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Theorem xadddilem 10863
Description: Lemma for xadddi 10864. (Contributed by Mario Carneiro, 20-Aug-2015.)
Assertion
Ref Expression
xadddilem  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  0  <  A )  ->  ( A x e ( B + e C ) )  =  ( ( A x e B ) + e ( A x e C ) ) )

Proof of Theorem xadddilem
StepHypRef Expression
1 simpl2 961 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  0  <  A )  ->  B  e.  RR* )
2 elxr 10706 . . 3  |-  ( B  e.  RR*  <->  ( B  e.  RR  \/  B  = 
+oo  \/  B  =  -oo ) )
31, 2sylib 189 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  0  <  A )  ->  ( B  e.  RR  \/  B  =  +oo  \/  B  =  -oo ) )
4 simpl3 962 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  0  <  A )  ->  C  e.  RR* )
5 elxr 10706 . . . . . 6  |-  ( C  e.  RR*  <->  ( C  e.  RR  \/  C  = 
+oo  \/  C  =  -oo ) )
64, 5sylib 189 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  0  <  A )  ->  ( C  e.  RR  \/  C  =  +oo  \/  C  =  -oo ) )
76adantr 452 . . . 4  |-  ( ( ( ( A  e.  RR  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  0  < 
A )  /\  B  e.  RR )  ->  ( C  e.  RR  \/  C  =  +oo  \/  C  =  -oo ) )
8 simpl1 960 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  0  <  A )  ->  A  e.  RR )
9 recn 9070 . . . . . . . . . 10  |-  ( A  e.  RR  ->  A  e.  CC )
10 recn 9070 . . . . . . . . . 10  |-  ( B  e.  RR  ->  B  e.  CC )
11 recn 9070 . . . . . . . . . 10  |-  ( C  e.  RR  ->  C  e.  CC )
12 adddi 9069 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  x.  ( B  +  C ) )  =  ( ( A  x.  B )  +  ( A  x.  C ) ) )
139, 10, 11, 12syl3an 1226 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  x.  ( B  +  C ) )  =  ( ( A  x.  B )  +  ( A  x.  C ) ) )
14133expa 1153 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  RR )  ->  ( A  x.  ( B  +  C
) )  =  ( ( A  x.  B
)  +  ( A  x.  C ) ) )
15 readdcl 9063 . . . . . . . . . 10  |-  ( ( B  e.  RR  /\  C  e.  RR )  ->  ( B  +  C
)  e.  RR )
16 rexmul 10840 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( B  +  C
)  e.  RR )  ->  ( A x e ( B  +  C ) )  =  ( A  x.  ( B  +  C )
) )
1715, 16sylan2 461 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  ( B  e.  RR  /\  C  e.  RR ) )  ->  ( A x e ( B  +  C ) )  =  ( A  x.  ( B  +  C )
) )
1817anassrs 630 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  RR )  ->  ( A x e ( B  +  C ) )  =  ( A  x.  ( B  +  C )
) )
19 remulcl 9065 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  x.  B
)  e.  RR )
2019adantr 452 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  RR )  ->  ( A  x.  B )  e.  RR )
21 remulcl 9065 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  C  e.  RR )  ->  ( A  x.  C
)  e.  RR )
2221adantlr 696 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  RR )  ->  ( A  x.  C )  e.  RR )
23 rexadd 10808 . . . . . . . . 9  |-  ( ( ( A  x.  B
)  e.  RR  /\  ( A  x.  C
)  e.  RR )  ->  ( ( A  x.  B ) + e ( A  x.  C ) )  =  ( ( A  x.  B )  +  ( A  x.  C ) ) )
2420, 22, 23syl2anc 643 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  RR )  ->  ( ( A  x.  B ) + e ( A  x.  C ) )  =  ( ( A  x.  B )  +  ( A  x.  C ) ) )
