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Theorem xrge0adddir 27916
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 11610 . . . 4  |-  ( 0 [,] +oo )  C_  RR*
2 simpl1 997 . . . 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 998 . . . 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 11631 . . . 4  |-  ( 0 [,) +oo )  C_  RR
7 simpr 459 . . . 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 11491 . . 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 1226 . 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 1033 . . . . . . . 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 1034 . . . . . . . 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 11496 . . . . . 6  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  /\  0  <  A )  ->  ( A +e B )  e.  RR* )
16 simpr 459 . . . . . . 7  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  /\  0  <  A )  ->  0  <  A )
17 xrge0addgt0 27915 . . . . . . 7  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )
)  /\  0  <  A )  ->  0  <  ( A +e B ) )
1811, 13, 16, 17syl21anc 1225 . . . . . 6  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  /\  0  <  A )  ->  0  <  ( A +e
B ) )
19 xmulpnf1 11469 . . . . . 6  |-  ( ( ( A +e
B )  e.  RR*  /\  0  <  ( A +e B ) )  ->  ( ( A +e B ) xe +oo )  = +oo )
2015, 18, 19syl2anc 659 . . . . 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 6278 . . . . . 6  |-  ( C  = +oo  ->  (
( A +e
B ) xe C )  =  ( ( A +e
B ) xe +oo ) )
2221ad2antlr 724 . . . . 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 1035 . . . . . . . 8  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  /\  0  <  A )  ->  C  e.  ( 0 [,] +oo ) )
24 ge0xmulcl 11638 . . . . . . . 8  |-  ( ( B  e.  ( 0 [,] +oo )  /\  C  e.  ( 0 [,] +oo ) )  ->  ( B xe C )  e.  ( 0 [,] +oo ) )
2513, 23, 24syl2anc 659 . . . . . . 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 27913 . . . . . . 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 11429 . . . . . 6  |-  ( ( ( B xe C )  e.  RR*  /\  ( B xe C )  =/= -oo )  ->  ( +oo +e ( B xe C ) )  = +oo )
3026, 28, 29syl2anc 659 . . . . 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 2505 . . . 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 6278 . . . . . . 7  |-  ( C  = +oo  ->  ( A xe C )  =  ( A xe +oo ) )
3332ad2antlr 724 . . . . . 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 11469 . . . . . . 7  |-  ( ( A  e.  RR*  /\  0  <  A )  ->  ( A xe +oo )  = +oo )
3512, 16, 34syl2anc 659 . . . . . 6  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  /\  0  <  A )  ->  ( A xe +oo )  = +oo )
3633, 35eqtrd 2495 . . . . 5  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  /\  0  <  A )  ->  ( A xe C )  = +oo )
3736oveq1d 6285 . . . 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 2498 . . 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 1035 . . . . . . . 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 11463 . . . . . . 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 6285 . . . . 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 6277 . . . . . . 7  |-  ( 0  =  A  ->  (
0 xe C )  =  ( A xe C ) )
4544adantl 464 . . . . . 6  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  /\  0  =  A )  ->  (
0 xe C )  =  ( A xe C ) )
4645oveq1d 6285 . . . . 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 1034 . . . . . . . 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 11497 . . . . . 6  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  /\  0  =  A )  ->  ( B xe C )  e.  RR* )
50 xaddid2 11442 . . . . . 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 2503 . . . 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 11442 . . . . . 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 6285 . . . 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 6277 . . . . . 6  |-  ( 0  =  A  ->  (
0 +e B )  =  ( A +e B ) )
5756oveq1d 6285 . . . . 5  |-  ( 0  =  A  ->  (
( 0 +e
B ) xe C )  =  ( ( A +e
B ) xe C ) )
5857adantl 464 . . . 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 2502 . . 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 9629 . . . . 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 997 . . . . 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 11324 . . . . . 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 11584 . . . . 5  |-  ( ( 0  e.  RR*  /\ +oo  e.  RR*  /\  A  e.  ( 0 [,] +oo ) )  ->  0  <_  A )
6761, 65, 62, 66syl3anc 1226 . . . 4  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  ->  0  <_  A )
68 xrleloe 11353 . . . . 5  |-  ( ( 0  e.  RR*  /\  A  e.  RR* )  ->  (
0  <_  A  <->  ( 0  <  A  \/  0  =  A ) ) )
6968biimpa 482 . . . 4  |-  ( ( ( 0  e.  RR*  /\  A  e.  RR* )  /\  0  <_  A )  ->  ( 0  < 
A  \/  0  =  A ) )
7061, 63, 67, 69syl21anc 1225 . . 3  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  ->  ( 0  <  A  \/  0  =  A ) )
7138, 59, 70mpjaodan 784 . 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 11343 . . . . 5  |-  0  <_ +oo
73 eliccelico 27822 . . . . 5  |-  ( ( 0  e.  RR*  /\ +oo  e.  RR*  /\  0  <_ +oo )  ->  ( C  e.  ( 0 [,] +oo )  <->  ( C  e.  ( 0 [,) +oo )  \/  C  = +oo ) ) )
7460, 64, 72, 73mp3an 1322 . . . 4  |-  ( C  e.  ( 0 [,] +oo )  <->  ( C  e.  ( 0 [,) +oo )  \/  C  = +oo ) )
75743anbi3i 1187 . . 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 1011 . 2  |-  ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  ( 0 [,] +oo ) )  ->  ( C  e.  ( 0 [,) +oo )  \/  C  = +oo ) )
7710, 71, 76mpjaodan 784 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 366    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649   class class class wbr 4439  (class class class)co 6270   RRcr 9480   0cc0 9481   +oocpnf 9614   -oocmnf 9615   RR*cxr 9616    < clt 9617    <_ cle 9618   +ecxad 11319   xecxmu 11320   [,)cico 11534   [,]cicc 11535
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-po 4789  df-so 4790  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-1st 6773  df-2nd 6774  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-xneg 11321  df-xadd 11322  df-xmul 11323  df-ico 11538  df-icc 11539
This theorem is referenced by:  xrge0adddi  27917  xrge0slmod  28069  esummulc1  28310
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