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Theorem xrge0slmod 26311
Description: The extended nonnegative real numbers form a semiring left module. One could also have used subringAlg to get the same structure. (Contributed by Thierry Arnoux, 6-Sep-2018.)
Hypotheses
Ref Expression
xrge0slmod.1  |-  G  =  ( RR*ss  ( 0 [,] +oo ) )
xrge0slmod.2  |-  W  =  ( Gv  ( 0 [,) +oo ) )
Assertion
Ref Expression
xrge0slmod  |-  W  e. SLMod

Proof of Theorem xrge0slmod
Dummy variables  r 
q  w  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xrge0slmod.1 . . . 4  |-  G  =  ( RR*ss  ( 0 [,] +oo ) )
2 xrge0cmn 17854 . . . 4  |-  ( RR*ss  ( 0 [,] +oo ) )  e. CMnd
31, 2eqeltri 2512 . . 3  |-  G  e. CMnd
4 ovex 6115 . . . 4  |-  ( 0 [,) +oo )  e. 
_V
5 xrge0slmod.2 . . . . 5  |-  W  =  ( Gv  ( 0 [,) +oo ) )
65resvcmn 26305 . . . 4  |-  ( ( 0 [,) +oo )  e.  _V  ->  ( G  e. CMnd  <-> 
W  e. CMnd ) )
74, 6ax-mp 5 . . 3  |-  ( G  e. CMnd 
<->  W  e. CMnd )
83, 7mpbi 208 . 2  |-  W  e. CMnd
9 rge0srg 17881 . 2  |-  (flds  ( 0 [,) +oo ) )  e. SRing
10 icossicc 26057 . . . . . . . 8  |-  ( 0 [,) +oo )  C_  ( 0 [,] +oo )
11 simplr 754 . . . . . . . 8  |-  ( ( ( q  e.  ( 0 [,) +oo )  /\  r  e.  (
0 [,) +oo )
)  /\  ( x  e.  ( 0 [,] +oo )  /\  w  e.  ( 0 [,] +oo )
) )  ->  r  e.  ( 0 [,) +oo ) )
1210, 11sseldi 3353 . . . . . . 7  |-  ( ( ( q  e.  ( 0 [,) +oo )  /\  r  e.  (
0 [,) +oo )
)  /\  ( x  e.  ( 0 [,] +oo )  /\  w  e.  ( 0 [,] +oo )
) )  ->  r  e.  ( 0 [,] +oo ) )
13 simprr 756 . . . . . . 7  |-  ( ( ( q  e.  ( 0 [,) +oo )  /\  r  e.  (
0 [,) +oo )
)  /\  ( x  e.  ( 0 [,] +oo )  /\  w  e.  ( 0 [,] +oo )
) )  ->  w  e.  ( 0 [,] +oo ) )
14 ge0xmulcl 11399 . . . . . . 7  |-  ( ( r  e.  ( 0 [,] +oo )  /\  w  e.  ( 0 [,] +oo ) )  ->  ( r xe w )  e.  ( 0 [,] +oo ) )
1512, 13, 14syl2anc 661 . . . . . 6  |-  ( ( ( q  e.  ( 0 [,) +oo )  /\  r  e.  (
0 [,) +oo )
)  /\  ( x  e.  ( 0 [,] +oo )  /\  w  e.  ( 0 [,] +oo )
) )  ->  (
r xe w )  e.  ( 0 [,] +oo ) )
16 simprl 755 . . . . . . 7  |-  ( ( ( q  e.  ( 0 [,) +oo )  /\  r  e.  (
0 [,) +oo )
)  /\  ( x  e.  ( 0 [,] +oo )  /\  w  e.  ( 0 [,] +oo )
) )  ->  x  e.  ( 0 [,] +oo ) )
17 xrge0adddi 26155 . . . . . . 7  |-  ( ( w  e.  ( 0 [,] +oo )  /\  x  e.  ( 0 [,] +oo )  /\  r  e.  ( 0 [,] +oo ) )  ->  ( r xe ( w +e x ) )  =  ( ( r xe w ) +e ( r xe x ) ) )
