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Theorem xadddilem 11580
Description: Lemma for xadddi 11581. (Contributed by Mario Carneiro, 20-Aug-2015.)
Assertion
Ref Expression
xadddilem  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  0  <  A )  ->  ( A xe ( B +e C ) )  =  ( ( A xe B ) +e ( A xe C ) ) )

Proof of Theorem xadddilem
StepHypRef Expression
1 simpl1 1008 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  0  <  A )  ->  A  e.  RR )
2 recn 9629 . . . . . . . 8  |-  ( A  e.  RR  ->  A  e.  CC )
3 recn 9629 . . . . . . . 8  |-  ( B  e.  RR  ->  B  e.  CC )
4 recn 9629 . . . . . . . 8  |-  ( C  e.  RR  ->  C  e.  CC )
5 adddi 9628 . . . . . . . 8  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  x.  ( B  +  C ) )  =  ( ( A  x.  B )  +  ( A  x.  C ) ) )
62, 3, 4, 5syl3an 1306 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  x.  ( B  +  C ) )  =  ( ( A  x.  B )  +  ( A  x.  C ) ) )
763expa 1205 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  RR )  ->  ( A  x.  ( B  +  C
) )  =  ( ( A  x.  B
)  +  ( A  x.  C ) ) )
8 readdcl 9622 . . . . . . . 8  |-  ( ( B  e.  RR  /\  C  e.  RR )  ->  ( B  +  C
)  e.  RR )
9 rexmul 11557 . . . . . . . 8  |-  ( ( A  e.  RR  /\  ( B  +  C
)  e.  RR )  ->  ( A xe ( B  +  C ) )  =  ( A  x.  ( B  +  C )
) )
108, 9sylan2 476 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( B  e.  RR  /\  C  e.  RR ) )  ->  ( A xe ( B  +  C ) )  =  ( A  x.  ( B  +  C
) ) )
1110anassrs 652 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  RR )  ->  ( A xe ( B  +  C ) )  =  ( A  x.  ( B  +  C )
) )
12 remulcl 9624 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  x.  B
)  e.  RR )
1312adantr 466 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  RR )  ->  ( A  x.  B )  e.  RR )
14 remulcl 9624 . . . . . . . 8  |-  ( ( A  e.  RR  /\  C  e.  RR )  ->  ( A  x.  C
)  e.  RR )
1514adantlr 719 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  RR )  ->  ( A  x.  C )  e.  RR )
16 rexadd 11525 . . . . . . 7  |-  ( ( ( A  x.  B
)  e.  RR  /\  ( A  x.  C
)  e.  RR )  ->  ( ( A  x.  B ) +e ( A  x.  C ) )  =  ( ( A  x.  B )  +  ( A  x.  C ) ) )
1713, 15, 16syl2anc 665 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  RR )  ->  ( ( A  x.  B ) +e ( A  x.  C ) )  =  ( ( A  x.  B )  +  ( A  x.  C ) ) )
187, 11, 173eqtr4d 2473 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  RR )  ->  ( A xe ( B  +  C ) )  =  ( ( A  x.  B ) +e
( A  x.  C
) ) )
19 rexadd 11525 . . . . . . 7  |-  ( ( B  e.  RR  /\  C  e.  RR )  ->  ( B +e
C )  =  ( B  +  C ) )
2019adantll 718 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  RR )  ->  ( B +e C )  =  ( B  +  C
) )
2120oveq2d 6317 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  RR )  ->  ( A xe ( B +e C ) )  =  ( A xe ( B  +  C ) ) )
22 rexmul 11557 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A xe B )  =  ( A  x.  B ) )
2322adantr 466 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  RR )  ->  ( A xe B )  =  ( A  x.  B
) )
24 rexmul 11557 . . . . . . 7  |-  ( ( A  e.  RR  /\  C  e.  RR )  ->  ( A xe C )  =  ( A  x.  C ) )
2524adantlr 719 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  RR )  ->  ( A xe C )  =  ( A  x.  C
) )
2623, 25oveq12d 6319 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  RR )  ->  ( ( A xe B ) +e ( A xe C ) )  =  ( ( A  x.  B ) +e ( A  x.  C ) ) )
