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Theorem xmulasslem 10869
Description: Lemma for xmulass 10871. (Contributed by Mario Carneiro, 20-Aug-2015.)
Hypotheses
Ref Expression
xmulasslem.1  |-  ( x  =  D  ->  ( ps 
<->  X  =  Y ) )
xmulasslem.2  |-  ( x  =  - e D  ->  ( ps  <->  E  =  F ) )
xmulasslem.x  |-  ( ph  ->  X  e.  RR* )
xmulasslem.y  |-  ( ph  ->  Y  e.  RR* )
xmulasslem.d  |-  ( ph  ->  D  e.  RR* )
xmulasslem.ps  |-  ( (
ph  /\  ( x  e.  RR*  /\  0  < 
x ) )  ->  ps )
xmulasslem.0  |-  ( ph  ->  ( x  =  0  ->  ps ) )
xmulasslem.e  |-  ( ph  ->  E  =  - e X )
xmulasslem.f  |-  ( ph  ->  F  =  - e Y )
Assertion
Ref Expression
xmulasslem  |-  ( ph  ->  X  =  Y )
Distinct variable groups:    x, D    x, E    x, F    ph, x    x, X    x, Y
Allowed substitution hint:    ps( x)

Proof of Theorem xmulasslem
StepHypRef Expression
1 xmulasslem.d . . 3  |-  ( ph  ->  D  e.  RR* )
2 0xr 9136 . . 3  |-  0  e.  RR*
3 xrltso 10739 . . . 4  |-  <  Or  RR*
4 solin 4529 . . . 4  |-  ( (  <  Or  RR*  /\  ( D  e.  RR*  /\  0  e.  RR* ) )  -> 
( D  <  0  \/  D  =  0  \/  0  <  D ) )
53, 4mpan 653 . . 3  |-  ( ( D  e.  RR*  /\  0  e.  RR* )  ->  ( D  <  0  \/  D  =  0  \/  0  <  D ) )
61, 2, 5sylancl 645 . 2  |-  ( ph  ->  ( D  <  0  \/  D  =  0  \/  0  <  D ) )
7 xlt0neg1 10810 . . . . . 6  |-  ( D  e.  RR*  ->  ( D  <  0  <->  0  <  - e D ) )
81, 7syl 16 . . . . 5  |-  ( ph  ->  ( D  <  0  <->  0  <  - e D ) )
9 xnegcl 10804 . . . . . . 7  |-  ( D  e.  RR*  ->  - e D  e.  RR* )
101, 9syl 16 . . . . . 6  |-  ( ph  -> 
- e D  e. 
RR* )
11 breq2 4219 . . . . . . . . 9  |-  ( x  =  - e D  ->  ( 0  < 
x  <->  0  <  - e D ) )
12 xmulasslem.2 . . . . . . . . 9  |-  ( x  =  - e D  ->  ( ps  <->  E  =  F ) )
1311, 12imbi12d 313 . . . . . . . 8  |-  ( x  =  - e D  ->  ( ( 0  <  x  ->  ps ) 
<->  ( 0  <  - e D  ->  E  =  F ) ) )
1413imbi2d 309 . . . . . . 7  |-  ( x  =  - e D  ->  ( ( ph  ->  ( 0  <  x  ->  ps ) )  <->  ( ph  ->  ( 0  <  - e D  ->  E  =  F ) ) ) )
15 xmulasslem.ps . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  RR*  /\  0  < 
x ) )  ->  ps )
1615exp32 590 . . . . . . . 8  |-  ( ph  ->  ( x  e.  RR*  ->  ( 0  <  x  ->  ps ) ) )
1716com12 30 . . . . . . 7  |-  ( x  e.  RR*  ->  ( ph  ->  ( 0  <  x  ->  ps ) ) )
1814, 17vtoclga 3019 . . . . . 6  |-  (  - e D  e.  RR*  ->  (
ph  ->  ( 0  <  - e D  ->  E  =  F ) ) )
1910, 18mpcom 35 . . . . 5  |-  ( ph  ->  ( 0  <  - e D  ->  E  =  F ) )