2514, 18, 243eqtr4d 2477 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  RR )  ->  ( A x e ( B  +  C ) )  =  ( ( A  x.  B ) + e
( A  x.  C
) ) )
26 rexadd 10808 . . . . . . . . 9  |-  ( ( B  e.  RR  /\  C  e.  RR )  ->  ( B + e C )  =  ( B  +  C ) )
2726adantll 695 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  RR )  ->  ( B + e C )  =  ( B  +  C ) )
2827oveq2d 6089 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  RR )  ->  ( A x e ( B + e C ) )  =  ( A x e ( B  +  C
) ) )
29 rexmul 10840 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A x e B )  =  ( A  x.  B ) )
3029adantr 452 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  RR )  ->  ( A x e B )  =  ( A  x.  B
) )
31 rexmul 10840 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  C  e.  RR )  ->  ( A x e C )  =  ( A  x.  C ) )
3231adantlr 696 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  RR )  ->  ( A x e C )  =  ( A  x.  C
) )
3330, 32oveq12d 6091 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  RR )  ->  ( ( A x e B ) + e ( A x e C ) )  =  ( ( A  x.  B ) + e ( A  x.  C ) ) )
3425, 28, 333eqtr4d 2477 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  RR )  ->  ( A x e ( B + e C ) )  =  ( ( A x e B ) + e ( A x e C ) ) )
358, 34sylanl1 632 . . . . 5  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  e.  RR )  /\  C  e.  RR )  ->  ( A x e ( B + e C ) )  =  ( ( A x e B ) + e ( A x e C ) ) )
36 rexr 9120 . . . . . . . . . . 11  |-  ( A  e.  RR  ->  A  e.  RR* )
37363ad2ant1 978 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  ->  A  e.  RR* )
38 xmulpnf1 10843 . . . . . . . . . 10  |-  ( ( A  e.  RR*  /\  0  <  A )  ->  ( A x e  +oo )  =  +oo )
3937, 38sylan 458 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  0  <  A )  ->  ( A x e  +oo )  =  +oo )
4039adantr 452 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  0  < 
A )  /\  B  e.  RR )  ->  ( A x e  +oo )  =  +oo )
4129, 19eqeltrd 2509 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A x e B )  e.  RR )
428, 41sylan 458 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  0  < 
A )  /\  B  e.  RR )  ->  ( A x e B )  e.  RR )
43 rexr 9120 . . . . . . . . . 10  |-  ( ( A x e B )  e.  RR  ->  ( A x e B )  e.  RR* )
44 renemnf 9123 . . . . . . . . . 10  |-  ( ( A x e B )  e.  RR  ->  ( A x e B )  =/=  -oo )
45 xaddpnf1 10802 . . . . . . . . . 10  |-  ( ( ( A x e B )  e.  RR*  /\  ( A x e B )  =/=  -oo )  ->  ( ( A x e B ) + e  +oo )  =  +oo )
4643, 44, 45syl2anc 643 . . . . . . . . 9  |-  ( ( A x e B )  e.  RR  ->  ( ( A x e B ) + e  +oo )  =  +oo )
4742, 46syl 16 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  0  < 
A )  /\  B  e.  RR )  ->  (
( A x e B ) + e  +oo )  =  +oo )
4840, 47eqtr4d 2470 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  0  < 
A )  /\  B  e.  RR )  ->  ( A x e  +oo )  =  ( ( A x e B ) + e  +oo )
)
4948adantr 452 . . . . . 6  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  e.  RR )  /\  C  =  +oo )  ->  ( A x e  +oo )  =  ( ( A x e B ) + e  +oo )
)
50 oveq2 6081 . . . . . . . 8  |-  ( C  =  +oo  ->  ( B + e C )  =  ( B + e  +oo ) )
51 rexr 9120 . . . . . . . . . 10  |-  ( B  e.  RR  ->  B  e.  RR* )
52 renemnf 9123 . . . . . . . . . 10  |-  ( B  e.  RR  ->  B  =/=  -oo )
53 xaddpnf1 10802 . . . . . . . . . 10  |-  ( ( B  e.  RR*  /\  B  =/=  -oo )  ->  ( B + e  +oo )  =  +oo )
