Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  xlt2addrd Structured version   Unicode version

Theorem xlt2addrd 27735
Description: If the right-hand side of a 'less than' relationship is an addition, then we can express the left-hand side as an addition, too, where each term is respectively less than each term of the original right side. (Contributed by Thierry Arnoux, 15-Mar-2017.)
Hypotheses
Ref Expression
xlt2addrd.1  |-  ( ph  ->  A  e.  RR )
xlt2addrd.2  |-  ( ph  ->  B  e.  RR* )
xlt2addrd.3  |-  ( ph  ->  C  e.  RR* )
xlt2addrd.4  |-  ( ph  ->  B  =/= -oo )
xlt2addrd.5  |-  ( ph  ->  C  =/= -oo )
xlt2addrd.6  |-  ( ph  ->  A  <  ( B +e C ) )
Assertion
Ref Expression
xlt2addrd  |-  ( ph  ->  E. b  e.  RR*  E. c  e.  RR*  ( A  =  ( b +e c )  /\  b  <  B  /\  c  <  C ) )
Distinct variable groups:    b, c, A    B, b, c    C, b, c
Allowed substitution hints:    ph( b, c)

Proof of Theorem xlt2addrd
StepHypRef Expression
1 xlt2addrd.1 . . . . . 6  |-  ( ph  ->  A  e.  RR )
21rexrd 9660 . . . . 5  |-  ( ph  ->  A  e.  RR* )
32ad2antrr 725 . . . 4  |-  ( ( ( ph  /\  B  = +oo )  /\  C  = +oo )  ->  A  e.  RR* )
4 0xr 9657 . . . . 5  |-  0  e.  RR*
54a1i 11 . . . 4  |-  ( ( ( ph  /\  B  = +oo )  /\  C  = +oo )  ->  0  e.  RR* )
6 xaddid1 11463 . . . . . 6  |-  ( A  e.  RR*  ->  ( A +e 0 )  =  A )
76eqcomd 2465 . . . . 5  |-  ( A  e.  RR*  ->  A  =  ( A +e 0 ) )
83, 7syl 16 . . . 4  |-  ( ( ( ph  /\  B  = +oo )  /\  C  = +oo )  ->  A  =  ( A +e 0 ) )
91ad2antrr 725 . . . . . 6  |-  ( ( ( ph  /\  B  = +oo )  /\  C  = +oo )  ->  A  e.  RR )
10 ltpnf 11356 . . . . . 6  |-  ( A  e.  RR  ->  A  < +oo )
119, 10syl 16 . . . . 5  |-  ( ( ( ph  /\  B  = +oo )  /\  C  = +oo )  ->  A  < +oo )
12 simplr 755 . . . . 5  |-  ( ( ( ph  /\  B  = +oo )  /\  C  = +oo )  ->  B  = +oo )
1311, 12breqtrrd 4482 . . . 4  |-  ( ( ( ph  /\  B  = +oo )  /\  C  = +oo )  ->  A  <  B )
14 0ltpnf 11357 . . . . 5  |-  0  < +oo
15 simpr 461 . . . . 5  |-  ( ( ( ph  /\  B  = +oo )  /\  C  = +oo )  ->  C  = +oo )
1614, 15syl5breqr 4492 . . . 4  |-  ( ( ( ph  /\  B  = +oo )  /\  C  = +oo )  ->  0  <  C )
17 oveq1 6303 . . . . . . 7  |-  ( b  =  A  ->  (
b +e c )  =  ( A +e c ) )
1817eqeq2d 2471 . . . . . 6  |-  ( b  =  A  ->  ( A  =  ( b +e c )  <-> 
A  =  ( A +e c ) ) )
19 breq1 4459 . . . . . 6  |-  ( b  =  A  ->  (
b  <  B  <->  A  <  B ) )
2018, 193anbi12d 1300 . . . . 5  |-  ( b  =  A  ->  (
( A  =  ( b +e c )  /\  b  < 
B  /\  c  <  C )  <->  ( A  =  ( A +e
c )  /\  A  <  B  /\  c  < 
C ) ) )
21 oveq2 6304 . . . . . . 7  |-  ( c  =  0  ->  ( A +e c )  =  ( A +e 0 ) )
2221eqeq2d 2471 . . . . . 6  |-  ( c  =  0  ->  ( A  =  ( A +e c )  <-> 
A  =  ( A +e 0 ) ) )
23 breq1 4459 . . . . . 6  |-  ( c  =  0  ->  (
c  <  C  <->  0  <  C ) )
2422, 233anbi13d 1301 . . . . 5  |-  ( c  =  0  ->  (
( A  =  ( A +e c )  /\  A  < 
B  /\  c  <  C )  <->  ( A  =  ( A +e 0 )  /\  A  <  B  /\  0  < 
C ) ) )
2520, 24rspc2ev 3221 . . . 4  |-  ( ( A  e.  RR*  /\  0  e.  RR*  /\  ( A  =  ( A +e 0 )  /\  A  <  B  /\  0  <  C ) )  ->  E. b  e.  RR*  E. c  e.  RR*  ( A  =  ( b +e
c )  /\  b  <  B  /\  c  < 
C ) )
263, 5, 8, 13, 16, 25syl113anc 1240 . . 3  |-  ( ( ( ph  /\  B  = +oo )  /\  C  = +oo )  ->  E. b  e.  RR*  E. c  e. 