1813, 16, 12, 17syl3anc 1218 . . . . . 6  |-  ( ( ( q  e.  ( 0 [,) +oo )  /\  r  e.  (
0 [,) +oo )
)  /\  ( x  e.  ( 0 [,] +oo )  /\  w  e.  ( 0 [,] +oo )
) )  ->  (
r xe ( w +e x ) )  =  ( ( r xe w ) +e
( r xe x ) ) )
19 rge0ssre 11392 . . . . . . . . . 10  |-  ( 0 [,) +oo )  C_  RR
20 simpll 753 . . . . . . . . . 10  |-  ( ( ( q  e.  ( 0 [,) +oo )  /\  r  e.  (
0 [,) +oo )
)  /\  ( x  e.  ( 0 [,] +oo )  /\  w  e.  ( 0 [,] +oo )
) )  ->  q  e.  ( 0 [,) +oo ) )
2119, 20sseldi 3353 . . . . . . . . 9  |-  ( ( ( q  e.  ( 0 [,) +oo )  /\  r  e.  (
0 [,) +oo )
)  /\  ( x  e.  ( 0 [,] +oo )  /\  w  e.  ( 0 [,] +oo )
) )  ->  q  e.  RR )
2219, 11sseldi 3353 . . . . . . . . 9  |-  ( ( ( q  e.  ( 0 [,) +oo )  /\  r  e.  (
0 [,) +oo )
)  /\  ( x  e.  ( 0 [,] +oo )  /\  w  e.  ( 0 [,] +oo )
) )  ->  r  e.  RR )
23 rexadd 11201 . . . . . . . . 9  |-  ( ( q  e.  RR  /\  r  e.  RR )  ->  ( q +e
r )  =  ( q  +  r ) )
2421, 22, 23syl2anc 661 . . . . . . . 8  |-  ( ( ( q  e.  ( 0 [,) +oo )  /\  r  e.  (
0 [,) +oo )
)  /\  ( x  e.  ( 0 [,] +oo )  /\  w  e.  ( 0 [,] +oo )
) )  ->  (
q +e r )  =  ( q  +  r ) )
2524oveq1d 6105 . . . . . . 7  |-  ( ( ( q  e.  ( 0 [,) +oo )  /\  r  e.  (
0 [,) +oo )
)  /\  ( x  e.  ( 0 [,] +oo )  /\  w  e.  ( 0 [,] +oo )
) )  ->  (
( q +e
r ) xe w )  =  ( ( q  +  r ) xe w ) )
2610, 20sseldi 3353 . . . . . . . 8  |-  ( ( ( q  e.  ( 0 [,) +oo )  /\  r  e.  (
0 [,) +oo )
)  /\  ( x  e.  ( 0 [,] +oo )  /\  w  e.  ( 0 [,] +oo )
) )  ->  q  e.  ( 0 [,] +oo ) )
27 xrge0adddir 26154 . . . . . . . 8  |-  ( ( q  e.  ( 0 [,] +oo )  /\  r  e.  ( 0 [,] +oo )  /\  w  e.  ( 0 [,] +oo ) )  ->  ( ( q +e r ) xe w )  =  ( ( q xe w ) +e ( r xe w ) ) )
2826, 12, 13, 27syl3anc 1218 . . . . . . 7  |-  ( ( ( q  e.  ( 0 [,) +oo )  /\  r  e.  (
0 [,) +oo )
)  /\  ( x  e.  ( 0 [,] +oo )  /\  w  e.  ( 0 [,] +oo )
) )  ->  (
( q +e
r ) xe w )  =  ( ( q xe w ) +e
( r xe w ) ) )
2925, 28eqtr3d 2476 . . . . . 6  |-  ( ( ( q  e.  ( 0 [,) +oo )  /\  r  e.  (
0 [,) +oo )
)  /\  ( x  e.  ( 0 [,] +oo )  /\  w  e.  ( 0 [,] +oo )
) )  ->  (
( q  +  r ) xe w )  =  ( ( q xe w ) +e ( r xe w ) ) )
3015, 18, 293jca 1168 . . . . 5  |-  ( ( ( q  e.  ( 0 [,) +oo )  /\  r  e.  (
0 [,) +oo )