5451, 52, 53syl2anc 643 . . . . . . . . 9  |-  ( B  e.  RR  ->  ( B + e  +oo )  =  +oo )
5554adantl 453 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  0  < 
A )  /\  B  e.  RR )  ->  ( B + e  +oo )  =  +oo )
5650, 55sylan9eqr 2489 . . . . . . 7  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  e.  RR )  /\  C  =  +oo )  ->  ( B + e C )  =  +oo )
5756oveq2d 6089 . . . . . 6  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  e.  RR )  /\  C  =  +oo )  ->  ( A x e ( B + e C ) )  =  ( A x e  +oo )
)
58 oveq2 6081 . . . . . . . 8  |-  ( C  =  +oo  ->  ( A x e C )  =  ( A x e  +oo ) )
5958, 40sylan9eqr 2489 . . . . . . 7  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  e.  RR )  /\  C  =  +oo )  ->  ( A x e C )  =  +oo )
6059oveq2d 6089 . . . . . 6  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  e.  RR )  /\  C  =  +oo )  ->  (
( A x e B ) + e
( A x e C ) )  =  ( ( A x e B ) + e  +oo ) )
6149, 57, 603eqtr4d 2477 . . . . 5  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  e.  RR )  /\  C  =  +oo )  ->  ( A x e ( B + e C ) )  =  ( ( A x e B ) + e ( A x e C ) ) )
62 xmulmnf1 10845 . . . . . . . . . 10  |-  ( ( A  e.  RR*  /\  0  <  A )  ->  ( A x e  -oo )  =  -oo )
6337, 62sylan 458 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  0  <  A )  ->  ( A x e  -oo )  =  -oo )
6463adantr 452 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  0  < 
A )  /\  B  e.  RR )  ->  ( A x e  -oo )  =  -oo )
6564adantr 452 . . . . . . 7  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  e.  RR )  /\  C  =  -oo )  ->  ( A x e  -oo )  =  -oo )
6642adantr 452 . . . . . . . 8  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  e.  RR )  /\  C  =  -oo )  ->  ( A x e B )  e.  RR )
67 renepnf 9122 . . . . . . . . 9  |-  ( ( A x e B )  e.  RR  ->  ( A x e B )  =/=  +oo )
68 xaddmnf1 10804 . . . . . . . . 9  |-  ( ( ( A x e B )  e.  RR*  /\  ( A x e B )  =/=  +oo )  ->  ( ( A x e B ) + e  -oo )  =  -oo )
6943, 67, 68syl2anc 643 . . . . . . . 8  |-  ( ( A x e B )  e.  RR  ->  ( ( A x e B ) + e  -oo )  =  -oo )
7066, 69syl 16 . . . . . . 7  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  e.  RR )  /\  C  =  -oo )  ->  (
( A x e B ) + e  -oo )  =  -oo )
7165, 70eqtr4d 2470 . . . . . 6  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  e.  RR )  /\  C  =  -oo )  ->  ( A x e  -oo )  =  ( ( A x e B ) + e  -oo )
)
72 oveq2 6081 . . . . . . . 8  |-  ( C  =  -oo  ->  ( B + e C )  =  ( B + e  -oo ) )
73 renepnf 9122 . . . . . . . . . 10  |-  ( B  e.  RR  ->  B  =/=  +oo )
74 xaddmnf1 10804 . . . . . . . . . 10  |-  ( ( B  e.  RR*  /\  B  =/=  +oo )  ->  ( B + e  -oo )  =  -oo )
7551, 73, 74syl2anc 643 . . . . . . . . 9  |-  ( B  e.  RR  ->  ( B + e  -oo )  =  -oo )
7675adantl 453 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  0  < 
A )  /\  B  e.  RR )  ->  ( B + e  -oo )  =  -oo )
7772, 76sylan9eqr 2489 . . . . . . 7  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  e.  RR )  /\  C  =  -oo )  ->  ( B + e C )  =  -oo )
7877oveq2d 6089 . . . . . 6  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  e.  RR )  /\  C  =  -oo )  ->  ( A x e ( B + e C ) )  =  ( A x e  -oo )
)