RR*  ( A  =  ( b +e
c )  /\  b  <  B  /\  c  < 
C ) )
272ad2antrr 725 . . . . 5  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  A  e.  RR* )
28 xlt2addrd.3 . . . . . . . 8  |-  ( ph  ->  C  e.  RR* )
2928ad2antrr 725 . . . . . . 7  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  C  e.  RR* )
30 1re 9612 . . . . . . . . . 10  |-  1  e.  RR
3130rexri 9663 . . . . . . . . 9  |-  1  e.  RR*
3231a1i 11 . . . . . . . 8  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  1  e.  RR* )
3332xnegcld 11517 . . . . . . 7  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  -e 1  e.  RR* )
3429, 33xaddcld 11518 . . . . . 6  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  ( C +e  -e 1 )  e.  RR* )
3534xnegcld 11517 . . . . 5  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  -e
( C +e  -e 1 )  e. 
RR* )
3627, 35xaddcld 11518 . . . 4  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  ( A +e  -e
( C +e  -e 1 ) )  e.  RR* )
371ad2antrr 725 . . . . . . 7  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  A  e.  RR )
3837renemnfd 9662 . . . . . 6  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  A  =/= -oo )
39 xrnepnf 11354 . . . . . . . . . . . . . . 15  |-  ( ( C  e.  RR*  /\  C  =/= +oo )  <->  ( C  e.  RR  \/  C  = -oo ) )
4039biimpi 194 . . . . . . . . . . . . . 14  |-  ( ( C  e.  RR*  /\  C  =/= +oo )  ->  ( C  e.  RR  \/  C  = -oo )
)
4129, 40sylancom 667 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  ( C  e.  RR  \/  C  = -oo )
)
4241orcomd 388 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  ( C  = -oo  \/  C  e.  RR ) )
43 xlt2addrd.5 . . . . . . . . . . . . . 14  |-  ( ph  ->  C  =/= -oo )
4443ad2antrr 725 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  C  =/= -oo )
4544neneqd 2659 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  -.  C  = -oo )
46 pm2.53 373 . . . . . . . . . . . 12  |-  ( ( C  = -oo  \/  C  e.  RR )  ->  ( -.  C  = -oo  ->  C  e.  RR ) )
4742, 45, 46sylc 60 . . . . . . . . . . 11  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  C  e.  RR )
48 rexsub 11457 . . . . . . . . . . 11  |-  ( ( C  e.  RR  /\  1  e.  RR )  ->  ( C +e  -e 1 )  =  ( C  -  1 ) )
4947, 30, 48sylancl 662 . . . . . . . . . 10  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  ( C +e  -e 1 )  =  ( C  -  1 ) )
50 resubcl 9902 . . . . . . . . . . 11  |-  ( ( C  e.  RR  /\  1  e.  RR )  ->  ( C  -  1 )  e.  RR )
5147, 30, 50sylancl 662 . . . . . . . . . 10  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  ( C  -  1 )  e.  RR )
5249, 51eqeltrd 2545 . . . . . . . . 9  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  ( C +e  -e 1 )  e.  RR )
53 rexneg 11435 . . . . . . . . 9  |-  ( ( C +e  -e 1 )  e.  RR  ->  -e ( C +e  -e 1 )  = 
-u ( C +e  -e 1 ) )
5452, 53syl 16 . . . . . . . 8  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  -e
( C +e  -e 1 )  = 
-u ( C +e  -e 1 ) )
5552renegcld 10007 . . . . . . . 8  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  -u ( C +e  -e 1 )  e.  RR )
5654, 55eqeltrd 2545 . . . . . . 7  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  -e
( C +e  -e 1 )  e.  RR )
5756renemnfd 9662 . . . . . 6  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  -e
( C +e  -e 1 )  =/= -oo )
5852renemnfd 9662 . . . . . 6  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  ( C +e  -e 1 )  =/= -oo )
59 xaddass 11466 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  A  =/= -oo )  /\  (  -e ( C +e  -e 1 )  e. 
