-  ( F : RR --> RR  ->  ran 
F  C_  RR )
1614, 15syl 16 . . . . . . . . . . 11  |-  ( ph  ->  ran  F  C_  RR )
17 ax-resscn 9344 . . . . . . . . . . 11  |-  RR  C_  CC
1816, 17syl6ss 3373 . . . . . . . . . 10  |-  ( ph  ->  ran  F  C_  CC )
19 i1ff 21159 . . . . . . . . . . . . 13  |-  ( G  e.  dom  S.1  ->  G : RR --> RR )
202, 19syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  G : RR --> RR )
21 frn 5570 . . . . . . . . . . . 12  |-  ( G : RR --> RR  ->  ran 
G  C_  RR )
2220, 21syl 16 . . . . . . . . . . 11  |-  ( ph  ->  ran  G  C_  RR )
2322, 17syl6ss 3373 . . . . . . . . . 10  |-  ( ph  ->  ran  G  C_  CC )
24 xpss12 4950 . . . . . . . . . 10  |-  ( ( ran  F  C_  CC  /\ 
ran  G  C_  CC )  ->  ( ran  F  X.  ran  G )  C_  ( CC  X.  CC ) )
2518, 23, 24syl2anc 661 . . . . . . . . 9  |-  ( ph  ->  ( ran  F  X.  ran  G )  C_  ( CC  X.  CC ) )
26 fnssres 5529 . . . . . . . . 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 663 . . . . . . . 8  |-  ( ph  ->  (  +  |`  ( ran  F  X.  ran  G
) )  Fn  ( ran  F  X.  ran  G
) )
28 itg1add.4 . . . . . . . . 9  |-  P  =  (  +  |`  ( ran  F  X.  ran  G
) )
2928fneq1i 5510 . . . . . . . 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 5631 . . . . . . 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 7602 . . . . . 6  |-  ( ( ( ran  F  X.  ran  G )  e.  Fin  /\  P : ( ran 
F  X.  ran  G
) -onto-> ran  P )  ->  ran  P  e.  Fin )
349, 32, 33syl2anc 661 . . . . 5  |-  ( ph  ->  ran  P  e.  Fin )
35 difss 3488 . . . . 5  |-  ( ran 
P  \  { 0 } )  C_  ran  P
36 ssfi 7538 . . . . 5  |-  ( ( ran  P  e.  Fin  /\  ( ran  P  \  { 0 } ) 
C_  ran  P )  ->  ( ran  P  \  { 0 } )  e.  Fin )
3734, 35, 36sylancl 662 . . . 4  |-  ( ph  ->  ( ran  P  \  { 0 } )  e.  Fin )
38 opelxpi 4876 . . . . . . . . . 10  |-  ( ( x  e.  ran  F  /\  y  e.  ran  G )  ->  <. x ,  y >.  e.  ( ran  F  X.  ran  G
) )
39 ffun 5566 . . . . . . . . . . . 12  |-  (  +  : ( CC  X.  CC ) --> CC  ->  Fun  +  )
4010, 39ax-mp 5 . . . . . . . . . . 11  |-  Fun  +
4110fdmi 5569 . . . . . . . . . . . 12  |-  dom  +  =  ( CC  X.  CC )
4225, 41syl6sseqr 3408 . . . . . . . . . . 11  |-  ( ph  ->  ( ran  F  X.  ran  G )  C_  dom  +  )
43 funfvima2 5958 . . . . . . . . . . 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 663 . . . . . . . . . 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 429 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ran  F  /\  y  e.  ran  G ) )  ->  (  +  `  <. x ,  y >.