208, 19sylbid 208 . . . 4  |-  ( ph  ->  ( D  <  0  ->  E  =  F ) )
21 xmulasslem.e . . . . . 6  |-  ( ph  ->  E  =  - e X )
22 xmulasslem.f . . . . . 6  |-  ( ph  ->  F  =  - e Y )
2321, 22eqeq12d 2452 . . . . 5  |-  ( ph  ->  ( E  =  F  <->  - e X  =  - e Y ) )
24 xmulasslem.x . . . . . 6  |-  ( ph  ->  X  e.  RR* )
25 xmulasslem.y . . . . . 6  |-  ( ph  ->  Y  e.  RR* )
26 xneg11 10806 . . . . . 6  |-  ( ( X  e.  RR*  /\  Y  e.  RR* )  ->  (  - e X  =  - e Y  <->  X  =  Y
) )
2724, 25, 26syl2anc 644 . . . . 5  |-  ( ph  ->  (  - e X  =  - e Y  <-> 
X  =  Y ) )
2823, 27bitrd 246 . . . 4  |-  ( ph  ->  ( E  =  F  <-> 
X  =  Y ) )
2920, 28sylibd 207 . . 3  |-  ( ph  ->  ( D  <  0  ->  X  =  Y ) )
30 eqeq1 2444 . . . . . . 7  |-  ( x  =  D  ->  (
x  =  0  <->  D  =  0 ) )
31 xmulasslem.1 . . . . . . 7  |-  ( x  =  D  ->  ( ps 
<->  X  =  Y ) )
3230, 31imbi12d 313 . . . . . 6  |-  ( x  =  D  ->  (
( x  =  0  ->  ps )  <->  ( D  =  0  ->  X  =  Y ) ) )
3332imbi2d 309 . . . . 5  |-  ( x  =  D  ->  (
( ph  ->  ( x  =  0  ->  ps ) )  <->  ( ph  ->  ( D  =  0  ->  X  =  Y ) ) ) )
34 xmulasslem.0 . . . . 5  |-  ( ph  ->  ( x  =  0  ->  ps ) )
3533, 34vtoclg 3013 . . . 4  |-  ( D  e.  RR*  ->  ( ph  ->  ( D  =  0  ->  X  =  Y ) ) )
361, 35mpcom 35 . . 3  |-  ( ph  ->  ( D  =  0  ->  X  =  Y ) )
37 breq2 4219 . . . . . . 7  |-  ( x  =  D  ->  (
0  <  x  <->  0  <  D ) )
3837, 31imbi12d 313 . . . . . 6  |-  ( x  =  D  ->  (
( 0  <  x  ->  ps )  <->  ( 0  <  D  ->  X  =  Y ) ) )
3938imbi2d 309 . . . . 5  |-  ( x  =  D  ->  (
( ph  ->  ( 0  <  x  ->  ps ) )  <->  ( ph  ->  ( 0  <  D  ->  X  =  Y ) ) ) )
4039, 17vtoclga 3019 . . . 4  |-  ( D  e.  RR*  ->  ( ph  ->  ( 0  <  D  ->  X  =  Y ) ) )
411, 40mpcom 35 . . 3  |-  ( ph  ->  ( 0  <  D  ->  X  =  Y ) )
4229, 36, 413jaod 1249 . 2  |-  ( ph  ->  ( ( D  <  0  \/  D  =  0  \/  0  < 
D )  ->  X  =  Y ) )
436, 42mpd 15 1  |-  ( ph  ->  X  =  Y )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    \/ w3o 936    = wceq 1653    e. wcel 1726   class class class wbr 4215    Or wor 4505   0cc0 8995   RR*cxr 9124    < clt 9125    - ecxne 10712
This theorem is referenced by:  xmulass  10871
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-po 4506  df-so 4507  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-riota 6552  df-er 6908  df-en 7113  df-dom 7114  df-sdom 7115  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-xneg 10715
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