RR*  /\  -e ( C +e  -e 1 )  =/= -oo )  /\  (
( C +e  -e 1 )  e. 
RR*  /\  ( C +e  -e 1 )  =/= -oo )
)  ->  ( ( A +e  -e
( C +e  -e 1 ) ) +e ( C +e  -e 1 ) )  =  ( A +e
(  -e ( C +e  -e 1 ) +e
( C +e  -e 1 ) ) ) )
6027, 38, 35, 57, 34, 58, 59syl222anc 1244 . . . . 5  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  (
( A +e  -e ( C +e  -e 1 ) ) +e ( C +e  -e 1 ) )  =  ( A +e (  -e
( C +e  -e 1 ) +e ( C +e  -e 1 ) ) ) )
61 xaddcom 11462 . . . . . . . 8  |-  ( ( 
-e ( C +e  -e 1 )  e.  RR*  /\  ( C +e  -e 1 )  e. 
RR* )  ->  (  -e ( C +e  -e 1 ) +e ( C +e  -e 1 ) )  =  ( ( C +e  -e 1 ) +e  -e
( C +e  -e 1 ) ) )
6235, 34, 61syl2anc 661 . . . . . . 7  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  (  -e ( C +e  -e 1 ) +e ( C +e  -e 1 ) )  =  ( ( C +e  -e 1 ) +e  -e
( C +e  -e 1 ) ) )
63 xnegid 11460 . . . . . . . 8  |-  ( ( C +e  -e 1 )  e. 
RR*  ->  ( ( C +e  -e 1 ) +e  -e ( C +e  -e 1 ) )  =  0 )
6434, 63syl 16 . . . . . . 7  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  (
( C +e  -e 1 ) +e  -e ( C +e  -e 1 ) )  =  0 )
6562, 64eqtrd 2498 . . . . . 6  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  (  -e ( C +e  -e 1 ) +e ( C +e  -e 1 ) )  =  0 )
6665oveq2d 6312 . . . . 5  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  ( A +e (  -e ( C +e  -e 1 ) +e ( C +e  -e 1 ) ) )  =  ( A +e 0 ) )
6727, 6syl 16 . . . . 5  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  ( A +e 0 )  =  A )
6860, 66, 673eqtrrd 2503 . . . 4  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  A  =  ( ( A +e  -e
( C +e  -e 1 ) ) +e ( C +e  -e 1 ) ) )
6937, 51resubcld 10008 . . . . . 6  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  ( A  -  ( C  -  1 ) )  e.  RR )
70 ltpnf 11356 . . . . . 6  |-  ( ( A  -  ( C  -  1 ) )  e.  RR  ->  ( A  -  ( C  -  1 ) )  < +oo )
7169, 70syl 16 . . . . 5  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  ( A  -  ( C  -  1 ) )  < +oo )
72 rexsub 11457 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( C +e  -e 1 )  e.  RR )  ->  ( A +e  -e
( C +e  -e 1 ) )  =  ( A  -  ( C +e  -e 1 ) ) )
7337, 52, 72syl2anc 661 . . . . . 6  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  ( A +e  -e
( C +e  -e 1 ) )  =  ( A  -  ( C +e  -e 1 ) ) )
7449oveq2d 6312 . . . . . 6  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  ( A  -  ( C +e  -e 1 ) )  =  ( A  -  ( C  -  1 ) ) )
7573, 74eqtrd 2498 . . . . 5  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  ( A +e  -e
( C +e  -e 1 ) )  =  ( A  -  ( C  -  1
) ) )
76 simplr 755 . . . . 5  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  B  = +oo )
7771, 75, 763brtr4d 4486 . . . 4  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  ( A +e  -e
( C +e  -e 1 ) )  <  B )
7847ltm1d 10498 . . . . 5  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  ( C  -  1 )  <  C )