)  /\  ( x  e.  ( 0 [,] +oo )  /\  w  e.  ( 0 [,] +oo )
) )  ->  (
( r xe w )  e.  ( 0 [,] +oo )  /\  ( r xe ( w +e
x ) )  =  ( ( r xe w ) +e ( r xe x ) )  /\  ( ( q  +  r ) xe w )  =  ( ( q xe w ) +e ( r xe w ) ) ) )
31 rexmul 11233 . . . . . . . . 9  |-  ( ( q  e.  RR  /\  r  e.  RR )  ->  ( q xe r )  =  ( q  x.  r ) )
3221, 22, 31syl2anc 661 . . . . . . . 8  |-  ( ( ( q  e.  ( 0 [,) +oo )  /\  r  e.  (
0 [,) +oo )
)  /\  ( x  e.  ( 0 [,] +oo )  /\  w  e.  ( 0 [,] +oo )
) )  ->  (
q xe r )  =  ( q  x.  r ) )
3332oveq1d 6105 . . . . . . 7  |-  ( ( ( q  e.  ( 0 [,) +oo )  /\  r  e.  (
0 [,) +oo )
)  /\  ( x  e.  ( 0 [,] +oo )  /\  w  e.  ( 0 [,] +oo )
) )  ->  (
( q xe r ) xe w )  =  ( ( q  x.  r
) xe w ) )
3421rexrd 9432 . . . . . . . 8  |-  ( ( ( q  e.  ( 0 [,) +oo )  /\  r  e.  (
0 [,) +oo )
)  /\  ( x  e.  ( 0 [,] +oo )  /\  w  e.  ( 0 [,] +oo )
) )  ->  q  e.  RR* )
3522rexrd 9432 . . . . . . . 8  |-  ( ( ( q  e.  ( 0 [,) +oo )  /\  r  e.  (
0 [,) +oo )
)  /\  ( x  e.  ( 0 [,] +oo )  /\  w  e.  ( 0 [,] +oo )
) )  ->  r  e.  RR* )
36 iccssxr 11377 . . . . . . . . 9  |-  ( 0 [,] +oo )  C_  RR*
3736, 13sseldi 3353 . . . . . . . 8  |-  ( ( ( q  e.  ( 0 [,) +oo )  /\  r  e.  (
0 [,) +oo )
)  /\  ( x  e.  ( 0 [,] +oo )  /\  w  e.  ( 0 [,] +oo )
) )  ->  w  e.  RR* )
38 xmulass 11249 . . . . . . . 8  |-  ( ( q  e.  RR*  /\  r  e.  RR*  /\  w  e. 
RR* )  ->  (
( q xe r ) xe w )  =  ( q xe ( r xe w ) ) )
3934, 35, 37, 38syl3anc 1218 . . . . . . 7  |-  ( ( ( q  e.  ( 0 [,) +oo )  /\  r  e.  (
0 [,) +oo )
)  /\  ( x  e.  ( 0 [,] +oo )  /\  w  e.  ( 0 [,] +oo )
) )  ->  (
( q xe r ) xe w )  =  ( q xe ( r xe w ) ) )
4033, 39eqtr3d 2476 . . . . . 6  |-  ( ( ( q  e.  ( 0 [,) +oo )  /\  r  e.  (
0 [,) +oo )
)  /\  ( x  e.  ( 0 [,] +oo )  /\  w  e.  ( 0 [,] +oo )
) )  ->  (
( q  x.  r
) xe w )  =  ( q xe ( r xe w ) ) )
41 xmulid2 11242 . . . . . . 7  |-  ( w  e.  RR*  ->  ( 1 xe w )  =  w )
4237, 41syl 16 . . . . . 6  |-  ( ( ( q  e.  ( 0 [,) +oo )  /\  r  e.  (
0 [,) +oo )
)  /\  ( x  e.  ( 0 [,] +oo )  /\  w  e.  ( 0 [,] +oo )
) )  ->  (
1 xe w )  =  w )
43 xmul02 11230 . . . . . . 7  |-  ( w  e.  RR*  ->  ( 0 xe w )  =  0 )
4437, 43syl 16 . . . . . 6  |-  ( ( ( q  e.  ( 0 [,) +oo )  /\  r  e.  (
0 [,) +oo )
)  /\  ( x  e.  ( 0 [,] +oo )  /\  w  e.  ( 0 [,] +oo )
