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Theorem xmulasslem 11246
Description: Lemma for xmulass 11248. (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 9428 . . 3  |-  0  e.  RR*
3 xrltso 11116 . . . 4  |-  <  Or  RR*
4 solin 4662 . . . 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 11187 . . . . . 6  |-  ( D  e.  RR*  ->  ( D  <  0  <->  0  <  -e D ) )
81, 7syl 16 . . . . 5  |-  ( ph  ->  ( D  <  0  <->  0  <  -e D ) )
9 xnegcl 11181 . . . . . . 7  |-  ( D  e.  RR*  ->  -e
D  e.  RR* )
101, 9syl 16 . . . . . 6  |-  ( ph  -> 
-e D  e. 
RR* )
11 breq2 4294 . . . . . . . . 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 3034 . . . . . 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 2455 . . . . 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 11183 . . . . . 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 2447 . . . . . . 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 3028 . . . 4  |-  ( D  e.  RR*  ->  ( ph  ->  ( D  =  0  ->  X  =  Y ) ) )
361, 35mpcom 36 . . 3  |-  ( ph  ->  ( D  =  0  ->  X  =  Y ) )
37 breq2 4294 . . . . . . 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 3034 . . . 4  |-  ( D  e.  RR*  ->  ( ph  ->  ( 0  <  D  ->  X  =  Y ) ) )
411, 40mpcom 36 . . 3  |-  ( ph  ->  ( 0  <  D  ->  X  =  Y ) )
4229, 36, 413jaod 1282 . 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 964    = wceq 1369    e. wcel 1756   class class class wbr 4290    Or wor 4638   0cc0 9280   RR*cxr 9415    < clt 9416    -ecxne 11084
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 2422  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-cnex 9336  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-addrcl 9341  ax-mulcl 9342  ax-mulrcl 9343  ax-mulcom 9344  ax-addass 9345  ax-mulass 9346  ax-distr 9347  ax-i2m1 9348  ax-1ne0 9349  ax-1rid 9350  ax-rnegex 9351  ax-rrecex 9352  ax-cnre 9353  ax-pre-lttri 9354  ax-pre-lttrn 9355  ax-pre-ltadd 9356
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 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-br 4291  df-opab 4349  df-mpt 4350  df-id 4634  df-po 4639  df-so 4640  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-er 7099  df-en 7309  df-dom 7310  df-sdom 7311  df-pnf 9418  df-mnf 9419  df-xr 9420  df-ltxr 9421  df-le 9422  df-sub 9595  df-neg 9596  df-xneg 11087
This theorem is referenced by:  xmulass  11248
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