7949, 78eqbrtrd 4476 . . . 4  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  ( C +e  -e 1 )  <  C
)
80 oveq1 6303 . . . . . . 7  |-  ( b  =  ( A +e  -e ( C +e  -e 1 ) )  -> 
( b +e
c )  =  ( ( A +e  -e ( C +e  -e 1 ) ) +e c ) )
8180eqeq2d 2471 . . . . . 6  |-  ( b  =  ( A +e  -e ( C +e  -e 1 ) )  -> 
( A  =  ( b +e c )  <->  A  =  (
( A +e  -e ( C +e  -e 1 ) ) +e c ) ) )
82 breq1 4459 . . . . . 6  |-  ( b  =  ( A +e  -e ( C +e  -e 1 ) )  -> 
( b  <  B  <->  ( A +e  -e ( C +e  -e 1 ) )  <  B ) )
8381, 823anbi12d 1300 . . . . 5  |-  ( b  =  ( A +e  -e ( C +e  -e 1 ) )  -> 
( ( A  =  ( b +e
c )  /\  b  <  B  /\  c  < 
C )  <->  ( A  =  ( ( A +e  -e
( C +e  -e 1 ) ) +e c )  /\  ( A +e  -e ( C +e  -e 1 ) )  < 
B  /\  c  <  C ) ) )
84 oveq2 6304 . . . . . . 7  |-  ( c  =  ( C +e  -e 1 )  ->  ( ( A +e  -e
( C +e  -e 1 ) ) +e c )  =  ( ( A +e  -e
( C +e  -e 1 ) ) +e ( C +e  -e 1 ) ) )
8584eqeq2d 2471 . . . . . 6  |-  ( c  =  ( C +e  -e 1 )  ->  ( A  =  ( ( A +e  -e ( C +e  -e 1 ) ) +e c )  <->  A  =  ( ( A +e  -e ( C +e  -e 1 ) ) +e ( C +e  -e 1 ) ) ) )
86 breq1 4459 . . . . . 6  |-  ( c  =  ( C +e  -e 1 )  ->  ( c  < 
C  <->  ( C +e  -e 1 )  <  C ) )
8785, 863anbi13d 1301 . . . . 5  |-  ( c  =  ( C +e  -e 1 )  ->  ( ( A  =  ( ( A +e  -e
( C +e  -e 1 ) ) +e c )  /\  ( A +e  -e ( C +e  -e 1 ) )  < 
B  /\  c  <  C )  <->  ( A  =  ( ( A +e  -e ( C +e  -e 1 ) ) +e ( C +e  -e 1 ) )  /\  ( A +e  -e
( C +e  -e 1 ) )  <  B  /\  ( C +e  -e 1 )  <  C
) ) )
8883, 87rspc2ev 3221 . . . 4  |-  ( ( ( A +e  -e ( C +e  -e 1 ) )  e.  RR*  /\  ( C +e  -e 1 )  e.  RR*  /\  ( A  =  ( ( A +e  -e ( C +e  -e 1 ) ) +e ( C +e  -e 1 ) )  /\  ( A +e  -e ( C +e  -e 1 ) )  < 
B  /\  ( C +e  -e 1 )  <  C ) )  ->  E. b  e.  RR*  E. c  e. 
RR*  ( A  =  ( b +e
c )  /\  b  <  B  /\  c  < 
C ) )
8936, 34, 68, 77, 79, 88syl113anc 1240 . . 3  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  E. b  e.  RR*  E. c  e. 
RR*  ( A  =  ( b +e
c )  /\  b  <  B  /\  c  < 
C ) )
9026, 89pm2.61dane 2775 . 2  |-  ( (
ph  /\  B  = +oo )  ->  E. b  e.  RR*  E. c  e. 
RR*  ( A  =  ( b +e
c )  /\  b  <  B  /\  c  < 
C ) )
91 xlt2addrd.2 . . . . . 6  |-  ( ph  ->  B  e.  RR* )
9291ad2antrr 725 . . . . 5  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  B  e.  RR* )
9331a1i 11 . . . . . 6  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  1  e.  RR* )
9493xnegcld 11517 . . . . 5  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  -e 1  e.  RR* )
9592, 94xaddcld 11518 . . . 4  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  ( B +e  -e 1 )  e.  RR* )
962ad2antrr 725 . . . . 5  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  A  e.  RR* )
9795xnegcld 11517 . . . . 5  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  -e
( B +e  -e 1 )  e. 
