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Theorem itg1addlem4 19544
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  o F  +  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 19540 . . . 4  |-  ( ph  ->  ( F  o F  +  G )  e. 
dom  S.1 )
4 i1frn 19522 . . . . . . . 8  |-  ( F  e.  dom  S.1  ->  ran 
F  e.  Fin )
51, 4syl 16 . . . . . . 7  |-  ( ph  ->  ran  F  e.  Fin )
6 i1frn 19522 . . . . . . . 8  |-  ( G  e.  dom  S.1  ->  ran 
G  e.  Fin )
72, 6syl 16 . . . . . . 7  |-  ( ph  ->  ran  G  e.  Fin )
8 xpfi 7337 . . . . . . 7  |-  ( ( ran  F  e.  Fin  /\ 
ran  G  e.  Fin )  ->  ( ran  F  X.  ran  G )  e. 
Fin )
95, 7, 8syl2anc 643 . . . . . 6  |-  ( ph  ->  ( ran  F  X.  ran  G )  e.  Fin )
10 ax-addf 9025 . . . . . . . . . 10  |-  +  :
( CC  X.  CC )
--> CC
11 ffn 5550 . . . . . . . . . 10  |-  (  +  : ( CC  X.  CC ) --> CC  ->  +  Fn  ( CC  X.  CC ) )
1210, 11ax-mp 8 . . . . . . . . 9  |-  +  Fn  ( CC  X.  CC )
13 i1ff 19521 . . . . . . . . . . . . 13  |-  ( F  e.  dom  S.1  ->  F : RR --> RR )
141, 13syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  F : RR --> RR )
15 frn 5556 . . . . . . . . . . . 12  |-  ( F : RR --> RR  ->  ran 
F  C_  RR )
1614, 15syl 16 . . . . . . . . . . 11  |-  ( ph  ->  ran  F  C_  RR )
17 ax-resscn 9003 . . . . . . . . . . 11  |-  RR  C_  CC
1816, 17syl6ss 3320 . . . . . . . . . 10  |-  ( ph  ->  ran  F  C_  CC )
19 i1ff 19521 . . . . . . . . . . . . 13  |-  ( G  e.  dom  S.1  ->  G : RR --> RR )
202, 19syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  G : RR --> RR )
21 frn 5556 . . . . . . . . . . . 12  |-  ( G : RR --> RR  ->  ran 
G  C_  RR )
2220, 21syl 16 . . . . . . . . . . 11  |-  ( ph  ->  ran  G  C_  RR )
2322, 17syl6ss 3320 . . . . . . . . . 10  |-  ( ph  ->  ran  G  C_  CC )
24 xpss12 4940 . . . . . . . . . 10  |-  ( ( ran  F  C_  CC  /\ 
ran  G  C_  CC )  ->  ( ran  F  X.  ran  G )  C_  ( CC  X.  CC ) )
2518, 23, 24syl2anc 643 . . . . . . . . 9  |-  ( ph  ->  ( ran  F  X.  ran  G )  C_  ( CC  X.  CC ) )
26 fnssres 5517 . . . . . . . . 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 645 . . . . . . . 8  |-  ( ph  ->  (  +  |`  ( ran  F  X.  ran  G
) )  Fn  ( ran  F  X.  ran  G
) )
28 itg1add.4 . . . . . . . . 9  |-  P  =  (  +  |`  ( ran  F  X.  ran  G
) )
2928fneq1i 5498 . . . . . . . 8  |-  ( P  Fn  ( ran  F  X.  ran  G )  <->  (  +  |`  ( ran  F  X.  ran  G ) )  Fn  ( ran  F  X.  ran  G ) )
3027, 29sylibr 204 . . . . . . 7  |-  ( ph  ->  P  Fn  ( ran 
F  X.  ran  G
) )
31 dffn4 5618 . . . . . . 7  |-  ( P  Fn  ( ran  F  X.  ran  G )  <->  P :
( ran  F  X.  ran  G ) -onto-> ran  P
)
3230, 31sylib 189 . . . . . 6  |-  ( ph  ->  P : ( ran 
F  X.  ran  G
) -onto-> ran  P )
33 fofi 7351 . . . . . 6  |-  ( ( ( ran  F  X.  ran  G )  e.  Fin  /\  P : ( ran 
F  X.  ran  G
) -onto-> ran  P )  ->  ran  P  e.  Fin )
349, 32, 33syl2anc 643 . . . . 5  |-  ( ph  ->  ran  P  e.  Fin )
35 difss 3434 . . . . 5  |-  ( ran 
P  \  { 0 } )  C_  ran  P
36 ssfi 7288 . . . . 5  |-  ( ( ran  P  e.  Fin  /\  ( ran  P  \  { 0 } ) 
C_  ran  P )  ->  ( ran  P  \  { 0 } )  e.  Fin )
3734, 35, 36sylancl 644 . . . 4  |-  ( ph  ->  ( ran  P  \  { 0 } )  e.  Fin )
38 opelxpi 4869 . . . . . . . . . 10  |-  ( ( x  e.  ran  F  /\  y  e.  ran  G )  ->  <. x ,  y >.  e.  ( ran  F  X.  ran  G
) )
39 ffun 5552 . . . . . . . . . . . 12  |-  (  +  : ( CC  X.  CC ) --> CC  ->  Fun  +  )
4010, 39ax-mp 8 . . . . . . . . . . 11  |-  Fun  +
4110fdmi 5555 . . . . . . . . . . . 12  |-  dom  +  =  ( CC  X.  CC )
4225, 41syl6sseqr 3355 . . . . . . . . . . 11  |-  ( ph  ->  ( ran  F  X.  ran  G )  C_  dom  +  )
43 funfvima2 5933 . . . . . . . . . . 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 645 . . . . . . . . . 10  |-  ( ph  ->  ( <. x ,  y
>.  e.  ( ran  F  X.  ran  G )  -> 
(  +  `  <. x ,  y >. )  e.  (  +  " ( ran  F  X.  ran  G
) ) ) )
4538, 44syl5 30 . . . . . . . . 9  |-  ( ph  ->  ( ( x  e. 