79 oveq2 6081 . . . . . . . 8  |-  ( C  =  -oo  ->  ( A x e C )  =  ( A x e  -oo ) )
8079, 64sylan9eqr 2489 . . . . . . 7  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  e.  RR )  /\  C  =  -oo )  ->  ( A x e C )  =  -oo )
8180oveq2d 6089 . . . . . 6  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  e.  RR )  /\  C  =  -oo )  ->  (
( A x e B ) + e
( A x e C ) )  =  ( ( A x e B ) + e  -oo ) )
8271, 78, 813eqtr4d 2477 . . . . 5  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  e.  RR )  /\  C  =  -oo )  ->  ( A x e ( B + e C ) )  =  ( ( A x e B ) + e ( A x e C ) ) )
8335, 61, 823jaodan 1250 . . . 4  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  e.  RR )  /\  ( C  e.  RR  \/  C  =  +oo  \/  C  =  -oo ) )  -> 
( A x e ( B + e C ) )  =  ( ( A x e B ) + e ( A x e C ) ) )
847, 83mpdan 650 . . 3  |-  ( ( ( ( A  e.  RR  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  0  < 
A )  /\  B  e.  RR )  ->  ( A x e ( B + e C ) )  =  ( ( A x e B ) + e ( A x e C ) ) )
856adantr 452 . . . 4  |-  ( ( ( ( A  e.  RR  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  0  < 
A )  /\  B  =  +oo )  ->  ( C  e.  RR  \/  C  =  +oo  \/  C  =  -oo ) )
8639ad2antrr 707 . . . . . . 7  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  =  +oo )  /\  C  e.  RR )  ->  ( A x e  +oo )  =  +oo )
878adantr 452 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  0  < 
A )  /\  B  =  +oo )  ->  A  e.  RR )
8831, 21eqeltrd 2509 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  C  e.  RR )  ->  ( A x e C )  e.  RR )
8987, 88sylan 458 . . . . . . . 8  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  =  +oo )  /\  C  e.  RR )  ->  ( A x e C )  e.  RR )
90 rexr 9120 . . . . . . . . 9  |-  ( ( A x e C )  e.  RR  ->  ( A x e C )  e.  RR* )
91 renemnf 9123 . . . . . . . . 9  |-  ( ( A x e C )  e.  RR  ->  ( A x e C )  =/=  -oo )
92 xaddpnf2 10803 . . . . . . . . 9  |-  ( ( ( A x e C )  e.  RR*  /\  ( A x e C )  =/=  -oo )  ->  (  +oo + e ( A x e C ) )  =  +oo )
9390, 91, 92syl2anc 643 . . . . . . . 8  |-  ( ( A x e C )  e.  RR  ->  ( 
+oo + e ( A x e C ) )  =  +oo )
9489, 93syl 16 . . . . . . 7  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  =  +oo )  /\  C  e.  RR )  ->  (  +oo + e ( A x e C ) )  =  +oo )
9586, 94eqtr4d 2470 . . . . . 6  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  =  +oo )  /\  C  e.  RR )  ->  ( A x e  +oo )  =  (  +oo + e
( A x e C ) ) )
96 simpr 448 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  0  < 
A )  /\  B  =  +oo )  ->  B  =  +oo )
9796oveq1d 6088 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  0  < 
A )  /\  B  =  +oo )  ->  ( B + e C )  =  (  +oo + e C ) )
98 rexr 9120 . . . . . . . . 9  |-  ( C  e.  RR  ->  C  e.  RR* )
99 renemnf 9123 . . . . . . . . 9  |-  ( C  e.  RR  ->  C  =/=  -oo )
100 xaddpnf2 10803 . . . . . . . . 9  |-  ( ( C  e.  RR*  /\  C  =/=  -oo )  ->  (  +oo + e C )  =  +oo )
10198, 99, 100syl2anc 643 . . . . . . . 8  |-  ( C  e.  RR  ->  (  +oo + e C )  =  +oo )
10297, 101sylan9eq 2487 . . . . . . 7  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  =  +oo )  /\  C  e.  RR )  ->  ( B + e C )  =  +oo )
103102oveq2d 6089 . . . . . 6  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  =  +oo )  /\  C  e.  RR )  ->  ( A x e ( B + e C ) )  =  ( A x e  +oo )