)  e.  (  + 
" ( ran  F  X.  ran  G ) ) )
47 df-ov 6099 . . . . . . . 8  |-  ( x  +  y )  =  (  +  `  <. x ,  y >. )
4828rneqi 5071 . . . . . . . . 9  |-  ran  P  =  ran  (  +  |`  ( ran  F  X.  ran  G
) )
49 df-ima 4858 . . . . . . . . 9  |-  (  + 
" ( ran  F  X.  ran  G ) )  =  ran  (  +  |`  ( ran  F  X.  ran  G ) )
5048, 49eqtr4i 2466 . . . . . . . 8  |-  ran  P  =  (  +  " ( ran  F  X.  ran  G
) )
5146, 47, 503eltr4g 2526 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ran  F  /\  y  e.  ran  G ) )  ->  ( x  +  y )  e.  ran  P )
52 ffn 5564 . . . . . . . . 9  |-  ( F : RR --> RR  ->  F  Fn  RR )
5314, 52syl 16 . . . . . . . 8  |-  ( ph  ->  F  Fn  RR )
54 dffn3 5571 . . . . . . . 8  |-  ( F  Fn  RR  <->  F : RR
--> ran  F )
5553, 54sylib 196 . . . . . . 7  |-  ( ph  ->  F : RR --> ran  F
)
56 ffn 5564 . . . . . . . . 9  |-  ( G : RR --> RR  ->  G  Fn  RR )
5720, 56syl 16 . . . . . . . 8  |-  ( ph  ->  G  Fn  RR )
58 dffn3 5571 . . . . . . . 8  |-  ( G  Fn  RR  <->  G : RR
--> ran  G )
5957, 58sylib 196 . . . . . . 7  |-  ( ph  ->  G : RR --> ran  G
)
60 reex 9378 . . . . . . . 8  |-  RR  e.  _V
6160a1i 11 . . . . . . 7  |-  ( ph  ->  RR  e.  _V )
62 inidm 3564 . . . . . . 7  |-  ( RR 
i^i  RR )  =  RR
6351, 55, 59, 61, 61, 62off 6339 . . . . . 6  |-  ( ph  ->  ( F  oF  +  G ) : RR --> ran  P )
64 frn 5570 . . . . . 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 3497 . . . 4  |-  ( ph  ->  ( ran  ( F  oF  +  G
)  \  { 0 } )  C_  ( ran  P  \  { 0 } ) )
6716sselda 3361 . . . . . . . . . 10  |-  ( (
ph  /\  y  e.  ran  F )  ->  y  e.  RR )
6822sselda 3361 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  ran  G )  ->  z  e.  RR )
6967, 68anim12dan 833 . . . . . . . . 9  |-  ( (
ph  /\  ( y  e.  ran  F  /\  z  e.  ran  G ) )  ->  ( y  e.  RR  /\  z  e.  RR ) )
70 readdcl 9370 . . . . . . . . 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 2813 . . . . . . 7  |-  ( ph  ->  A. y  e.  ran  F A. z  e.  ran  G ( y  +  z )  e.  RR )
73 funimassov 6245 . . . . . . . 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 663 . . . . . . 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 3405 . . . . 5  |-  ( ph  ->  ran  P  C_  RR )
7776ssdifd 3497 . . . 4  |-  ( ph  ->  ( ran  P  \  { 0 } ) 
C_  ( RR  \  { 0 } ) )
78 itg1val2 21167 . . . 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 1220 . . 3  |-  ( ph  ->  ( S.1 `  ( F  oF  +  G
) )  =  sum_ w  e.  ( ran  P  \  { 0 } ) ( w  x.  ( vol `  ( `' ( F  oF  +  G ) " {
w } ) ) ) )
8020adantr 465 . . . . . . . 8  |-  ( (
ph  /\  w  e.  ( ran  P  \  {
0 } ) )  ->  G : RR --> RR )
817adantr 465 . . . . . . . 8  |-  ( (
ph  /\  w  e.  ( ran  P  \  {
0 } ) )  ->  ran  G  e.  Fin )
82 inss2 3576 . . . . . . . . 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 21161 . . . . . . . . . . 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 21161 . . . . . . . . . . 11  |-  ( G  e.  dom  S.1  ->  ( `' G " { z } )  e.  dom  vol )
872, 86syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( `' G " { z } )  e.  dom  vol )
88 inmbl 21028 . . . . . . . . . 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 661 . . . . . . . . 9  |-  ( ph  ->  ( ( `' F " { ( w  -  z ) } )  i^i  ( `' G " { z } ) )  e.  dom  vol )
9089ad2antrr 725 . . . . . . . 8  |-  ( ( ( ph  /\  w  e.  ( ran  P  \  { 0 } ) )  /\  z  e. 
