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Theorem tdeglem4 19462
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 2567 . . . . 5  |-  ( E. x  e.  I  -.  ( X `  x )  =  0  <->  -.  A. x  e.  I  ( X `  x )  =  0 )
2 df-ne 2461 . . . . . . 7  |-  ( ( X `  x )  =/=  0  <->  -.  ( X `  x )  =  0 )
3 oveq2 5882 . . . . . . . . . . . 12  |-  ( h  =  X  ->  (fld  gsumg  h )  =  (fld  gsumg  X ) )
4 tdeglem.h . . . . . . . . . . . 12  |-  H  =  ( h  e.  A  |->  (fld 
gsumg  h ) )
5 ovex 5899 . . . . . . . . . . . 12  |-  (fld  gsumg  X )  e.  _V
63, 4, 5fvmpt 5618 . . . . . . . . . . 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 16129 . . . . . . . . . . . . 13  |-  ( ( I  e.  V  /\  X  e.  A )  ->  X : I --> NN0 )
109feqmptd 5591 . . . . . . . . . . . 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 5890 . . . . . . . . . 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 16399 . . . . . . . . . . 11  |-  CC  =  ( Base ` fld )
14 cnfld0 16414 . . . . . . . . . . 11  |-  0  =  ( 0g ` fld )
15 cnfldadd 16400 . . . . . . . . . . 11  |-  +  =  ( +g  ` fld )
16 cnrng 16412 . . . . . . . . . . . 12  |-fld  e.  Ring
17 rngcmn 15387 . . . . . . . . . . . 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 5679 . . . . . . . . . . . . 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 10036 . . . . . . . . . . 11  |-  ( ( ( ( I  e.  V  /\  X  e.  A )  /\  (
x  e.  I  /\  ( X `  x )  =/=  0 ) )  /\  y  e.  I
)  ->  ( X `  y )  e.  CC )
2411cnveqd 4873 . . . . . . . . . . . . 13  |-  ( ( ( I  e.  V  /\  X  e.  A
)  /\  ( x  e.  I  /\  ( X `  x )  =/=  0 ) )  ->  `' X  =  `' ( y  e.  I  |->  ( X `  y
) ) )
2524imaeq1d 5027 . . . . . . . . . . . 12  |-  ( ( ( I  e.  V  /\  X  e.  A
)  /\  ( x  e.  I  /\  ( X `  x )  =/=  0 ) )  -> 
( `' X "
( _V  \  {
0 } ) )  =  ( `' ( y  e.  I  |->  ( X `  y ) ) " ( _V 
\  { 0 } ) ) )
268psrbagsuppfi 16262 . . . . . . . . . . . . . 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 2371 . . . . . . . . . . 11  |-  ( ( ( I  e.  V  /\  X  e.  A
)  /\  ( x  e.  I  /\  ( X `  x )  =/=  0 ) )  -> 
( `' ( y  e.  I  |->  ( X `
 y ) )
" ( _V  \  { 0 } ) )  e.  Fin )
30 incom 3374 . . . . . . . . . . . . 13  |-  ( ( I  \  { x } )  i^i  {
x } )  =  ( { x }  i^i  ( I  \  {
x } ) )
31 disjdif 3539 . . . . . . . . . . . . 13  |-  ( { x }  i^i  (
I  \  { x } ) )  =  (/)
3230, 31eqtri 2316 . . . . . . . . . . . 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 3777 . . . . . . . . . . . . 13  |-  ( x  e.  I  ->  (
( I  \  {
x } )  u. 
{ x } )  =  I )
3534eqcomd 2301 . . . . . . . . . . . 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 15224 . . . . . . . . . 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 2332 . . . . . . . . 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 4178 . . . . . . . . . . . . 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 16443 . . . . . . . . . . . . 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 3311 . . . . . . . . . . . . . 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 2296 . . . . . . . . . . . . 13  |-  ( y  e.  ( I  \  { x } ) 
|->  ( X `  y
) )  =  ( y  e.  ( I 
\  { x }
)  |->  ( X `  y ) )
4644, 45fmptd 5700 . . . . . . . . . . . 12  |-  ( ( ( I  e.  V  /\  X  e.  A
)  /\  ( x  e.  I  /\  ( X `  x )  =/=  0 ) )  -> 
( y  e.  ( I  \  { x } )  |->  ( X `
 y ) ) : ( I  \  { x } ) --> NN0 )
47 difss 3316 . . . . . . . . . . . . . . . 16  |-  ( I 
\  { x }
)  C_  I
48 resmpt 5016 . . . . . . . . . . . . . . . 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 4995 . . . . . . . . . . . . . . 15  |-  ( ( y  e.  I  |->  ( X `  y ) )  |`  ( I  \  { x } ) )  C_  ( y  e.  I  |->  ( X `
 y ) )
5149, 50eqsstr3i 3222 . . . . . . . . . . . . . 14  |-  ( y  e.  ( I  \  { x } ) 
|->  ( X `  y
) )  C_  (
y  e.  I  |->  ( X `  y ) )