) )  ->  (
0 xe w )  =  0 )
4540, 42, 443jca 1168 . . . . 5  |-  ( ( ( q  e.  ( 0 [,) +oo )  /\  r  e.  (
0 [,) +oo )
)  /\  ( x  e.  ( 0 [,] +oo )  /\  w  e.  ( 0 [,] +oo )
) )  ->  (
( ( q  x.  r ) xe w )  =  ( q xe ( r xe w ) )  /\  (
1 xe w )  =  w  /\  ( 0 xe w )  =  0 ) )
4630, 45jca 532 . . . 4  |-  ( ( ( q  e.  ( 0 [,) +oo )  /\  r  e.  (
0 [,) +oo )
)  /\  ( x  e.  ( 0 [,] +oo )  /\  w  e.  ( 0 [,] +oo )
) )  ->  (
( ( r xe w )  e.  ( 0 [,] +oo )  /\  ( r xe ( w +e x ) )  =  ( ( r xe w ) +e ( r xe x ) )  /\  ( ( q  +  r ) xe w )  =  ( ( q xe w ) +e ( r xe w ) ) )  /\  (
( ( q  x.  r ) xe w )  =  ( q xe ( r xe w ) )  /\  (
1 xe w )  =  w  /\  ( 0 xe w )  =  0 ) ) )
4746ralrimivva 2807 . . 3  |-  ( ( q  e.  ( 0 [,) +oo )  /\  r  e.  ( 0 [,) +oo ) )  ->  A. x  e.  ( 0 [,] +oo ) A. w  e.  (
0 [,] +oo )
( ( ( r xe w )  e.  ( 0 [,] +oo )  /\  (
r xe ( w +e x ) )  =  ( ( r xe w ) +e
( r xe x ) )  /\  ( ( q  +  r ) xe w )  =  ( ( q xe w ) +e
( r xe w ) ) )  /\  ( ( ( q  x.  r ) xe w )  =  ( q xe ( r xe w ) )  /\  ( 1 xe w )  =  w  /\  ( 0 xe w )  =  0 ) ) )
4847rgen2a 2781 . 2  |-  A. q  e.  ( 0 [,) +oo ) A. r  e.  ( 0 [,) +oo ) A. x  e.  (
0 [,] +oo ) A. w  e.  (
0 [,] +oo )
( ( ( r xe w )  e.  ( 0 [,] +oo )  /\  (
r xe ( w +e x ) )  =  ( ( r xe w ) +e
( r xe x ) )  /\  ( ( q  +  r ) xe w )  =  ( ( q xe w ) +e
( r xe w ) ) )  /\  ( ( ( q  x.  r ) xe w )  =  ( q xe ( r xe w ) )  /\  ( 1 xe w )  =  w  /\  ( 0 xe w )  =  0 ) )
49 xrge0base 26145 . . . . . 6  |-  ( 0 [,] +oo )  =  ( Base `  ( RR*ss  ( 0 [,] +oo ) ) )
501fveq2i 5693 . . . . . 6  |-  ( Base `  G )  =  (
Base `  ( RR*ss  ( 0 [,] +oo ) ) )
5149, 50eqtr4i 2465 . . . . 5  |-  ( 0 [,] +oo )  =  ( Base `  G
)
525, 51resvbas 26299 . . . 4  |-  ( ( 0 [,) +oo )  e.  _V  ->  ( 0 [,] +oo )  =  ( Base `  W
) )
534, 52ax-mp 5 . . 3  |-  ( 0 [,] +oo )  =  ( Base `  W
)
54 xrge0plusg 26147 . . . . . 6  |-  +e 
=  ( +g  `  ( RR*ss  ( 0 [,] +oo ) ) )
551fveq2i 5693 . . . . . 6  |-  ( +g  `  G )  =  ( +g  `  ( RR*ss  ( 0 [,] +oo ) ) )
5654, 55eqtr4i 2465 . . . . 5  |-  +e 
=  ( +g  `  G
)
575, 56resvplusg 26300 . . . 4  |-  ( ( 0 [,) +oo )  e.  _V  ->  +e 
=  ( +g  `  W
) )
584, 57ax-mp 5 . . 3  |-  +e 
=  ( +g  `  W
)
59 ovex 6115 . . . . . 6  |-  ( 0 [,] +oo )  e. 
