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Theorem tdeglem4 19936
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 2677 . . . . 5  |-  ( E. x  e.  I  -.  ( X `  x )  =  0  <->  -.  A. x  e.  I  ( X `  x )  =  0 )
2 df-ne 2569 . . . . . . 7  |-  ( ( X `  x )  =/=  0  <->  -.  ( X `  x )  =  0 )
3 oveq2 6048 . . . . . . . . . . . 12  |-  ( h  =  X  ->  (fld  gsumg  h )  =  (fld  gsumg  X ) )
4 tdeglem.h . . . . . . . . . . . 12  |-  H  =  ( h  e.  A  |->  (fld 
gsumg  h ) )
5 ovex 6065 . . . . . . . . . . . 12  |-  (fld  gsumg  X )  e.  _V
63, 4, 5fvmpt 5765 . . . . . . . . . . 11  |-  ( X  e.  A  ->  ( H `  X )  =  (fld 
gsumg  X ) )
76ad2antlr 708 . . . . . . . . . 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 16387 . . . . . . . . . . . . 13  |-  ( ( I  e.  V  /\  X  e.  A )  ->  X : I --> NN0 )
109feqmptd 5738 . . . . . . . . . . . 12  |-  ( ( I  e.  V  /\  X  e.  A )  ->  X  =  ( y  e.  I  |->  ( X `
 y ) ) )
1110adantr 452 . . . . . . . . . . 11  |-  ( ( ( I  e.  V  /\  X  e.  A
)  /\  ( x  e.  I  /\  ( X `  x )  =/=  0 ) )  ->  X  =  ( y  e.  I  |->  ( X `
 y ) ) )
1211oveq2d 6056 . . . . . . . . . 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 16662 . . . . . . . . . . 11  |-  CC  =  ( Base ` fld )
14 cnfld0 16680 . . . . . . . . . . 11  |-  0  =  ( 0g ` fld )
15 cnfldadd 16663 . . . . . . . . . . 11  |-  +  =  ( +g  ` fld )
16 cnrng 16678 . . . . . . . . . . . 12  |-fld  e.  Ring
17 rngcmn 15649 . . . . . . . . . . . 12  |-  (fld  e.  Ring  ->fld  e. CMnd )
1816, 17mp1i 12 . . . . . . . . . . 11  |-  ( ( ( I  e.  V  /\  X  e.  A
)  /\  ( x  e.  I  /\  ( X `  x )  =/=  0 ) )  ->fld  e. CMnd )
19 simpll 731 . . . . . . . . . . 11  |-  ( ( ( I  e.  V  /\  X  e.  A
)  /\  ( x  e.  I  /\  ( X `  x )  =/=  0 ) )  ->  I  e.  V )
209adantr 452 . . . . . . . . . . . . 13  |-  ( ( ( I  e.  V  /\  X  e.  A
)  /\  ( x  e.  I  /\  ( X `  x )  =/=  0 ) )  ->  X : I --> NN0 )
2120ffvelrnda 5829 . . . . . . . . . . . 12  |-  ( ( ( ( I  e.  V  /\  X  e.  A )  /\  (
x  e.  I  /\  ( X `  x )  =/=  0 ) )  /\  y  e.  I
)  ->  ( X `  y )  e.  NN0 )
2221nn0cnd 10232 . . . . . . . . . . 11  |-  ( ( ( ( I  e.  V  /\  X  e.  A )  /\  (
x  e.  I  /\  ( X `  x )  =/=  0 ) )  /\  y  e.  I
)  ->  ( X `  y )  e.  CC )
2311cnveqd 5007 . . . . . . . . . . . . 13  |-  ( ( ( I  e.  V  /\  X  e.  A
)  /\  ( x  e.  I  /\  ( X `  x )  =/=  0 ) )  ->  `' X  =  `' ( y  e.  I  |->  ( X `  y
) ) )
2423imaeq1d 5161 . . . . . . . . . . . 12  |-  ( ( ( I  e.  V  /\  X  e.  A
)  /\  ( x  e.  I  /\  ( X `  x )  =/=  0 ) )  -> 
( `' X "
( _V  \  {
0 } ) )  =  ( `' ( y  e.  I  |->  ( X `  y ) ) " ( _V 
\  { 0 } ) ) )
258psrbagsuppfi 16520 . . . . . . . . . . . . . 14  |-  ( ( X  e.  A  /\  I  e.  V )  ->  ( `' X "
( _V  \  {
0 } ) )  e.  Fin )
2625ancoms 440 . . . . . . . . . . . . 13  |-  ( ( I  e.  V  /\  X  e.  A )  ->  ( `' X "