52 cnvss 4870 . . . . . . . . . . . . . 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 5064 . . . . . . . . . . . . . 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 7099 . . . . . . . . . . . . 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 15217 . . . . . . . . . . 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 15366 . . . . . . . . . . . . . 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 5679 . . . . . . . . . . . . . . 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 10036 . . . . . . . . . . . . 13  |-  ( ( ( I  e.  V  /\  X  e.  A
)  /\  ( x  e.  I  /\  ( X `  x )  =/=  0 ) )  -> 
( X `  x
)  e.  CC )
64 fveq2 5541 . . . . . . . . . . . . . 14  |-  ( y  =  x  ->  ( X `  y )  =  ( X `  x ) )
6513, 64gsumsn 15236 . . . . . . . . . . . . 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 9983 . . . . . . . . . . . . . 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 2370 . . . . . . . . . . 11  |-  ( ( ( I  e.  V  /\  X  e.  A
)  /\  ( x  e.  I  /\  ( X `  x )  =/=  0 ) )  -> 
(fld  gsumg  ( y  e.  { x }  |->  ( X `  y ) ) )  e.  NN )
74 nn0nnaddcl 10012 . . . . . . . . . . 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 9806 . . . . . . . . 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 2477 . . . . . . . 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 2680 . . . . 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 2521 . . 3  |-  ( ( I  e.  V  /\  X  e.  A )  ->  ( ( H `  X )  =  0  ->  A. x  e.  I 
( X `  x
)  =  0 ) )
83 ffn 5405 . . . . . 6  |-  ( X : I --> NN0  ->  X  Fn  I )
849, 83syl 15 . . . . 5  |-  ( ( I  e.  V  /\  X  e.  A )  ->  X  Fn  I )
85 0nn0 9996 . . . . . 6  |-  0  e.  NN0
86 fnconstg 5445 . . . . . 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 5638 . . . . 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 8848 . . . . . . 7  |-  0  e.  _V
9190fvconst2 5745 . . . . . 6  |-  ( x  e.  I  ->  (
( I  X.  {
0 } ) `  x )  =  0 )
9291eqeq2d 2307 . . . . 5  |-  ( x  e.  I  ->  (
( X `  x
)  =  ( ( I  X.  { 0 } ) `  x
)  <->  ( X `  x )  =  0 ) )
9392ralbiia 2588 . . . 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 16251 . . . . . 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 5882 . . . . . 6  |-  ( h  =  ( I  X.  { 0 } )  ->  (fld 
gsumg  h )  =  (fld  gsumg  ( I  X.  { 0 } ) ) )
99 ovex 5899 . . . . . 6  |-  (fld  gsumg  ( I  X.  {
0 } ) )  e.  _V
10098, 4, 99fvmpt 5618 . . . . 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 4748 . . . . . 6  |-  ( I  X.  { 0 } )  =  ( x  e.  I  |->  0 )
103102oveq2i 5885 . . . . 5  |-  (fld  gsumg  ( I  X.  {
0 } ) )  =  (fld 
gsumg  ( x  e.  I  |->  0 ) )
10416, 58ax-mp 8 . . . . . . 7  |-fld  e.  Mnd
10514gsumz 14474 . . . . . . 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 2340 . . . 4  |-  ( ( I  e.  V  /\  X  e.  A )  ->  (fld 
gsumg  ( I  X.  { 0 } ) )  =  0 )
109101, 108eqtrd 2328 . . 3  |-  ( ( I  e.  V  /\  X  e.  A )  ->  ( H `  (
I  X.  { 0 } ) )  =  0 )
110 fveq2 5541 . . . 4  |-  ( X  =  ( I  X.  { 0 } )  ->  ( H `  X )  =  ( H `  ( I  X.  { 0 } ) ) )
111110eqeq1d 2304 . . 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 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   E.wrex 2557   {crab 2560   _Vcvv 2801    \ cdif 3162    u. cun 3163    i^i cin 3164    C_ wss 3165   (/)c0 3468   {csn 3653    e. cmpt 4093    X. cxp 4703   `'ccnv 4704    |` cres 4707   "cima 4708    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874    ^m cmap 6788   Fincfn 6879   CCcc 8751   0cc0 8753    + caddc 8756   NNcn 9762   NN0cn0 9981    gsumg cgsu 13417   Mndcmnd 14377  SubMndcsubmnd 14430  CMndccmn 15105   Ringcrg 15353  ℂfldccnfld 16393
This theorem is referenced by:  mdegle0  19479
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-addf 8832  ax-mulf 8833
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-oi 7241  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-fz 10799  df-fzo 10887  df-seq 11063  df-hash 11354  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-starv 13239  df-tset 13243  df-ple 13244  df-ds 13246  df-0g 13420  df-gsum 13421  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-submnd 14432  df-grp 14505  df-minusg 14506  df-mulg 14508  df-cntz 14809  df-cmn 15107  df-abl 15108  df-mgp 15342  df-rng 15356  df-cring 15357  df-ur 15358  df-cnfld 16394
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