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Theorem tdeglem4 19446
Description: There is only one multi-index with total degree 0. (Contributed by Stefan O'Rear, 29-Mar-2015.)
Hypotheses
Ref Expression
tdeglem.a  |-  A  =  { m  e.  ( NN0  ^m  I )  |  ( `' m " NN )  e.  Fin }
tdeglem.h  |-  H  =  ( h  e.  A  |->  (fld 
gsumg  h ) )
Assertion
Ref Expression
tdeglem4  |-  ( ( I  e.  V  /\  X  e.  A )  ->  ( ( H `  X )  =  0  <-> 
X  =  ( I  X.  { 0 } ) ) )
Distinct variable groups:    A, h    h, I, m    h, V   
h, X, m
Allowed substitution hints:    A( m)    H( h, m)    V( m)

Proof of Theorem tdeglem4
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rexnal 2554 . . . . 5  |-  ( E. x  e.  I  -.  ( X `  x )  =  0  <->  -.  A. x  e.  I  ( X `  x )  =  0 )
2 df-ne 2448 . . . . . . 7  |-  ( ( X `  x )  =/=  0  <->  -.  ( X `  x )  =  0 )
3 oveq2 5866 . . . . . . . . . . . 12  |-  ( h  =  X  ->  (fld  gsumg  h )  =  (fld  gsumg  X ) )
4 tdeglem.h . . . . . . . . . . . 12  |-  H  =  ( h  e.  A  |->  (fld 
gsumg  h ) )
5 ovex 5883 . . . . . . . . . . . 12  |-  (fld  gsumg  X )  e.  _V
63, 4, 5fvmpt 5602 . . . . . . . . . . 11  |-  ( X  e.  A  ->  ( H `  X )  =  (fld 
gsumg  X ) )
76ad2antlr 707 . . . . . . . . . 10  |-  ( ( ( I  e.  V  /\  X  e.  A
)  /\  ( x  e.  I  /\  ( X `  x )  =/=  0 ) )  -> 
( H `  X
)  =  (fld  gsumg  X ) )
8 tdeglem.a . . . . . . . . . . . . . 14  |-  A  =  { m  e.  ( NN0  ^m  I )  |  ( `' m " NN )  e.  Fin }
98psrbagf 16113 . . . . . . . . . . . . 13  |-  ( ( I  e.  V  /\  X  e.  A )  ->  X : I --> NN0 )
109feqmptd 5575 . . . . . . . . . . . 12  |-  ( ( I  e.  V  /\  X  e.  A )  ->  X  =  ( y  e.  I  |->  ( X `
 y ) ) )
1110adantr 451 . . . . . . . . . . 11  |-  ( ( ( I  e.  V  /\  X  e.  A
)  /\  ( x  e.  I  /\  ( X `  x )  =/=  0 ) )  ->  X  =  ( y  e.  I  |->  ( X `
 y ) ) )
1211oveq2d 5874 . . . . . . . . . 10  |-  ( ( ( I  e.  V  /\  X  e.  A
)  /\  ( x  e.  I  /\  ( X `  x )  =/=  0 ) )  -> 
(fld  gsumg  X )  =  (fld  gsumg  ( y  e.  I  |->  ( X `  y
) ) ) )
13 cnfldbas 16383 . . . . . . . . . . 11  |-  CC  =  ( Base ` fld )
14 cnfld0 16398 . . . . . . . . . . 11  |-  0  =  ( 0g ` fld )
15 cnfldadd 16384 . . . . . . . . . . 11  |-  +  =  ( +g  ` fld )
16 cnrng 16396 . . . . . . . . . . . 12  |-fld  e.  Ring
17 rngcmn 15371 . . . . . . . . . . . 12  |-  (fld  e.  Ring  ->fld  e. CMnd )
