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Theorem xlt2addrd 27392
Description: If the right-side of a 'less-than' relationship is an addition, then we can express the left-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 9655 . . . . 5  |-  ( ph  ->  A  e.  RR* )
32ad2antrr 725 . . . 4  |-  ( ( ( ph  /\  B  = +oo )  /\  C  = +oo )  ->  A  e.  RR* )
4 0xr 9652 . . . . 5  |-  0  e.  RR*
54a1i 11 . . . 4  |-  ( ( ( ph  /\  B  = +oo )  /\  C  = +oo )  ->  0  e.  RR* )
6 xaddid1 11450 . . . . . 6  |-  ( A  e.  RR*  ->  ( A +e 0 )  =  A )
76eqcomd 2475 . . . . 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 11343 . . . . . 6  |-  ( A  e.  RR  ->  A  < +oo )
119, 10syl 16 . . . . 5  |-  ( ( ( ph  /\  B  = +oo )  /\  C  = +oo )  ->  A  < +oo )
12 simplr 754 . . . . 5  |-  ( ( ( ph  /\  B  = +oo )  /\  C  = +oo )  ->  B  = +oo )
1311, 12breqtrrd 4479 . . . 4  |-  ( ( ( ph  /\  B  = +oo )  /\  C  = +oo )  ->  A  <  B )
14 0ltpnf 11344 . . . . . 6  |-  0  < +oo
1514a1i 11 . . . . 5  |-  ( ( ( ph  /\  B  = +oo )  /\  C  = +oo )  ->  0  < +oo )
16 simpr 461 . . . . 5  |-  ( ( ( ph  /\  B  = +oo )  /\  C  = +oo )  ->  C  = +oo )
1715, 16breqtrrd 4479 . . . 4  |-  ( ( ( ph  /\  B  = +oo )  /\  C  = +oo )  ->  0  <  C )
18 oveq1 6302 . . . . . . 7  |-  ( b  =  A  ->  (
b +e c )  =  ( A +e c ) )
1918eqeq2d 2481 . . . . . 6  |-  ( b  =  A  ->  ( A  =  ( b +e c )  <-> 
A  =  ( A +e c ) ) )
20 breq1 4456 . . . . . 6  |-  ( b  =  A  ->  (
b  <  B  <->  A  <  B ) )
2119, 203anbi12d 1300 . . . . 5  |-  ( b  =  A  ->  (
( A  =  ( b +e c )  /\  b  < 
B  /\  c  <  C )  <->  ( A  =  ( A +e
c )  /\  A  <  B  /\  c  < 
C ) ) )
22 oveq2 6303 . . . . . . 7  |-  ( c  =  0  ->  ( A +e c )  =  ( A +e 0 ) )
2322eqeq2d 2481 . . . . . 6  |-  ( c  =  0  ->  ( A  =  ( A +e c )  <-> 
A  =  ( A +e 0 ) ) )
24 breq1 4456 . . . . . 6  |-  ( c  =  0  ->  (
c  <  C  <->  0  <  C ) )
2523, 243anbi13d 1301 . . . . 5  |-  ( c  =  0  ->  (
( A  =  ( A +e c )  /\  A  < 
B  /\  c  <  C )  <->  ( A  =  ( A +e 0 )  /\  A  <  B  /\  0  < 
C ) ) )
2621, 25rspc2ev 3230 . . . 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 ) )
273, 5, 8, 13, 17, 26syl113anc 1240 . . 3  |-  ( ( ( ph  /\  B  = +oo )  /\  C  = +oo )  ->  E. b  e.  RR*  E. c  e. 
RR*  ( A  =  ( b +e
c )  /\  b  <  B  /\  c  < 
C ) )
282ad2antrr 725 . . . . 5  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  A  e.  RR* )
29 xlt2addrd.3 . . . . . . . 8  |-  ( ph  ->  C  e.  RR* )
3029ad2antrr 725 . . . . . . 7  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  C  e.  RR* )
31 1re 9607 . . . . . . . . . 10  |-  1  e.  RR
3231rexri 9658 . . . . . . . . 9  |-  1  e.  RR*
3332a1i 11 . . . . . . . 8  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  1  e.  RR* )
3433xnegcld 11504 . . . . . . 7  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  -e 1  e.  RR* )
3530, 34xaddcld 11505 . . . . . 6  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  ( C +e  -e 1 )  e.  RR* )
3635xnegcld 11504 . . . . 5  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  -e
( C +e  -e 1 )  e. 
