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Theorem xrge0adddir 27360
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 simpl1 999 . . . 4  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  e.  ( 0 [,) +oo ) )  ->  A  e.  ( 0 [,] +oo ) )
2 simpl2 1000 . . . 4  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  e.  ( 0 [,) +oo ) )  ->  B  e.  ( 0 [,] +oo ) )
3 simpr 461 . . . 4  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  e.  ( 0 [,) +oo ) )  ->  C  e.  ( 0 [,) +oo ) )
4 iccssxr 11606 . . . . 5  |-  ( 0 [,] +oo )  C_  RR*
5 simp1 996 . . . . 5  |-  ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  ( 0 [,) +oo ) )  ->  A  e.  ( 0 [,] +oo )
)
64, 5sseldi 3502 . . . 4  |-  ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  ( 0 [,) +oo ) )  ->  A  e.  RR* )
71, 2, 3, 6syl3anc 1228 . . 3  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  e.  ( 0 [,) +oo ) )  ->  A  e.  RR* )
8 simp2 997 . . . . 5  |-  ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  ( 0 [,) +oo ) )  ->  B  e.  ( 0 [,] +oo )
)
94, 8sseldi 3502 . . . 4  |-  ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  ( 0 [,) +oo ) )  ->  B  e.  RR* )
101, 2, 3, 9syl3anc 1228 . . 3  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  e.  ( 0 [,) +oo ) )  ->  B  e.  RR* )
11 rge0ssre 11627 . . . . 5  |-  ( 0 [,) +oo )  C_  RR
12 simp3 998 . . . . 5  |-  ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  ( 0 [,) +oo ) )  ->  C  e.  ( 0 [,) +oo )
)
1311, 12sseldi 3502 . . . 4  |-  ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  ( 0 [,) +oo ) )  ->  C  e.  RR )
141, 2, 3, 13syl3anc 1228 . . 3  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  e.  ( 0 [,) +oo ) )  ->  C  e.  RR )
15 xadddir 11487 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  ->  (
( A +e
B ) xe C )  =  ( ( A xe C ) +e
( B xe C ) ) )
167, 10, 14, 15syl3anc 1228 . 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 ) ) )
17 simpll1 1035 . . . . . . . 8  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  /\  0  <  A )  ->  A  e.  ( 0 [,] +oo ) )
184, 17sseldi 3502 . . . . . . 7  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  /\  0  <  A )  ->  A  e.  RR* )
19 simpll2 1036 . . . . . . . 8  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  /\  0  <  A )  ->  B  e.  ( 0 [,] +oo ) )
204, 19sseldi 3502 . . . . . . 7  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  /\  0  <  A )  ->  B  e.  RR* )
2118, 20xaddcld 11492 . . . . . 6  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  /\  0  <  A )  ->  ( A +e B )  e.  RR* )
22 simpr 461 . . . . . . 7  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  /\  0  <  A )  ->  0  <  A )
23 xrge0addgt0 27359 . . . . . . 7  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )
)  /\  0  <  A )  ->  0  <  ( A +e B ) )
2417, 19, 22, 23syl21anc 1227 . . . . . 6  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  /\  0  <  A )  ->  0  <  ( A +e
B ) )
25 xmulpnf1 11465 . . . . . 6  |-  ( ( ( A +e
B )  e.  RR*  /\  0  <  ( A +e B ) )  ->  ( ( A +e B ) xe +oo )  = +oo )
2621, 24, 25syl2anc 661 . . . . 5  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  /\  0  <  A )  ->  (
( A +e
B ) xe +oo )  = +oo )
27 oveq2 6291 . . . . . 6  |-  ( C  = +oo  ->  (
( A +e
B ) xe C )  =  ( ( A +e
B ) xe +oo ) )
2827ad2antlr 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 ) )
29 simpll3 1037 . . . . . . . 8  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  /\  0  <  A )  ->  C  e.  ( 0 [,] +oo ) )
30 ge0xmulcl 11634 . . . . . . . 8  |-  ( ( B  e.  ( 0 [,] +oo )  /\  C  e.  ( 0 [,] +oo ) )  ->  ( B xe C )  e.  ( 0 [,] +oo ) )
3119, 29, 30syl2anc 661 . . . . . . 7  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  /\  0  <  A )  ->  ( B xe C )  e.  ( 0 [,] +oo ) )