ran  G )  -> 
( ( `' F " { ( w  -  z ) } )  i^i  ( `' G " { z } ) )  e.  dom  vol )
9135, 76syl5ss 3372 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ran  P  \  { 0 } ) 
C_  RR )
9291sselda 3361 . . . . . . . . . . . 12  |-  ( (
ph  /\  w  e.  ( ran  P  \  {
0 } ) )  ->  w  e.  RR )
9392adantr 465 . . . . . . . . . . 11  |-  ( ( ( ph  /\  w  e.  ( ran  P  \  { 0 } ) )  /\  z  e. 
ran  G )  ->  w  e.  RR )
9468adantlr 714 . . . . . . . . . . 11  |-  ( ( ( ph  /\  w  e.  ( ran  P  \  { 0 } ) )  /\  z  e. 
ran  G )  -> 
z  e.  RR )
9593, 94resubcld 9781 . . . . . . . . . 10  |-  ( ( ( ph  /\  w  e.  ( ran  P  \  { 0 } ) )  /\  z  e. 
ran  G )  -> 
( w  -  z
)  e.  RR )
9693recnd 9417 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  w  e.  ( ran  P  \  { 0 } ) )  /\  z  e. 
ran  G )  ->  w  e.  CC )
9794recnd 9417 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  w  e.  ( ran  P  \  { 0 } ) )  /\  z  e. 
ran  G )  -> 
z  e.  CC )
9896, 97npcand 9728 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  w  e.  ( ran  P  \  { 0 } ) )  /\  z  e. 
ran  G )  -> 
( ( w  -  z )  +  z )  =  w )
99 eldifsni 4006 . . . . . . . . . . . . 13  |-  ( w  e.  ( ran  P  \  { 0 } )  ->  w  =/=  0
)
10099ad2antlr 726 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  w  e.  ( ran  P  \  { 0 } ) )  /\  z  e. 
ran  G )  ->  w  =/=  0 )
10198, 100eqnetrd 2631 . . . . . . . . . . 11  |-  ( ( ( ph  /\  w  e.  ( ran  P  \  { 0 } ) )  /\  z  e. 
ran  G )  -> 
( ( w  -  z )  +  z )  =/=  0 )
102 oveq12 6105 . . . . . . . . . . . . 13  |-  ( ( ( w  -  z
)  =  0  /\  z  =  0 )  ->  ( ( w  -  z )  +  z )  =  ( 0  +  0 ) )
103 00id 9549 . . . . . . . . . . . . 13  |-  ( 0  +  0 )  =  0
104102, 103syl6eq 2491 . . . . . . . . . . . 12  |-  ( ( ( w  -  z
)  =  0  /\  z  =  0 )  ->  ( ( w  -  z )  +  z )  =  0 )
105104necon3ai 2656 . . . . . . . . . . 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 21181 . . . . . . . . . 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 1217 . . . . . . . . 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 21180 . . . . . . . . . . 11  |-  ( ph  ->  I : ( RR 
X.  RR ) --> RR )
111110ad2antrr 725 . . . . . . . . . 10  |-  ( ( ( ph  /\  w  e.  ( ran  P  \  { 0 } ) )  /\  z  e. 
ran  G )  ->  I : ( RR  X.  RR ) --> RR )
112111, 95, 94fovrnd 6240 . . . . . . . . 9  |-  ( ( ( ph  /\  w  e.  ( ran  P  \  { 0 } ) )  /\  z  e. 
ran  G )  -> 
( ( w  -  z ) I z )  e.  RR )
113109, 112eqeltrrd 2518 . . . . . . . 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 21175 . . . . . . 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 9417 . . . . . . . . 9  |-  ( (
ph  /\  w  e.  ( ran  P  \  {
0 } ) )  ->  w  e.  CC )
1161, 2i1faddlem 21176 . . . . . . . . 9  |-  ( (
ph  /\  w  e.  CC )  ->  ( `' ( F  oF  +  G ) " { w } )  =  U_ z  e. 
ran  G ( ( `' F " { ( w  -  z ) } )  i^i  ( `' G " { z } ) ) )
117115, 116syldan 470 . . . . . . . 8  |-  ( (
ph  /\  w  e.  ( ran  P  \  {
0 } ) )  ->  ( `' ( F  oF  +  G ) " {
w } )  = 
U_ z  e.  ran  G ( ( `' F " { ( w  -  z ) } )  i^i  ( `' G " { z } ) ) )
118117fveq2d 5700 . . . . . . 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 13185 . . . . . . 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 2485 . . . . . 6  |-  ( (
ph  /\  w  e.  ( ran  P  \  {
0 } ) )  ->  ( vol `  ( `' ( F  oF  +  G ) " { w } ) )  =  sum_ z  e.  ran  G ( ( w  -  z ) I z ) )
121120oveq2d 6112 . . . . 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 9417 . . . . . 6  |-  ( ( ( ph  /\  w  e.  ( ran  P  \  { 0 } ) )  /\  z  e. 