_V
60 ax-xrsvsca 26134 . . . . . . 7  |-  xe  =  ( .s `  RR*s )
611, 60ressvsca 14316 . . . . . 6  |-  ( ( 0 [,] +oo )  e.  _V  ->  xe 
=  ( .s `  G ) )
6259, 61ax-mp 5 . . . . 5  |-  xe  =  ( .s `  G )
635, 62resvvsca 26301 . . . 4  |-  ( ( 0 [,) +oo )  e.  _V  ->  xe 
=  ( .s `  W ) )
644, 63ax-mp 5 . . 3  |-  xe  =  ( .s `  W )
65 xrge00 26146 . . . . . 6  |-  0  =  ( 0g `  ( RR*ss  ( 0 [,] +oo ) ) )
661fveq2i 5693 . . . . . 6  |-  ( 0g
`  G )  =  ( 0g `  ( RR*ss  ( 0 [,] +oo ) ) )
6765, 66eqtr4i 2465 . . . . 5  |-  0  =  ( 0g `  G )
685, 67resv0g 26303 . . . 4  |-  ( ( 0 [,) +oo )  e.  _V  ->  0  =  ( 0g `  W ) )
694, 68ax-mp 5 . . 3  |-  0  =  ( 0g `  W )
70 df-refld 18034 . . . . . 6  |- RRfld  =  (flds  RR )
7170oveq1i 6100 . . . . 5  |-  (RRfld ↾s  ( 0 [,) +oo ) )  =  ( (flds  RR )s  ( 0 [,) +oo ) )
72 reex 9372 . . . . . 6  |-  RR  e.  _V
73 ressress 14234 . . . . . 6  |-  ( ( RR  e.  _V  /\  ( 0 [,) +oo )  e.  _V )  ->  ( (flds  RR )s  ( 0 [,) +oo ) )  =  (flds  ( RR 
i^i  ( 0 [,) +oo ) ) ) )
7472, 4, 73mp2an 672 . . . . 5  |-  ( (flds  RR )s  ( 0 [,) +oo )
)  =  (flds  ( RR  i^i  ( 0 [,) +oo ) ) )
7571, 74eqtri 2462 . . . 4  |-  (RRfld ↾s  ( 0 [,) +oo ) )  =  (flds  ( RR  i^i  ( 0 [,) +oo ) ) )
76 ax-xrssca 26133 . . . . . . . . 9  |-  (flds  RR )  =  (Scalar `  RR*s )
771, 76resssca 14315 . . . . . . . 8  |-  ( ( 0 [,] +oo )  e.  _V  ->  (flds  RR )  =  (Scalar `  G ) )
7859, 77ax-mp 5 . . . . . . 7  |-  (flds  RR )  =  (Scalar `  G )
7970, 78eqtri 2462 . . . . . 6  |- RRfld  =  (Scalar `  G )
80 rebase 18035 . . . . . 6  |-  RR  =  ( Base ` RRfld )
815, 79, 80resvsca 26297 . . . . 5  |-  ( ( 0 [,) +oo )  e.  _V  ->  (RRfld ↾s  ( 0 [,) +oo ) )  =  (Scalar `  W ) )
824, 81ax-mp 5 . . . 4  |-  (RRfld ↾s  ( 0 [,) +oo ) )  =  (Scalar `  W
)
83 incom 3542 . . . . . 6  |-  ( ( 0 [,) +oo )  i^i  RR )  =  ( RR  i^i  ( 0 [,) +oo ) )
84 df-ss 3341 . . . . . . 7  |-  ( ( 0 [,) +oo )  C_  RR  <->  ( ( 0 [,) +oo )  i^i 
RR )  =  ( 0 [,) +oo )
)
8519, 84mpbi 208 . . . . . 6  |-  ( ( 0 [,) +oo )  i^i  RR )  =  ( 0 [,) +oo )
8683, 85eqtr3i 2464 . . . . 5  |-  ( RR 
i^i  ( 0 [,) +oo ) )  =  ( 0 [,) +oo )
8786oveq2i 6101 . . . 4  |-  (flds  ( RR  i^i  ( 0 [,) +oo ) ) )  =  (flds  ( 0 [,) +oo )
)
8875, 82, 873eqtr3ri 2471 . . 3  |-  (flds  ( 0 [,) +oo ) )  =  (Scalar `  W )