324, 31sseldi 3502 . . . . . 6  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  /\  0  <  A )  ->  ( B xe C )  e.  RR* )
33 xrge0neqmnf 27357 . . . . . . 7  |-  ( ( B xe C )  e.  ( 0 [,] +oo )  -> 
( B xe C )  =/= -oo )
3431, 33syl 16 . . . . . 6  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  /\  0  <  A )  ->  ( B xe C )  =/= -oo )
35 xaddpnf2 11425 . . . . . 6  |-  ( ( ( B xe C )  e.  RR*  /\  ( B xe C )  =/= -oo )  ->  ( +oo +e ( B xe C ) )  = +oo )
3632, 34, 35syl2anc 661 . . . . 5  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  /\  0  <  A )  ->  ( +oo +e ( B xe C ) )  = +oo )
3726, 28, 363eqtr4d 2518 . . . 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 ) ) )
38 oveq2 6291 . . . . . . 7  |-  ( C  = +oo  ->  ( A xe C )  =  ( A xe +oo ) )
3938ad2antlr 726 . . . . . 6  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  /\  0  <  A )  ->  ( A xe C )  =  ( A xe +oo ) )
40 xmulpnf1 11465 . . . . . . 7  |-  ( ( A  e.  RR*  /\  0  <  A )  ->  ( A xe +oo )  = +oo )
4118, 22, 40syl2anc 661 . . . . . 6  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  /\  0  <  A )  ->  ( A xe +oo )  = +oo )
4239, 41eqtrd 2508 . . . . 5  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  /\  0  <  A )  ->  ( A xe C )  = +oo )
4342oveq1d 6298 . . . 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 ) ) )
4437, 43eqtr4d 2511 . . 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 ) ) )
45 simpll3 1037 . . . . . . . . 9  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  /\  0  =  A )  ->  C  e.  ( 0 [,] +oo ) )
464, 45sseldi 3502 . . . . . . . 8  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  /\  0  =  A )  ->  C  e.  RR* )
47 xmul02 11459 . . . . . . . 8  |-  ( C  e.  RR*  ->  ( 0 xe C )  =  0 )
4846, 47syl 16 . . . . . . 7  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  /\  0  =  A )  ->  (
0 xe C )  =  0 )
4948oveq1d 6298 . . . . . 6  |-  ( ( ( ( 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 ) ) )
50 oveq1 6290 . . . . . . . 8  |-  ( 0  =  A  ->  (
0 xe C )  =  ( A xe C ) )
5150adantl 466 . . . . . . 7  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  /\  0  =  A )  ->  (
0 xe C )  =  ( A xe C ) )
5251oveq1d 6298 . . . . . 6  |-  ( ( ( ( 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 ) ) )
53 simpll2 1036 . . . . . . . . 9  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  /\  0  =  A )  ->  B  e.  ( 0 [,] +oo ) )
544, 53sseldi 3502 . . . . . . . 8  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  /\  0  =  A )  ->  B  e.  RR* )
5554, 46xmulcld 11493 . . . . . . 7  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  /\  0  =  A )  ->  ( B xe C )  e.  RR* )
56 xaddid2 11438 . . . . . . 7  |-  ( ( B xe C )  e.  RR*  ->  ( 0 +e ( B xe C ) )  =  ( B xe C ) )
5755, 56syl 16 . . . . . 6  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  /\  0  =  A )  ->  (
0 +e ( B xe C ) )  =  ( B xe C ) )
5849, 52, 573eqtr3d 2516 . . . . 5  |-  ( ( ( ( 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 ) )
59 xaddid2 11438 . . . . . . 7  |-  ( B  e.  RR*  ->  ( 0 +e B )  =  B )
6054, 59syl 16 . . . . . 6  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  /\  0  =  A )  ->  (
0 +e B )  =  B )
6160oveq1d 6298 . . . . 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 ) )
6258, 52, 613eqtr4rd 2519 . . . 4  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  /\  0  =  A )  ->  (
( 0 +e
B ) xe C )  =  ( ( 0 xe C ) +e
( B xe C ) ) )
63 oveq1 6290 . . . . . 6  |-  ( 0  =  A  ->  (
0 +e B )  =  ( A +e B ) )
6463oveq1d 6298 . . . . 5  |-  ( 0  =  A  ->  (
( 0 +e
B ) xe C )  =  ( ( A +e
B ) xe C ) )
6564adantl 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 ) )
6662, 65, 523eqtr3d 2516 . . 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 ) ) )
67 0xr 9639 . . . . 5  |-  0  e.  RR*
6867a1i 11 . . . 4  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  ->  0  e. 