2718, 21, 263eqtr4d 2473 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  RR )  ->  ( A xe ( B +e C ) )  =  ( ( A xe B ) +e ( A xe C ) ) )
281, 27sylanl1 654 . . 3  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  e.  RR )  /\  C  e.  RR )  ->  ( A xe ( B +e C ) )  =  ( ( A xe B ) +e ( A xe C ) ) )
29 rexr 9686 . . . . . . . . 9  |-  ( A  e.  RR  ->  A  e.  RR* )
30293ad2ant1 1026 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  ->  A  e.  RR* )
31 xmulpnf1 11560 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  0  <  A )  ->  ( A xe +oo )  = +oo )
3230, 31sylan 473 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  0  <  A )  ->  ( A xe +oo )  = +oo )
3332adantr 466 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  0  < 
A )  /\  B  e.  RR )  ->  ( A xe +oo )  = +oo )
3422, 12eqeltrd 2510 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A xe B )  e.  RR )
351, 34sylan 473 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  0  < 
A )  /\  B  e.  RR )  ->  ( A xe B )  e.  RR )
36 rexr 9686 . . . . . . . 8  |-  ( ( A xe B )  e.  RR  ->  ( A xe B )  e.  RR* )
37 renemnf 9689 . . . . . . . 8  |-  ( ( A xe B )  e.  RR  ->  ( A xe B )  =/= -oo )
38 xaddpnf1 11519 . . . . . . . 8  |-  ( ( ( A xe B )  e.  RR*  /\  ( A xe B )  =/= -oo )  ->  ( ( A xe B ) +e +oo )  = +oo )
3936, 37, 38syl2anc 665 . . . . . . 7  |-  ( ( A xe B )  e.  RR  ->  ( ( A xe B ) +e +oo )  = +oo )
4035, 39syl 17 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  0  < 
A )  /\  B  e.  RR )  ->  (
( A xe B ) +e +oo )  = +oo )
4133, 40eqtr4d 2466 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  0  < 
A )  /\  B  e.  RR )  ->  ( A xe +oo )  =  ( ( A xe B ) +e +oo )
)
4241adantr 466 . . . 4  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  e.  RR )  /\  C  = +oo )  ->  ( A xe +oo )  =  ( ( A xe B ) +e +oo )
)
43 oveq2 6309 . . . . . 6  |-  ( C  = +oo  ->  ( B +e C )  =  ( B +e +oo ) )
44 rexr 9686 . . . . . . . 8  |-  ( B  e.  RR  ->  B  e.  RR* )
45 renemnf 9689 . . . . . . . 8  |-  ( B  e.  RR  ->  B  =/= -oo )
46 xaddpnf1 11519 . . . . . . . 8  |-  ( ( B  e.  RR*  /\  B  =/= -oo )  ->  ( B +e +oo )  = +oo )
4744, 45, 46syl2anc 665 . . . . . . 7  |-  ( B  e.  RR  ->  ( B +e +oo )  = +oo )
4847adantl 467 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  0  < 
A )  /\  B  e.  RR )  ->  ( B +e +oo )  = +oo )
4943, 48sylan9eqr 2485 . . . . 5  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  e.  RR )  /\  C  = +oo )  ->  ( B +e C )  = +oo )
5049oveq2d 6317 . . . 4  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  e.  RR )  /\  C  = +oo )  ->  ( A xe ( B +e C ) )  =  ( A xe +oo )
)
51 oveq2 6309 . . . . . 6  |-  ( C  = +oo  ->  ( A xe C )  =  ( A xe +oo ) )
5251, 33sylan9eqr 2485 . . . . 5  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  e.  RR )  /\  C  = +oo )  ->  ( A xe C )  = +oo )
5352oveq2d 6317 . . . 4  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  e.  RR )  /\  C  = +oo )  ->  (
( A xe B ) +e
( A xe C ) )  =  ( ( A xe B ) +e +oo ) )
5442, 50, 533eqtr4d 2473 . . 3  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  e.  RR )  /\  C  = +oo )  ->  ( A xe ( B +e C ) )  =  ( ( A xe B ) +e ( A xe C ) ) )
55 xmulmnf1 11562 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  0  <  A )  ->  ( A xe -oo )  = -oo )
5630, 55sylan 473 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  0  <  A )  ->  ( A xe -oo )  = -oo )