RR* )
9896, 97xaddcld 11518 . . . 4  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  ( A +e  -e
( B +e  -e 1 ) )  e.  RR* )
99 xaddcom 11462 . . . . . 6  |-  ( ( ( B +e  -e 1 )  e. 
RR*  /\  ( A +e  -e ( B +e  -e 1 ) )  e.  RR* )  ->  (
( B +e  -e 1 ) +e ( A +e  -e ( B +e  -e 1 ) ) )  =  ( ( A +e  -e
( B +e  -e 1 ) ) +e ( B +e  -e 1 ) ) )
10095, 98, 99syl2anc 661 . . . . 5  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  (
( B +e  -e 1 ) +e ( A +e  -e ( B +e  -e 1 ) ) )  =  ( ( A +e  -e
( B +e  -e 1 ) ) +e ( B +e  -e 1 ) ) )
1011ad2antrr 725 . . . . . . 7  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  A  e.  RR )
102101renemnfd 9662 . . . . . 6  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  A  =/= -oo )
103 simplr 755 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  B  =/= +oo )
104 xrnepnf 11354 . . . . . . . . . . . . . . 15  |-  ( ( B  e.  RR*  /\  B  =/= +oo )  <->  ( B  e.  RR  \/  B  = -oo ) )
105104biimpi 194 . . . . . . . . . . . . . 14  |-  ( ( B  e.  RR*  /\  B  =/= +oo )  ->  ( B  e.  RR  \/  B  = -oo )
)
10692, 103, 105syl2anc 661 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  ( B  e.  RR  \/  B  = -oo )
)
107106orcomd 388 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  ( B  = -oo  \/  B  e.  RR ) )
108 xlt2addrd.4 . . . . . . . . . . . . . 14  |-  ( ph  ->  B  =/= -oo )
109108ad2antrr 725 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  B  =/= -oo )
110109neneqd 2659 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  -.  B  = -oo )
111 pm2.53 373 . . . . . . . . . . . 12  |-  ( ( B  = -oo  \/  B  e.  RR )  ->  ( -.  B  = -oo  ->  B  e.  RR ) )
112107, 110, 111sylc 60 . . . . . . . . . . 11  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  B  e.  RR )
113 rexsub 11457 . . . . . . . . . . 11  |-  ( ( B  e.  RR  /\  1  e.  RR )  ->  ( B +e  -e 1 )  =  ( B  -  1 ) )
114112, 30, 113sylancl 662 . . . . . . . . . 10  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  ( B +e  -e 1 )  =  ( B  -  1 ) )
115 resubcl 9902 . . . . . . . . . . 11  |-  ( ( B  e.  RR  /\  1  e.  RR )  ->  ( B  -  1 )  e.  RR )
116112, 30, 115sylancl 662 . . . . . . . . . 10  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  ( B  -  1 )  e.  RR )
117114, 116eqeltrd 2545 . . . . . . . . 9  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  ( B +e  -e 1 )  e.  RR )
118 rexneg 11435 . . . . . . . . 9  |-  ( ( B +e  -e 1 )  e.  RR  ->  -e ( B +e  -e 1 )  = 
-u ( B +e  -e 1 ) )
119117, 118syl 16 . . . . . . . 8  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  -e
( B +e  -e 1 )  = 
-u ( B +e  -e 1 ) )
120117renegcld 10007 . . . . . . . 8  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  -u ( B +e  -e 1 )  e.  RR )
121119, 120eqeltrd 2545 . . . . . . 7  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  -e
( B +e  -e 1 )  e.  RR )
122121renemnfd 9662 . . . . . 6  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  -e
( B +e  -e 1 )  =/= -oo )
123117renemnfd 9662 . . . . . 6  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  ( B +e  -e 1 )  =/= -oo )
124 xaddass 11466 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  A  =/= -oo )  /\  (  -e ( B +e  -e 1 )  e. 
RR*  /\  -e ( B +e  -e 1 )  =/= -oo )  /\  (
( B +e  -e 1 )  e. 
RR*  /\  ( B +e  -e 1 )  =/= -oo )
)  ->  ( ( A +e  -e
( B +e  -e 1 ) ) +e ( B +e  -e 1 ) )  =  ( A +e
(  -e ( B +e  -e 1 ) +e
( B +e  -e 1 ) ) ) )
12596, 102, 97, 122, 95, 123, 124syl222anc 1244 . . . . 5  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  (
( A +e  -e ( B +e  -e 1 ) ) +e ( B +e  -e 1 ) )  =  ( A +e (  -e
( B +e  -e 1 ) +e ( B +e  -e 1 ) ) ) )
126 xaddcom 11462 . . . . . . . . 9  |-  ( ( 
-e ( B +e  -e 1 )  e.  RR*  /\  ( B +e  -e 1 )  e. 
