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Theorem xrge0slmod 27485
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 18223 . . . 4  |-  ( RR*ss  ( 0 [,] +oo ) )  e. CMnd
31, 2eqeltri 2546 . . 3  |-  G  e. CMnd
4 ovex 6302 . . . 4  |-  ( 0 [,) +oo )  e. 
_V
5 xrge0slmod.2 . . . . 5  |-  W  =  ( Gv  ( 0 [,) +oo ) )
65resvcmn 27479 . . . 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 18250 . 2  |-  (flds  ( 0 [,) +oo ) )  e. SRing
10 icossicc 11602 . . . . . . . 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 3497 . . . . . . 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 11626 . . . . . . 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 27333 . . . . . . 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 1223 . . . . . 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 11619 . . . . . . . . . 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 3497 . . . . . . . . 9  |-  ( ( ( q  e.  ( 0 [,) +oo )  /\  r  e.  (
0 [,) +oo )
)  /\  ( x  e.  ( 0 [,] +oo )  /\  w  e.  ( 0 [,] +oo )
) )  ->  q  e.  RR )
2219, 11sseldi 3497 . . . . . . . . 9  |-  ( ( ( q  e.  ( 0 [,) +oo )  /\  r  e.  (
0 [,) +oo )
)  /\  ( x  e.  ( 0 [,] +oo )  /\  w  e.  ( 0 [,] +oo )
) )  ->  r  e.  RR )
23 rexadd 11422 . . . . . . . . 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 6292 . . . . . . 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 3497 . . . . . . . 8  |-  ( ( ( q  e.  ( 0 [,) +oo )  /\  r  e.  (
0 [,) +oo )
)  /\  ( x  e.  ( 0 [,] +oo )  /\  w  e.  ( 0 [,] +oo )
) )  ->  q  e.  ( 0 [,] +oo ) )
27 xrge0adddir 27332 . . . . . . . 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 1223 . . . . . . 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 2505 . . . . . 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 1171 . . . . 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 11454 . . . . . . . . 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 6292 . . . . . . 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 9634 . . . . . . . 8  |-  ( ( ( q  e.  ( 0 [,) +oo )  /\  r  e.  (
0 [,) +oo )
)  /\  ( x  e.  ( 0 [,] +oo )  /\  w  e.  ( 0 [,] +oo )
) )  ->  q  e.  RR* )
3522rexrd 9634 . . . . . . . 8  |-  ( ( ( q  e.  ( 0 [,) +oo )  /\  r  e.  (
0 [,) +oo )
)  /\  ( x  e.  ( 0 [,] +oo )  /\  w  e.  ( 0 [,] +oo )
) )  ->  r  e.  RR* )
36 iccssxr 11598 . . . . . . . . 9  |-  ( 0 [,] +oo )  C_  RR*
3736, 13sseldi 3497 . . . . . . . 8  |-  ( ( ( q  e.  ( 0 [,) +oo )  /\  r  e.  (
0 [,) +oo )
)  /\  ( x  e.  ( 0 [,] +oo )  /\  w  e.  ( 0 [,] +oo )
) )  ->  w  e.  RR* )
38 xmulass 11470 . . . . . . . 8  |-  ( ( q  e.  RR*  /\  r  e.  RR*  /\  w  e. 
RR* )  ->  (
( q xe r ) xe w )  =  ( q xe ( r xe w ) ) )
3934, 35, 37, 38syl3anc 1223 . . . . . . 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 2505 . . . . . 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 11463 . . . . . . 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 11451 . . . . . . 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 1171 . . . . 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 2880 . . 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 2886 . 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 27323 . . . . . 6  |-  ( 0 [,] +oo )  =  ( Base `  ( RR*ss  ( 0 [,] +oo ) ) )
501fveq2i 5862 . . . . . 6  |-  ( Base `  G )  =  (
Base `  ( RR*ss  ( 0 [,] +oo ) ) )
5149, 50eqtr4i 2494 . . . . 5  |-  ( 0 [,] +oo )  =  ( Base `  G
)
525, 51resvbas 27473 . . . 4  |-  ( ( 0 [,) +oo )  e.  _V  ->  ( 0 [,] +oo )  =  ( Base `  W
) )
534, 52ax-mp 5 . . 3  |-  ( 0 [,] +oo )  =  ( Base `  W
)
54 xrge0plusg 27325 . . . . . 6  |-  +e 
=  ( +g  `  ( RR*ss  ( 0 [,] +oo ) ) )
551fveq2i 5862 . . . . . 6  |-  ( +g  `  G )  =  ( +g  `  ( RR*ss  ( 0 [,] +oo ) ) )
5654, 55eqtr4i 2494 . . . . 5  |-  +e 
=  ( +g  `  G
)
575, 56resvplusg 27474 . . . 4  |-  ( ( 0 [,) +oo )  e.  _V  ->  +e 
=  ( +g  `  W
) )
584, 57ax-mp 5 . . 3  |-  +e 
=  ( +g  `  W
)
59 ovex 6302 . . . . . 6  |-  ( 0 [,] +oo )  e. 