ran  F  /\  y  e.  ran  G )  -> 
(  +  `  <. x ,  y >. )  e.  (  +  " ( ran  F  X.  ran  G
) ) ) )
4645imp 419 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ran  F  /\  y  e.  ran  G ) )  ->  (  +  `  <. x ,  y >.
)  e.  (  + 
" ( ran  F  X.  ran  G ) ) )
47 df-ov 6043 . . . . . . . 8  |-  ( x  +  y )  =  (  +  `  <. x ,  y >. )
4828rneqi 5055 . . . . . . . . 9  |-  ran  P  =  ran  (  +  |`  ( ran  F  X.  ran  G
) )
49 df-ima 4850 . . . . . . . . 9  |-  (  + 
" ( ran  F  X.  ran  G ) )  =  ran  (  +  |`  ( ran  F  X.  ran  G ) )
5048, 49eqtr4i 2427 . . . . . . . 8  |-  ran  P  =  (  +  " ( ran  F  X.  ran  G
) )
5146, 47, 503eltr4g 2487 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ran  F  /\  y  e.  ran  G ) )  ->  ( x  +  y )  e.  ran  P )
52 ffn 5550 . . . . . . . . 9  |-  ( F : RR --> RR  ->  F  Fn  RR )
5314, 52syl 16 . . . . . . . 8  |-  ( ph  ->  F  Fn  RR )
54 dffn3 5557 . . . . . . . 8  |-  ( F  Fn  RR  <->  F : RR
--> ran  F )
5553, 54sylib 189 . . . . . . 7  |-  ( ph  ->  F : RR --> ran  F
)
56 ffn 5550 . . . . . . . . 9  |-  ( G : RR --> RR  ->  G  Fn  RR )
5720, 56syl 16 . . . . . . . 8  |-  ( ph  ->  G  Fn  RR )
58 dffn3 5557 . . . . . . . 8  |-  ( G  Fn  RR  <->  G : RR
--> ran  G )
5957, 58sylib 189 . . . . . . 7  |-  ( ph  ->  G : RR --> ran  G
)
60 reex 9037 . . . . . . . 8  |-  RR  e.  _V
6160a1i 11 . . . . . . 7  |-  ( ph  ->  RR  e.  _V )
62 inidm 3510 . . . . . . 7  |-  ( RR 
i^i  RR )  =  RR
6351, 55, 59, 61, 61, 62off 6279 . . . . . 6  |-  ( ph  ->  ( F  o F  +  G ) : RR --> ran  P )
64 frn 5556 . . . . . 6  |-  ( ( F  o F  +  G ) : RR --> ran  P  ->  ran  ( F  o F  +  G
)  C_  ran  P )
6563, 64syl 16 . . . . 5  |-  ( ph  ->  ran  ( F  o F  +  G )  C_ 
ran  P )
6665ssdifd 3443 . . . 4  |-  ( ph  ->  ( ran  ( F  o F  +  G
)  \  { 0 } )  C_  ( ran  P  \  { 0 } ) )
6716sselda 3308 . . . . . . . . . 10  |-  ( (
ph  /\  y  e.  ran  F )  ->  y  e.  RR )
6822sselda 3308 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  ran  G )  ->  z  e.  RR )
6967, 68anim12dan 811 . . . . . . . . 9  |-  ( (
ph  /\  ( y  e.  ran  F  /\  z  e.  ran  G ) )  ->  ( y  e.  RR  /\  z  e.  RR ) )
70 readdcl 9029 . . . . . . . . 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 2758 . . . . . . 7  |-  ( ph  ->  A. y  e.  ran  F A. z  e.  ran  G ( y  +  z )  e.  RR )
73 funimassov 6182 . . . . . . . 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 645 . . . . . . 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 224 . . . . . 6  |-  ( ph  ->  (  +  " ( ran  F  X.  ran  G
) )  C_  RR )
7650, 75syl5eqss 3352 . . . . 5  |-  ( ph  ->  ran  P  C_  RR )
7776ssdifd 3443 . . . 4  |-  ( ph  ->  ( ran  P  \  { 0 } ) 
C_  ( RR  \  { 0 } ) )
78 itg1val2 19529 . . . 4  |-  ( ( ( F  o F  +  G )  e. 
dom  S.1  /\  ( ( ran  P  \  {
0 } )  e. 