RR* )  ->  (  -e ( B +e  -e 1 ) +e ( B +e  -e 1 ) )  =  ( ( B +e  -e 1 ) +e  -e
( B +e  -e 1 ) ) )
12797, 95, 126syl2anc 661 . . . . . . . 8  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  (  -e ( B +e  -e 1 ) +e ( B +e  -e 1 ) )  =  ( ( B +e  -e 1 ) +e  -e
( B +e  -e 1 ) ) )
128 xnegid 11460 . . . . . . . . 9  |-  ( ( B +e  -e 1 )  e. 
RR*  ->  ( ( B +e  -e 1 ) +e  -e ( B +e  -e 1 ) )  =  0 )
12995, 128syl 16 . . . . . . . 8  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  (
( B +e  -e 1 ) +e  -e ( B +e  -e 1 ) )  =  0 )
130127, 129eqtrd 2498 . . . . . . 7  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  (  -e ( B +e  -e 1 ) +e ( B +e  -e 1 ) )  =  0 )
131130oveq2d 6312 . . . . . 6  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  ( A +e (  -e ( B +e  -e 1 ) +e ( B +e  -e 1 ) ) )  =  ( A +e 0 ) )
13296, 6syl 16 . . . . . 6  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  ( A +e 0 )  =  A )
133131, 132eqtrd 2498 . . . . 5  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  ( A +e (  -e ( B +e  -e 1 ) +e ( B +e  -e 1 ) ) )  =  A )
134100, 125, 1333eqtrrd 2503 . . . 4  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  A  =  ( ( B +e  -e 1 ) +e
( A +e  -e ( B +e  -e 1 ) ) ) )
135112ltm1d 10498 . . . . 5  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  ( B  -  1 )  <  B )
136114, 135eqbrtrd 4476 . . . 4  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  ( B +e  -e 1 )  <  B
)
137101, 116resubcld 10008 . . . . . 6  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  ( A  -  ( B  -  1 ) )  e.  RR )
138 ltpnf 11356 . . . . . 6  |-  ( ( A  -  ( B  -  1 ) )  e.  RR  ->  ( A  -  ( B  -  1 ) )  < +oo )
139137, 138syl 16 . . . . 5  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  ( A  -  ( B  -  1 ) )  < +oo )
140 rexsub 11457 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( B +e  -e 1 )  e.  RR )  ->  ( A +e  -e
( B +e  -e 1 ) )  =  ( A  -  ( B +e  -e 1 ) ) )
141101, 117, 140syl2anc 661 . . . . . 6  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  ( A +e  -e
( B +e  -e 1 ) )  =  ( A  -  ( B +e  -e 1 ) ) )
142114oveq2d 6312 . . . . . 6  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  ( A  -  ( B +e  -e 1 ) )  =  ( A  -  ( B  -  1 ) ) )
143141, 142eqtrd 2498 . . . . 5  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  ( A +e  -e
( B +e  -e 1 ) )  =  ( A  -  ( B  -  1
) ) )
144 simpr 461 . . . . 5  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  C  = +oo )
145139, 143, 1443brtr4d 4486 . . . 4  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  ( A +e  -e
( B +e  -e 1 ) )  <  C )
146 oveq1 6303 . . . . . . 7  |-  ( b  =  ( B +e  -e 1 )  ->  ( b +e c )  =  ( ( B +e  -e 1 ) +e c ) )
147146eqeq2d 2471 . . . . . 6  |-  ( b  =  ( B +e  -e 1 )  ->  ( A  =  ( b +e
c )  <->  A  =  ( ( B +e  -e 1 ) +e c ) ) )
148 breq1 4459 . . . . . 6  |-  ( b  =  ( B +e  -e 1 )  ->  ( b  < 
B  <->  ( B +e  -e 1 )  <  B ) )
149147, 1483anbi12d 1300 . . . . 5  |-  ( b  =  ( B +e  -e 1 )  ->  ( ( A  =  ( b +e c )  /\  b  <  B  /\  c  <  C )  <->  ( A  =  ( ( B +e  -e 1 ) +e
c )  /\  ( B +e  -e 1 )  <  B  /\  c  <  C ) ) )