1816, 17mp1i 11 . . . . . . . . . . 11  |-  ( ( ( I  e.  V  /\  X  e.  A
)  /\  ( x  e.  I  /\  ( X `  x )  =/=  0 ) )  ->fld  e. CMnd )
19 simpll 730 . . . . . . . . . . 11  |-  ( ( ( I  e.  V  /\  X  e.  A
)  /\  ( x  e.  I  /\  ( X `  x )  =/=  0 ) )  ->  I  e.  V )
209adantr 451 . . . . . . . . . . . . 13  |-  ( ( ( I  e.  V  /\  X  e.  A
)  /\  ( x  e.  I  /\  ( X `  x )  =/=  0 ) )  ->  X : I --> NN0 )
21 ffvelrn 5663 . . . . . . . . . . . . 13  |-  ( ( X : I --> NN0  /\  y  e.  I )  ->  ( X `  y
)  e.  NN0 )
2220, 21sylan 457 . . . . . . . . . . . 12  |-  ( ( ( ( I  e.  V  /\  X  e.  A )  /\  (
x  e.  I  /\  ( X `  x )  =/=  0 ) )  /\  y  e.  I
)  ->  ( X `  y )  e.  NN0 )
2322nn0cnd 10020 . . . . . . . . . . 11  |-  ( ( ( ( I  e.  V  /\  X  e.  A )  /\  (
x  e.  I  /\  ( X `  x )  =/=  0 ) )  /\  y  e.  I
)  ->  ( X `  y )  e.  CC )
2411cnveqd 4857 . . . . . . . . . . . . 13  |-  ( ( ( I  e.  V  /\  X  e.  A
)  /\  ( x  e.  I  /\  ( X `  x )  =/=  0 ) )  ->  `' X  =  `' ( y  e.  I  |->  ( X `  y
) ) )
2524imaeq1d 5011 . . . . . . . . . . . 12  |-  ( ( ( I  e.  V  /\  X  e.  A
)  /\  ( x  e.  I  /\  ( X `  x )  =/=  0 ) )  -> 
( `' X "
( _V  \  {
0 } ) )  =  ( `' ( y  e.  I  |->  ( X `  y ) ) " ( _V 
\  { 0 } ) ) )
268psrbagsuppfi 16246 . . . . . . . . . . . . . 14  |-  ( ( X  e.  A  /\  I  e.  V )  ->  ( `' X "
( _V  \  {
0 } ) )  e.  Fin )
2726ancoms 439 . . . . . . . . . . . . 13  |-  ( ( I  e.  V  /\  X  e.  A )  ->  ( `' X "
( _V  \  {
0 } ) )  e.  Fin )
2827adantr 451 . . . . . . . . . . . 12  |-  ( ( ( I  e.  V  /\  X  e.  A
)  /\  ( x  e.  I  /\  ( X `  x )  =/=  0 ) )  -> 
( `' X "
( _V  \  {
0 } ) )  e.  Fin )
2925, 28eqeltrrd 2358 . . . . . . . . . . 11  |-  ( ( ( I  e.  V  /\  X  e.  A
)  /\  ( x  e.  I  /\  ( X `  x )  =/=  0 ) )  -> 
( `' ( y  e.  I  |->  ( X `
 y ) )
" ( _V  \  { 0 } ) )  e.  Fin )
30 incom 3361 . . . . . . . . . . . . 13  |-  ( ( I  \  { x } )  i^i  {
x } )  =  ( { x }  i^i  ( I  \  {
x } ) )
31 disjdif 3526 . . . . . . . . . . . . 13  |-  ( { x }  i^i  (
I  \  { x } ) )  =  (/)
3230, 31eqtri 2303 . . . . . . . . . . . 12  |-  ( ( I  \  { x } )  i^i  {
x } )  =  (/)
3332a1i 10 . . . . . . . . . . 11  |-  ( ( ( I  e.  V  /\  X  e.  A
)  /\  ( x  e.  I  /\  ( X `  x )  =/=  0 ) )  -> 
( ( I  \  { x } )  i^i  { x }
)  =  (/) )
34 difsnid 3761 . . . . . . . . . . . . 13  |-  ( x  e.  I  ->  (
( I  \  {
x } )  u. 