5756adantr 466 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  0  < 
A )  /\  B  e.  RR )  ->  ( A xe -oo )  = -oo )
5857adantr 466 . . . . 5  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  e.  RR )  /\  C  = -oo )  ->  ( A xe -oo )  = -oo )
5935adantr 466 . . . . . 6  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  e.  RR )  /\  C  = -oo )  ->  ( A xe B )  e.  RR )
60 renepnf 9688 . . . . . . 7  |-  ( ( A xe B )  e.  RR  ->  ( A xe B )  =/= +oo )
61 xaddmnf1 11521 . . . . . . 7  |-  ( ( ( A xe B )  e.  RR*  /\  ( A xe B )  =/= +oo )  ->  ( ( A xe B ) +e -oo )  = -oo )
6236, 60, 61syl2anc 665 . . . . . 6  |-  ( ( A xe B )  e.  RR  ->  ( ( A xe B ) +e -oo )  = -oo )
6359, 62syl 17 . . . . 5  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  e.  RR )  /\  C  = -oo )  ->  (
( A xe B ) +e -oo )  = -oo )
6458, 63eqtr4d 2466 . . . 4  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  e.  RR )  /\  C  = -oo )  ->  ( A xe -oo )  =  ( ( A xe B ) +e -oo )
)
65 oveq2 6309 . . . . . 6  |-  ( C  = -oo  ->  ( B +e C )  =  ( B +e -oo ) )
66 renepnf 9688 . . . . . . . 8  |-  ( B  e.  RR  ->  B  =/= +oo )
67 xaddmnf1 11521 . . . . . . . 8  |-  ( ( B  e.  RR*  /\  B  =/= +oo )  ->  ( B +e -oo )  = -oo )
6844, 66, 67syl2anc 665 . . . . . . 7  |-  ( B  e.  RR  ->  ( B +e -oo )  = -oo )
6968adantl 467 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  0  < 
A )  /\  B  e.  RR )  ->  ( B +e -oo )  = -oo )
7065, 69sylan9eqr 2485 . . . . 5  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  e.  RR )  /\  C  = -oo )  ->  ( B +e C )  = -oo )
7170oveq2d 6317 . . . 4  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  e.  RR )  /\  C  = -oo )  ->  ( A xe ( B +e C ) )  =  ( A xe -oo )
)
72 oveq2 6309 . . . . . 6  |-  ( C  = -oo  ->  ( A xe C )  =  ( A xe -oo ) )
7372, 57sylan9eqr 2485 . . . . 5  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  e.  RR )  /\  C  = -oo )  ->  ( A xe C )  = -oo )
7473oveq2d 6317 . . . 4  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  e.  RR )  /\  C  = -oo )  ->  (
( A xe B ) +e
( A xe C ) )  =  ( ( A xe B ) +e -oo ) )
7564, 71, 743eqtr4d 2473 . . 3  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  e.  RR )  /\  C  = -oo )  ->  ( A xe ( B +e C ) )  =  ( ( A xe B ) +e ( A xe C ) ) )
76 simpl3 1010 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  0  <  A )  ->  C  e.  RR* )
77 elxr 11416 . . . . 5  |-  ( C  e.  RR*  <->  ( C  e.  RR  \/  C  = +oo  \/  C  = -oo ) )
7876, 77sylib 199 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  0  <  A )  ->  ( C  e.  RR  \/  C  = +oo  \/  C  = -oo ) )
7978adantr 466 . . 3  |-  ( ( ( ( A  e.  RR  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  0  < 
A )  /\  B  e.  RR )  ->  ( C  e.  RR  \/  C  = +oo  \/  C  = -oo ) )
8028, 54, 75, 79mpjao3dan 1331 . 2  |-  ( ( ( ( A  e.  RR  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  0  < 
A )  /\  B  e.  RR )  ->  ( A xe ( B +e C ) )  =  ( ( A xe B ) +e ( A xe C ) ) )
8132ad2antrr 730 . . . . 5  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  = +oo )  /\  C  e.  RR )  ->  ( A xe +oo )  = +oo )
821adantr 466 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  0  < 
A )  /\  B  = +oo )  ->  A  e.  RR )
8324, 14eqeltrd 2510 . . . . . . 7  |-  ( ( A  e.  RR  /\  C  e.  RR )  ->  ( A xe C )  e.  RR )
8482, 83sylan 473 . . . . . 6  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  = +oo )  /\  C  e.  RR )  ->  ( A xe C )  e.  RR )
