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Theorem xrge0adddir 27555
Description: Right-distributivity of extended nonnegative real multiplication over addition. (Contributed by Thierry Arnoux, 30-Jun-2017.)
Assertion
Ref Expression
xrge0adddir  |-  ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  ( 0 [,] +oo ) )  ->  ( ( A +e B ) xe C )  =  ( ( A xe C ) +e ( B xe C ) ) )

Proof of Theorem xrge0adddir
StepHypRef Expression
1 iccssxr 11616 . . . 4  |-  ( 0 [,] +oo )  C_  RR*
2 simpl1 1000 . . . 4  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  e.  ( 0 [,) +oo ) )  ->  A  e.  ( 0 [,] +oo ) )
31, 2sseldi 3487 . . 3  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  e.  ( 0 [,) +oo ) )  ->  A  e.  RR* )
4 simpl2 1001 . . . 4  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  e.  ( 0 [,) +oo ) )  ->  B  e.  ( 0 [,] +oo ) )
51, 4sseldi 3487 . . 3  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  e.  ( 0 [,) +oo ) )  ->  B  e.  RR* )
6 rge0ssre 11637 . . . 4  |-  ( 0 [,) +oo )  C_  RR
7 simpr 461 . . . 4  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  e.  ( 0 [,) +oo ) )  ->  C  e.  ( 0 [,) +oo ) )
86, 7sseldi 3487 . . 3  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  e.  ( 0 [,) +oo ) )  ->  C  e.  RR )
9 xadddir 11497 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  ->  (
( A +e
B ) xe C )  =  ( ( A xe C ) +e
( B xe C ) ) )
103, 5, 8, 9syl3anc 1229 . 2  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  e.  ( 0 [,) +oo ) )  ->  (
( A +e
B ) xe C )  =  ( ( A xe C ) +e
( B xe C ) ) )
11 simpll1 1036 . . . . . . . 8  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  /\  0  <  A )  ->  A  e.  ( 0 [,] +oo ) )
121, 11sseldi 3487 . . . . . . 7  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  /\  0  <  A )  ->  A  e.  RR* )
13 simpll2 1037 . . . . . . . 8  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  /\  0  <  A )  ->  B  e.  ( 0 [,] +oo ) )
141, 13sseldi 3487 . . . . . . 7  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  /\  0  <  A )  ->  B  e.  RR* )
1512, 14xaddcld 11502 . . . . . 6  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  /\  0  <  A )  ->  ( A +e B )  e.  RR* )
16 simpr 461 . . . . . . 7  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  /\  0  <  A )  ->  0  <  A )
17 xrge0addgt0 27554 . . . . . . 7  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )
)  /\  0  <  A )  ->  0  <  ( A +e B ) )
1811, 13, 16, 17syl21anc 1228 . . . . . 6  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  /\  0  <  A )  ->  0  <  ( A +e
B ) )
19 xmulpnf1 11475 . . . . . 6  |-  ( ( ( A +e
B )  e.  RR*  /\  0  <  ( A +e B ) )  ->  ( ( A +e B ) xe +oo )  = +oo )
2015, 18, 19syl2anc 661 . . . . 5  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  /\  0  <  A )  ->  (
( A +e
B ) xe +oo )  = +oo )
21 oveq2 6289 . . . . . 6  |-  ( C  = +oo  ->  (
( A +e
B ) xe C )  =  ( ( A +e
B ) xe +oo ) )
2221ad2antlr 726 . . . . 5  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  /\  0  <  A )  ->  (
( A +e
B ) xe C )  =  ( ( A +e
B ) xe +oo ) )
23 simpll3 1038 . . . . . . . 8  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  /\  0  <  A )  ->  C  e.  ( 0 [,] +oo ) )
24 ge0xmulcl 11644 . . . . . . . 8  |-  ( ( B  e.  ( 0 [,] +oo )  /\  C  e.  ( 0 [,] +oo ) )  ->  ( B xe C )  e.  ( 0 [,] +oo ) )
2513, 23, 24syl2anc 661 . . . . . . 7  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  /\  0  <  A )  ->  ( B xe C )  e.  ( 0 [,] +oo ) )
261, 25sseldi 3487 . . . . . 6  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  /\  0  <  A )  ->  ( B xe C )  e.  RR* )
27 xrge0neqmnf 27552 . . . . . . 7  |-  ( ( B xe C )  e.  ( 0 [,] +oo )  -> 
( B xe C )  =/= -oo )
2825, 27syl 16 . . . . . 6  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  /\  0  <  A )  ->  ( B xe C )  =/= -oo )