)
104 oveq2 6081 . . . . . . . . 9  |-  ( B  =  +oo  ->  ( A x e B )  =  ( A x e  +oo ) )
105104, 39sylan9eqr 2489 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  0  < 
A )  /\  B  =  +oo )  ->  ( A x e B )  =  +oo )
106105adantr 452 . . . . . . 7  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  =  +oo )  /\  C  e.  RR )  ->  ( A x e B )  =  +oo )
107106oveq1d 6088 . . . . . 6  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  =  +oo )  /\  C  e.  RR )  ->  (
( A x e B ) + e
( A x e C ) )  =  (  +oo + e
( A x e C ) ) )
10895, 103, 1073eqtr4d 2477 . . . . 5  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  =  +oo )  /\  C  e.  RR )  ->  ( A x e ( B + e C ) )  =  ( ( A x e B ) + e ( A x e C ) ) )
109 pnfxr 10703 . . . . . . . . 9  |-  +oo  e.  RR*
110 pnfnemnf 10707 . . . . . . . . 9  |-  +oo  =/=  -oo
111 xaddpnf1 10802 . . . . . . . . 9  |-  ( ( 
+oo  e.  RR*  /\  +oo  =/=  -oo )  ->  (  +oo + e  +oo )  =  +oo )
112109, 110, 111mp2an 654 . . . . . . . 8  |-  (  +oo + e  +oo )  = 
+oo
11339, 39oveq12d 6091 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  0  <  A )  ->  (
( A x e 
+oo ) + e
( A x e 
+oo ) )  =  (  +oo + e  +oo ) )
114112, 113, 393eqtr4a 2493 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  0  <  A )  ->  (
( A x e 
+oo ) + e
( A x e 
+oo ) )  =  ( A x e 
+oo ) )
115114ad2antrr 707 . . . . . 6  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  =  +oo )  /\  C  =  +oo )  ->  (
( A x e 
+oo ) + e
( A x e 
+oo ) )  =  ( A x e 
+oo ) )
116104, 58oveqan12d 6092 . . . . . . 7  |-  ( ( B  =  +oo  /\  C  =  +oo )  -> 
( ( A x e B ) + e ( A x e C ) )  =  ( ( A x e  +oo ) + e ( A x e  +oo ) ) )
117116adantll 695 . . . . . 6  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  =  +oo )  /\  C  =  +oo )  ->  (
( A x e B ) + e
( A x e C ) )  =  ( ( A x e  +oo ) + e ( A x e  +oo ) ) )
118 oveq12 6082 . . . . . . . . 9  |-  ( ( B  =  +oo  /\  C  =  +oo )  -> 
( B + e C )  =  ( 
+oo + e  +oo ) )
119118, 112syl6eq 2483 . . . . . . . 8  |-  ( ( B  =  +oo  /\  C  =  +oo )  -> 
( B + e C )  =  +oo )
120119oveq2d 6089 . . . . . . 7  |-  ( ( B  =  +oo  /\  C  =  +oo )  -> 
( A x e ( B + e C ) )  =  ( A x e 
+oo ) )
121120adantll 695 . . . . . 6  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  =  +oo )  /\  C  =  +oo )  ->  ( A x e ( B + e C ) )  =  ( A x e  +oo )
)
122115, 117, 1213eqtr4rd 2478 . . . . 5  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  =  +oo )  /\  C  =  +oo )  ->  ( A x e ( B + e C ) )  =  ( ( A x e B ) + e ( A x e C ) ) )
123 pnfaddmnf 10806 . . . . . . . 8  |-  (  +oo + e  -oo )  =  0
12439, 63oveq12d 6091 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  0  <  A )  ->  (
( A x e 
+oo ) + e
( A x e 
-oo ) )  =  (  +oo + e  -oo ) )
125 xmul01 10836 . . . . . . . . 9  |-  ( A  e.  RR*  ->  ( A x e 0 )  =  0 )
1268, 36, 1253syl 19 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  0  <  A )  ->  ( A x e 0 )  =  0 )
127123, 124, 1263eqtr4a 2493 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  0  <  A )  ->  (
( A x e 
+oo ) + e
( A x e 
-oo ) )  =  ( A x e 0 ) )
128127ad2antrr 707 . . . . . 6  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  =  +oo )  /\  C  =  -oo )  ->  (