89 ax-resscn 9338 . . . . 5  |-  RR  C_  CC
9019, 89sstri 3364 . . . 4  |-  ( 0 [,) +oo )  C_  CC
91 eqid 2442 . . . . 5  |-  (flds  ( 0 [,) +oo ) )  =  (flds  ( 0 [,) +oo ) )
92 cnfldbas 17821 . . . . 5  |-  CC  =  ( Base ` fld )
9391, 92ressbas2 14228 . . . 4  |-  ( ( 0 [,) +oo )  C_  CC  ->  ( 0 [,) +oo )  =  ( Base `  (flds  ( 0 [,) +oo ) ) ) )
9490, 93ax-mp 5 . . 3  |-  ( 0 [,) +oo )  =  ( Base `  (flds  ( 0 [,) +oo ) ) )
95 cnfldadd 17822 . . . . 5  |-  +  =  ( +g  ` fld )
9691, 95ressplusg 14279 . . . 4  |-  ( ( 0 [,) +oo )  e.  _V  ->  +  =  ( +g  `  (flds  ( 0 [,) +oo ) ) ) )
974, 96ax-mp 5 . . 3  |-  +  =  ( +g  `  (flds  ( 0 [,) +oo ) ) )
98 cnfldmul 17823 . . . . 5  |-  x.  =  ( .r ` fld )
9991, 98ressmulr 14290 . . . 4  |-  ( ( 0 [,) +oo )  e.  _V  ->  x.  =  ( .r `  (flds  ( 0 [,) +oo ) ) ) )
1004, 99ax-mp 5 . . 3  |-  x.  =  ( .r `  (flds  ( 0 [,) +oo ) ) )
101 cndrng 17844 . . . . 5  |-fld  e.  DivRing
102 drngrng 16838 . . . . 5  |-  (fld  e.  DivRing  ->fld  e.  Ring )
103101, 102ax-mp 5 . . . 4  |-fld  e.  Ring
104 1re 9384 . . . . . 6  |-  1  e.  RR
105 0le1 9862 . . . . . 6  |-  0  <_  1
106 ltpnf 11101 . . . . . . 7  |-  ( 1  e.  RR  ->  1  < +oo )
107104, 106ax-mp 5 . . . . . 6  |-  1  < +oo
108104, 105, 1073pm3.2i 1166 . . . . 5  |-  ( 1  e.  RR  /\  0  <_  1  /\  1  < +oo )
109 0re 9385 . . . . . 6  |-  0  e.  RR
110 pnfxr 11091 . . . . . 6  |- +oo  e.  RR*
111 elico2 11358 . . . . . 6  |-  ( ( 0  e.  RR  /\ +oo  e.  RR* )  ->  (
1  e.  ( 0 [,) +oo )  <->  ( 1  e.  RR  /\  0  <_  1  /\  1  < +oo ) ) )
112109, 110, 111mp2an 672 . . . . 5  |-  ( 1  e.  ( 0 [,) +oo )  <->  ( 1  e.  RR  /\  0  <_ 
1  /\  1  < +oo ) )
113108, 112mpbir 209 . . . 4  |-  1  e.  ( 0 [,) +oo )
114 cnfld1 17840 . . . . 5  |-  1  =  ( 1r ` fld )
11591, 92, 114ress1r 26256 . . . 4  |-  ( (fld  e. 
Ring  /\  1  e.  ( 0 [,) +oo )  /\  ( 0 [,) +oo )  C_  CC )  -> 
1  =  ( 1r
`  (flds  ( 0 [,) +oo )
) ) )
116103, 113, 90, 115mp3an 1314 . . 3  |-  1  =  ( 1r `  (flds  (
0 [,) +oo )
) )
117 rngmnd 16653 . . . . 5  |-  (fld  e.  Ring  ->fld  e.  Mnd )
118101, 102, 117mp2b 10 . . . 4  |-fld  e.  Mnd
119 0e0icopnf 11394 . . . 4  |-  0  e.  ( 0 [,) +oo )
120 cnfld0 17839 . . . . 5  |-  0  =  ( 0g ` fld )
12191, 92, 120ress0g 15449 . . . 4  |-  ( (fld  e. 