29 xaddpnf2 11435 . . . . . 6  |-  ( ( ( B xe C )  e.  RR*  /\  ( B xe C )  =/= -oo )  ->  ( +oo +e ( B xe C ) )  = +oo )
3026, 28, 29syl2anc 661 . . . . 5  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  /\  0  <  A )  ->  ( +oo +e ( B xe C ) )  = +oo )
3120, 22, 303eqtr4d 2494 . . . 4  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  /\  0  <  A )  ->  (
( A +e
B ) xe C )  =  ( +oo +e ( B xe C ) ) )
32 oveq2 6289 . . . . . . 7  |-  ( C  = +oo  ->  ( A xe C )  =  ( A xe +oo ) )
3332ad2antlr 726 . . . . . 6  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  /\  0  <  A )  ->  ( A xe C )  =  ( A xe +oo ) )
34 xmulpnf1 11475 . . . . . . 7  |-  ( ( A  e.  RR*  /\  0  <  A )  ->  ( A xe +oo )  = +oo )
3512, 16, 34syl2anc 661 . . . . . 6  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  /\  0  <  A )  ->  ( A xe +oo )  = +oo )
3633, 35eqtrd 2484 . . . . 5  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  /\  0  <  A )  ->  ( A xe C )  = +oo )
3736oveq1d 6296 . . . 4  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  /\  0  <  A )  ->  (
( A xe C ) +e
( B xe C ) )  =  ( +oo +e
( B xe C ) ) )
3831, 37eqtr4d 2487 . . 3  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  /\  0  <  A )  ->  (
( A +e
B ) xe C )  =  ( ( A xe C ) +e
( B xe C ) ) )
39 simpll3 1038 . . . . . . . 8  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  /\  0  =  A )  ->  C  e.  ( 0 [,] +oo ) )
401, 39sseldi 3487 . . . . . . 7  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  /\  0  =  A )  ->  C  e.  RR* )
41 xmul02 11469 . . . . . . 7  |-  ( C  e.  RR*  ->  ( 0 xe C )  =  0 )
4240, 41syl 16 . . . . . 6  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  /\  0  =  A )  ->  (
0 xe C )  =  0 )
4342oveq1d 6296 . . . . 5  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  /\  0  =  A )  ->  (
( 0 xe C ) +e
( B xe C ) )  =  ( 0 +e
( B xe C ) ) )
44 oveq1 6288 . . . . . . 7  |-  ( 0  =  A  ->  (
0 xe C )  =  ( A xe C ) )
4544adantl 466 . . . . . 6  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  /\  0  =  A )  ->  (
0 xe C )  =  ( A xe C ) )
4645oveq1d 6296 . . . . 5  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  /\  0  =  A )  ->  (
( 0 xe C ) +e
( B xe C ) )  =  ( ( A xe C ) +e ( B xe C ) ) )
47 simpll2 1037 . . . . . . . 8  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  /\  0  =  A )  ->  B  e.  ( 0 [,] +oo ) )
481, 47sseldi 3487 . . . . . . 7  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  /\  0  =  A )  ->  B  e.  RR* )
4948, 40xmulcld 11503 . . . . . 6  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  /\  0  =  A )  ->  ( B xe C )  e.  RR* )
50 xaddid2 11448 . . . . . 6  |-  ( ( B xe C )  e.  RR*  ->  ( 0 +e ( B xe C ) )  =  ( B xe C ) )
5149, 50syl 16 . . . . 5  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  /\  0  =  A )  ->  (
0 +e ( B xe C ) )  =  ( B xe C ) )
5243, 46, 513eqtr3d 2492 . . . 4  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  /\  0  =  A )  ->  (
( A xe C ) +e
( B xe C ) )  =  ( B xe C ) )
53 xaddid2 11448 . . . . . 6  |-  ( B  e.  RR*  ->  ( 0 +e B )  =  B )
5448, 53syl 16 . . . . 5  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  /\  0  =  A )  ->  (
0 +e B )  =  B )
5554oveq1d 6296 . . . 4  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  /\  0  =  A )  ->  (
( 0 +e
B ) xe C )  =  ( B xe C ) )
56 oveq1 6288 . . . . . 6  |-  ( 0  =  A  ->  (
0 +e B )  =  ( A +e B ) )
5756oveq1d 6296 . . . . 5  |-  ( 0  =  A  ->  (
( 0 +e
B ) xe C )  =  ( ( A +e
B ) xe C ) )
5857adantl 466 . . . 4  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  /\  0  =  A )  ->  (
( 0 +e
B ) xe C )  =  ( ( A +e
B ) xe C ) )
5952, 55, 583eqtr2rd 2491 . . 3  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  /\  0  =  A )  ->  (
( A +e
B ) xe C )  =  ( ( A xe C ) +e
( B xe C ) ) )
60 0xr 9643 . . . . 5  |-  0  e.  RR*
6160a1i 11 . . . 4  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  ->  0  e. 