( _V  \  {
0 } ) )  e.  Fin )
2726adantr 452 . . . . . . . . . . . 12  |-  ( ( ( I  e.  V  /\  X  e.  A
)  /\  ( x  e.  I  /\  ( X `  x )  =/=  0 ) )  -> 
( `' X "
( _V  \  {
0 } ) )  e.  Fin )
2824, 27eqeltrrd 2479 . . . . . . . . . . 11  |-  ( ( ( I  e.  V  /\  X  e.  A
)  /\  ( x  e.  I  /\  ( X `  x )  =/=  0 ) )  -> 
( `' ( y  e.  I  |->  ( X `
 y ) )
" ( _V  \  { 0 } ) )  e.  Fin )
29 incom 3493 . . . . . . . . . . . . 13  |-  ( ( I  \  { x } )  i^i  {
x } )  =  ( { x }  i^i  ( I  \  {
x } ) )
30 disjdif 3660 . . . . . . . . . . . . 13  |-  ( { x }  i^i  (
I  \  { x } ) )  =  (/)
3129, 30eqtri 2424 . . . . . . . . . . . 12  |-  ( ( I  \  { x } )  i^i  {
x } )  =  (/)
3231a1i 11 . . . . . . . . . . 11  |-  ( ( ( I  e.  V  /\  X  e.  A
)  /\  ( x  e.  I  /\  ( X `  x )  =/=  0 ) )  -> 
( ( I  \  { x } )  i^i  { x }
)  =  (/) )
33 difsnid 3904 . . . . . . . . . . . . 13  |-  ( x  e.  I  ->  (
( I  \  {
x } )  u. 
{ x } )  =  I )
3433eqcomd 2409 . . . . . . . . . . . 12  |-  ( x  e.  I  ->  I  =  ( ( I 
\  { x }
)  u.  { x } ) )
3534ad2antrl 709 . . . . . . . . . . 11  |-  ( ( ( I  e.  V  /\  X  e.  A
)  /\  ( x  e.  I  /\  ( X `  x )  =/=  0 ) )  ->  I  =  ( (
I  \  { x } )  u.  {
x } ) )
3613, 14, 15, 18, 19, 22, 28, 32, 35gsumsplit2 15486 . . . . . . . . . 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 ) ) ) ) )
377, 12, 363eqtrd 2440 . . . . . . . . 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 ) ) ) ) )
38 difexg 4311 . . . . . . . . . . . . 13  |-  ( I  e.  V  ->  (
I  \  { x } )  e.  _V )
3919, 38syl 16 . . . . . . . . . . . 12  |-  ( ( ( I  e.  V  /\  X  e.  A
)  /\  ( x  e.  I  /\  ( X `  x )  =/=  0 ) )  -> 
( I  \  {
x } )  e. 
_V )
40 nn0subm 16709 . . . . . . . . . . . . 13  |-  NN0  e.  (SubMnd ` fld )
4140a1i 11 . . . . . . . . . . . 12  |-  ( ( ( I  e.  V  /\  X  e.  A
)  /\  ( x  e.  I  /\  ( X `  x )  =/=  0 ) )  ->  NN0  e.  (SubMnd ` fld ) )
42 eldifi 3429 . . . . . . . . . . . . . 14  |-  ( y  e.  ( I  \  { x } )  ->  y  e.  I
)
43 ffvelrn 5827 . . . . . . . . . . . . . 14  |-  ( ( X : I --> NN0  /\  y  e.  I )  ->  ( X `  y
)  e.  NN0 )
4420, 42, 43syl2an 464 . . . . . . . . . . . . 13  |-  ( ( ( ( I  e.  V  /\  X  e.  A )  /\  (
x  e.  I  /\  ( X `  x )  =/=  0 ) )  /\  y  e.  ( I  \  { x } ) )  -> 
( X `  y
)  e.  NN0 )
45 eqid 2404 . . . . . . . . . . . . 13  |-  ( y  e.  ( I  \  { x } ) 
|->  ( X `  y
) )  =  ( y  e.  ( I 
\  { x }
)  |->  ( X `  y ) )
4644, 45fmptd 5852 . . . . . . . . . . . 12  |-  ( ( ( I  e.  V  /\  X  e.  A
)  /\  ( x  e.  I  /\  ( X `  x )  =/=  0 ) )  -> 
( y  e.  ( I  \  { x } )  |->  ( X `
 y ) ) : ( I  \  { x } ) --> NN0 )
47 difss 3434 . . . . . . . . . . . . . . . 16  |-  ( I 
\  { x }
)  C_  I
48 resmpt 5150 . . . . . . . . . . . . . . . 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 5129 . . . . . . . . . . . . . . 15  |-  ( ( y  e.  I  |->  ( X `  y ) )  |`  ( I  \  { x } ) )  C_  ( y  e.  I  |->  ( X `