85 rexr 9686 . . . . . . 7  |-  ( ( A xe C )  e.  RR  ->  ( A xe C )  e.  RR* )
86 renemnf 9689 . . . . . . 7  |-  ( ( A xe C )  e.  RR  ->  ( A xe C )  =/= -oo )
87 xaddpnf2 11520 . . . . . . 7  |-  ( ( ( A xe C )  e.  RR*  /\  ( A xe C )  =/= -oo )  ->  ( +oo +e ( A xe C ) )  = +oo )
8885, 86, 87syl2anc 665 . . . . . 6  |-  ( ( A xe C )  e.  RR  ->  ( +oo +e ( A xe C ) )  = +oo )
8984, 88syl 17 . . . . 5  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  = +oo )  /\  C  e.  RR )  ->  ( +oo +e ( A xe C ) )  = +oo )
9081, 89eqtr4d 2466 . . . 4  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  = +oo )  /\  C  e.  RR )  ->  ( A xe +oo )  =  ( +oo +e
( A xe C ) ) )
91 simpr 462 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  0  < 
A )  /\  B  = +oo )  ->  B  = +oo )
9291oveq1d 6316 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  0  < 
A )  /\  B  = +oo )  ->  ( B +e C )  =  ( +oo +e C ) )
93 rexr 9686 . . . . . . 7  |-  ( C  e.  RR  ->  C  e.  RR* )
94 renemnf 9689 . . . . . . 7  |-  ( C  e.  RR  ->  C  =/= -oo )
95 xaddpnf2 11520 . . . . . . 7  |-  ( ( C  e.  RR*  /\  C  =/= -oo )  ->  ( +oo +e C )  = +oo )
9693, 94, 95syl2anc 665 . . . . . 6  |-  ( C  e.  RR  ->  ( +oo +e C )  = +oo )
9792, 96sylan9eq 2483 . . . . 5  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  = +oo )  /\  C  e.  RR )  ->  ( B +e C )  = +oo )
9897oveq2d 6317 . . . 4  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  = +oo )  /\  C  e.  RR )  ->  ( A xe ( B +e C ) )  =  ( A xe +oo )
)
99 oveq2 6309 . . . . . . 7  |-  ( B  = +oo  ->  ( A xe B )  =  ( A xe +oo ) )
10099, 32sylan9eqr 2485 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  0  < 
A )  /\  B  = +oo )  ->  ( A xe B )  = +oo )
101100adantr 466 . . . . 5  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  = +oo )  /\  C  e.  RR )  ->  ( A xe B )  = +oo )
102101oveq1d 6316 . . . 4  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  = +oo )  /\  C  e.  RR )  ->  (
( A xe B ) +e
( A xe C ) )  =  ( +oo +e
( A xe C ) ) )
10390, 98, 1023eqtr4d 2473 . . 3  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  = +oo )  /\  C  e.  RR )  ->  ( A xe ( B +e C ) )  =  ( ( A xe B ) +e ( A xe C ) ) )
104 pnfxr 11412 . . . . . . 7  |- +oo  e.  RR*
105 pnfnemnf 11417 . . . . . . 7  |- +oo  =/= -oo
106 xaddpnf1 11519 . . . . . . 7  |-  ( ( +oo  e.  RR*  /\ +oo  =/= -oo )  ->  ( +oo +e +oo )  = +oo )
107104, 105, 106mp2an 676 . . . . . 6  |-  ( +oo +e +oo )  = +oo
10832, 32oveq12d 6319 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  0  <  A )  ->  (
( A xe +oo ) +e
( A xe +oo ) )  =  ( +oo +e +oo ) )
109107, 108, 323eqtr4a 2489 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  0  <  A )  ->  (
( A xe +oo ) +e
( A xe +oo ) )  =  ( A xe +oo ) )
110109ad2antrr 730 . . . 4  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  = +oo )  /\  C  = +oo )  ->  (
( A xe +oo ) +e
( A xe +oo ) )  =  ( A xe +oo ) )
11199, 51oveqan12d 6320 . . . . 5  |-  ( ( B  = +oo  /\  C  = +oo )  ->  ( ( A xe B ) +e ( A xe C ) )  =  ( ( A xe +oo ) +e ( A xe +oo )
) )
112111adantll 718 . . . 4  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  = +oo )  /\  C  = +oo )  ->  (
( A xe B ) +e
( A xe C ) )  =  ( ( A xe +oo ) +e ( A xe +oo ) ) )
113 oveq12 6310 . . . . . . 7  |-  ( ( B  = +oo  /\  C  = +oo )  ->  ( B +e
C )  =  ( +oo +e +oo ) )
