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Theorem xrge0slmod 27707
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 18334 . . . 4  |-  ( RR*ss  ( 0 [,] +oo ) )  e. CMnd
31, 2eqeltri 2527 . . 3  |-  G  e. CMnd
4 ovex 6309 . . . 4  |-  ( 0 [,) +oo )  e. 
_V
5 xrge0slmod.2 . . . . 5  |-  W  =  ( Gv  ( 0 [,) +oo ) )
65resvcmn 27701 . . . 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 18361 . 2  |-  (flds  ( 0 [,) +oo ) )  e. SRing
10 icossicc 11620 . . . . . . . 8  |-  ( 0 [,) +oo )  C_  ( 0 [,] +oo )
11 simplr 755 . . . . . . . 8  |-  ( ( ( q  e.  ( 0 [,) +oo )  /\  r  e.  (
0 [,) +oo )
)  /\  ( x  e.  ( 0 [,] +oo )  /\  w  e.  ( 0 [,] +oo )
) )  ->  r  e.  ( 0 [,) +oo ) )
1210, 11sseldi 3487 . . . . . . 7  |-  ( ( ( q  e.  ( 0 [,) +oo )  /\  r  e.  (
0 [,) +oo )
)  /\  ( x  e.  ( 0 [,] +oo )  /\  w  e.  ( 0 [,] +oo )
) )  ->  r  e.  ( 0 [,] +oo ) )
13 simprr 757 . . . . . . 7  |-  ( ( ( q  e.  ( 0 [,) +oo )  /\  r  e.  (
0 [,) +oo )
)  /\  ( x  e.  ( 0 [,] +oo )  /\  w  e.  ( 0 [,] +oo )
) )  ->  w  e.  ( 0 [,] +oo ) )
14 ge0xmulcl 11644 . . . . . . 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 756 . . . . . . 7  |-  ( ( ( q  e.  ( 0 [,) +oo )  /\  r  e.  (
0 [,) +oo )
)  /\  ( x  e.  ( 0 [,] +oo )  /\  w  e.  ( 0 [,] +oo )
) )  ->  x  e.  ( 0 [,] +oo ) )
17 xrge0adddi 27556 . . . . . . 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 1229 . . . . . 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 11637 . . . . . . . . . 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 3487 . . . . . . . . 9  |-  ( ( ( q  e.  ( 0 [,) +oo )  /\  r  e.  (
0 [,) +oo )
)  /\  ( x  e.  ( 0 [,] +oo )  /\  w  e.  ( 0 [,] +oo )
) )  ->  q  e.  RR )
2219, 11sseldi 3487 . . . . . . . . 9  |-  ( ( ( q  e.  ( 0 [,) +oo )  /\  r  e.  (
0 [,) +oo )
)  /\  ( x  e.  ( 0 [,] +oo )  /\  w  e.  ( 0 [,] +oo )
) )  ->  r  e.  RR )
23 rexadd 11440 . . . . . . . . 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 6296 . . . . . . 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 3487 . . . . . . . 8  |-  ( ( ( q  e.  ( 0 [,) +oo )  /\  r  e.  (
0 [,) +oo )
)  /\  ( x  e.  ( 0 [,] +oo )  /\  w  e.  ( 0 [,] +oo )
) )  ->  q  e.  ( 0 [,] +oo ) )
27 xrge0adddir 27555 . . . . . . . 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 1229 . . . . . . 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 2486 . . . . . 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 1177 . . . . 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 11472 . . . . . . . . 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 6296 . . . . . . 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 9646 . . . . . . . 8  |-  ( ( ( q  e.  ( 0 [,) +oo )  /\  r  e.  (
0 [,) +oo )
)  /\  ( x  e.  ( 0 [,] +oo )  /\  w  e.  ( 0 [,] +oo )
) )  ->  q  e.  RR* )
3522rexrd 9646 . . . . . . . 8  |-  ( ( ( q  e.  ( 0 [,) +oo )  /\  r  e.  (
0 [,) +oo )
)  /\  ( x  e.  ( 0 [,] +oo )  /\  w  e.  ( 0 [,] +oo )
) )  ->  r  e.  RR* )
36 iccssxr 11616 . . . . . . . . 9  |-  ( 0 [,] +oo )  C_  RR*
3736, 13sseldi 3487 . . . . . . . 8  |-  ( ( ( q  e.  ( 0 [,) +oo )  /\  r  e.  (
0 [,) +oo )
)  /\  ( x  e.  ( 0 [,] +oo )  /\  w  e.  ( 0 [,] +oo )
) )  ->  w  e.  RR* )
38 xmulass 11488 . . . . . . . 8  |-  ( ( q  e.  RR*  /\  r  e.  RR*  /\  w  e. 