( A x e 
+oo ) + e
( A x e 
-oo ) )  =  ( A x e 0 ) )
129104, 79oveqan12d 6092 . . . . . . 7  |-  ( ( B  =  +oo  /\  C  =  -oo )  -> 
( ( A x e B ) + e ( A x e C ) )  =  ( ( A x e  +oo ) + e ( A x e  -oo ) ) )
130129adantll 695 . . . . . 6  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  =  +oo )  /\  C  =  -oo )  ->  (
( A x e B ) + e
( A x e C ) )  =  ( ( A x e  +oo ) + e ( A x e  -oo ) ) )
131 oveq12 6082 . . . . . . . . 9  |-  ( ( B  =  +oo  /\  C  =  -oo )  -> 
( B + e C )  =  ( 
+oo + e  -oo ) )
132131, 123syl6eq 2483 . . . . . . . 8  |-  ( ( B  =  +oo  /\  C  =  -oo )  -> 
( B + e C )  =  0 )
133132oveq2d 6089 . . . . . . 7  |-  ( ( B  =  +oo  /\  C  =  -oo )  -> 
( A x e ( B + e C ) )  =  ( A x e 0 ) )
134133adantll 695 . . . . . 6  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  =  +oo )  /\  C  =  -oo )  ->  ( A x e ( B + e C ) )  =  ( A x e 0 ) )
135128, 130, 1343eqtr4rd 2478 . . . . 5  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  =  +oo )  /\  C  =  -oo )  ->  ( A x e ( B + e C ) )  =  ( ( A x e B ) + e ( A x e C ) ) )
136108, 122, 1353jaodan 1250 . . . 4  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  =  +oo )  /\  ( C  e.  RR  \/  C  =  +oo  \/  C  =  -oo ) )  -> 
( A x e ( B + e C ) )  =  ( ( A x e B ) + e ( A x e C ) ) )
13785, 136mpdan 650 . . 3  |-  ( ( ( ( A  e.  RR  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  0  < 
A )  /\  B  =  +oo )  ->  ( A x e ( B + e C ) )  =  ( ( A x e B ) + e ( A x e C ) ) )
1386adantr 452 . . . 4  |-  ( ( ( ( A  e.  RR  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  0  < 
A )  /\  B  =  -oo )  ->  ( C  e.  RR  \/  C  =  +oo  \/  C  =  -oo ) )
13963ad2antrr 707 . . . . . . 7  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  =  -oo )  /\  C  e.  RR )  ->  ( A x e  -oo )  =  -oo )
1408adantr 452 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  0  < 
A )  /\  B  =  -oo )  ->  A  e.  RR )
141140, 88sylan 458 . . . . . . . 8  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  =  -oo )  /\  C  e.  RR )  ->  ( A x e C )  e.  RR )
142 renepnf 9122 . . . . . . . . 9  |-  ( ( A x e C )  e.  RR  ->  ( A x e C )  =/=  +oo )
143 xaddmnf2 10805 . . . . . . . . 9  |-  ( ( ( A x e C )  e.  RR*  /\  ( A x e C )  =/=  +oo )  ->  (  -oo + e ( A x e C ) )  =  -oo )
14490, 142, 143syl2anc 643 . . . . . . . 8  |-  ( ( A x e C )  e.  RR  ->  ( 
-oo + e ( A x e C ) )  =  -oo )
145141, 144syl 16 . . . . . . 7  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  =  -oo )  /\  C  e.  RR )  ->  (  -oo + e ( A x e C ) )  =  -oo )
146139, 145eqtr4d 2470 . . . . . 6  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  =  -oo )  /\  C  e.  RR )  ->  ( A x e  -oo )  =  (  -oo + e
( A x e C ) ) )
147 simpr 448 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  0  < 
A )  /\  B  =  -oo )  ->  B  =  -oo )
148147oveq1d 6088 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  0  < 
A )  /\  B  =  -oo )  ->  ( B + e C )  =  (  -oo + e C ) )
149 renepnf 9122 . . . . . . . . 9  |-  ( C  e.  RR  ->  C  =/=  +oo )
150 xaddmnf2 10805 . . . . . . . . 9  |-  ( ( C  e.  RR*  /\  C  =/=  +oo )  ->  (  -oo + e C )  =  -oo )
15198, 149, 150syl2anc 643 . . . . . . . 8  |-  ( C  e.  RR  ->  (  -oo + e C )  =  -oo )