{ x } )  =  I )
3534eqcomd 2288 . . . . . . . . . . . 12  |-  ( x  e.  I  ->  I  =  ( ( I 
\  { x }
)  u.  { x } ) )
3635ad2antrl 708 . . . . . . . . . . 11  |-  ( ( ( I  e.  V  /\  X  e.  A
)  /\  ( x  e.  I  /\  ( X `  x )  =/=  0 ) )  ->  I  =  ( (
I  \  { x } )  u.  {
x } ) )
3713, 14, 15, 18, 19, 23, 29, 33, 36gsumsplit2 15208 . . . . . . . . . 10  |-  ( ( ( I  e.  V  /\  X  e.  A
)  /\  ( x  e.  I  /\  ( X `  x )  =/=  0 ) )  -> 
(fld  gsumg  ( y  e.  I  |->  ( X `  y ) ) )  =  ( (fld 
gsumg  ( y  e.  ( I  \  { x } )  |->  ( X `
 y ) ) )  +  (fld  gsumg  ( y  e.  {
x }  |->  ( X `
 y ) ) ) ) )
387, 12, 373eqtrd 2319 . . . . . . . . 9  |-  ( ( ( I  e.  V  /\  X  e.  A
)  /\  ( x  e.  I  /\  ( X `  x )  =/=  0 ) )  -> 
( H `  X
)  =  ( (fld  gsumg  ( y  e.  ( I  \  { x } ) 
|->  ( X `  y
) ) )  +  (fld 
gsumg  ( y  e.  {
x }  |->  ( X `
 y ) ) ) ) )
39 difexg 4162 . . . . . . . . . . . . 13  |-  ( I  e.  V  ->  (
I  \  { x } )  e.  _V )
4019, 39syl 15 . . . . . . . . . . . 12  |-  ( ( ( I  e.  V  /\  X  e.  A
)  /\  ( x  e.  I  /\  ( X `  x )  =/=  0 ) )  -> 
( I  \  {
x } )  e. 
_V )
41 nn0subm 16427 . . . . . . . . . . . . 13  |-  NN0  e.  (SubMnd ` fld )
4241a1i 10 . . . . . . . . . . . 12  |-  ( ( ( I  e.  V  /\  X  e.  A
)  /\  ( x  e.  I  /\  ( X `  x )  =/=  0 ) )  ->  NN0  e.  (SubMnd ` fld ) )
43 eldifi 3298 . . . . . . . . . . . . . 14  |-  ( y  e.  ( I  \  { x } )  ->  y  e.  I
)
4420, 43, 21syl2an 463 . . . . . . . . . . . . 13  |-  ( ( ( ( I  e.  V  /\  X  e.  A )  /\  (
x  e.  I  /\  ( X `  x )  =/=  0 ) )  /\  y  e.  ( I  \  { x } ) )  -> 
( X `  y
)  e.  NN0 )
45 eqid 2283 . . . . . . . . . . . . 13  |-  ( y  e.  ( I  \  { x } ) 
|->  ( X `  y
) )  =  ( y  e.  ( I 
\  { x }
)  |->  ( X `  y ) )
4644, 45fmptd 5684 . . . . . . . . . . . 12  |-  ( ( ( I  e.  V  /\  X  e.  A
)  /\  ( x  e.  I  /\  ( X `  x )  =/=  0 ) )  -> 
( y  e.  ( I  \  { x } )  |->  ( X `
 y ) ) : ( I  \  { x } ) --> NN0 )
47 difss 3303 . . . . . . . . . . . . . . . 16  |-  ( I 
\  { x }
)  C_  I
48 resmpt 5000 . . . . . . . . . . . . . . . 16  |-  ( ( I  \  { x } )  C_  I  ->  ( ( y  e.  I  |->  ( X `  y ) )  |`  ( I  \  { x } ) )  =  ( y  e.  ( I  \  { x } )  |->  ( X `
 y ) ) )
4947, 48ax-mp 8 . . . . . . . . . . . . . . 15  |-  ( ( y  e.  I  |->  ( X `  y ) )  |`  ( I  \  { x } ) )  =  ( y  e.  ( I  \  { x } ) 
|->  ( X `  y
) )
50 resss 4979 . . . . . . . . . . . . . . 15  |-  ( ( y  e.  I  |->  ( X `  y ) )  |`  ( I  \  { x } ) )  C_  ( y  e.  I  |->  ( X `
 y ) )