RR* )
3728, 36xaddcld 11505 . . . 4  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  ( A +e  -e
( C +e  -e 1 ) )  e.  RR* )
381ad2antrr 725 . . . . . . 7  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  A  e.  RR )
3938renemnfd 9657 . . . . . 6  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  A  =/= -oo )
40 xrnepnf 11341 . . . . . . . . . . . . . . 15  |-  ( ( C  e.  RR*  /\  C  =/= +oo )  <->  ( C  e.  RR  \/  C  = -oo ) )
4140biimpi 194 . . . . . . . . . . . . . 14  |-  ( ( C  e.  RR*  /\  C  =/= +oo )  ->  ( C  e.  RR  \/  C  = -oo )
)
4230, 41sylancom 667 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  ( C  e.  RR  \/  C  = -oo )
)
4342orcomd 388 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  ( C  = -oo  \/  C  e.  RR ) )
44 xlt2addrd.5 . . . . . . . . . . . . . 14  |-  ( ph  ->  C  =/= -oo )
4544ad2antrr 725 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  C  =/= -oo )
4645neneqd 2669 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  -.  C  = -oo )
47 pm2.53 373 . . . . . . . . . . . 12  |-  ( ( C  = -oo  \/  C  e.  RR )  ->  ( -.  C  = -oo  ->  C  e.  RR ) )
4843, 46, 47sylc 60 . . . . . . . . . . 11  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  C  e.  RR )
49 rexsub 11444 . . . . . . . . . . 11  |-  ( ( C  e.  RR  /\  1  e.  RR )  ->  ( C +e  -e 1 )  =  ( C  -  1 ) )
5048, 31, 49sylancl 662 . . . . . . . . . 10  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  ( C +e  -e 1 )  =  ( C  -  1 ) )
51 resubcl 9895 . . . . . . . . . . 11  |-  ( ( C  e.  RR  /\  1  e.  RR )  ->  ( C  -  1 )  e.  RR )
5248, 31, 51sylancl 662 . . . . . . . . . 10  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  ( C  -  1 )  e.  RR )
5350, 52eqeltrd 2555 . . . . . . . . 9  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  ( C +e  -e 1 )  e.  RR )
54 rexneg 11422 . . . . . . . . 9  |-  ( ( C +e  -e 1 )  e.  RR  ->  -e ( C +e  -e 1 )  = 
-u ( C +e  -e 1 ) )
5553, 54syl 16 . . . . . . . 8  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  -e
( C +e  -e 1 )  = 
-u ( C +e  -e 1 ) )
5653renegcld 9998 . . . . . . . 8  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  -u ( C +e  -e 1 )  e.  RR )
5755, 56eqeltrd 2555 . . . . . . 7  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  -e
( C +e  -e 1 )  e.  RR )
5857renemnfd 9657 . . . . . 6  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  -e
( C +e  -e 1 )  =/= -oo )
5953renemnfd 9657 . . . . . 6  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  ( C +e  -e 1 )  =/= -oo )
60 xaddass 11453 . . . . . 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 ) ) ) )
6128, 39, 36, 58, 35, 59, 60syl222anc 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 ) ) ) )
62 xaddcom 11449 . . . . . . . 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 ) ) )
6336, 35, 62syl2anc 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 ) ) )
64 xnegid 11447 . . . . . . . 8  |-  ( ( C +e  -e 1 )  e. 
RR*  ->  ( ( C +e  -e 1 ) +e  -e ( C +e  -e 1 ) )  =  0 )
6535, 64syl 16 . . . . . . 7  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  (
( C +e  -e 1 ) +e  -e ( C +e  -e 1 ) )  =  0 )
6663, 65eqtrd 2508 . . . . . 6  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  (  -e ( C +e  -e 1 ) +e ( C +e  -e 1 ) )  =  0 )
6766oveq2d 6311 . . . . 5  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  ( A +e (  -e ( C +e  -e 1 ) +e ( C +e  -e 1 ) ) )  =  ( A +e 0 ) )
6828, 6syl 16 . . . . 5  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  ( A +e 0 )  =  A )
6961, 67, 683eqtrrd 2513 . . . 4  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  A  =  ( ( A +e  -e
( C +e  -e 1 ) ) +e ( C +e  -e 1 ) ) )
7038, 52resubcld 9999 . . . . . 6  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  ( A  -  ( C  -  1 ) )  e.  RR )
71 ltpnf 11343 . . . . . 6  |-  ( ( A  -  ( C  -  1 ) )  e.  RR  ->  ( A  -  ( C  -  1 ) )  < +oo )
7270, 71syl 16 . . . . 5  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  ( A  -  ( C  -  1 ) )  < +oo )
73 rexsub 11444 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( C +e  -e 1 )  e.  RR )  ->  ( A +e  -e
( C +e  -e 1 ) )  =  ( A  -  ( C +e  -e 1 ) ) )
7438, 53, 73syl2anc 661 . . . . . 6  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  ( A +e  -e
( C +e  -e 1 ) )  =  ( A  -  ( C +e  -e 1 ) ) )
7550oveq2d 6311 . . . . . 6  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  ( A  -  ( C +e  -e 1 ) )  =  ( A  -  ( C  -  1 ) ) )
7674, 75eqtrd 2508 . . . . 5  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  ( A +e  -e
( C +e  -e 1 ) )  =  ( A  -  ( C  -  1
) ) )
77 simplr 754 . . . . 5  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  B  = +oo )
7872, 76, 773brtr4d 4483 . . . 4  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  ( A +e  -e
( C +e  -e 1 ) )  <  B )
7948ltm1d 10490 . . . . 5  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  ( C  -  1 )  <  C )
8050, 79eqbrtrd 4473 . . . 4  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  ( C +e  -e 1 )  <  C
)
81 oveq1 6302 . . . . . . 7  |-  ( b  =  ( A +e  -e ( C +e  -e 1 ) )  -> 
( b +e
c )  =  ( ( A +e  -e ( C +e  -e 1 ) ) +e c ) )
8281eqeq2d 2481 . . . . . 6  |-  ( b  =  ( A +e  -e ( C +e  -e 1 ) )  -> 
( A  =  ( b +e c )  <->  A  =  (
( A +e  -e ( C +e  -e 1 ) ) +e c ) ) )
83 breq1 4456 . . . . . 6  |-  ( b  =  ( A +e  -e ( C +e  -e 1 ) )  -> 
( b  <  B  <->  ( A +e  -e ( C +e  -e 1 ) )  <  B ) )
8482, 833anbi12d 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 ) ) )
85 oveq2 6303 . . . . . . 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 ) ) )
8685eqeq2d 2481 . . . . . 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 ) ) ) )
87 breq1 4456 . . . . . 6  |-  ( c  =  ( C +e  -e 1 )  ->  ( c  < 
C  <->  ( C +e  -e 1 )  <  C ) )
8886, 873anbi13d 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
) ) )
8984, 88rspc2ev 3230 . . . 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 ) )
9037, 35, 69, 78, 80, 89syl113anc 1240 . . 3  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  E. b  e.  RR*  E. c  e. 
