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Theorem xmulasslem 11476
Description: Lemma for xmulass 11478. (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 9639 . . 3  |-  0  e.  RR*
3 xrltso 11346 . . . 4  |-  <  Or  RR*
4 solin 4823 . . . 4  |-  ( (  <  Or  RR*  /\  ( D  e.  RR*  /\  0  e.  RR* ) )  -> 
( D  <  0  \/  D  =  0  \/  0  <  D ) )
53, 4mpan 670 . . 3  |-  ( ( D  e.  RR*  /\  0  e.  RR* )  ->  ( D  <  0  \/  D  =  0  \/  0  <  D ) )
61, 2, 5sylancl 662 . 2  |-  ( ph  ->  ( D  <  0  \/  D  =  0  \/  0  <  D ) )
7 xlt0neg1 11417 . . . . . 6  |-  ( D  e.  RR*  ->  ( D  <  0  <->  0  <  -e D ) )
81, 7syl 16 . . . . 5  |-  ( ph  ->  ( D  <  0  <->  0  <  -e D ) )
9 xnegcl 11411 . . . . . . 7  |-  ( D  e.  RR*  ->  -e
D  e.  RR* )
101, 9syl 16 . . . . . 6  |-  ( ph  -> 
-e D  e. 
RR* )
11 breq2 4451 . . . . . . . . 9  |-  ( x  =  -e D  ->  ( 0  < 
x  <->  0  <  -e
D ) )
12 xmulasslem.2 . . . . . . . . 9  |-  ( x  =  -e D  ->  ( ps  <->  E  =  F ) )
1311, 12imbi12d 320 . . . . . . . 8  |-  ( x  =  -e D  ->  ( ( 0  <  x  ->  ps ) 
<->  ( 0  <  -e
D  ->  E  =  F ) ) )
1413imbi2d 316 . . . . . . 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 605 . . . . . . . 8  |-  ( ph  ->  ( x  e.  RR*  ->  ( 0  <  x  ->  ps ) ) )
1716com12 31 . . . . . . 7  |-  ( x  e.  RR*  ->  ( ph  ->  ( 0  <  x  ->  ps ) ) )
1814, 17vtoclga 3177 . . . . . 6  |-  (  -e D  e.  RR*  ->  (
ph  ->  ( 0  <  -e D  ->  E  =  F )
) )
1910, 18mpcom 36 . . . . 5  |-  ( ph  ->  ( 0  <  -e
D  ->  E  =  F ) )
208, 19sylbid 215 . . . 4  |-  ( ph  ->  ( D  <  0  ->  E  =  F ) )
21 xmulasslem.e . . . . . 6  |-  ( ph  ->  E  =  -e
X )
22 xmulasslem.f . . . . . 6  |-  ( ph  ->  F  =  -e
Y )
2321, 22eqeq12d 2489 . . . . 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 11413 . . . . . 6  |-  ( ( X  e.  RR*  /\  Y  e.  RR* )  ->  (  -e X  =  -e Y  <->  X  =  Y
) )
2724, 25, 26syl2anc 661 . . . . 5  |-  ( ph  ->  (  -e X  =  -e Y  <-> 
X  =  Y ) )
2823, 27bitrd 253 . . . 4  |-  ( ph  ->  ( E  =  F  <-> 
X  =  Y ) )
2920, 28sylibd 214 . . 3  |-  ( ph  ->  ( D  <  0  ->  X  =  Y ) )
30 eqeq1 2471 . . . . . . 7  |-  ( x  =  D  ->  (
x  =  0  <->  D  =  0 ) )
31 xmulasslem.1 . . . . . . 7  |-  ( x  =  D  ->  ( ps 
<->  X  =  Y ) )
3230, 31imbi12d 320 . . . . . 6  |-  ( x  =  D  ->  (
( x  =  0  ->  ps )  <->  ( D  =  0  ->  X  =  Y ) ) )
3332imbi2d 316 . . . . 5  |-  ( x  =  D  ->  (
( ph  ->  ( x  =  0  ->  ps ) )  <->  ( ph  ->  ( D  =  0  ->  X  =  Y ) ) ) )
34 xmulasslem.0 . . . . 5  |-  ( ph  ->  ( x  =  0  ->  ps ) )
3533, 34vtoclg 3171 . . . 4  |-  ( D  e.  RR*  ->  ( ph  ->  ( D  =  0  ->  X  =  Y ) ) )
361, 35mpcom 36 . . 3  |-  ( ph  ->  ( D  =  0  ->  X  =  Y ) )
37 breq2 4451 . . . . . . 7  |-  ( x  =  D  ->  (
0  <  x  <->  0  <  D ) )
3837, 31imbi12d 320 . . . . . 6  |-  ( x  =  D  ->  (
( 0  <  x  ->  ps )  <->  ( 0  <  D  ->  X  =  Y ) ) )
3938imbi2d 316 . . . . 5  |-  ( x  =  D  ->  (
( ph  ->  ( 0  <  x  ->  ps ) )  <->  ( ph  ->  ( 0  <  D  ->  X  =  Y ) ) ) )
4039, 17vtoclga 3177 . . . 4  |-  ( D  e.  RR*  ->  ( ph  ->  ( 0  <  D  ->  X  =  Y ) ) )
411, 40mpcom 36 . . 3  |-  ( ph  ->  ( 0  <  D  ->  X  =  Y ) )
4229, 36, 413jaod 1292 . 2  |-  ( ph  ->  ( ( D  <  0  \/  D  =  0  \/  0  < 
D )  ->  X  =  Y ) )
436, 42mpd 15 1  |-  ( ph  ->  X  =  Y )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    \/ w3o 972    = wceq 1379    e. wcel 1767   class class class wbr 4447    Or wor 4799   0cc0 9491   RR*cxr 9626    < clt 9627    -ecxne 11314
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
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-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-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
This theorem is referenced by:  xmulass  11478
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