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Theorem xrge0adddir 26155
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 991 . . . 4  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  e.  ( 0 [,) +oo ) )  ->  A  e.  ( 0 [,] +oo ) )
2 simpl2 992 . . . 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 11378 . . . . 5  |-  ( 0 [,] +oo )  C_  RR*
5 simp1 988 . . . . 5  |-  ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  ( 0 [,) +oo ) )  ->  A  e.  ( 0 [,] +oo )
)
64, 5sseldi 3354 . . . 4  |-  ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  ( 0 [,) +oo ) )  ->  A  e.  RR* )
71, 2, 3, 6syl3anc 1218 . . 3  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  e.  ( 0 [,) +oo ) )  ->  A  e.  RR* )
8 simp2 989 . . . . 5  |-  ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  ( 0 [,) +oo ) )  ->  B  e.  ( 0 [,] +oo )
)
94, 8sseldi 3354 . . . 4  |-  ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  ( 0 [,) +oo ) )  ->  B  e.  RR* )
101, 2, 3, 9syl3anc 1218 . . 3  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  e.  ( 0 [,) +oo ) )  ->  B  e.  RR* )
11 rge0ssre 11393 . . . . 5  |-  ( 0 [,) +oo )  C_  RR
12 simp3 990 . . . . 5  |-  ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  ( 0 [,) +oo ) )  ->  C  e.  ( 0 [,) +oo )
)
1311, 12sseldi 3354 . . . 4  |-  ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  ( 0 [,) +oo ) )  ->  C  e.  RR )
141, 2, 3, 13syl3anc 1218 . . 3  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  e.  ( 0 [,) +oo ) )  ->  C  e.  RR )
15 xadddir 11259 . . 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 1218 . 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 1027 . . . . . . . 8  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  /\  0  <  A )  ->  A  e.  ( 0 [,] +oo ) )
184, 17sseldi 3354 . . . . . . 7  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  /\  0  <  A )  ->  A  e.  RR* )
19 simpll2 1028 . . . . . . . 8  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  /\  0  <  A )  ->  B  e.  ( 0 [,] +oo ) )
204, 19sseldi 3354 . . . . . . 7  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  /\  0  <  A )  ->  B  e.  RR* )
2118, 20xaddcld 11264 . . . . . 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 26154 . . . . . . 7  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )
)  /\  0  <  A )  ->  0  <  ( A +e B ) )
2417, 19, 22, 23syl21anc 1217 . . . . . 6  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  /\  0  <  A )  ->  0  <  ( A +e
B ) )
25 xmulpnf1 11237 . . . . . 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 6099 . . . . . 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 1029 . . . . . . . 8  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  /\  0  <  A )  ->  C  e.  ( 0 [,] +oo ) )
30 ge0xmulcl 11400 . . . . . . . 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 3354 . . . . . 6  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  /\  0  <  A )  ->  ( B xe C )  e.  RR* )
33 xrge0neqmnf 26152 . . . . . . 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 11197 . . . . . 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 2485 . . . 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 6099 . . . . . . 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 11237 . . . . . . 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 2475 . . . . 5  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  /\  0  <  A )  ->  ( A xe C )  = +oo )
4342oveq1d 6106 . . . 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 2478 . . 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 1029 . . . . . . . . 9  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  /\  0  =  A )  ->  C  e.  ( 0 [,] +oo ) )
464, 45sseldi 3354 . . . . . . . 8  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  /\  0  =  A )  ->  C  e.  RR* )
47 xmul02 11231 . . . . . . . 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 6106 . . . . . 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 6098 . . . . . . . 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 6106 . . . . . 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 1028 . . . . . . . . 9  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  /\  0  =  A )  ->  B  e.  ( 0 [,] +oo ) )
544, 53sseldi 3354 . . . . . . . 8  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  /\  0  =  A )  ->  B  e.  RR* )
5554, 46xmulcld 11265 . . . . . . 7  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  /\  0  =  A )  ->  ( B xe C )  e.  RR* )
56 xaddid2 11210 . . . . . . 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 2483 . . . . 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 11210 . . . . . . 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 6106 . . . . 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 2486 . . . 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 6098 . . . . . 6  |-  ( 0  =  A  ->  (
0 +e B )  =  ( A +e B ) )
6463oveq1d 6106 . . . . 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 2483 . . 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 9430 . . . . 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 991 . . . . 5  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  ->  A  e.  ( 0 [,] +oo ) )
704, 69sseldi 3354 . . . 4  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  ->  A  e. 
RR* )
71 pnfxr 11092 . . . . . 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 11352 . . . . 5  |-  ( ( 0  e.  RR*  /\ +oo  e.  RR*  /\  A  e.  ( 0 [,] +oo ) )  ->  0  <_  A )
7468, 72, 69, 73syl3anc 1218 . . . 4  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  C  e.  (
0 [,] +oo )
)  /\  C  = +oo )  ->  0  <_  A )
75 xrleloe 11121 . . . . 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 1217 . . 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 11111 . . . . 5  |-  0  <_ +oo
80 eliccelico 26067 . . . . 5  |-  ( ( 0  e.  RR*  /\ +oo  e.  RR*  /\  0  <_ +oo )  ->  ( C  e.  ( 0 [,] +oo )  <->  ( C  e.  ( 0 [,) +oo )  \/  C  = +oo ) ) )
8167, 71, 79, 80mp3an 1314 . . . 4  |-  ( C  e.  ( 0 [,] +oo )  <->  ( C  e.  ( 0 [,) +oo )  \/  C  = +oo ) )
82813anbi3i 1180 . . 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 1005 . 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 965    = wceq 1369    e. wcel 1756    =/= wne 2606   class class class wbr 4292  (class class class)co 6091   RRcr 9281   0cc0 9282   +oocpnf 9415   -oocmnf 9416   RR*cxr 9417    < clt 9418    <_ cle 9419   +ecxad 11087   xecxmu 11088   [,)cico 11302   [,]cicc 11303
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-po 4641  df-so 4642  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-1st 6577  df-2nd 6578  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-xneg 11089  df-xadd 11090  df-xmul 11091  df-ico 11306  df-icc 11307
This theorem is referenced by:  xrge0adddi  26156  xrge0slmod  26312  esummulc1  26530
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