RR* )
69 simpl1 999 . . . . 5  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  ->  A  e.  ( 0 [,] +oo ) )
704, 69sseldi 3502 . . . 4  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  ->  A  e. 
RR* )
71 pnfxr 11320 . . . . . 6  |- +oo  e.  RR*
7271a1i 11 . . . . 5  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  -> +oo  e.  RR* )
73 iccgelb 11580 . . . . 5  |-  ( ( 0  e.  RR*  /\ +oo  e.  RR*  /\  A  e.  ( 0 [,] +oo ) )  ->  0  <_  A )
7468, 72, 69, 73syl3anc 1228 . . . 4  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  ->  0  <_  A )
75 xrleloe 11349 . . . . 5  |-  ( ( 0  e.  RR*  /\  A  e.  RR* )  ->  (
0  <_  A  <->  ( 0  <  A  \/  0  =  A ) ) )
7675biimpa 484 . . . 4  |-  ( ( ( 0  e.  RR*  /\  A  e.  RR* )  /\  0  <_  A )  ->  ( 0  < 
A  \/  0  =  A ) )
7768, 70, 74, 76syl21anc 1227 . . 3  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  ->  ( 0  <  A  \/  0  =  A ) )
7844, 66, 77mpjaodan 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 ) ) )
79 0lepnf 11339 . . . . 5  |-  0  <_ +oo
80 eliccelico 27272 . . . . 5  |-  ( ( 0  e.  RR*  /\ +oo  e.  RR*  /\  0  <_ +oo )  ->  ( C  e.  ( 0 [,] +oo )  <->  ( C  e.  ( 0 [,) +oo )  \/  C  = +oo ) ) )
8167, 71, 79, 80mp3an 1324 . . . 4  |-  ( C  e.  ( 0 [,] +oo )  <->  ( C  e.  ( 0 [,) +oo )  \/  C  = +oo ) )
82813anbi3i 1189 . . 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 ) ) )
8382simp3bi 1013 . 2  |-  ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  ( 0 [,] +oo ) )  ->  ( C  e.  ( 0 [,) +oo )  \/  C  = +oo ) )
8416, 78, 83mpjaodan 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 368    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   class class class wbr 4447  (class class class)co 6283   RRcr 9490   0cc0 9491   +oocpnf 9624   -oocmnf 9625   RR*cxr 9626    < clt 9627    <_ cle 9628   +ecxad 11315   xecxmu 11316   [,)cico 11530   [,]cicc 11531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575  ax-cnex 9547  ax-resscn 9548  ax-1cn 9549  ax-icn 9550  ax-addcl 9551  ax-addrcl 9552  ax-mulcl 9553  ax-mulrcl 9554  ax-mulcom 9555  ax-addass 9556  ax-mulass 9557  ax-distr 9558  ax-i2m1 9559  ax-1ne0 9560  ax-1rid 9561  ax-rnegex 9562  ax-rrecex 9563  ax-cnre 9564  ax-pre-lttri 9565  ax-pre-lttrn 9566  ax-pre-ltadd 9567  ax-pre-mulgt0 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-riota 6244  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-1st 6784  df-2nd 6785  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-pnf 9629  df-mnf 9630  df-xr 9631  df-ltxr 9632  df-le 9633  df-sub 9806  df-neg 9807  df-xneg 11317  df-xadd 11318  df-xmul 11319  df-ico 11534  df-icc 11535
This theorem is referenced by:  xrge0adddi  27361  xrge0slmod  27513  esummulc1  27743
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