150 oveq2 6304 . . . . . . 7  |-  ( c  =  ( A +e  -e ( B +e  -e 1 ) )  -> 
( ( B +e  -e 1 ) +e c )  =  ( ( B +e  -e 1 ) +e
( A +e  -e ( B +e  -e 1 ) ) ) )
151150eqeq2d 2471 . . . . . 6  |-  ( c  =  ( A +e  -e ( B +e  -e 1 ) )  -> 
( A  =  ( ( B +e  -e 1 ) +e c )  <->  A  =  ( ( B +e  -e 1 ) +e ( A +e  -e
( B +e  -e 1 ) ) ) ) )
152 breq1 4459 . . . . . 6  |-  ( c  =  ( A +e  -e ( B +e  -e 1 ) )  -> 
( c  <  C  <->  ( A +e  -e ( B +e  -e 1 ) )  <  C ) )
153151, 1523anbi13d 1301 . . . . 5  |-  ( c  =  ( A +e  -e ( B +e  -e 1 ) )  -> 
( ( A  =  ( ( B +e  -e 1 ) +e c )  /\  ( B +e  -e 1 )  <  B  /\  c  <  C )  <->  ( A  =  ( ( B +e  -e 1 ) +e
( A +e  -e ( B +e  -e 1 ) ) )  /\  ( B +e  -e 1 )  <  B  /\  ( A +e  -e ( B +e  -e 1 ) )  <  C ) ) )
154149, 153rspc2ev 3221 . . . 4  |-  ( ( ( B +e  -e 1 )  e. 
RR*  /\  ( A +e  -e ( B +e  -e 1 ) )  e.  RR*  /\  ( A  =  ( ( B +e  -e 1 ) +e
( A +e  -e ( B +e  -e 1 ) ) )  /\  ( B +e  -e 1 )  <  B  /\  ( A +e  -e ( B +e  -e 1 ) )  <  C ) )  ->  E. b  e.  RR*  E. c  e. 
RR*  ( A  =  ( b +e
c )  /\  b  <  B  /\  c  < 
C ) )
15595, 98, 134, 136, 145, 154syl113anc 1240 . . 3  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  E. b  e.  RR*  E. c  e. 
RR*  ( A  =  ( b +e
c )  /\  b  <  B  /\  c  < 
C ) )
1561ad2antrr 725 . . . . . 6  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  A  e.  RR )
15791ad2antrr 725 . . . . . . . . 9  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  B  e.  RR* )
158 simplr 755 . . . . . . . . 9  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  B  =/= +oo )
159157, 158, 105syl2anc 661 . . . . . . . 8  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  ( B  e.  RR  \/  B  = -oo )
)
160159orcomd 388 . . . . . . 7  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  ( B  = -oo  \/  B  e.  RR ) )
161108ad2antrr 725 . . . . . . . 8  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  B  =/= -oo )
162161neneqd 2659 . . . . . . 7  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  -.  B  = -oo )
163160, 162, 111sylc 60 . . . . . 6  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  B  e.  RR )
16428ad2antrr 725 . . . . . . . . 9  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  C  e.  RR* )
165164, 40sylancom 667 . . . . . . . 8  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  ( C  e.  RR  \/  C  = -oo )
)
166165orcomd 388 . . . . . . 7  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  ( C  = -oo  \/  C  e.  RR ) )
16743ad2antrr 725 . . . . . . . 8  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  C  =/= -oo )
168167neneqd 2659 . . . . . . 7  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  -.  C  = -oo )
169166, 168, 46sylc 60 . . . . . 6  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  C  e.  RR )
170 xlt2addrd.6 . . . . . . . 8  |-  ( ph  ->  A  <  ( B +e C ) )
171170ad2antrr 725 . . . . . . 7  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  A  <  ( B +e
C ) )
172 rexadd 11456 . . . . . . . 8  |-  ( ( B  e.  RR  /\  C  e.  RR )  ->  ( B +e
C )  =  ( B  +  C ) )
173163, 169, 172syl2anc 661 . . . . . . 7  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  ( B +e C )  =  ( B  +  C ) )