RR* )  ->  (
( q xe r ) xe w )  =  ( q xe ( r xe w ) ) )
3934, 35, 37, 38syl3anc 1229 . . . . . . 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 2486 . . . . . 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 11481 . . . . . . 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 11469 . . . . . . 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 1177 . . . . 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 2864 . . 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 2870 . 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 27546 . . . . . 6  |-  ( 0 [,] +oo )  =  ( Base `  ( RR*ss  ( 0 [,] +oo ) ) )
501fveq2i 5859 . . . . . 6  |-  ( Base `  G )  =  (
Base `  ( RR*ss  ( 0 [,] +oo ) ) )
5149, 50eqtr4i 2475 . . . . 5  |-  ( 0 [,] +oo )  =  ( Base `  G
)
525, 51resvbas 27695 . . . 4  |-  ( ( 0 [,) +oo )  e.  _V  ->  ( 0 [,] +oo )  =  ( Base `  W
) )
534, 52ax-mp 5 . . 3  |-  ( 0 [,] +oo )  =  ( Base `  W
)
54 xrge0plusg 27548 . . . . . 6  |-  +e 
=  ( +g  `  ( RR*ss  ( 0 [,] +oo ) ) )
551fveq2i 5859 . . . . . 6  |-  ( +g  `  G )  =  ( +g  `  ( RR*ss  ( 0 [,] +oo ) ) )
5654, 55eqtr4i 2475 . . . . 5  |-  +e 
=  ( +g  `  G
)
575, 56resvplusg 27696 . . . 4  |-  ( ( 0 [,) +oo )  e.  _V  ->  +e 
=  ( +g  `  W
) )
584, 57ax-mp 5 . . 3  |-  +e 
=  ( +g  `  W
)
59 ovex 6309 . . . . . 6  |-  ( 0 [,] +oo )  e. 
_V
60 ax-xrsvsca 27535 . . . . . . 7  |-  xe  =  ( .s `  RR*s )
611, 60ressvsca 14653 . . . . . 6  |-  ( ( 0 [,] +oo )  e.  _V  ->  xe 
=  ( .s `  G ) )
6259, 61ax-mp 5 . . . . 5  |-  xe  =  ( .s `  G )
635, 62resvvsca 27697 . . . 4  |-  ( ( 0 [,) +oo )  e.  _V  ->  xe 
=  ( .s `  W ) )
644, 63ax-mp 5 . . 3  |-  xe  =  ( .s `  W )
65 xrge00 27547 . . . . . 6  |-  0  =  ( 0g `  ( RR*ss  ( 0 [,] +oo ) ) )
661fveq2i 5859 . . . . . 6  |-  ( 0g
`  G )  =  ( 0g `  ( RR*ss  ( 0 [,] +oo ) ) )
6765, 66eqtr4i 2475 . . . . 5  |-  0  =  ( 0g `  G )
685, 67resv0g 27699 . . . 4  |-  ( ( 0 [,) +oo )  e.  _V  ->  0  =  ( 0g `  W ) )
694, 68ax-mp 5 . . 3  |-  0  =  ( 0g `  W )
70 df-refld 18514 . . . . . 6  |- RRfld  =  (flds  RR )
7170oveq1i 6291 . . . . 5  |-  (RRfld ↾s  ( 0 [,) +oo ) )  =  ( (flds  RR )s  ( 0 [,) +oo ) )
72 reex 9586 . . . . . 6  |-  RR  e.  _V
73 ressress 14571 . . . . . 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 2472 . . . 4  |-  (RRfld ↾s  ( 0 [,) +oo ) )  =  (flds  ( RR  i^i  ( 0 [,) +oo ) ) )
76 ax-xrssca 27534 . . . . . . . . 9  |-  (flds  RR )  =  (Scalar `  RR*s )