114113, 107syl6eq 2479 . . . . . 6  |-  ( ( B  = +oo  /\  C  = +oo )  ->  ( B +e
C )  = +oo )
115114oveq2d 6317 . . . . 5  |-  ( ( B  = +oo  /\  C  = +oo )  ->  ( A xe ( B +e
C ) )  =  ( A xe +oo ) )
116115adantll 718 . . . 4  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  = +oo )  /\  C  = +oo )  ->  ( A xe ( B +e C ) )  =  ( A xe +oo )
)
117110, 112, 1163eqtr4rd 2474 . . 3  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  = +oo )  /\  C  = +oo )  ->  ( A xe ( B +e C ) )  =  ( ( A xe B ) +e ( A xe C ) ) )
118 pnfaddmnf 11523 . . . . . 6  |-  ( +oo +e -oo )  =  0
11932, 56oveq12d 6319 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  0  <  A )  ->  (
( A xe +oo ) +e
( A xe -oo ) )  =  ( +oo +e -oo ) )
120 xmul01 11553 . . . . . . 7  |-  ( A  e.  RR*  ->  ( A xe 0 )  =  0 )
1211, 29, 1203syl 18 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  0  <  A )  ->  ( A xe 0 )  =  0 )
122118, 119, 1213eqtr4a 2489 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  0  <  A )  ->  (
( A xe +oo ) +e
( A xe -oo ) )  =  ( A xe 0 ) )
123122ad2antrr 730 . . . 4  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  = +oo )  /\  C  = -oo )  ->  (
( A xe +oo ) +e
( A xe -oo ) )  =  ( A xe 0 ) )
12499, 72oveqan12d 6320 . . . . 5  |-  ( ( B  = +oo  /\  C  = -oo )  ->  ( ( A xe B ) +e ( A xe C ) )  =  ( ( A xe +oo ) +e ( A xe -oo )
) )
125124adantll 718 . . . 4  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  = +oo )  /\  C  = -oo )  ->  (
( A xe B ) +e
( A xe C ) )  =  ( ( A xe +oo ) +e ( A xe -oo ) ) )
126 oveq12 6310 . . . . . . 7  |-  ( ( B  = +oo  /\  C  = -oo )  ->  ( B +e
C )  =  ( +oo +e -oo ) )
127126, 118syl6eq 2479 . . . . . 6  |-  ( ( B  = +oo  /\  C  = -oo )  ->  ( B +e
C )  =  0 )
128127oveq2d 6317 . . . . 5  |-  ( ( B  = +oo  /\  C  = -oo )  ->  ( A xe ( B +e
C ) )  =  ( A xe 0 ) )
129128adantll 718 . . . 4  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  = +oo )  /\  C  = -oo )  ->  ( A xe ( B +e C ) )  =  ( A xe 0 ) )
130123, 125, 1293eqtr4rd 2474 . . 3  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  = +oo )  /\  C  = -oo )  ->  ( A xe ( B +e C ) )  =  ( ( A xe B ) +e ( A xe C ) ) )
13178adantr 466 . . 3  |-  ( ( ( ( A  e.  RR  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  0  < 
A )  /\  B  = +oo )  ->  ( C  e.  RR  \/  C  = +oo  \/  C  = -oo ) )
132103, 117, 130, 131mpjao3dan 1331 . 2  |-  ( ( ( ( A  e.  RR  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  0  < 
A )  /\  B  = +oo )  ->  ( A xe ( B +e C ) )  =  ( ( A xe B ) +e ( A xe C ) ) )
13356ad2antrr 730 . . . . 5  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  = -oo )  /\  C  e.  RR )  ->  ( A xe -oo )  = -oo )
1341adantr 466 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  0  < 
A )  /\  B  = -oo )  ->  A  e.  RR )
135134, 83sylan 473 . . . . . 6  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  = -oo )  /\  C  e.  RR )  ->  ( A xe C )  e.  RR )
136 renepnf 9688 . . . . . . 7  |-  ( ( A xe C )  e.  RR  ->  ( A xe C )  =/= +oo )
137 xaddmnf2 11522 . . . . . . 7  |-  ( ( ( A xe C )  e.  RR*  /\  ( A xe C )  =/= +oo )  ->  ( -oo +e ( A xe C ) )  = -oo )
13885, 136, 137syl2anc 665 . . . . . 6  |-  ( ( A xe C )  e.  RR  ->  ( -oo +e ( A xe C ) )  = -oo )
139135, 138syl 17 . . . . 5  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  = -oo )  /\  C  e.  RR )  ->  ( -oo +e ( A xe C ) )  = -oo )