Fin  /\  ( ran  ( F  o F  +  G )  \  {
0 } )  C_  ( ran  P  \  {
0 } )  /\  ( ran  P  \  {
0 } )  C_  ( RR  \  { 0 } ) ) )  ->  ( S.1 `  ( F  o F  +  G
) )  =  sum_ w  e.  ( ran  P  \  { 0 } ) ( w  x.  ( vol `  ( `' ( F  o F  +  G ) " {
w } ) ) ) )
793, 37, 66, 77, 78syl13anc 1186 . . 3  |-  ( ph  ->  ( S.1 `  ( F  o F  +  G
) )  =  sum_ w  e.  ( ran  P  \  { 0 } ) ( w  x.  ( vol `  ( `' ( F  o F  +  G ) " {
w } ) ) ) )
8020adantr 452 . . . . . . . 8  |-  ( (
ph  /\  w  e.  ( ran  P  \  {
0 } ) )  ->  G : RR --> RR )
817adantr 452 . . . . . . . 8  |-  ( (
ph  /\  w  e.  ( ran  P  \  {
0 } ) )  ->  ran  G  e.  Fin )
82 inss2 3522 . . . . . . . . 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 19523 . . . . . . . . . . 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 19523 . . . . . . . . . . 11  |-  ( G  e.  dom  S.1  ->  ( `' G " { z } )  e.  dom  vol )
872, 86syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( `' G " { z } )  e.  dom  vol )
88 inmbl 19389 . . . . . . . . . 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 643 . . . . . . . . 9  |-  ( ph  ->  ( ( `' F " { ( w  -  z ) } )  i^i  ( `' G " { z } ) )  e.  dom  vol )
9089ad2antrr 707 . . . . . . . 8  |-  ( ( ( ph  /\  w  e.  ( ran  P  \  { 0 } ) )  /\  z  e. 
ran  G )  -> 
( ( `' F " { ( w  -  z ) } )  i^i  ( `' G " { z } ) )  e.  dom  vol )
9135, 76syl5ss 3319 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ran  P  \  { 0 } ) 
C_  RR )
9291sselda 3308 . . . . . . . . . . . 12  |-  ( (
ph  /\  w  e.  ( ran  P  \  {
0 } ) )  ->  w  e.  RR )
9392adantr 452 . . . . . . . . . . 11  |-  ( ( ( ph  /\  w  e.  ( ran  P  \  { 0 } ) )  /\  z  e. 
ran  G )  ->  w  e.  RR )
9468adantlr 696 . . . . . . . . . . 11  |-  ( ( ( ph  /\  w  e.  ( ran  P  \  { 0 } ) )  /\  z  e. 
ran  G )  -> 
z  e.  RR )
9593, 94resubcld 9421 . . . . . . . . . 10  |-  ( ( ( ph  /\  w  e.  ( ran  P  \  { 0 } ) )  /\  z  e. 
ran  G )  -> 
( w  -  z
)  e.  RR )
9693recnd 9070 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  w  e.  ( ran  P  \  { 0 } ) )  /\  z  e. 
ran  G )  ->  w  e.  CC )
9794recnd 9070 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  w  e.  ( ran  P  \  { 0 } ) )  /\  z  e. 
ran  G )  -> 
z  e.  CC )
9896, 97npcand 9371 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  w  e.  ( ran  P  \  { 0 } ) )  /\  z  e. 
ran  G )  -> 
( ( w  -  z )  +  z )  =  w )
99 eldifsni 3888 . . . . . . . . . . . . 13  |-  ( w  e.  ( ran  P  \  { 0 } )  ->  w  =/=  0
)
10099ad2antlr 708 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  w  e.  ( ran  P  \  { 0 } ) )  /\  z  e. 
ran  G )  ->  w  =/=  0 )
10198, 100eqnetrd 2585 . . . . . . . . . . 11  |-  ( ( ( ph  /\  w  e.  ( ran  P  \  { 0 } ) )  /\  z  e. 
ran  G )  -> 
( ( w  -  z )  +  z )  =/=  0 )
102 oveq12 6049 . . . . . . . . . . . . 13  |-  ( ( ( w  -  z
)  =  0  /\  z  =  0 )  ->  ( ( w  -  z )  +  z )  =  ( 0  +  0 ) )
103 00id 9197 . . . . . . . . . . . . 13  |-  ( 0  +  0 )  =  0
104102, 103syl6eq 2452 . . . . . . . . . . . 12  |-  ( ( ( w  -  z
)  =  0  /\  z  =  0 )  ->  ( ( w  -  z )  +  z )  =  0 )
105104necon3ai 2607 . . . . . . . . . . 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 19543 . . . . . . . . . 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 1183 . . . . . . . . 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 19542 . . . . . . . . . . 11  |-  ( ph  ->  I : ( RR 
X.  RR ) --> RR )
111110ad2antrr 707 . . . . . . . . . 10  |-  ( ( ( ph  /\  w  e.  ( ran  P  \  { 0 } ) )  /\  z  e. 
ran  G )  ->  I : ( RR  X.  RR ) --> RR )
112111, 95, 94fovrnd 6177 . . . . . . . . 9  |-  ( ( ( ph  /\  w  e.  ( ran  P  \  { 0 } ) )  /\  z  e. 