ran  G )  -> 
( ( w  -  z ) I z )  e.  CC )
12381, 115, 122fsummulc2 13256 . . . . 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 2475 . . . 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 13185 . . 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 9411 . . . . 5  |-  ( ( ( ph  /\  w  e.  ( ran  P  \  { 0 } ) )  /\  z  e. 
ran  G )  -> 
( w  x.  (
( w  -  z
) I z ) )  e.  CC )
127126anasss 647 . . . 4  |-  ( (
ph  /\  ( w  e.  ( ran  P  \  { 0 } )  /\  z  e.  ran  G ) )  ->  (
w  x.  ( ( w  -  z ) I z ) )  e.  CC )
12837, 7, 127fsumcom 13247 . . 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 2479 . 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 6103 . . . . . . 7  |-  ( y  =  ( w  -  z )  ->  (
y  +  z )  =  ( ( w  -  z )  +  z ) )
131 oveq1 6103 . . . . . . 7  |-  ( y  =  ( w  -  z )  ->  (
y I z )  =  ( ( w  -  z ) I z ) )
132130, 131oveq12d 6114 . . . . . 6  |-  ( y  =  ( w  -  z )  ->  (
( y  +  z )  x.  ( y I z ) )  =  ( ( ( w  -  z )  +  z )  x.  ( ( w  -  z ) I z ) ) )
13334adantr 465 . . . . . 6  |-  ( (
ph  /\  z  e.  ran  G )  ->  ran  P  e.  Fin )
13476adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  z  e.  ran  G )  ->  ran  P 
C_  RR )
135134sselda 3361 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  v  e.  ran  P )  ->  v  e.  RR )
13668adantr 465 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  v  e.  ran  P )  ->  z  e.  RR )
137135, 136resubcld 9781 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  v  e.  ran  P )  ->  ( v  -  z )  e.  RR )
138137ex 434 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ran  G )  ->  (
v  e.  ran  P  ->  ( v  -  z
)  e.  RR ) )
139135recnd 9417 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  v  e.  ran  P )  ->  v  e.  CC )
140139adantrr 716 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( v  e.  ran  P  /\  y  e.  ran  P ) )  ->  v  e.  CC )
14176sselda 3361 . . . . . . . . . . . 12  |-  ( (
ph  /\  y  e.  ran  P )  ->  y  e.  RR )
142141ad2ant2rl 748 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( v  e.  ran  P  /\  y  e.  ran  P ) )  ->  y  e.  RR )
143142recnd 9417 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( v  e.  ran  P  /\  y  e.  ran  P ) )  ->  y  e.  CC )
14468recnd 9417 . . . . . . . . . . 11  |-  ( (
ph  /\  z  e.  ran  G )  ->  z  e.  CC )
145144adantr 465 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( v  e.  ran  P  /\  y  e.  ran  P ) )  ->  z  e.  CC )
146140, 143, 145subcan2ad 9769 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( v  e.  ran  P  /\  y  e.  ran  P ) )  ->  (
( v  -  z
)  =  ( y  -  z )  <->  v  =  y ) )
147146ex 434 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ran  G )  ->  (
( v  e.  ran  P  /\  y  e.  ran  P )  ->  ( (
v  -  z )  =  ( y  -  z )  <->  v  =  y ) ) )
148138, 147dom2lem 7354 . . . . . . 7  |-  ( (
ph  /\  z  e.  ran  G )  ->  (
v  e.  ran  P  |->  ( v  -  z
) ) : ran  P
-1-1-> RR )
149 f1f1orn 5657 . . . . . . 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 6103 . . . . . . . 8  |-  ( v  =  w  ->  (
v  -  z )  =  ( w  -  z ) )
152 eqid 2443 . . . . . . . 8  |-  ( v  e.  ran  P  |->  ( v  -  z ) )  =  ( v  e.  ran  P  |->  ( v  -  z ) )
153 ovex 6121 . . . . . . . 8  |-  ( w  -  z )  e. 