_V
60 ax-xrsvsca 27312 . . . . . . 7  |-  xe  =  ( .s `  RR*s )
611, 60ressvsca 14625 . . . . . 6  |-  ( ( 0 [,] +oo )  e.  _V  ->  xe 
=  ( .s `  G ) )
6259, 61ax-mp 5 . . . . 5  |-  xe  =  ( .s `  G )
635, 62resvvsca 27475 . . . 4  |-  ( ( 0 [,) +oo )  e.  _V  ->  xe 
=  ( .s `  W ) )
644, 63ax-mp 5 . . 3  |-  xe  =  ( .s `  W )
65 xrge00 27324 . . . . . 6  |-  0  =  ( 0g `  ( RR*ss  ( 0 [,] +oo ) ) )
661fveq2i 5862 . . . . . 6  |-  ( 0g
`  G )  =  ( 0g `  ( RR*ss  ( 0 [,] +oo ) ) )
6765, 66eqtr4i 2494 . . . . 5  |-  0  =  ( 0g `  G )
685, 67resv0g 27477 . . . 4  |-  ( ( 0 [,) +oo )  e.  _V  ->  0  =  ( 0g `  W ) )
694, 68ax-mp 5 . . 3  |-  0  =  ( 0g `  W )
70 df-refld 18403 . . . . . 6  |- RRfld  =  (flds  RR )
7170oveq1i 6287 . . . . 5  |-  (RRfld ↾s  ( 0 [,) +oo ) )  =  ( (flds  RR )s  ( 0 [,) +oo ) )
72 reex 9574 . . . . . 6  |-  RR  e.  _V
73 ressress 14543 . . . . . 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 2491 . . . 4  |-  (RRfld ↾s  ( 0 [,) +oo ) )  =  (flds  ( RR  i^i  ( 0 [,) +oo ) ) )
76 ax-xrssca 27311 . . . . . . . . 9  |-  (flds  RR )  =  (Scalar `  RR*s )
771, 76resssca 14624 . . . . . . . 8  |-  ( ( 0 [,] +oo )  e.  _V  ->  (flds  RR )  =  (Scalar `  G ) )
7859, 77ax-mp 5 . . . . . . 7  |-  (flds  RR )  =  (Scalar `  G )
7970, 78eqtri 2491 . . . . . 6  |- RRfld  =  (Scalar `  G )
80 rebase 18404 . . . . . 6  |-  RR  =  ( Base ` RRfld )
815, 79, 80resvsca 27471 . . . . 5  |-  ( ( 0 [,) +oo )  e.  _V  ->  (RRfld ↾s  ( 0 [,) +oo ) )  =  (Scalar `  W ) )
824, 81ax-mp 5 . . . 4  |-  (RRfld ↾s  ( 0 [,) +oo ) )  =  (Scalar `  W
)
83 incom 3686 . . . . . 6  |-  ( ( 0 [,) +oo )  i^i  RR )  =  ( RR  i^i  ( 0 [,) +oo ) )
84 df-ss 3485 . . . . . . 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 2493 . . . . 5  |-  ( RR 
i^i  ( 0 [,) +oo ) )  =  ( 0 [,) +oo )
8786oveq2i 6288 . . . 4  |-  (flds  ( RR  i^i  ( 0 [,) +oo ) ) )  =  (flds  ( 0 [,) +oo )
)
8875, 82, 873eqtr3ri 2500 . . 3  |-  (flds  ( 0 [,) +oo ) )  =  (Scalar `  W )
89 ax-resscn 9540 . . . . 5  |-  RR  C_  CC
9019, 89sstri 3508 . . . 4  |-  ( 0 [,) +oo )  C_  CC
91 eqid 2462 . . . . 5  |-  (flds  ( 0 [,) +oo ) )  =  (flds  ( 0 [,) +oo ) )
92 cnfldbas 18190 . . . . 5  |-  CC  =  ( Base ` fld )
9391, 92ressbas2 14537 . . . 4  |-  ( ( 0 [,) +oo )  C_  CC  ->  ( 0 [,) +oo )  =  ( Base `  (flds  ( 0 [,) +oo ) ) ) )
9490, 93ax-mp 5 . . 3  |-  ( 0 [,) +oo )  =  ( Base `  (flds  ( 0 [,) +oo ) ) )
95 cnfldadd 18191 . . . . 5  |-  +  =  ( +g  ` fld )
9691, 95ressplusg 14588 . . . 4  |-  ( ( 0 [,) +oo )  e.  _V  ->  +  =  ( +g  `  (flds  ( 0 [,) +oo ) ) ) )
974, 96ax-mp 5 . . 3  |-  +  =  ( +g  `  (flds  ( 0 [,) +oo ) ) )
98 cnfldmul 18192 . . . . 5  |-  x.  =  ( .r ` fld )
9991, 98ressmulr 14599 . . . 4  |-  ( ( 0 [,) +oo )  e.  _V  ->  x.  =  ( .r `  (flds  ( 0 [,) +oo ) ) ) )
1004, 99ax-mp 5 . . 3  |-  x.  =  ( .r `  (flds  ( 0 [,) +oo ) ) )
101 cndrng 18213 . . . . 5  |-fld  e.  DivRing
102 drngrng 17181 . . . . 5  |-  (fld  e.  DivRing  ->fld  e.  Ring )
103101, 102ax-mp 5 . . . 4  |-fld  e.  Ring
104 1re 9586 . . . . . 6  |-  1  e.  RR
105 0le1 10067 . . . . . 6  |-  0  <_  1
106 ltpnf 11322 . . . . . . 7  |-  ( 1  e.  RR  ->  1  < +oo )
107104, 106ax-mp 5 . . . . . 6  |-  1  < +oo
108104, 105, 1073pm3.2i 1169 . . . . 5  |-  ( 1  e.  RR  /\  0  <_  1  /\  1  < +oo )
109 0re 9587 . . . . . 6  |-  0  e.  RR
110 pnfxr 11312 . . . . . 6  |- +oo  e.  RR*
111 elico2 11579 . . . . . 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 18209 . . . . 5  |-  1  =  ( 1r ` fld )
11591, 92, 114ress1r 27430 . . . 4  |-  ( (fld  e. 