Mnd  /\  0  e.  ( 0 [,) +oo )  /\  ( 0 [,) +oo )  C_  CC )  ->  0  =  ( 0g `  (flds  ( 0 [,) +oo ) ) ) )
122118, 119, 90, 121mp3an 1314 . . 3  |-  0  =  ( 0g `  (flds  (
0 [,) +oo )
) )
12353, 58, 64, 69, 88, 94, 97, 100, 116, 122isslmd 26217 . 2  |-  ( W  e. SLMod 
<->  ( W  e. CMnd  /\  (flds  (
0 [,) +oo )
)  e. SRing  /\  A. q  e.  ( 0 [,) +oo ) A. r  e.  ( 0 [,) +oo ) A. x  e.  (
0 [,] +oo ) A. w  e.  (
0 [,] +oo )
( ( ( r xe w )  e.  ( 0 [,] +oo )  /\  (
r xe ( w +e x ) )  =  ( ( r xe w ) +e
( r xe x ) )  /\  ( ( q  +  r ) xe w )  =  ( ( q xe w ) +e
( r xe w ) ) )  /\  ( ( ( q  x.  r ) xe w )  =  ( q xe ( r xe w ) )  /\  ( 1 xe w )  =  w  /\  ( 0 xe w )  =  0 ) ) ) )
1248, 9, 48, 123mpbir3an 1170 1  |-  W  e. SLMod
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2714   _Vcvv 2971    i^i cin 3326    C_ wss 3327   class class class wbr 4291   ` cfv 5417  (class class class)co 6090   CCcc 9279   RRcr 9280   0cc0 9281   1c1 9282    + caddc 9284    x. cmul 9286   +oocpnf 9414   RR*cxr 9416    < clt 9417    <_ cle 9418   +ecxad 11086   xecxmu 11087   [,)cico 11301   [,]cicc 11302   Basecbs 14173   ↾s cress 14174   +g cplusg 14237   .rcmulr 14238  Scalarcsca 14240   .scvsca 14241   0gc0g 14377   RR*scxrs 14437   Mndcmnd 15408  CMndccmn 16276   1rcur 16602  SRingcsrg 16606   Ringcrg 16644   DivRingcdr 16831  ℂfldccnfld 17817  RRfldcrefld 18033  SLModcslmd 26215   ↾v cresv 26291
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-cnex 9337  ax-resscn 9338  ax-1cn 9339  ax-icn 9340  ax-addcl 9341  ax-addrcl 9342  ax-mulcl 9343  ax-mulrcl 9344  ax-mulcom 9345  ax-addass 9346  ax-mulass 9347  ax-distr 9348  ax-i2m1 9349  ax-1ne0 9350  ax-1rid 9351  ax-rnegex 9352  ax-rrecex 9353  ax-cnre 9354  ax-pre-lttri 9355  ax-pre-lttrn 9356  ax-pre-ltadd 9357  ax-pre-mulgt0 9358  ax-addf 9360  ax-mulf 9361  ax-xrssca 26133  ax-xrsvsca 26134
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-int 4128  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-tpos 6744  df-recs 6831  df-rdg 6865  df-1o 6919  df-oadd 6923  df-er 7100  df-en 7310  df-dom 7311  df-sdom 7312  df-fin 7313  df-pnf 9419  df-mnf 9420  df-xr 9421  df-ltxr 9422  df-le 9423  df-sub 9596  df-neg 9597  df-div 9993  df-nn 10322  df-2 10379  df-3 10380  df-4 10381  df-5 10382  df-6 10383  df-7 10384  df-8 10385  df-9 10386  df-10 10387  df-n0 10579  df-z 10646  df-dec 10755  df-uz 10861  df-xneg 11088  df-xadd 11089  df-xmul 11090  df-ico 11305  df-icc 11306  df-fz 11437  df-struct 14175  df-ndx 14176  df-slot 14177  df-base 14178  df-sets 14179  df-ress 14180  df-plusg 14250  df-mulr 14251  df-starv 14252  df-sca 14253  df-vsca 14254  df-tset 14256  df-ple 14257  df-ds 14259  df-unif 14260  df-0g 14379  df-xrs 14439  df-mnd 15414  df-submnd 15464  df-grp 15544  df-minusg 15545  df-cmn 16278  df-abl 16279  df-mgp 16591  df-ur 16603  df-srg 16607  df-rng 16646  df-cring 16647  df-oppr 16714  df-dvdsr 16732  df-unit 16733  df-invr 16763  df-dvr 16774  df-drng 16833  df-cnfld 17818  df-refld 18034  df-slmd 26216  df-resv 26292
This theorem is referenced by: (None)
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