5149, 50eqsstr3i 3209 . . . . . . . . . . . . . 14  |-  ( y  e.  ( I  \  { x } ) 
|->  ( X `  y
) )  C_  (
y  e.  I  |->  ( X `  y ) )
52 cnvss 4854 . . . . . . . . . . . . . 14  |-  ( ( y  e.  ( I 
\  { x }
)  |->  ( X `  y ) )  C_  ( y  e.  I  |->  ( X `  y
) )  ->  `' ( y  e.  ( I  \  { x } )  |->  ( X `
 y ) ) 
C_  `' ( y  e.  I  |->  ( X `
 y ) ) )
53 imass1 5048 . . . . . . . . . . . . . 14  |-  ( `' ( y  e.  ( I  \  { x } )  |->  ( X `
 y ) ) 
C_  `' ( y  e.  I  |->  ( X `
 y ) )  ->  ( `' ( y  e.  ( I 
\  { x }
)  |->  ( X `  y ) ) "
( _V  \  {
0 } ) ) 
C_  ( `' ( y  e.  I  |->  ( X `  y ) ) " ( _V 
\  { 0 } ) ) )
5451, 52, 53mp2b 9 . . . . . . . . . . . . 13  |-  ( `' ( y  e.  ( I  \  { x } )  |->  ( X `
 y ) )
" ( _V  \  { 0 } ) )  C_  ( `' ( y  e.  I  |->  ( X `  y
) ) " ( _V  \  { 0 } ) )
55 ssfi 7083 . . . . . . . . . . . . 13  |-  ( ( ( `' ( y  e.  I  |->  ( X `
 y ) )
" ( _V  \  { 0 } ) )  e.  Fin  /\  ( `' ( y  e.  ( I  \  {
x } )  |->  ( X `  y ) ) " ( _V 
\  { 0 } ) )  C_  ( `' ( y  e.  I  |->  ( X `  y ) ) "
( _V  \  {
0 } ) ) )  ->  ( `' ( y  e.  ( I  \  { x } )  |->  ( X `
 y ) )
" ( _V  \  { 0 } ) )  e.  Fin )
5629, 54, 55sylancl 643 . . . . . . . . . . . 12  |-  ( ( ( I  e.  V  /\  X  e.  A
)  /\  ( x  e.  I  /\  ( X `  x )  =/=  0 ) )  -> 
( `' ( y  e.  ( I  \  { x } ) 
|->  ( X `  y
) ) " ( _V  \  { 0 } ) )  e.  Fin )
5714, 18, 40, 42, 46, 56gsumsubmcl 15201 . . . . . . . . . . 11  |-  ( ( ( I  e.  V  /\  X  e.  A
)  /\  ( x  e.  I  /\  ( X `  x )  =/=  0 ) )  -> 
(fld  gsumg  ( y  e.  ( I 
\  { x }
)  |->  ( X `  y ) ) )  e.  NN0 )
58 rngmnd 15350 . . . . . . . . . . . . . 14  |-  (fld  e.  Ring  ->fld  e.  Mnd )
5916, 58mp1i 11 . . . . . . . . . . . . 13  |-  ( ( ( I  e.  V  /\  X  e.  A
)  /\  ( x  e.  I  /\  ( X `  x )  =/=  0 ) )  ->fld  e.  Mnd )
60 simprl 732 . . . . . . . . . . . . 13  |-  ( ( ( I  e.  V  /\  X  e.  A
)  /\  ( x  e.  I  /\  ( X `  x )  =/=  0 ) )  ->  x  e.  I )
61 ffvelrn 5663 . . . . . . . . . . . . . . 15  |-  ( ( X : I --> NN0  /\  x  e.  I )  ->  ( X `  x
)  e.  NN0 )
6220, 60, 61syl2anc 642 . . . . . . . . . . . . . 14  |-  ( ( ( I  e.  V  /\  X  e.  A
)  /\  ( x  e.  I  /\  ( X `  x )  =/=  0 ) )  -> 
( X `  x
)  e.  NN0 )
6362nn0cnd 10020 . . . . . . . . . . . . 13  |-  ( ( ( I  e.  V  /\  X  e.  A
)  /\  ( x  e.  I  /\  ( X `  x )  =/=  0 ) )  -> 
( X `  x
)  e.  CC )
64 fveq2 5525 . . . . . . . . . . . . . 14  |-  ( y  =  x  ->  ( X `  y )  =  ( X `  x ) )
6513, 64gsumsn 15220 . . . . . . . . . . . . 13  |-  ( (fld  e. 