140133, 139eqtr4d 2466 . . . 4  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  = -oo )  /\  C  e.  RR )  ->  ( A xe -oo )  =  ( -oo +e
( A xe C ) ) )
141 simpr 462 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  0  < 
A )  /\  B  = -oo )  ->  B  = -oo )
142141oveq1d 6316 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  0  < 
A )  /\  B  = -oo )  ->  ( B +e C )  =  ( -oo +e C ) )
143 renepnf 9688 . . . . . . 7  |-  ( C  e.  RR  ->  C  =/= +oo )
144 xaddmnf2 11522 . . . . . . 7  |-  ( ( C  e.  RR*  /\  C  =/= +oo )  ->  ( -oo +e C )  = -oo )
14593, 143, 144syl2anc 665 . . . . . 6  |-  ( C  e.  RR  ->  ( -oo +e C )  = -oo )
146142, 145sylan9eq 2483 . . . . 5  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  = -oo )  /\  C  e.  RR )  ->  ( B +e C )  = -oo )
147146oveq2d 6317 . . . 4  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  = -oo )  /\  C  e.  RR )  ->  ( A xe ( B +e C ) )  =  ( A xe -oo )
)
148 oveq2 6309 . . . . . . 7  |-  ( B  = -oo  ->  ( A xe B )  =  ( A xe -oo ) )
149148, 56sylan9eqr 2485 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  0  < 
A )  /\  B  = -oo )  ->  ( A xe B )  = -oo )
150149adantr 466 . . . . 5  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  = -oo )  /\  C  e.  RR )  ->  ( A xe B )  = -oo )
151150oveq1d 6316 . . . 4  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  = -oo )  /\  C  e.  RR )  ->  (
( A xe B ) +e
( A xe C ) )  =  ( -oo +e
( A xe C ) ) )
152140, 147, 1513eqtr4d 2473 . . 3  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  = -oo )  /\  C  e.  RR )  ->  ( A xe ( B +e C ) )  =  ( ( A xe B ) +e ( A xe C ) ) )
15356, 32oveq12d 6319 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  0  <  A )  ->  (
( A xe -oo ) +e
( A xe +oo ) )  =  ( -oo +e +oo ) )
154 mnfaddpnf 11524 . . . . . . 7  |-  ( -oo +e +oo )  =  0
155153, 154syl6eq 2479 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  0  <  A )  ->  (
( A xe -oo ) +e
( A xe +oo ) )  =  0 )
156121, 155eqtr4d 2466 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  0  <  A )  ->  ( A xe 0 )  =  ( ( A xe -oo ) +e ( A xe +oo )
) )
157156ad2antrr 730 . . . 4  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  = -oo )  /\  C  = +oo )  ->  ( A xe 0 )  =  ( ( A xe -oo ) +e ( A xe +oo )
) )
158 oveq12 6310 . . . . . . 7  |-  ( ( B  = -oo  /\  C  = +oo )  ->  ( B +e
C )  =  ( -oo +e +oo ) )
159158, 154syl6eq 2479 . . . . . 6  |-  ( ( B  = -oo  /\  C  = +oo )  ->  ( B +e
C )  =  0 )
160159oveq2d 6317 . . . . 5  |-  ( ( B  = -oo  /\  C  = +oo )  ->  ( A xe ( B +e
C ) )  =  ( A xe 0 ) )
161160adantll 718 . . . 4  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  = -oo )  /\  C  = +oo )  ->  ( A xe ( B +e C ) )  =  ( A xe 0 ) )
162148, 51oveqan12d 6320 . . . . 5  |-  ( ( B  = -oo  /\  C  = +oo )  ->  ( ( A xe B ) +e ( A xe C ) )  =  ( ( A xe -oo ) +e ( A xe +oo )
) )
163162adantll 718 . . . 4  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  = -oo )  /\  C  = +oo )  ->  (
( A xe B ) +e
( A xe C ) )  =  ( ( A xe -oo ) +e ( A xe +oo ) ) )
164157, 161, 1633eqtr4d 2473 . . 3  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  = -oo )  /\  C  = +oo )  ->  ( A xe ( B +e C ) )  =  ( ( A xe B ) +e ( A xe C ) ) )
165 mnfxr 11414 . . . . . . 7  |- -oo  e.  RR*
166 mnfnepnf 11418 . . . . . . 7  |- -oo  =/= +oo
167 xaddmnf1 11521 . . . . . . 7  |-  ( ( -oo  e.  RR*  /\ -oo  =/= +oo )  ->  ( -oo +e -oo )  = -oo )
168165, 166, 167mp2an 676 . . . . . 6  |-  ( -oo +e -oo )  = -oo