RR* )
62 simpl1 1000 . . . . 5  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  ->  A  e.  ( 0 [,] +oo ) )
631, 62sseldi 3487 . . . 4  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  ->  A  e. 
RR* )
64 pnfxr 11330 . . . . . 6  |- +oo  e.  RR*
6564a1i 11 . . . . 5  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  -> +oo  e.  RR* )
66 iccgelb 11590 . . . . 5  |-  ( ( 0  e.  RR*  /\ +oo  e.  RR*  /\  A  e.  ( 0 [,] +oo ) )  ->  0  <_  A )
6761, 65, 62, 66syl3anc 1229 . . . 4  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  ->  0  <_  A )
68 xrleloe 11359 . . . . 5  |-  ( ( 0  e.  RR*  /\  A  e.  RR* )  ->  (
0  <_  A  <->  ( 0  <  A  \/  0  =  A ) ) )
6968biimpa 484 . . . 4  |-  ( ( ( 0  e.  RR*  /\  A  e.  RR* )  /\  0  <_  A )  ->  ( 0  < 
A  \/  0  =  A ) )
7061, 63, 67, 69syl21anc 1228 . . 3  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  ->  ( 0  <  A  \/  0  =  A ) )
7138, 59, 70mpjaodan 786 . 2  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  ->  ( ( A +e B ) xe C )  =  ( ( A xe C ) +e ( B xe C ) ) )
72 0lepnf 11349 . . . . 5  |-  0  <_ +oo
73 eliccelico 27460 . . . . 5  |-  ( ( 0  e.  RR*  /\ +oo  e.  RR*  /\  0  <_ +oo )  ->  ( C  e.  ( 0 [,] +oo )  <->  ( C  e.  ( 0 [,) +oo )  \/  C  = +oo ) ) )
7460, 64, 72, 73mp3an 1325 . . . 4  |-  ( C  e.  ( 0 [,] +oo )  <->  ( C  e.  ( 0 [,) +oo )  \/  C  = +oo ) )
75743anbi3i 1190 . . 3  |-  ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  ( 0 [,] +oo ) )  <-> 
( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  ( C  e.  ( 0 [,) +oo )  \/  C  = +oo ) ) )
7675simp3bi 1014 . 2  |-  ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  ( 0 [,] +oo ) )  ->  ( C  e.  ( 0 [,) +oo )  \/  C  = +oo ) )
7710, 71, 76mpjaodan 786 1  |-  ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  ( 0 [,] +oo ) )  ->  ( ( A +e B ) xe C )  =  ( ( A xe C ) +e ( B xe C ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 974    = wceq 1383    e. wcel 1804    =/= wne 2638   class class class wbr 4437  (class class class)co 6281   RRcr 9494   0cc0 9495   +oocpnf 9628   -oocmnf 9629   RR*cxr 9630    < clt 9631    <_ cle 9632   +ecxad 11325   xecxmu 11326   [,)cico 11540   [,]cicc 11541
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-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
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-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-po 4790  df-so 4791  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-1st 6785  df-2nd 6786  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-xneg 11327  df-xadd 11328  df-xmul 11329  df-ico 11544  df-icc 11545
This theorem is referenced by:  xrge0adddi  27556  xrge0slmod  27707  esummulc1  27960
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