771, 76resssca 14652 . . . . . . . 8  |-  ( ( 0 [,] +oo )  e.  _V  ->  (flds  RR )  =  (Scalar `  G ) )
7859, 77ax-mp 5 . . . . . . 7  |-  (flds  RR )  =  (Scalar `  G )
7970, 78eqtri 2472 . . . . . 6  |- RRfld  =  (Scalar `  G )
80 rebase 18515 . . . . . 6  |-  RR  =  ( Base ` RRfld )
815, 79, 80resvsca 27693 . . . . 5  |-  ( ( 0 [,) +oo )  e.  _V  ->  (RRfld ↾s  ( 0 [,) +oo ) )  =  (Scalar `  W ) )
824, 81ax-mp 5 . . . 4  |-  (RRfld ↾s  ( 0 [,) +oo ) )  =  (Scalar `  W
)
83 incom 3676 . . . . . 6  |-  ( ( 0 [,) +oo )  i^i  RR )  =  ( RR  i^i  ( 0 [,) +oo ) )
84 df-ss 3475 . . . . . . 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 2474 . . . . 5  |-  ( RR 
i^i  ( 0 [,) +oo ) )  =  ( 0 [,) +oo )
8786oveq2i 6292 . . . 4  |-  (flds  ( RR  i^i  ( 0 [,) +oo ) ) )  =  (flds  ( 0 [,) +oo )
)
8875, 82, 873eqtr3ri 2481 . . 3  |-  (flds  ( 0 [,) +oo ) )  =  (Scalar `  W )
89 ax-resscn 9552 . . . . 5  |-  RR  C_  CC
9019, 89sstri 3498 . . . 4  |-  ( 0 [,) +oo )  C_  CC
91 eqid 2443 . . . . 5  |-  (flds  ( 0 [,) +oo ) )  =  (flds  ( 0 [,) +oo ) )
92 cnfldbas 18298 . . . . 5  |-  CC  =  ( Base ` fld )
9391, 92ressbas2 14565 . . . 4  |-  ( ( 0 [,) +oo )  C_  CC  ->  ( 0 [,) +oo )  =  ( Base `  (flds  ( 0 [,) +oo ) ) ) )
9490, 93ax-mp 5 . . 3  |-  ( 0 [,) +oo )  =  ( Base `  (flds  ( 0 [,) +oo ) ) )
95 cnfldadd 18299 . . . . 5  |-  +  =  ( +g  ` fld )
9691, 95ressplusg 14616 . . . 4  |-  ( ( 0 [,) +oo )  e.  _V  ->  +  =  ( +g  `  (flds  ( 0 [,) +oo ) ) ) )
974, 96ax-mp 5 . . 3  |-  +  =  ( +g  `  (flds  ( 0 [,) +oo ) ) )
98 cnfldmul 18300 . . . . 5  |-  x.  =  ( .r ` fld )
9991, 98ressmulr 14627 . . . 4  |-  ( ( 0 [,) +oo )  e.  _V  ->  x.  =  ( .r `  (flds  ( 0 [,) +oo ) ) ) )
1004, 99ax-mp 5 . . 3  |-  x.  =  ( .r `  (flds  ( 0 [,) +oo ) ) )
101 cnring 18314 . . . 4  |-fld  e.  Ring
102 1re 9598 . . . . 5  |-  1  e.  RR
103 0le1 10082 . . . . 5  |-  0  <_  1
104 ltpnf 11340 . . . . . 6  |-  ( 1  e.  RR  ->  1  < +oo )
105102, 104ax-mp 5 . . . . 5  |-  1  < +oo
106 0re 9599 . . . . . 6  |-  0  e.  RR
107 pnfxr 11330 . . . . . 6  |- +oo  e.  RR*
108 elico2 11597 . . . . . 6  |-  ( ( 0  e.  RR  /\ +oo  e.  RR* )  ->  (
1  e.  ( 0 [,) +oo )  <->  ( 1  e.  RR  /\  0  <_  1  /\  1  < +oo ) ) )
109106, 107, 108mp2an 672 . . . . 5  |-  ( 1  e.  ( 0 [,) +oo )  <->  ( 1  e.  RR  /\  0  <_ 
1  /\  1  < +oo ) )
110102, 103, 105, 109mpbir3an 1179 . . . 4  |-  1  e.  ( 0 [,) +oo )
111 cnfld1 18317 . . . . 5  |-  1  =  ( 1r ` fld )
11291, 92, 111ress1r 27652 . . . 4  |-  ( (fld  e. 