152148, 151sylan9eq 2487 . . . . . . 7  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  =  -oo )  /\  C  e.  RR )  ->  ( B + e C )  =  -oo )
153152oveq2d 6089 . . . . . 6  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  =  -oo )  /\  C  e.  RR )  ->  ( A x e ( B + e C ) )  =  ( A x e  -oo )
)
154 oveq2 6081 . . . . . . . . 9  |-  ( B  =  -oo  ->  ( A x e B )  =  ( A x e  -oo ) )
155154, 63sylan9eqr 2489 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  0  < 
A )  /\  B  =  -oo )  ->  ( A x e B )  =  -oo )
156155adantr 452 . . . . . . 7  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  =  -oo )  /\  C  e.  RR )  ->  ( A x e B )  =  -oo )
157156oveq1d 6088 . . . . . 6  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  =  -oo )  /\  C  e.  RR )  ->  (
( A x e B ) + e
( A x e C ) )  =  (  -oo + e
( A x e C ) ) )
158146, 153, 1573eqtr4d 2477 . . . . 5  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  =  -oo )  /\  C  e.  RR )  ->  ( A x e ( B + e C ) )  =  ( ( A x e B ) + e ( A x e C ) ) )
15963, 39oveq12d 6091 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  0  <  A )  ->  (
( A x e 
-oo ) + e
( A x e 
+oo ) )  =  (  -oo + e  +oo ) )
160 mnfaddpnf 10807 . . . . . . . . 9  |-  (  -oo + e  +oo )  =  0
161159, 160syl6eq 2483 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  0  <  A )  ->  (
( A x e 
-oo ) + e
( A x e 
+oo ) )  =  0 )
162126, 161eqtr4d 2470 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  0  <  A )  ->  ( A x e 0 )  =  ( ( A x e  -oo ) + e ( A x e  +oo ) ) )
163162ad2antrr 707 . . . . . 6  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  =  -oo )  /\  C  =  +oo )  ->  ( A x e 0 )  =  ( ( A x e  -oo ) + e ( A x e  +oo ) ) )
164 oveq12 6082 . . . . . . . . 9  |-  ( ( B  =  -oo  /\  C  =  +oo )  -> 
( B + e C )  =  ( 
-oo + e  +oo ) )
165164, 160syl6eq 2483 . . . . . . . 8  |-  ( ( B  =  -oo  /\  C  =  +oo )  -> 
( B + e C )  =  0 )
166165oveq2d 6089 . . . . . . 7  |-  ( ( B  =  -oo  /\  C  =  +oo )  -> 
( A x e ( B + e C ) )  =  ( A x e 0 ) )
167166adantll 695 . . . . . 6  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  =  -oo )  /\  C  =  +oo )  ->  ( A x e ( B + e C ) )  =  ( A x e 0 ) )
168154, 58oveqan12d 6092 . . . . . . 7  |-  ( ( B  =  -oo  /\  C  =  +oo )  -> 
( ( A x e B ) + e ( A x e C ) )  =  ( ( A x e  -oo ) + e ( A x e  +oo ) ) )
169168adantll 695 . . . . . 6  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  =  -oo )  /\  C  =  +oo )  ->  (
( A x e B ) + e
( A x e C ) )  =  ( ( A x e  -oo ) + e ( A x e  +oo ) ) )
170163, 167, 1693eqtr4d 2477 . . . . 5  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  =  -oo )  /\  C  =  +oo )  ->  ( A x e ( B + e C ) )  =  ( ( A x e B ) + e ( A x e C ) ) )
171 mnfxr 10704 . . . . . . . . 9  |-  -oo  e.  RR*
172110necomi 2680 . . . . . . . . 9  |-  -oo  =/=  +oo
173 xaddmnf1 10804 . . . . . . . . 9  |-  ( ( 
-oo  e.  RR*  /\  -oo  =/=  +oo )  ->  (  -oo + e  -oo )  =  -oo )
174171, 172, 173mp2an 654 . . . . . . . 8  |-  (  -oo + e  -oo )  = 
-oo
17563, 63oveq12d 6091 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  0  <  A )  ->  (
( A x e 
-oo ) + e
( A x e 
-oo ) )  =  (  -oo + e  -oo ) )