ran  G )  -> 
( ( w  -  z ) I z )  e.  RR )
113109, 112eqeltrrd 2479 . . . . . . . 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 19537 . . . . . . 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 9070 . . . . . . . . 9  |-  ( (
ph  /\  w  e.  ( ran  P  \  {
0 } ) )  ->  w  e.  CC )
1161, 2i1faddlem 19538 . . . . . . . . 9  |-  ( (
ph  /\  w  e.  CC )  ->  ( `' ( F  o F  +  G ) " { w } )  =  U_ z  e. 
ran  G ( ( `' F " { ( w  -  z ) } )  i^i  ( `' G " { z } ) ) )
117115, 116syldan 457 . . . . . . . 8  |-  ( (
ph  /\  w  e.  ( ran  P  \  {
0 } ) )  ->  ( `' ( F  o F  +  G ) " {
w } )  = 
U_ z  e.  ran  G ( ( `' F " { ( w  -  z ) } )  i^i  ( `' G " { z } ) ) )
118117fveq2d 5691 . . . . . . 7  |-  ( (
ph  /\  w  e.  ( ran  P  \  {
0 } ) )  ->  ( vol `  ( `' ( F  o F  +  G ) " { w } ) )  =  ( vol `  U_ z  e.  ran  G ( ( `' F " { ( w  -  z ) } )  i^i  ( `' G " { z } ) ) ) )
119109sumeq2dv 12452 . . . . . . 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 2446 . . . . . 6  |-  ( (
ph  /\  w  e.  ( ran  P  \  {
0 } ) )  ->  ( vol `  ( `' ( F  o F  +  G ) " { w } ) )  =  sum_ z  e.  ran  G ( ( w  -  z ) I z ) )
121120oveq2d 6056 . . . . 5  |-  ( (
ph  /\  w  e.  ( ran  P  \  {
0 } ) )  ->  ( w  x.  ( vol `  ( `' ( F  o F  +  G ) " { w } ) ) )  =  ( w  x.  sum_ z  e.  ran  G ( ( w  -  z ) I z ) ) )
122112recnd 9070 . . . . . 6  |-  ( ( ( ph  /\  w  e.  ( ran  P  \  { 0 } ) )  /\  z  e. 
ran  G )  -> 
( ( w  -  z ) I z )  e.  CC )
12381, 115, 122fsummulc2 12522 . . . . 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 2436 . . . 4  |-  ( (
ph  /\  w  e.  ( ran  P  \  {
0 } ) )  ->  ( w  x.  ( vol `  ( `' ( F  o F  +  G ) " { w } ) ) )  =  sum_ z  e.  ran  G ( w  x.  ( ( w  -  z ) I z ) ) )
125124sumeq2dv 12452 . . 3  |-  ( ph  -> 
sum_ w  e.  ( ran  P  \  { 0 } ) ( w  x.  ( vol `  ( `' ( F  o F  +  G ) " { w } ) ) )  =  sum_ w  e.  ( ran  P  \  { 0 } )
sum_ z  e.  ran  G ( w  x.  (
( w  -  z
) I z ) ) )
12696, 122mulcld 9064 . . . . 5  |-  ( ( ( ph  /\  w  e.  ( ran  P  \  { 0 } ) )  /\  z  e. 
ran  G )  -> 
( w  x.  (
( w  -  z
) I z ) )  e.  CC )
127126anasss 629 . . . 4  |-  ( (
ph  /\  ( w  e.  ( ran  P  \  { 0 } )  /\  z  e.  ran  G ) )  ->  (
w  x.  ( ( w  -  z ) I z ) )  e.  CC )
12837, 7, 127fsumcom 12514 . . 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 2440 . 2  |-  ( ph  ->  ( S.1 `  ( F  o F  +  G
) )  =  sum_ z  e.  ran  G sum_ w  e.  ( ran  P  \  { 0 } ) ( w  x.  (
( w  -  z
) I z ) ) )
130 oveq1 6047 . . . . . . 7  |-  ( y  =  ( w  -  z )  ->  (
y  +  z )  =  ( ( w  -  z )  +  z ) )
131 oveq1 6047 . . . . . . 7  |-  ( y  =  ( w  -  z )  ->  (
y I z )  =  ( ( w  -  z ) I z ) )
132130, 131oveq12d 6058 . . . . . 6  |-  ( y  =  ( w  -  z )  ->  (
( y  +  z )  x.  ( y I z ) )  =  ( ( ( w  -  z )  +  z )  x.  ( ( w  -  z ) I z ) ) )
13334adantr 452 . . . . . 6  |-  ( (
ph  /\  z  e.  ran  G )  ->  ran  P  e.  Fin )
13476adantr 452 . . . . . . . . . . 11  |-  ( (
ph  /\  z  e.  ran  G )  ->  ran  P 
C_  RR )
135134sselda 3308 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  v  e.  ran  P )  ->  v  e.  RR )
13668adantr 452 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  v  e.  ran  P )  ->  z  e.  RR )
137135, 136resubcld 9421 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  v  e.  ran  P )  ->  ( v  -  z )  e.  RR )
138137ex 424 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ran  G )  ->  (
v  e.  ran  P  ->  ( v  -  z
)  e.  RR ) )
139135recnd 9070 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  v  e.  ran  P )  ->  v  e.  CC )
140139adantrr 698 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( v  e.  ran  P  /\  y  e.  ran  P ) )  ->  v  e.  CC )
14176sselda 3308 . . . . . . . . . . . 12  |-  ( (
ph  /\  y  e.  ran  P )  ->  y  e.  RR )
142141ad2ant2rl 730 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( v  e.  ran  P  /\  y  e.  ran  P ) )  ->  y  e.  RR )
143142recnd 9070 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( v  e.  ran  P  /\  y  e.  ran  P ) )  ->  y  e.  CC )
14468recnd 9070 . . . . . . . . . . 11  |-  ( (
ph  /\  z  e.  ran  G )  ->  z  e.  CC )