16956, 56oveq12d 6319 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  0  <  A )  ->  (
( A xe -oo ) +e
( A xe -oo ) )  =  ( -oo +e -oo ) )
170168, 169, 563eqtr4a 2489 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  0  <  A )  ->  (
( A xe -oo ) +e
( A xe -oo ) )  =  ( A xe -oo ) )
171170ad2antrr 730 . . . 4  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  = -oo )  /\  C  = -oo )  ->  (
( A xe -oo ) +e
( A xe -oo ) )  =  ( A xe -oo ) )
172148, 72oveqan12d 6320 . . . . 5  |-  ( ( B  = -oo  /\  C  = -oo )  ->  ( ( A xe B ) +e ( A xe C ) )  =  ( ( A xe -oo ) +e ( A xe -oo )
) )
173172adantll 718 . . . 4  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  = -oo )  /\  C  = -oo )  ->  (
( A xe B ) +e
( A xe C ) )  =  ( ( A xe -oo ) +e ( A xe -oo ) ) )
174 oveq12 6310 . . . . . . 7  |-  ( ( B  = -oo  /\  C  = -oo )  ->  ( B +e
C )  =  ( -oo +e -oo ) )
175174, 168syl6eq 2479 . . . . . 6  |-  ( ( B  = -oo  /\  C  = -oo )  ->  ( B +e
C )  = -oo )
176175oveq2d 6317 . . . . 5  |-  ( ( B  = -oo  /\  C  = -oo )  ->  ( A xe ( B +e
C ) )  =  ( A xe -oo ) )
177176adantll 718 . . . 4  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  = -oo )  /\  C  = -oo )  ->  ( A xe ( B +e C ) )  =  ( A xe -oo )
)
178171, 173, 1773eqtr4rd 2474 . . 3  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  0  <  A )  /\  B  = -oo )  /\  C  = -oo )  ->  ( A xe ( B +e C ) )  =  ( ( A xe B ) +e ( A xe C ) ) )
17978adantr 466 . . 3  |-  ( ( ( ( A  e.  RR  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  0  < 
A )  /\  B  = -oo )  ->  ( C  e.  RR  \/  C  = +oo  \/  C  = -oo ) )
180152, 164, 178, 179mpjao3dan 1331 . 2  |-  ( ( ( ( A  e.  RR  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  0  < 
A )  /\  B  = -oo )  ->  ( A xe ( B +e C ) )  =  ( ( A xe B ) +e ( A xe C ) ) )
181 simpl2 1009 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  0  <  A )  ->  B  e.  RR* )
182 elxr 11416 . . 3  |-  ( B  e.  RR*  <->  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )
183181, 182sylib 199 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  0  <  A )  ->  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )
18480, 132, 180, 183mpjao3dan 1331 1  |-  ( ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  /\  0  <  A )  ->  ( A xe ( B +e C ) )  =  ( ( A xe B ) +e ( A xe C ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    \/ w3o 981    /\ w3a 982    = wceq 1437    e. wcel 1868    =/= wne 2618   class class class wbr 4420  (class class class)co 6301   CCcc 9537   RRcr 9538   0cc0 9539    + caddc 9542    x. cmul 9544   +oocpnf 9672   -oocmnf 9673   RR*cxr 9674    < clt 9675   +ecxad 11407   xecxmu 11408
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4551  ax-pow 4598  ax-pr 4656  ax-un 6593  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-op 4003  df-uni 4217  df-br 4421  df-opab 4480  df-mpt 4481  df-id 4764  df-po 4770  df-so 4771  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-rn 4860  df-res 4861  df-ima 4862  df-iota 5561  df-fun 5599  df-fn 5600  df-f 5601  df-f1 5602  df-fo 5603  df-f1o 5604  df-fv 5605  df-riota 6263  df-ov 6304  df-oprab 6305  df-mpt2 6306  df-er 7367  df-en 7574  df-dom 7575  df-sdom 7576  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-xneg 11409  df-xadd 11410  df-xmul 11411
This theorem is referenced by:  xadddi  11581
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