Mnd  /\  x  e.  I  /\  ( X `  x )  e.  CC )  ->  (fld 
gsumg  ( y  e.  {
x }  |->  ( X `
 y ) ) )  =  ( X `
 x ) )
6659, 60, 63, 65syl3anc 1182 . . . . . . . . . . . 12  |-  ( ( ( I  e.  V  /\  X  e.  A
)  /\  ( x  e.  I  /\  ( X `  x )  =/=  0 ) )  -> 
(fld  gsumg  ( y  e.  { x }  |->  ( X `  y ) ) )  =  ( X `  x ) )
67 simprr 733 . . . . . . . . . . . . . 14  |-  ( ( ( I  e.  V  /\  X  e.  A
)  /\  ( x  e.  I  /\  ( X `  x )  =/=  0 ) )  -> 
( X `  x
)  =/=  0 )
6867, 2sylib 188 . . . . . . . . . . . . 13  |-  ( ( ( I  e.  V  /\  X  e.  A
)  /\  ( x  e.  I  /\  ( X `  x )  =/=  0 ) )  ->  -.  ( X `  x
)  =  0 )
69 elnn0 9967 . . . . . . . . . . . . . 14  |-  ( ( X `  x )  e.  NN0  <->  ( ( X `
 x )  e.  NN  \/  ( X `
 x )  =  0 ) )
7062, 69sylib 188 . . . . . . . . . . . . 13  |-  ( ( ( I  e.  V  /\  X  e.  A
)  /\  ( x  e.  I  /\  ( X `  x )  =/=  0 ) )  -> 
( ( X `  x )  e.  NN  \/  ( X `  x
)  =  0 ) )
71 orel2 372 . . . . . . . . . . . . 13  |-  ( -.  ( X `  x
)  =  0  -> 
( ( ( X `
 x )  e.  NN  \/  ( X `
 x )  =  0 )  ->  ( X `  x )  e.  NN ) )
7268, 70, 71sylc 56 . . . . . . . . . . . 12  |-  ( ( ( I  e.  V  /\  X  e.  A
)  /\  ( x  e.  I  /\  ( X `  x )  =/=  0 ) )  -> 
( X `  x
)  e.  NN )
7366, 72eqeltrd 2357 . . . . . . . . . . 11  |-  ( ( ( I  e.  V  /\  X  e.  A
)  /\  ( x  e.  I  /\  ( X `  x )  =/=  0 ) )  -> 
(fld  gsumg  ( y  e.  { x }  |->  ( X `  y ) ) )  e.  NN )
74 nn0nnaddcl 9996 . . . . . . . . . . 11  |-  ( ( (fld 
gsumg  ( y  e.  ( I  \  { x } )  |->  ( X `
 y ) ) )  e.  NN0  /\  (fld  gsumg  (
y  e.  { x }  |->  ( X `  y ) ) )  e.  NN )  -> 
( (fld 
gsumg  ( y  e.  ( I  \  { x } )  |->  ( X `
 y ) ) )  +  (fld  gsumg  ( y  e.  {
x }  |->  ( X `
 y ) ) ) )  e.  NN )
7557, 73, 74syl2anc 642 . . . . . . . . . 10  |-  ( ( ( I  e.  V  /\  X  e.  A
)  /\  ( x  e.  I  /\  ( X `  x )  =/=  0 ) )  -> 
( (fld 
gsumg  ( y  e.  ( I  \  { x } )  |->  ( X `
 y ) ) )  +  (fld  gsumg  ( y  e.  {
x }  |->  ( X `
 y ) ) ) )  e.  NN )
7675nnne0d 9790 . . . . . . . . 9  |-  ( ( ( I  e.  V  /\  X  e.  A
)  /\  ( x  e.  I  /\  ( X `  x )  =/=  0 ) )  -> 
( (fld 
gsumg  ( y  e.  ( I  \  { x } )  |->  ( X `
 y ) ) )  +  (fld  gsumg  ( y  e.  {
x }  |->  ( X `
 y ) ) ) )  =/=  0
)
7738, 76eqnetrd 2464 . . . . . . . 8  |-  ( ( ( I  e.  V  /\  X  e.  A
)  /\  ( x  e.  I  /\  ( X `  x )  =/=  0 ) )  -> 
( H `  X
)  =/=  0 )
7877expr 598 . . . . . . 7  |-  ( ( ( I  e.  V  /\  X  e.  A