145144adantr 452 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( v  e.  ran  P  /\  y  e.  ran  P ) )  ->  z  e.  CC )
146140, 143, 145subcan2ad 9412 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( v  e.  ran  P  /\  y  e.  ran  P ) )  ->  (
( v  -  z
)  =  ( y  -  z )  <->  v  =  y ) )
147146ex 424 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ran  G )  ->  (
( v  e.  ran  P  /\  y  e.  ran  P )  ->  ( (
v  -  z )  =  ( y  -  z )  <->  v  =  y ) ) )
148138, 147dom2lem 7106 . . . . . . 7  |-  ( (
ph  /\  z  e.  ran  G )  ->  (
v  e.  ran  P  |->  ( v  -  z
) ) : ran  P
-1-1-> RR )
149 f1f1orn 5644 . . . . . . 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 6047 . . . . . . . 8  |-  ( v  =  w  ->  (
v  -  z )  =  ( w  -  z ) )
152 eqid 2404 . . . . . . . 8  |-  ( v  e.  ran  P  |->  ( v  -  z ) )  =  ( v  e.  ran  P  |->  ( v  -  z ) )
153 ovex 6065 . . . . . . . 8  |-  ( w  -  z )  e. 
_V
154151, 152, 153fvmpt 5765 . . . . . . 7  |-  ( w  e.  ran  P  -> 
( ( v  e. 
ran  P  |->  ( v  -  z ) ) `
 w )  =  ( w  -  z
) )
155154adantl 453 . . . . . 6  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  w  e.  ran  P )  ->  ( ( v  e.  ran  P  |->  ( v  -  z ) ) `  w )  =  ( w  -  z ) )
156 f1f 5598 . . . . . . . . . . 11  |-  ( ( v  e.  ran  P  |->  ( v  -  z
) ) : ran  P
-1-1-> RR  ->  ( v  e.  ran  P  |->  ( v  -  z ) ) : ran  P --> RR )
157 frn 5556 . . . . . . . . . . 11  |-  ( ( v  e.  ran  P  |->  ( v  -  z
) ) : ran  P --> RR  ->  ran  ( v  e.  ran  P  |->  ( v  -  z ) )  C_  RR )
158148, 156, 1573syl 19 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  ran  G )  ->  ran  ( v  e.  ran  P 
|->  ( v  -  z
) )  C_  RR )
159158sselda 3308 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  ( v  e.  ran  P  |->  ( v  -  z ) ) )  ->  y  e.  RR )
16068adantr 452 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  ( v  e.  ran  P  |->  ( v  -  z ) ) )  ->  z  e.  RR )
161159, 160readdcld 9071 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  ( v  e.  ran  P  |->  ( v  -  z ) ) )  ->  (
y  +  z )  e.  RR )
162110ad2antrr 707 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  ( v  e.  ran  P  |->  ( v  -  z ) ) )  ->  I : ( RR  X.  RR ) --> RR )
163162, 159, 160fovrnd 6177 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  ( v  e.  ran  P  |->  ( v  -  z ) ) )  ->  (
y I z )  e.  RR )
164161, 163remulcld 9072 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  ( v  e.  ran  P  |->  ( v  -  z ) ) )  ->  (
( y  +  z )  x.  ( y I z ) )  e.  RR )
165164recnd 9070 . . . . . 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 12472 . . . . 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 3308 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  w  e.  ran  P )  ->  w  e.  RR )
168167recnd 9070 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  w  e.  ran  P )  ->  w  e.  CC )
169144adantr 452 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  w  e.  ran  P )  ->  z  e.  CC )
170168, 169npcand 9371 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  w  e.  ran  P )  ->  ( ( w  -  z )  +  z )  =  w )
171170oveq1d 6055 . . . . . 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 12452 . . . . 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 2436 . . . 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 707 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  ( ran  F  X.  ran  G )  C_  dom  +  )
175 simpr 448 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  y  e.  ran  F )
176 simplr 732 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  z  e.  ran  G )
177 opelxpi 4869 . . . . . . . . . . . 12  |-  ( ( y  e.  ran  F  /\  z  e.  ran  G )  ->  <. y ,  z >.  e.  ( ran  F  X.  ran  G
) )
178175, 176, 177syl2anc 643 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  <. y ,  z
>.  e.  ( ran  F  X.  ran  G ) )
179 funfvima2 5933 . . . . . . . . . . . 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 652 . . . . . . . . . . 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 58 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  (  +  `  <. y ,  z >.