 y ) )
5149, 50eqsstr3i 3339 . . . . . . . . . . . . . 14  |-  ( y  e.  ( I  \  { x } ) 
|->  ( X `  y
) )  C_  (
y  e.  I  |->  ( X `  y ) )
52 cnvss 5004 . . . . . . . . . . . . . 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 5198 . . . . . . . . . . . . . 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 10 . . . . . . . . . . . . 13  |-  ( `' ( y  e.  ( I  \  { x } )  |->  ( X `
 y ) )
" ( _V  \  { 0 } ) )  C_  ( `' ( y  e.  I  |->  ( X `  y
) ) " ( _V  \  { 0 } ) )
55 ssfi 7288 . . . . . . . . . . . . 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 )
5628, 54, 55sylancl 644 . . . . . . . . . . . 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, 39, 41, 46, 56gsumsubmcl 15479 . . . . . . . . . . 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 15628 . . . . . . . . . . . . . 14  |-  (fld  e.  Ring  ->fld  e.  Mnd )
5916, 58mp1i 12 . . . . . . . . . . . . 13  |-  ( ( ( I  e.  V  /\  X  e.  A
)  /\  ( x  e.  I  /\  ( X `  x )  =/=  0 ) )  ->fld  e.  Mnd )
60 simprl 733 . . . . . . . . . . . . 13  |-  ( ( ( I  e.  V  /\  X  e.  A
)  /\  ( x  e.  I  /\  ( X `  x )  =/=  0 ) )  ->  x  e.  I )
6120, 60ffvelrnd 5830 . . . . . . . . . . . . . 14  |-  ( ( ( I  e.  V  /\  X  e.  A
)  /\  ( x  e.  I  /\  ( X `  x )  =/=  0 ) )  -> 
( X `  x
)  e.  NN0 )
6261nn0cnd 10232 . . . . . . . . . . . . 13  |-  ( ( ( I  e.  V  /\  X  e.  A
)  /\  ( x  e.  I  /\  ( X `  x )  =/=  0 ) )  -> 
( X `  x
)  e.  CC )
63 fveq2 5687 . . . . . . . . . . . . . 14  |-  ( y  =  x  ->  ( X `  y )  =  ( X `  x ) )
6413, 63gsumsn 15498 . . . . . . . . . . . . 13  |-  ( (fld  e. 
Mnd  /\  x  e.  I  /\  ( X `  x )  e.  CC )  ->  (fld 
gsumg  ( y  e.  {
x }  |->  ( X `
 y ) ) )  =  ( X `
 x ) )
6559, 60, 62, 64syl3anc 1184 . . . . . . . . . . . 12  |-  ( ( ( I  e.  V  /\  X  e.  A
)  /\  ( x  e.  I  /\  ( X `  x )  =/=  0 ) )  -> 
(fld  gsumg  ( y  e.  { x }  |->  ( X `  y ) ) )  =  ( X `  x ) )
66 simprr 734 . . . . . . . . . . . . . 14  |-  ( ( ( I  e.  V  /\  X  e.  A
)  /\  ( x  e.  I  /\  ( X `  x )  =/=  0 ) )  -> 
( X `  x
)  =/=  0 )
6766, 2sylib 189 . . . . . . . . . . . . 13  |-  ( ( ( I  e.  V  /\  X  e.  A
)  /\  ( x  e.  I  /\  ( X `  x )  =/=  0 ) )  ->  -.  ( X `  x
)  =  0 )
68 elnn0 10179 . . . . . . . . . . . . . 14  |-  ( ( X `  x )  e.  NN0  <->  ( ( X `
 x )  e.  NN  \/  ( X `
 x )  =  0 ) )
6961, 68sylib 189 . . . . . . . . . . . . 13  |-  ( ( ( I  e.  V  /\  X  e.  A
)  /\  ( x  e.  I  /\  ( X `  x )  =/=  0 ) )  -> 
( ( X `  x )  e.  NN  \/  ( X `  x
)  =  0 ) )
70 orel2 373 . . . . . . . . . . . . 13  |-  ( -.  ( X `  x
)  =  0  -> 
( ( ( X `
 x )  e.  NN  \/  ( X `
 x )  =  0 )  ->  ( X `  x )  e.  NN ) )
7167, 69, 70sylc 58 . . . . . . . . . . . 12  |-  ( ( ( I  e.  V  /\  X  e.  A