_V
154151, 152, 153fvmpt 5779 . . . . . . 7  |-  ( w  e.  ran  P  -> 
( ( v  e. 
ran  P  |->  ( v  -  z ) ) `
 w )  =  ( w  -  z
) )
155154adantl 466 . . . . . 6  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  w  e.  ran  P )  ->  ( ( v  e.  ran  P  |->  ( v  -  z ) ) `  w )  =  ( w  -  z ) )
156 f1f 5611 . . . . . . . . . . 11  |-  ( ( v  e.  ran  P  |->  ( v  -  z
) ) : ran  P
-1-1-> RR  ->  ( v  e.  ran  P  |->  ( v  -  z ) ) : ran  P --> RR )
157 frn 5570 . . . . . . . . . . 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 3361 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  ( v  e.  ran  P  |->  ( v  -  z ) ) )  ->  y  e.  RR )
16068adantr 465 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  ( v  e.  ran  P  |->  ( v  -  z ) ) )  ->  z  e.  RR )
161159, 160readdcld 9418 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  ( v  e.  ran  P  |->  ( v  -  z ) ) )  ->  (
y  +  z )  e.  RR )
162110ad2antrr 725 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  ( v  e.  ran  P  |->  ( v  -  z ) ) )  ->  I : ( RR  X.  RR ) --> RR )
163162, 159, 160fovrnd 6240 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  ( v  e.  ran  P  |->  ( v  -  z ) ) )  ->  (
y I z )  e.  RR )
164161, 163remulcld 9419 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  ( v  e.  ran  P  |->  ( v  -  z ) ) )  ->  (
( y  +  z )  x.  ( y I z ) )  e.  RR )
165164recnd 9417 . . . . . 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 13205 . . . . 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 3361 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  w  e.  ran  P )  ->  w  e.  RR )
168167recnd 9417 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  w  e.  ran  P )  ->  w  e.  CC )
169144adantr 465 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  w  e.  ran  P )  ->  z  e.  CC )
170168, 169npcand 9728 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  w  e.  ran  P )  ->  ( ( w  -  z )  +  z )  =  w )
171170oveq1d 6111 . . . . . 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 13185 . . . . 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 2475 . . . 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 725 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  ( ran  F  X.  ran  G )  C_  dom  +  )
175 simpr 461 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  y  e.  ran  F )
176 simplr 754 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  z  e.  ran  G )
177 opelxpi 4876 . . . . . . . . . . . 12  |-  ( ( y  e.  ran  F  /\  z  e.  ran  G )  ->  <. y ,  z >.  e.  ( ran  F  X.  ran  G
) )
178175, 176, 177syl2anc 661 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  <. y ,  z
>.  e.  ( ran  F  X.  ran  G ) )
179 funfvima2 5958 . . . . . . . . . . . 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 670 . . . . . . . . . . 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 6099 . . . . . . . . . 10  |-  ( y  +  z )  =  (  +  `  <. y ,  z >. )
183181, 182, 503eltr4g 2526 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  ( y  +  z )  e.  ran  P )
18467adantlr 714 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  y  e.  RR )
185184recnd 9417 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  y  e.  CC )
186144adantr 465 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  z  e.  CC )