)  e.  (  + 
" ( ran  F  X.  ran  G ) ) )
182 df-ov 6043 . . . . . . . . . 10  |-  ( y  +  z )  =  (  +  `  <. y ,  z >. )
183181, 182, 503eltr4g 2487 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  ( y  +  z )  e.  ran  P )
18467adantlr 696 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  y  e.  RR )
185184recnd 9070 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  y  e.  CC )
186144adantr 452 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  z  e.  CC )
187185, 186pncand 9368 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  ( ( y  +  z )  -  z )  =  y )
188187eqcomd 2409 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  y  =  ( ( y  +  z )  -  z ) )
189 oveq1 6047 . . . . . . . . . . 11  |-  ( v  =  ( y  +  z )  ->  (
v  -  z )  =  ( ( y  +  z )  -  z ) )
190189eqeq2d 2415 . . . . . . . . . 10  |-  ( v  =  ( y  +  z )  ->  (
y  =  ( v  -  z )  <->  y  =  ( ( y  +  z )  -  z
) ) )
191190rspcev 3012 . . . . . . . . 9  |-  ( ( ( y  +  z )  e.  ran  P  /\  y  =  (
( y  +  z )  -  z ) )  ->  E. v  e.  ran  P  y  =  ( v  -  z
) )
192183, 188, 191syl2anc 643 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  E. v  e.  ran  P  y  =  ( v  -  z ) )
193192ralrimiva 2749 . . . . . . 7  |-  ( (
ph  /\  z  e.  ran  G )  ->  A. y  e.  ran  F E. v  e.  ran  P  y  =  ( v  -  z
) )
194 ssabral 3374 . . . . . . 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 204 . . . . . 6  |-  ( (
ph  /\  z  e.  ran  G )  ->  ran  F 
C_  { y  |  E. v  e.  ran  P  y  =  ( v  -  z ) } )
196152rnmpt 5075 . . . . . 6  |-  ran  (
v  e.  ran  P  |->  ( v  -  z
) )  =  {
y  |  E. v  e.  ran  P  y  =  ( v  -  z
) }
197195, 196syl6sseqr 3355 . . . . 5  |-  ( (
ph  /\  z  e.  ran  G )  ->  ran  F 
C_  ran  ( v  e.  ran  P  |->  ( v  -  z ) ) )
19868adantr 452 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  z  e.  RR )
199184, 198readdcld 9071 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  ( y  +  z )  e.  RR )
200110ad2antrr 707 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  I : ( RR  X.  RR ) --> RR )
201200, 184, 198fovrnd 6177 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  ( y I z )  e.  RR )
202199, 201remulcld 9072 . . . . . 6  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  ( ( y  +  z )  x.  ( y I z ) )  e.  RR )
203202recnd 9070 . . . . 5  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  ( ( y  +  z )  x.  ( y I z ) )  e.  CC )
204158ssdifd 3443 . . . . . . 7  |-  ( (
ph  /\  z  e.  ran  G )  ->  ( ran  ( v  e.  ran  P 
|->  ( v  -  z
) )  \  ran  F )  C_  ( RR  \  ran  F ) )
205204sselda 3308 . . . . . 6  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ( ran  ( v  e.  ran  P 
|->  ( v  -  z
) )  \  ran  F ) )  ->  y  e.  ( RR  \  ran  F ) )
206 eldifi 3429 . . . . . . . . . . . . 13  |-  ( y  e.  ( RR  \  ran  F )  ->  y  e.  RR )
207206ad2antrl 709 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( y  e.  ( RR  \  ran  F
)  /\  -.  (
y  =  0  /\  z  =  0 ) ) )  ->  y  e.  RR )
20868adantr 452 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( y  e.  ( RR  \  ran  F
)  /\  -.  (
y  =  0  /\  z  =  0 ) ) )  ->  z  e.  RR )
209 simprr 734 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( y  e.  ( RR  \  ran  F
)  /\  -.  (
y  =  0  /\  z  =  0 ) ) )  ->  -.  ( y  =  0  /\  z  =  0 ) )
2101, 2, 107itg1addlem3 19543 . . . . . . . . . . . 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 1183 . . . . . . . . . . 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 3521 . . . . . . . . . . . . . . 15  |-  ( ( `' F " { y } )  i^i  ( `' G " { z } ) )  C_  ( `' F " { y } )
213 eldifn 3430 . . . . . . . . . . . . . . . . . . 19  |-  ( y  e.  ( RR  \  ran  F )  ->  -.  y  e.  ran  F )
214213ad2antrl 709 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( y  e.  ( RR  \  ran  F
)  /\  -.  (
y  =  0  /\  z  =  0 ) ) )  ->  -.  y  e.  ran  F )
215 vex 2919 . . . . . . . . . . . . . . . . . . . 20  |-  y  e. 
_V
216 vex 2919 . . . . . . . . . . . . . . . . . . . . 21  |-  v  e. 