)  /\  x  e.  I )  ->  (
( X `  x
)  =/=  0  -> 
( H `  X
)  =/=  0 ) )
792, 78syl5bir 209 . . . . . 6  |-  ( ( ( I  e.  V  /\  X  e.  A
)  /\  x  e.  I )  ->  ( -.  ( X `  x
)  =  0  -> 
( H `  X
)  =/=  0 ) )
8079rexlimdva 2667 . . . . 5  |-  ( ( I  e.  V  /\  X  e.  A )  ->  ( E. x  e.  I  -.  ( X `
 x )  =  0  ->  ( H `  X )  =/=  0
) )
811, 80syl5bir 209 . . . 4  |-  ( ( I  e.  V  /\  X  e.  A )  ->  ( -.  A. x  e.  I  ( X `  x )  =  0  ->  ( H `  X )  =/=  0
) )
8281necon4bd 2508 . . 3  |-  ( ( I  e.  V  /\  X  e.  A )  ->  ( ( H `  X )  =  0  ->  A. x  e.  I 
( X `  x
)  =  0 ) )
83 ffn 5389 . . . . . 6  |-  ( X : I --> NN0  ->  X  Fn  I )
849, 83syl 15 . . . . 5  |-  ( ( I  e.  V  /\  X  e.  A )  ->  X  Fn  I )
85 0nn0 9980 . . . . . 6  |-  0  e.  NN0
86 fnconstg 5429 . . . . . 6  |-  ( 0  e.  NN0  ->  ( I  X.  { 0 } )  Fn  I )
8785, 86mp1i 11 . . . . 5  |-  ( ( I  e.  V  /\  X  e.  A )  ->  ( I  X.  {
0 } )  Fn  I )
88 eqfnfv 5622 . . . . 5  |-  ( ( X  Fn  I  /\  ( I  X.  { 0 } )  Fn  I
)  ->  ( X  =  ( I  X.  { 0 } )  <->  A. x  e.  I 
( X `  x
)  =  ( ( I  X.  { 0 } ) `  x
) ) )
8984, 87, 88syl2anc 642 . . . 4  |-  ( ( I  e.  V  /\  X  e.  A )  ->  ( X  =  ( I  X.  { 0 } )  <->  A. x  e.  I  ( X `  x )  =  ( ( I  X.  {
0 } ) `  x ) ) )
90 c0ex 8832 . . . . . . 7  |-  0  e.  _V
9190fvconst2 5729 . . . . . 6  |-  ( x  e.  I  ->  (
( I  X.  {
0 } ) `  x )  =  0 )
9291eqeq2d 2294 . . . . 5  |-  ( x  e.  I  ->  (
( X `  x
)  =  ( ( I  X.  { 0 } ) `  x
)  <->  ( X `  x )  =  0 ) )
9392ralbiia 2575 . . . 4  |-  ( A. x  e.  I  ( X `  x )  =  ( ( I  X.  { 0 } ) `  x )  <->  A. x  e.  I 
( X `  x
)  =  0 )
9489, 93syl6bb 252 . . 3  |-  ( ( I  e.  V  /\  X  e.  A )  ->  ( X  =  ( I  X.  { 0 } )  <->  A. x  e.  I  ( X `  x )  =  0 ) )
9582, 94sylibrd 225 . 2  |-  ( ( I  e.  V  /\  X  e.  A )  ->  ( ( H `  X )  =  0  ->  X  =  ( I  X.  { 0 } ) ) )
968psrbag0 16235 . . . . . 6  |-  ( I  e.  V  ->  (
I  X.  { 0 } )  e.  A
)
9796adantr 451 . . . . 5  |-  ( ( I  e.  V  /\  X  e.  A )  ->  ( I  X.  {
0 } )  e.  A )
98 oveq2 5866 . . . . . 6  |-  ( h  =  ( I  X.  { 0 } )  ->  (fld 
gsumg  h )  =  (fld  gsumg  ( I  X.  { 0 } ) ) )
99 ovex 5883 . . . . . 6  |-  (fld  gsumg  ( I  X.  {
0 } ) )  e.  _V
10098, 4, 99fvmpt 5602 . . . . 5  |-  ( ( I  X.  { 0 } )  e.  A  ->  ( H `  (
I  X.  { 0 } ) )  =  (fld 
gsumg  ( I  X.  { 0 } ) ) )
10197, 100syl 15 . . . 4  |-  ( ( I  e.  V  /\  X  e.  A )  ->  ( H `  (