176174, 175, 633eqtr4a 2493 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  0  <  A )  ->  (
( A x e 
-oo ) + e
( A x e 
-oo ) )  =  ( A x e 
-oo ) )
177176ad2antrr 707 . . . . . 6  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  =  -oo )  /\  C  =  -oo )  ->  (
( A x e 
-oo ) + e
( A x e 
-oo ) )  =  ( A x e 
-oo ) )
178154, 79oveqan12d 6092 . . . . . . 7  |-  ( ( B  =  -oo  /\  C  =  -oo )  -> 
( ( A x e B ) + e ( A x e C ) )  =  ( ( A x e  -oo ) + e ( A x e  -oo ) ) )
179178adantll 695 . . . . . 6  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  =  -oo )  /\  C  =  -oo )  ->  (
( A x e B ) + e
( A x e C ) )  =  ( ( A x e  -oo ) + e ( A x e  -oo ) ) )
180 oveq12 6082 . . . . . . . . 9  |-  ( ( B  =  -oo  /\  C  =  -oo )  -> 
( B + e C )  =  ( 
-oo + e  -oo ) )
181180, 174syl6eq 2483 . . . . . . . 8  |-  ( ( B  =  -oo  /\  C  =  -oo )  -> 
( B + e C )  =  -oo )
182181oveq2d 6089 . . . . . . 7  |-  ( ( B  =  -oo  /\  C  =  -oo )  -> 
( A x e ( B + e C ) )  =  ( A x e 
-oo ) )
183182adantll 695 . . . . . 6  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  =  -oo )  /\  C  =  -oo )  ->  ( A x e ( B + e C ) )  =  ( A x e  -oo )
)
184177, 179, 1833eqtr4rd 2478 . . . . 5  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  =  -oo )  /\  C  =  -oo )  ->  ( A x e ( B + e C ) )  =  ( ( A x e B ) + e ( A x e C ) ) )
185158, 170, 1843jaodan 1250 . . . 4  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  =  -oo )  /\  ( C  e.  RR  \/  C  =  +oo  \/  C  =  -oo ) )  -> 
( A x e ( B + e C ) )  =  ( ( A x e B ) + e ( A x e C ) ) )
186138, 185mpdan 650 . . 3  |-  ( ( ( ( A  e.  RR  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  0  < 
A )  /\  B  =  -oo )  ->  ( A x e ( B + e C ) )  =  ( ( A x e B ) + e ( A x e C ) ) )
18784, 137, 1863jaodan 1250 . 2  |-  ( ( ( ( A  e.  RR  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  0  < 
A )  /\  ( B  e.  RR  \/  B  =  +oo  \/  B  =  -oo ) )  -> 
( A x e ( B + e C ) )  =  ( ( A x e B ) + e ( A x e C ) ) )
1883, 187mpdan 650 1  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  0  <  A )  ->  ( A x e ( B + e C ) )  =  ( ( A x e B ) + e ( A x e C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    \/ w3o 935    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   class class class wbr 4204  (class class class)co 6073   CCcc 8978   RRcr 8979   0cc0 8980    + caddc 8983    x. cmul 8985    +oocpnf 9107    -oocmnf 9108   RR*cxr 9109    < clt 9110   + ecxad 10698   x ecxmu 10699
This theorem is referenced by:  xadddi  10864
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9036  ax-resscn 9037  ax-1cn 9038  ax-icn 9039  ax-addcl 9040  ax-addrcl 9041  ax-mulcl 9042  ax-mulrcl 9043  ax-mulcom 9044  ax-addass 9045  ax-mulass 9046  ax-distr 9047  ax-i2m1 9048  ax-1ne0 9049  ax-1rid 9050  ax-rnegex 9051  ax-rrecex 9052  ax-cnre 9053  ax-pre-lttri 9054  ax-pre-lttrn 9055  ax-pre-ltadd 9056
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-po 4495  df-so 4496  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-riota 6541  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9112  df-mnf 9113  df-xr 9114  df-ltxr 9115  df-le 9116  df-sub 9283  df-neg 9284  df-xneg 10700  df-xadd 10701  df-xmul 10702
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