_V
217216eliniseg 5192 . . . . . . . . . . . . . . . . . . . 20  |-  ( y  e.  _V  ->  (
v  e.  ( `' F " { y } )  <->  v F
y ) )
218215, 217ax-mp 8 . . . . . . . . . . . . . . . . . . 19  |-  ( v  e.  ( `' F " { y } )  <-> 
v F y )
219216, 215brelrn 5059 . . . . . . . . . . . . . . . . . . 19  |-  ( v F y  ->  y  e.  ran  F )
220218, 219sylbi 188 . . . . . . . . . . . . . . . . . 18  |-  ( v  e.  ( `' F " { y } )  ->  y  e.  ran  F )
221214, 220nsyl 115 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( y  e.  ( RR  \  ran  F
)  /\  -.  (
y  =  0  /\  z  =  0 ) ) )  ->  -.  v  e.  ( `' F " { y } ) )
222221pm2.21d 100 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( y  e.  ( RR  \  ran  F
)  /\  -.  (
y  =  0  /\  z  =  0 ) ) )  ->  (
v  e.  ( `' F " { y } )  ->  v  e.  (/) ) )
223222ssrdv 3314 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( y  e.  ( RR  \  ran  F
)  /\  -.  (
y  =  0  /\  z  =  0 ) ) )  ->  ( `' F " { y } )  C_  (/) )
224212, 223syl5ss 3319 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( y  e.  ( RR  \  ran  F
)  /\  -.  (
y  =  0  /\  z  =  0 ) ) )  ->  (
( `' F " { y } )  i^i  ( `' G " { z } ) )  C_  (/) )
225 ss0 3618 . . . . . . . . . . . . . 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 5691 . . . . . . . . . . . 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 19387 . . . . . . . . . . . . . 14  |-  (/)  e.  dom  vol
229 mblvol 19379 . . . . . . . . . . . . . 14  |-  ( (/)  e.  dom  vol  ->  ( vol `  (/) )  =  ( vol * `  (/) ) )
230228, 229ax-mp 8 . . . . . . . . . . . . 13  |-  ( vol `  (/) )  =  ( vol * `  (/) )
231 ovol0 19342 . . . . . . . . . . . . 13  |-  ( vol
* `  (/) )  =  0
232230, 231eqtri 2424 . . . . . . . . . . . 12  |-  ( vol `  (/) )  =  0
233227, 232syl6eq 2452 . . . . . . . . . . 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 2436 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( y  e.  ( RR  \  ran  F
)  /\  -.  (
y  =  0  /\  z  =  0 ) ) )  ->  (
y I z )  =  0 )
235234oveq2d 6056 . . . . . . . . 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 9071 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( y  e.  ( RR  \  ran  F
)  /\  -.  (
y  =  0  /\  z  =  0 ) ) )  ->  (
y  +  z )  e.  RR )
237236recnd 9070 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( y  e.  ( RR  \  ran  F
)  /\  -.  (
y  =  0  /\  z  =  0 ) ) )  ->  (
y  +  z )  e.  CC )
238237mul01d 9221 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( y  e.  ( RR  \  ran  F
)  /\  -.  (
y  =  0  /\  z  =  0 ) ) )  ->  (
( y  +  z )  x.  0 )  =  0 )
239235, 238eqtrd 2436 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( y  e.  ( RR  \  ran  F
)  /\  -.  (
y  =  0  /\  z  =  0 ) ) )  ->  (
( y  +  z )  x.  ( y I z ) )  =  0 )
240239expr 599 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ( RR  \  ran  F ) )  ->  ( -.  (
y  =  0  /\  z  =  0 )  ->  ( ( y  +  z )  x.  ( y I z ) )  =  0 ) )
241 oveq12 6049 . . . . . . . . . 10  |-  ( ( y  =  0  /\  z  =  0 )  ->  ( y  +  z )  =  ( 0  +  0 ) )
242241, 103syl6eq 2452 . . . . . . . . 9  |-  ( ( y  =  0  /\  z  =  0 )  ->  ( y  +  z )  =  0 )
243 oveq12 6049 . . . . . . . . . 10  |-  ( ( y  =  0  /\  z  =  0 )  ->  ( y I z )  =  ( 0 I 0 ) )
244 0re 9047 . . . . . . . . . . 11  |-  0  e.  RR
245 iftrue 3705 . . . . . . . . . . . 12  |-  ( ( i  =  0  /\  j  =  0 )  ->  if ( ( i  =  0  /\  j  =  0 ) ,  0 ,  ( vol `  ( ( `' F " { i } )  i^i  ( `' G " { j } ) ) ) )  =  0 )
246 c0ex 9041 . . . . . . . . . . . 12  |-  0  e.  _V
247245, 107, 246ovmpt2a 6163 . . . . . . . . . . 11  |-  ( ( 0  e.  RR  /\  0  e.  RR )  ->  ( 0 I 0 )  =  0 )
248244, 244, 247mp2an 654 . . . . . . . . . 10  |-  ( 0 I 0 )  =  0
249243, 248syl6eq 2452 . . . . . . . . 9  |-  ( ( y  =  0  /\  z  =  0 )  ->  ( y I z )  =  0 )
250242, 249oveq12d 6058 . . . . . . . 8  |-  ( ( y  =  0  /\  z  =  0 )  ->  ( ( y  +  z )  x.  ( y I z ) )  =  ( 0  x.  0 ) )
251 0cn 9040 . . . . . . . . 9  |-  0  e.  CC
252251mul01i 9212 . . . . . . . 8  |-  ( 0  x.  0 )  =  0
253250, 252syl6eq 2452 . . . . . . 7  |-  ( ( y  =  0  /\  z  =  0 )  ->  ( ( y  +  z )  x.  ( y I z ) )  =  0 )
254240, 253pm2.61d2 154 . . . . . 6  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ( RR  \  ran  F ) )  ->  ( ( y  +  z )  x.  ( y I z ) )  =  0 )