187185, 186pncand 9725 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  ( ( y  +  z )  -  z )  =  y )
188187eqcomd 2448 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  y  =  ( ( y  +  z )  -  z ) )
189 oveq1 6103 . . . . . . . . . . 11  |-  ( v  =  ( y  +  z )  ->  (
v  -  z )  =  ( ( y  +  z )  -  z ) )
190189eqeq2d 2454 . . . . . . . . . 10  |-  ( v  =  ( y  +  z )  ->  (
y  =  ( v  -  z )  <->  y  =  ( ( y  +  z )  -  z
) ) )
191190rspcev 3078 . . . . . . . . 9  |-  ( ( ( y  +  z )  e.  ran  P  /\  y  =  (
( y  +  z )  -  z ) )  ->  E. v  e.  ran  P  y  =  ( v  -  z
) )
192183, 188, 191syl2anc 661 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  E. v  e.  ran  P  y  =  ( v  -  z ) )
193192ralrimiva 2804 . . . . . . 7  |-  ( (
ph  /\  z  e.  ran  G )  ->  A. y  e.  ran  F E. v  e.  ran  P  y  =  ( v  -  z
) )
194 ssabral 3428 . . . . . . 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 5090 . . . . . 6  |-  ran  (
v  e.  ran  P  |->  ( v  -  z
) )  =  {
y  |  E. v  e.  ran  P  y  =  ( v  -  z
) }
197195, 196syl6sseqr 3408 . . . . 5  |-  ( (
ph  /\  z  e.  ran  G )  ->  ran  F 
C_  ran  ( v  e.  ran  P  |->  ( v  -  z ) ) )
19868adantr 465 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  z  e.  RR )
199184, 198readdcld 9418 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  ( y  +  z )  e.  RR )
200110ad2antrr 725 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  I : ( RR  X.  RR ) --> RR )
201200, 184, 198fovrnd 6240 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  ( y I z )  e.  RR )
202199, 201remulcld 9419 . . . . . 6  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  ( ( y  +  z )  x.  ( y I z ) )  e.  RR )
203202recnd 9417 . . . . 5  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  ( ( y  +  z )  x.  ( y I z ) )  e.  CC )
204158ssdifd 3497 . . . . . . 7  |-  ( (
ph  /\  z  e.  ran  G )  ->  ( ran  ( v  e.  ran  P 
|->  ( v  -  z
) )  \  ran  F )  C_  ( RR  \  ran  F ) )
205204sselda 3361 . . . . . 6  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ( ran  ( v  e.  ran  P 
|->  ( v  -  z
) )  \  ran  F ) )  ->  y  e.  ( RR  \  ran  F ) )
206 eldifi 3483 . . . . . . . . . . . . 13  |-  ( y  e.  ( RR  \  ran  F )  ->  y  e.  RR )
207206ad2antrl 727 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( y  e.  ( RR  \  ran  F
)  /\  -.  (
y  =  0  /\  z  =  0 ) ) )  ->  y  e.  RR )
20868adantr 465 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( y  e.  ( RR  \  ran  F
)  /\  -.  (
y  =  0  /\  z  =  0 ) ) )  ->  z  e.  RR )
209 simprr 756 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( y  e.  ( RR  \  ran  F
)  /\  -.  (
y  =  0  /\  z  =  0 ) ) )  ->  -.  ( y  =  0  /\  z  =  0 ) )
2101, 2, 107itg1addlem3 21181 . . . . . . . . . . . 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 1217 . . . . . . . . . . 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 3575 . . . . . . . . . . . . . . 15  |-  ( ( `' F " { y } )  i^i  ( `' G " { z } ) )  C_  ( `' F " { y } )
213 eldifn 3484 . . . . . . . . . . . . . . . . . . 19  |-  ( y  e.  ( RR  \  ran  F )  ->  -.  y  e.  ran  F )
214213ad2antrl 727 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( y  e.  ( RR  \  ran  F
)  /\  -.  (
y  =  0  /\  z  =  0 ) ) )  ->  -.  y  e.  ran  F )
215 vex 2980 . . . . . . . . . . . . . . . . . . . 20  |-  y  e. 
_V
216 vex 2980 . . . . . . . . . . . . . . . . . . . . 21  |-  v  e. 