I  X.  { 0 } ) )  =  (fld 
gsumg  ( I  X.  { 0 } ) ) )
102 fconstmpt 4732 . . . . . 6  |-  ( I  X.  { 0 } )  =  ( x  e.  I  |->  0 )
103102oveq2i 5869 . . . . 5  |-  (fld  gsumg  ( I  X.  {
0 } ) )  =  (fld 
gsumg  ( x  e.  I  |->  0 ) )
10416, 58ax-mp 8 . . . . . . 7  |-fld  e.  Mnd
10514gsumz 14458 . . . . . . 7  |-  ( (fld  e. 
Mnd  /\  I  e.  V )  ->  (fld  gsumg  ( x  e.  I  |->  0 ) )  =  0 )
106104, 105mpan 651 . . . . . 6  |-  ( I  e.  V  ->  (fld  gsumg  ( x  e.  I  |->  0 ) )  =  0 )
107106adantr 451 . . . . 5  |-  ( ( I  e.  V  /\  X  e.  A )  ->  (fld 
gsumg  ( x  e.  I  |->  0 ) )  =  0 )
108103, 107syl5eq 2327 . . . 4  |-  ( ( I  e.  V  /\  X  e.  A )  ->  (fld 
gsumg  ( I  X.  { 0 } ) )  =  0 )
109101, 108eqtrd 2315 . . 3  |-  ( ( I  e.  V  /\  X  e.  A )  ->  ( H `  (
I  X.  { 0 } ) )  =  0 )
110 fveq2 5525 . . . 4  |-  ( X  =  ( I  X.  { 0 } )  ->  ( H `  X )  =  ( H `  ( I  X.  { 0 } ) ) )
111110eqeq1d 2291 . . 3  |-  ( X  =  ( I  X.  { 0 } )  ->  ( ( H `
 X )  =  0  <->  ( H `  ( I  X.  { 0 } ) )  =  0 ) )
112109, 111syl5ibrcom 213 . 2  |-  ( ( I  e.  V  /\  X  e.  A )  ->  ( X  =  ( I  X.  { 0 } )  ->  ( H `  X )  =  0 ) )
11395, 112impbid 183 1  |-  ( ( I  e.  V  /\  X  e.  A )  ->  ( ( H `  X )  =  0  <-> 
X  =  ( I  X.  { 0 } ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544   {crab 2547   _Vcvv 2788    \ cdif 3149    u. cun 3150    i^i cin 3151    C_ wss 3152   (/)c0 3455   {csn 3640    e. cmpt 4077    X. cxp 4687   `'ccnv 4688    |` cres 4691   "cima 4692    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858    ^m cmap 6772   Fincfn 6863   CCcc 8735   0cc0 8737    + caddc 8740   NNcn 9746   NN0cn0 9965    gsumg cgsu 13401   Mndcmnd 14361  SubMndcsubmnd 14414  CMndccmn 15089   Ringcrg 15337  ℂfldccnfld 16377
This theorem is referenced by:  mdegle0  19463
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-addf 8816  ax-mulf 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-fz 10783  df-fzo 10871  df-seq 11047  df-hash 11338  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-tset 13227  df-ple 13228  df-ds 13230  df-0g 13404  df-gsum 13405  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-grp 14489  df-minusg 14490  df-mulg 14492  df-cntz 14793  df-cmn 15091  df-abl 15092  df-mgp 15326  df-rng 15340  df-cring 15341  df-ur 15342  df-cnfld 16378
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