Ring  /\  1  e.  ( 0 [,) +oo )  /\  ( 0 [,) +oo )  C_  CC )  -> 
1  =  ( 1r
`  (flds  ( 0 [,) +oo )
) ) )
116103, 113, 90, 115mp3an 1319 . . 3  |-  1  =  ( 1r `  (flds  (
0 [,) +oo )
) )
117 rngmnd 16990 . . . . 5  |-  (fld  e.  Ring  ->fld  e.  Mnd )
118101, 102, 117mp2b 10 . . . 4  |-fld  e.  Mnd
119 0e0icopnf 11621 . . . 4  |-  0  e.  ( 0 [,) +oo )
120 cnfld0 18208 . . . . 5  |-  0  =  ( 0g ` fld )
12191, 92, 120ress0g 15758 . . . 4  |-  ( (fld  e. 
Mnd  /\  0  e.  ( 0 [,) +oo )  /\  ( 0 [,) +oo )  C_  CC )  ->  0  =  ( 0g `  (flds  ( 0 [,) +oo ) ) ) )
122118, 119, 90, 121mp3an 1319 . . 3  |-  0  =  ( 0g `  (flds  (
0 [,) +oo )
) )
12353, 58, 64, 69, 88, 94, 97, 100, 116, 122isslmd 27395 . 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 1173 1  |-  W  e. SLMod
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762   A.wral 2809   _Vcvv 3108    i^i cin 3470    C_ wss 3471   class class class wbr 4442   ` cfv 5581  (class class class)co 6277   CCcc 9481   RRcr 9482   0cc0 9483   1c1 9484    + caddc 9486    x. cmul 9488   +oocpnf 9616   RR*cxr 9618    < clt 9619    <_ cle 9620   +ecxad 11307   xecxmu 11308   [,)cico 11522   [,]cicc 11523   Basecbs 14481   ↾s cress 14482   +g cplusg 14546   .rcmulr 14547  Scalarcsca 14549   .scvsca 14550   0gc0g 14686   RR*scxrs 14746   Mndcmnd 15717  CMndccmn 16589   1rcur 16938  SRingcsrg 16942   Ringcrg 16981   DivRingcdr 17174  ℂfldccnfld 18186  RRfldcrefld 18402  SLModcslmd 27393   ↾v cresv 27465
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560  ax-addf 9562  ax-mulf 9563  ax-xrssca 27311  ax-xrsvsca 27312
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-int 4278  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6674  df-1st 6776  df-2nd 6777  df-tpos 6947  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-en 7509  df-dom 7510  df-sdom 7511  df-fin 7512  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-div 10198  df-nn 10528  df-2 10585  df-3 10586  df-4 10587  df-5 10588  df-6 10589  df-7 10590  df-8 10591  df-9 10592  df-10 10593  df-n0 10787  df-z 10856  df-dec 10968  df-uz 11074  df-xneg 11309  df-xadd 11310  df-xmul 11311  df-ico 11526  df-icc 11527  df-fz 11664  df-struct 14483  df-ndx 14484  df-slot 14485  df-base 14486  df-sets 14487  df-ress 14488  df-plusg 14559  df-mulr 14560  df-starv 14561  df-sca 14562  df-vsca 14563  df-tset 14565  df-ple 14566  df-ds 14568  df-unif 14569  df-0g 14688  df-xrs 14748  df-mnd 15723  df-submnd 15773  df-grp 15853  df-minusg 15854  df-cmn 16591  df-abl 16592  df-mgp 16927  df-ur 16939  df-srg 16943  df-rng 16983  df-cring 16984  df-oppr 17051  df-dvdsr 17069  df-unit 17070  df-invr 17100  df-dvr 17111  df-drng 17176  df-cnfld 18187  df-refld 18403  df-slmd 27394  df-resv 27466
This theorem is referenced by: (None)
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