255205, 254syldan 457 . . . . 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 5640 . . . . . . 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 7351 . . . . . 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 643 . . . . 5  |-  ( (
ph  /\  z  e.  ran  G )  ->  ran  ( v  e.  ran  P 
|->  ( v  -  z
) )  e.  Fin )
260197, 203, 255, 259fsumss 12474 . . . 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 780 . . . . 5  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  w  e.  ( ran  P 
\  { 0 } ) )  ->  (
w  x.  ( ( w  -  z ) I z ) )  e.  CC )
263 dfin4 3541 . . . . . . . 8  |-  ( ran 
P  i^i  { 0 } )  =  ( ran  P  \  ( ran  P  \  { 0 } ) )
264 inss2 3522 . . . . . . . 8  |-  ( ran 
P  i^i  { 0 } )  C_  { 0 }
265263, 264eqsstr3i 3339 . . . . . . 7  |-  ( ran 
P  \  ( ran  P 
\  { 0 } ) )  C_  { 0 }
266265sseli 3304 . . . . . 6  |-  ( w  e.  ( ran  P  \  ( ran  P  \  { 0 } ) )  ->  w  e.  { 0 } )
267 elsni 3798 . . . . . . . . 9  |-  ( w  e.  { 0 }  ->  w  =  0 )
268267adantl 453 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  w  e.  { 0 } )  ->  w  =  0 )
269268oveq1d 6055 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  w  e.  { 0 } )  ->  (
w  x.  ( ( w  -  z ) I z ) )  =  ( 0  x.  ( ( w  -  z ) I z ) ) )
270110ad2antrr 707 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  w  e.  { 0 } )  ->  I : ( RR  X.  RR ) --> RR )
271268, 244syl6eqel 2492 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  w  e.  { 0 } )  ->  w  e.  RR )
27268adantr 452 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  w  e.  { 0 } )  ->  z  e.  RR )
273271, 272resubcld 9421 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  w  e.  { 0 } )  ->  (
w  -  z )  e.  RR )
274270, 273, 272fovrnd 6177 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  w  e.  { 0 } )  ->  (
( w  -  z
) I z )  e.  RR )
275274recnd 9070 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  w  e.  { 0 } )  ->  (
( w  -  z
) I z )  e.  CC )
276275mul02d 9220 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  w  e.  { 0 } )  ->  (
0  x.  ( ( w  -  z ) I z ) )  =  0 )
277269, 276eqtrd 2436 . . . . . 6  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  w  e.  { 0 } )  ->  (
w  x.  ( ( w  -  z ) I z ) )  =  0 )
278266, 277sylan2 461 . . . . 5  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  w  e.  ( ran  P 
\  ( ran  P  \  { 0 } ) ) )  ->  (
w  x.  ( ( w  -  z ) I z ) )  =  0 )
279261, 262, 278, 133fsumss 12474 . . . 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 2446 . . 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 12452 . 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 629 . . 3  |-  ( (
ph  /\  ( z  e.  ran  G  /\  y  e.  ran  F ) )  ->  ( ( y  +  z )  x.  ( y I z ) )  e.  CC )
2837, 5, 282fsumcom 12514 . 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 2442 1  |-  ( ph  ->  ( S.1 `  ( F  o F  +  G
) )  =  sum_ y  e.  ran  F sum_ z  e.  ran  G ( ( y  +  z )  x.  ( y I z ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   {cab 2390    =/= wne 2567   A.wral 2666   E.wrex 2667   _Vcvv 2916    \ cdif 3277    i^i cin 3279    C_ wss 3280   (/)c0 3588   ifcif 3699   {csn 3774   <.cop 3777   U_ciun 4053   class class class wbr 4172    e. cmpt 4226    X. cxp 4835   `'ccnv 4836   dom cdm 4837   ran crn 4838    |` cres 4839   "cima 4840   Fun wfun 5407    Fn wfn 5408   -->wf 5409   -1-1->wf1 5410   -onto->wfo 5411   -1-1-onto->wf1o 5412   ` cfv 5413  (class class class)co 6040    e. cmpt2 6042    o Fcof 6262   Fincfn 7068   CCcc 8944   RRcr 8945   0cc0 8946    + caddc 8949    x. cmul 8951    - cmin 9247   sum_csu 12434   vol
*covol 19312   volcvol 19313   S.1citg1 19460
This theorem is referenced by:  itg1addlem5  19545
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-disj 4143  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-oi 7435  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-q 10531  df-rp 10569  df-xadd 10667  df-ioo 10876  df-ico 10878  df-icc 10879  df-fz 11000  df-fzo 11091  df-fl 11157  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-clim 12237  df-sum 12435  df-xmet 16650  df-met 16651  df-ovol 19314  df-vol 19315  df-mbf 19465  df-itg1 19466
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