_V
217216eliniseg 5203 . . . . . . . . . . . . . . . . . . . 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 5075 . . . . . . . . . . . . . . . . . . 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 3367 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( y  e.  ( RR  \  ran  F
)  /\  -.  (
y  =  0  /\  z  =  0 ) ) )  ->  ( `' F " { y } )  C_  (/) )
224212, 223syl5ss 3372 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( y  e.  ( RR  \  ran  F
)  /\  -.  (
y  =  0  /\  z  =  0 ) ) )  ->  (
( `' F " { y } )  i^i  ( `' G " { z } ) )  C_  (/) )
225 ss0 3673 . . . . . . . . . . . . . 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 5700 . . . . . . . . . . . 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 21026 . . . . . . . . . . . . . 14  |-  (/)  e.  dom  vol
229 mblvol 21018 . . . . . . . . . . . . . 14  |-  ( (/)  e.  dom  vol  ->  ( vol `  (/) )  =  ( vol* `  (/) ) )
230228, 229ax-mp 5 . . . . . . . . . . . . 13  |-  ( vol `  (/) )  =  ( vol* `  (/) )
231 ovol0 20981 . . . . . . . . . . . . 13  |-  ( vol* `  (/) )  =  0
232230, 231eqtri 2463 . . . . . . . . . . . 12  |-  ( vol `  (/) )  =  0
233227, 232syl6eq 2491 . . . . . . . . . . 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 2475 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( y  e.  ( RR  \  ran  F
)  /\  -.  (
y  =  0  /\  z  =  0 ) ) )  ->  (
y I z )  =  0 )
235234oveq2d 6112 . . . . . . . . 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 9418 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( y  e.  ( RR  \  ran  F
)  /\  -.  (
y  =  0  /\  z  =  0 ) ) )  ->  (
y  +  z )  e.  RR )
237236recnd 9417 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( y  e.  ( RR  \  ran  F
)  /\  -.  (
y  =  0  /\  z  =  0 ) ) )  ->  (
y  +  z )  e.  CC )
238237mul01d 9573 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( y  e.  ( RR  \  ran  F
)  /\  -.  (
y  =  0  /\  z  =  0 ) ) )  ->  (
( y  +  z )  x.  0 )  =  0 )
239235, 238eqtrd 2475 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( y  e.  ( RR  \  ran  F
)  /\  -.  (
y  =  0  /\  z  =  0 ) ) )  ->  (
( y  +  z )  x.  ( y I z ) )  =  0 )
240239expr 615 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ( RR  \  ran  F ) )  ->  ( -.  (
y  =  0  /\  z  =  0 )  ->  ( ( y  +  z )  x.  ( y I z ) )  =  0 ) )
241 oveq12 6105 . . . . . . . . . 10  |-  ( ( y  =  0  /\  z  =  0 )  ->  ( y  +  z )  =  ( 0  +  0 ) )
242241, 103syl6eq 2491 . . . . . . . . 9  |-  ( ( y  =  0  /\  z  =  0 )  ->  ( y  +  z )  =  0 )
243 oveq12 6105 . . . . . . . . . 10  |-  ( ( y  =  0  /\  z  =  0 )  ->  ( y I z )  =  ( 0 I 0 ) )
244 0re 9391 . . . . . . . . . . 11  |-  0  e.  RR
245 iftrue 3802 . . . . . . . . . . . 12  |-  ( ( i  =  0  /\  j  =  0 )  ->  if ( ( i  =  0  /\  j  =  0 ) ,  0 ,  ( vol `  ( ( `' F " { i } )  i^i  ( `' G " { j } ) ) ) )  =  0 )
246 c0ex 9385 . . . . . . . . . . . 12  |-  0  e.  _V
247245, 107, 246ovmpt2a 6226 . . . . . . . . . . 11  |-  ( ( 0  e.  RR  /\  0  e.  RR )  ->  ( 0 I 0 )  =  0 )
248244, 244, 247mp2an 672 . . . . . . . . . 10  |-  ( 0 I 0 )  =  0
249243, 248syl6eq 2491 . . . . . . . . 9  |-  ( ( y  =  0  /\  z  =  0 )  ->  ( y I z )  =  0 )
250242, 249oveq12d 6114 . . . . . . . 8  |-  ( ( y  =  0  /\  z  =  0 )  ->  ( ( y  +  z )  x.  ( y I z ) )  =  ( 0  x.  0 ) )
251 0cn 9383 . . . . . . . . 9  |-  0  e.  CC
252251mul01i 9564 . . . . . . . 8  |-  ( 0  x.  0 )  =  0
253250, 252syl6eq 2491 . . . . . . 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 470 . . . . 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 5653 . . . . . . 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 7602 . . . . . 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 661 . . . . 5  |-  ( (
ph  /\  z  e.  ran  G )  ->  ran  ( v  e.  ran  P 
|->  ( v  -  z
) )  e.  Fin )
260197, 203, 255, 259fsumss 13207 . . . 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 3595 . . . . . . . 8  |-  ( ran 
P  i^i  { 0 } )  =  ( ran  P  \  ( ran  P  \  { 0 } ) )
264 inss2 3576 . . . . . . . 8  |-  ( ran 
P  i^i  { 0 } )  C_  { 0 }