Ring  /\  1  e.  ( 0 [,) +oo )  /\  ( 0 [,) +oo )  C_  CC )  -> 
1  =  ( 1r
`  (flds  ( 0 [,) +oo )
) ) )
113101, 110, 90, 112mp3an 1325 . . 3  |-  1  =  ( 1r `  (flds  (
0 [,) +oo )
) )
114 cndrng 18321 . . . . 5  |-fld  e.  DivRing
115 drngring 17277 . . . . 5  |-  (fld  e.  DivRing  ->fld  e.  Ring )
116 ringmnd 17081 . . . . 5  |-  (fld  e.  Ring  ->fld  e.  Mnd )
117114, 115, 116mp2b 10 . . . 4  |-fld  e.  Mnd
118 0e0icopnf 11639 . . . 4  |-  0  e.  ( 0 [,) +oo )
119 cnfld0 18316 . . . . 5  |-  0  =  ( 0g ` fld )
12091, 92, 119ress0g 15823 . . . 4  |-  ( (fld  e. 
Mnd  /\  0  e.  ( 0 [,) +oo )  /\  ( 0 [,) +oo )  C_  CC )  ->  0  =  ( 0g `  (flds  ( 0 [,) +oo ) ) ) )
121117, 118, 90, 120mp3an 1325 . . 3  |-  0  =  ( 0g `  (flds  (
0 [,) +oo )
) )
12253, 58, 64, 69, 88, 94, 97, 100, 113, 121isslmd 27618 . 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 ) ) ) )
1238, 9, 48, 122mpbir3an 1179 1  |-  W  e. SLMod
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    /\ w3a 974    = wceq 1383    e. wcel 1804   A.wral 2793   _Vcvv 3095    i^i cin 3460    C_ wss 3461   class class class wbr 4437   ` cfv 5578  (class class class)co 6281   CCcc 9493   RRcr 9494   0cc0 9495   1c1 9496    + caddc 9498    x. cmul 9500   +oocpnf 9628   RR*cxr 9630    < clt 9631    <_ cle 9632   +ecxad 11325   xecxmu 11326   [,)cico 11540   [,]cicc 11541   Basecbs 14509   ↾s cress 14510   +g cplusg 14574   .rcmulr 14575  Scalarcsca 14577   .scvsca 14578   0gc0g 14714   RR*scxrs 14774   Mndcmnd 15793  CMndccmn 16672   1rcur 17027  SRingcsrg 17031   Ringcrg 17072   DivRingcdr 17270  ℂfldccnfld 18294  RRfldcrefld 18513  SLModcslmd 27616   ↾v cresv 27687
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-addf 9574  ax-mulf 9575  ax-xrssca 27534  ax-xrsvsca 27535
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-tpos 6957  df-recs 7044  df-rdg 7078  df-1o 7132  df-oadd 7136  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10213  df-nn 10543  df-2 10600  df-3 10601  df-4 10602  df-5 10603  df-6 10604  df-7 10605  df-8 10606  df-9 10607  df-10 10608  df-n0 10802  df-z 10871  df-dec 10985  df-uz 11091  df-xneg 11327  df-xadd 11328  df-xmul 11329  df-ico 11544  df-icc 11545  df-fz 11682  df-struct 14511  df-ndx 14512  df-slot 14513  df-base 14514  df-sets 14515  df-ress 14516  df-plusg 14587  df-mulr 14588  df-starv 14589  df-sca 14590  df-vsca 14591  df-tset 14593  df-ple 14594  df-ds 14596  df-unif 14597  df-0g 14716  df-xrs 14776  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-submnd 15841  df-grp 15931  df-minusg 15932  df-cmn 16674  df-abl 16675  df-mgp 17016  df-ur 17028  df-srg 17032  df-ring 17074  df-cring 17075  df-oppr 17146  df-dvdsr 17164  df-unit 17165  df-invr 17195  df-dvr 17206  df-drng 17272  df-cnfld 18295  df-refld 18514  df-slmd 27617  df-resv 27688
This theorem is referenced by: (None)
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