174171, 173breqtrd 4480 . . . . . 6  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  A  <  ( B  +  C
) )
175156, 163, 169, 174lt2addrd 27720 . . . . 5  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  E. b  e.  RR  E. c  e.  RR  ( A  =  ( b  +  c )  /\  b  < 
B  /\  c  <  C ) )
176 rexadd 11456 . . . . . . . 8  |-  ( ( b  e.  RR  /\  c  e.  RR )  ->  ( b +e
c )  =  ( b  +  c ) )
177176eqeq2d 2471 . . . . . . 7  |-  ( ( b  e.  RR  /\  c  e.  RR )  ->  ( A  =  ( b +e c )  <->  A  =  (
b  +  c ) ) )
1781773anbi1d 1303 . . . . . 6  |-  ( ( b  e.  RR  /\  c  e.  RR )  ->  ( ( A  =  ( b +e
c )  /\  b  <  B  /\  c  < 
C )  <->  ( A  =  ( b  +  c )  /\  b  <  B  /\  c  < 
C ) ) )
1791782rexbiia 2973 . . . . 5  |-  ( E. b  e.  RR  E. c  e.  RR  ( A  =  ( b +e c )  /\  b  <  B  /\  c  <  C )  <->  E. b  e.  RR  E. c  e.  RR  ( A  =  ( b  +  c )  /\  b  <  B  /\  c  <  C ) )
180175, 179sylibr 212 . . . 4  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  E. b  e.  RR  E. c  e.  RR  ( A  =  ( b +e
c )  /\  b  <  B  /\  c  < 
C ) )
181 ressxr 9654 . . . . . 6  |-  RR  C_  RR*
182 ssrexv 3561 . . . . . 6  |-  ( RR  C_  RR*  ->  ( E. c  e.  RR  ( A  =  ( b +e c )  /\  b  <  B  /\  c  <  C )  ->  E. c  e.  RR*  ( A  =  (
b +e c )  /\  b  < 
B  /\  c  <  C ) ) )
183181, 182ax-mp 5 . . . . 5  |-  ( E. c  e.  RR  ( A  =  ( b +e c )  /\  b  <  B  /\  c  <  C )  ->  E. c  e.  RR*  ( A  =  (
b +e c )  /\  b  < 
B  /\  c  <  C ) )
184183reximi 2925 . . . 4  |-  ( E. b  e.  RR  E. c  e.  RR  ( A  =  ( b +e c )  /\  b  <  B  /\  c  <  C )  ->  E. b  e.  RR  E. c  e.  RR*  ( A  =  ( b +e c )  /\  b  <  B  /\  c  <  C ) )
185 ssrexv 3561 . . . . 5  |-  ( RR  C_  RR*  ->  ( E. b  e.  RR  E. c  e.  RR*  ( A  =  ( b +e
c )  /\  b  <  B  /\  c  < 
C )  ->  E. b  e.  RR*  E. c  e. 
RR*  ( A  =  ( b +e
c )  /\  b  <  B  /\  c  < 
C ) ) )
186181, 185ax-mp 5 . . . 4  |-  ( E. b  e.  RR  E. c  e.  RR*  ( A  =  ( b +e c )  /\  b  <  B  /\  c  <  C )  ->  E. b  e.  RR*  E. c  e. 
RR*  ( A  =  ( b +e
c )  /\  b  <  B  /\  c  < 
C ) )
187180, 184, 1863syl 20 . . 3  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  E. b  e.  RR*  E. c  e. 
RR*  ( A  =  ( b +e
c )  /\  b  <  B  /\  c  < 
C ) )
188155, 187pm2.61dane 2775 . 2  |-  ( (
ph  /\  B  =/= +oo )  ->  E. b  e.  RR*  E. c  e. 
RR*  ( A  =  ( b +e
c )  /\  b  <  B  /\  c  < 
C ) )
18990, 188pm2.61dane 2775 1  |-  ( ph  ->  E. b  e.  RR*  E. c  e.  RR*  ( A  =  ( b +e c )  /\  b  <  B  /\  c  <  C ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 368    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819    =/= wne 2652   E.wrex 2808    C_ wss 3471   class class class wbr 4456  (class class class)co 6296   RRcr 9508   0cc0 9509   1c1 9510    + caddc 9512   +oocpnf 9642   -oocmnf 9643   RR*cxr 9644    < clt 9645    - cmin 9824   -ucneg 9825    -ecxne 11340   +ecxad 11341
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-po 4809  df-so 4810  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-2 10615  df-rp 11246  df-xneg 11343  df-xadd 11344
This theorem is referenced by:  xrofsup  27742
  Copyright terms: Public domain W3C validator