)  /\  ( x  e.  I  /\  ( X `  x )  =/=  0 ) )  -> 
( X `  x
)  e.  NN )
7265, 71eqeltrd 2478 . . . . . . . . . . 11  |-  ( ( ( I  e.  V  /\  X  e.  A
)  /\  ( x  e.  I  /\  ( X `  x )  =/=  0 ) )  -> 
(fld  gsumg  ( y  e.  { x }  |->  ( X `  y ) ) )  e.  NN )
73 nn0nnaddcl 10208 . . . . . . . . . . 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 )
7457, 72, 73syl2anc 643 . . . . . . . . . 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 )
7574nnne0d 10000 . . . . . . . . 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
)
7637, 75eqnetrd 2585 . . . . . . . 8  |-  ( ( ( I  e.  V  /\  X  e.  A
)  /\  ( x  e.  I  /\  ( X `  x )  =/=  0 ) )  -> 
( H `  X
)  =/=  0 )
7776expr 599 . . . . . . 7  |-  ( ( ( I  e.  V  /\  X  e.  A
)  /\  x  e.  I )  ->  (
( X `  x
)  =/=  0  -> 
( H `  X
)  =/=  0 ) )
782, 77syl5bir 210 . . . . . 6  |-  ( ( ( I  e.  V  /\  X  e.  A
)  /\  x  e.  I )  ->  ( -.  ( X `  x
)  =  0  -> 
( H `  X
)  =/=  0 ) )
7978rexlimdva 2790 . . . . 5  |-  ( ( I  e.  V  /\  X  e.  A )  ->  ( E. x  e.  I  -.  ( X `
 x )  =  0  ->  ( H `  X )  =/=  0
) )
801, 79syl5bir 210 . . . 4  |-  ( ( I  e.  V  /\  X  e.  A )  ->  ( -.  A. x  e.  I  ( X `  x )  =  0  ->  ( H `  X )  =/=  0
) )
8180necon4bd 2629 . . 3  |-  ( ( I  e.  V  /\  X  e.  A )  ->  ( ( H `  X )  =  0  ->  A. x  e.  I 
( X `  x
)  =  0 ) )
82 ffn 5550 . . . . . 6  |-  ( X : I --> NN0  ->  X  Fn  I )
839, 82syl 16 . . . . 5  |-  ( ( I  e.  V  /\  X  e.  A )  ->  X  Fn  I )
84 0nn0 10192 . . . . . 6  |-  0  e.  NN0
85 fnconstg 5590 . . . . . 6  |-  ( 0  e.  NN0  ->  ( I  X.  { 0 } )  Fn  I )
8684, 85mp1i 12 . . . . 5  |-  ( ( I  e.  V  /\  X  e.  A )  ->  ( I  X.  {
0 } )  Fn  I )
87 eqfnfv 5786 . . . . 5  |-  ( ( X  Fn  I  /\  ( I  X.  { 0 } )  Fn  I
)  ->  ( X  =  ( I  X.  { 0 } )  <->  A. x  e.  I 
( X `  x
)  =  ( ( I  X.  { 0 } ) `  x
) ) )
8883, 86, 87syl2anc 643 . . . 4  |-  ( ( I  e.  V  /\  X  e.  A )  ->  ( X  =  ( I  X.  { 0 } )  <->  A. x  e.  I  ( X `  x )  =  ( ( I  X.  {
0 } ) `  x ) ) )
89 c0ex 9041 . . . . . . 7  |-  0  e.  _V
9089fvconst2 5906 . . . . . 6  |-  ( x  e.  I  ->  (
( I  X.  {
0 } ) `  x )  =  0 )
9190eqeq2d 2415 . . . . 5  |-  ( x  e.  I  ->  (
( X `  x
)  =  ( ( I  X.  { 0 } ) `  x
)  <->  ( X `  x )  =  0 ) )
9291ralbiia 2698 . . . 4  |-  ( A. x  e.  I  ( X `  x )  =  ( ( I  X.  { 0 } ) `  x )  <->  A. x  e.  I 
( X `  x
)  =  0 )
9388, 92syl6bb 253 . . 3  |-  ( ( I  e.  V  /\  X  e.  A )  ->  ( X  =  ( I  X.  { 0 } )  <->  A. x  e.  I  ( X `  x )  =  0 ) )
9481, 93sylibrd 226 . 2  |-  ( ( I  e.  V  /\  X  e.  A )  ->  ( ( H `  X )  =  0  ->  X  =  ( I  X.  { 0 } ) ) )
958psrbag0 16509 . . . . . 6  |-  ( I  e.  V  ->  (
I  X.  { 0 } )  e.  A
)
9695adantr 452 . . . . 5  |-  ( ( I  e.  V  /\  X  e.  A )  ->  ( I  X.  {
0 } )  e.  A )
97 oveq2 6048 . . . . . 6  |-  ( h  =  ( I  X.  { 0 } )  ->  (fld 
gsumg  h )  =  (fld  gsumg  ( I  X.  { 0 } ) ) )