265263, 264eqsstr3i 3392 . . . . . . 7  |-  ( ran 
P  \  ( ran  P 
\  { 0 } ) )  C_  { 0 }
266265sseli 3357 . . . . . 6  |-  ( w  e.  ( ran  P  \  ( ran  P  \  { 0 } ) )  ->  w  e.  { 0 } )
267 elsni 3907 . . . . . . . . 9  |-  ( w  e.  { 0 }  ->  w  =  0 )
268267adantl 466 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  w  e.  { 0 } )  ->  w  =  0 )
269268oveq1d 6111 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  w  e.  { 0 } )  ->  (
w  x.  ( ( w  -  z ) I z ) )  =  ( 0  x.  ( ( w  -  z ) I z ) ) )
270110ad2antrr 725 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  w  e.  { 0 } )  ->  I : ( RR  X.  RR ) --> RR )
271268, 244syl6eqel 2531 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  w  e.  { 0 } )  ->  w  e.  RR )
27268adantr 465 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  w  e.  { 0 } )  ->  z  e.  RR )
273271, 272resubcld 9781 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  w  e.  { 0 } )  ->  (
w  -  z )  e.  RR )
274270, 273, 272fovrnd 6240 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  w  e.  { 0 } )  ->  (
( w  -  z
) I z )  e.  RR )
275274recnd 9417 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  w  e.  { 0 } )  ->  (
( w  -  z
) I z )  e.  CC )
276275mul02d 9572 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  w  e.  { 0 } )  ->  (
0  x.  ( ( w  -  z ) I z ) )  =  0 )
277269, 276eqtrd 2475 . . . . . 6  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  w  e.  { 0 } )  ->  (
w  x.  ( ( w  -  z ) I z ) )  =  0 )
278266, 277sylan2 474 . . . . 5  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  w  e.  ( ran  P 
\  ( ran  P  \  { 0 } ) ) )  ->  (
w  x.  ( ( w  -  z ) I z ) )  =  0 )
279261, 262, 278, 133fsumss 13207 . . . 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 2485 . . 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 13185 . 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 647 . . 3  |-  ( (
ph  /\  ( z  e.  ran  G  /\  y  e.  ran  F ) )  ->  ( ( y  +  z )  x.  ( y I z ) )  e.  CC )
2837, 5, 282fsumcom 13247 . 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 2481 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 369    = wceq 1369    e. wcel 1756   {cab 2429    =/= wne 2611   A.wral 2720   E.wrex 2721   _Vcvv 2977    \ cdif 3330    i^i cin 3332    C_ wss 3333   (/)c0 3642   ifcif 3796   {csn 3882   <.cop 3888   U_ciun 4176   class class class wbr 4297    e. cmpt 4355    X. cxp 4843   `'ccnv 4844   dom cdm 4845   ran crn 4846    |` cres 4847   "cima 4848   Fun wfun 5417    Fn wfn 5418   -->wf 5419   -1-1->wf1 5420   -onto->wfo 5421   -1-1-onto->wf1o 5422   ` cfv 5423  (class class class)co 6096    e. cmpt2 6098    oFcof 6323   Fincfn 7315   CCcc 9285   RRcr 9286   0cc0 9287    + caddc 9290    x. cmul 9292    - cmin 9600   sum_csu 13168   vol*covol 20951   volcvol 20952   S.1citg1 21100
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-inf2 7852  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364  ax-pre-sup 9365  ax-addf 9366
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-disj 4268  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-se 4685  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-isom 5432  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-of 6325  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-1o 6925  df-2o 6926  df-oadd 6929  df-er 7106  df-map 7221  df-pm 7222  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-sup 7696  df-oi 7729  df-card 8114  df-cda 8342  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-div 9999  df-nn 10328  df-2 10385  df-3 10386  df-n0 10585  df-z 10652  df-uz 10867  df-q 10959  df-rp 10997  df-xadd 11095  df-ioo 11309  df-ico 11311  df-icc 11312  df-fz 11443  df-fzo 11554  df-fl 11647  df-seq 11812  df-exp 11871  df-hash 12109  df-cj 12593  df-re 12594  df-im 12595  df-sqr 12729  df-abs 12730  df-clim 12971  df-sum 13169  df-xmet 17815  df-met 17816  df-ovol 20953  df-vol 20954  df-mbf 21104  df-itg1 21105
This theorem is referenced by:  itg1addlem5  21183
  Copyright terms: Public domain W3C validator