98 ovex 6065 . . . . . 6  |-  (fld  gsumg  ( I  X.  {
0 } ) )  e.  _V
9997, 4, 98fvmpt 5765 . . . . 5  |-  ( ( I  X.  { 0 } )  e.  A  ->  ( H `  (
I  X.  { 0 } ) )  =  (fld 
gsumg  ( I  X.  { 0 } ) ) )
10096, 99syl 16 . . . 4  |-  ( ( I  e.  V  /\  X  e.  A )  ->  ( H `  (
I  X.  { 0 } ) )  =  (fld 
gsumg  ( I  X.  { 0 } ) ) )
101 fconstmpt 4880 . . . . . 6  |-  ( I  X.  { 0 } )  =  ( x  e.  I  |->  0 )
102101oveq2i 6051 . . . . 5  |-  (fld  gsumg  ( I  X.  {
0 } ) )  =  (fld 
gsumg  ( x  e.  I  |->  0 ) )
10316, 58ax-mp 8 . . . . . . 7  |-fld  e.  Mnd
10414gsumz 14736 . . . . . . 7  |-  ( (fld  e. 
Mnd  /\  I  e.  V )  ->  (fld  gsumg  ( x  e.  I  |->  0 ) )  =  0 )
105103, 104mpan 652 . . . . . 6  |-  ( I  e.  V  ->  (fld  gsumg  ( x  e.  I  |->  0 ) )  =  0 )
106105adantr 452 . . . . 5  |-  ( ( I  e.  V  /\  X  e.  A )  ->  (fld 
gsumg  ( x  e.  I  |->  0 ) )  =  0 )
107102, 106syl5eq 2448 . . . 4  |-  ( ( I  e.  V  /\  X  e.  A )  ->  (fld 
gsumg  ( I  X.  { 0 } ) )  =  0 )
108100, 107eqtrd 2436 . . 3  |-  ( ( I  e.  V  /\  X  e.  A )  ->  ( H `  (
I  X.  { 0 } ) )  =  0 )
109 fveq2 5687 . . . 4  |-  ( X  =  ( I  X.  { 0 } )  ->  ( H `  X )  =  ( H `  ( I  X.  { 0 } ) ) )
110109eqeq1d 2412 . . 3  |-  ( X  =  ( I  X.  { 0 } )  ->  ( ( H `
 X )  =  0  <->  ( H `  ( I  X.  { 0 } ) )  =  0 ) )
111108, 110syl5ibrcom 214 . 2  |-  ( ( I  e.  V  /\  X  e.  A )  ->  ( X  =  ( I  X.  { 0 } )  ->  ( H `  X )  =  0 ) )
11294, 111impbid 184 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 177    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2567   A.wral 2666   E.wrex 2667   {crab 2670   _Vcvv 2916    \ cdif 3277    u. cun 3278    i^i cin 3279    C_ wss 3280   (/)c0 3588   {csn 3774    e. cmpt 4226    X. cxp 4835   `'ccnv 4836    |` cres 4839   "cima 4840    Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6040    ^m cmap 6977   Fincfn 7068   CCcc 8944   0cc0 8946    + caddc 8949   NNcn 9956   NN0cn0 10177    gsumg cgsu 13679   Mndcmnd 14639  SubMndcsubmnd 14692  CMndccmn 15367   Ringcrg 15615  ℂfldccnfld 16658
This theorem is referenced by:  mdegle0  19953
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-addf 9025  ax-mulf 9026
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-iin 4056  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-oadd 6687  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-oi 7435  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-fz 11000  df-fzo 11091  df-seq 11279  df-hash 11574  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-0g 13682  df-gsum 13683  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-submnd 14694  df-grp 14767  df-minusg 14768  df-mulg 14770  df-cntz 15071  df-cmn 15369  df-abl 15370  df-mgp 15604  df-rng 15618  df-cring 15619  df-ur 15620  df-cnfld 16659
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