RR*  ( A  =  ( b +e
c )  /\  b  <  B  /\  c  < 
C ) )
9127, 90pm2.61dane 2785 . 2  |-  ( (
ph  /\  B  = +oo )  ->  E. b  e.  RR*  E. c  e. 
RR*  ( A  =  ( b +e
c )  /\  b  <  B  /\  c  < 
C ) )
92 xlt2addrd.2 . . . . . 6  |-  ( ph  ->  B  e.  RR* )
9392ad2antrr 725 . . . . 5  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  B  e.  RR* )
9432a1i 11 . . . . . 6  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  1  e.  RR* )
9594xnegcld 11504 . . . . 5  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  -e 1  e.  RR* )
9693, 95xaddcld 11505 . . . 4  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  ( B +e  -e 1 )  e.  RR* )
972ad2antrr 725 . . . . 5  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  A  e.  RR* )
9896xnegcld 11504 . . . . 5  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  -e
( B +e  -e 1 )  e. 
RR* )
9997, 98xaddcld 11505 . . . 4  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  ( A +e  -e
( B +e  -e 1 ) )  e.  RR* )
100 xaddcom 11449 . . . . . 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 ) ) )
10196, 99, 100syl2anc 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 ) ) )
1021ad2antrr 725 . . . . . . 7  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  A  e.  RR )
103102renemnfd 9657 . . . . . 6  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  A  =/= -oo )
104 simplr 754 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  B  =/= +oo )
105 xrnepnf 11341 . . . . . . . . . . . . . . 15  |-  ( ( B  e.  RR*  /\  B  =/= +oo )  <->  ( B  e.  RR  \/  B  = -oo ) )
106105biimpi 194 . . . . . . . . . . . . . 14  |-  ( ( B  e.  RR*  /\  B  =/= +oo )  ->  ( B  e.  RR  \/  B  = -oo )
)
10793, 104, 106syl2anc 661 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  ( B  e.  RR  \/  B  = -oo )
)
108107orcomd 388 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  ( B  = -oo  \/  B  e.  RR ) )
109 xlt2addrd.4 . . . . . . . . . . . . . 14  |-  ( ph  ->  B  =/= -oo )
110109ad2antrr 725 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  B  =/= -oo )
111110neneqd 2669 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  -.  B  = -oo )
112 pm2.53 373 . . . . . . . . . . . 12  |-  ( ( B  = -oo  \/  B  e.  RR )  ->  ( -.  B  = -oo  ->  B  e.  RR ) )
113108, 111, 112sylc 60 . . . . . . . . . . 11  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  B  e.  RR )
114 rexsub 11444 . . . . . . . . . . 11  |-  ( ( B  e.  RR  /\  1  e.  RR )  ->  ( B +e  -e 1 )  =  ( B  -  1 ) )
115113, 31, 114sylancl 662 . . . . . . . . . 10  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  ( B +e  -e 1 )  =  ( B  -  1 ) )
116 resubcl 9895 . . . . . . . . . . 11  |-  ( ( B  e.  RR  /\  1  e.  RR )  ->  ( B  -  1 )  e.  RR )
117113, 31, 116sylancl 662 . . . . . . . . . 10  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  ( B  -  1 )  e.  RR )
118115, 117eqeltrd 2555 . . . . . . . . 9  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  ( B +e  -e 1 )  e.  RR )
119 rexneg 11422 . . . . . . . . 9  |-  ( ( B +e  -e 1 )  e.  RR  ->  -e ( B +e  -e 1 )  = 
-u ( B +e  -e 1 ) )
120118, 119syl 16 . . . . . . . 8  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  -e
( B +e  -e 1 )  = 
-u ( B +e  -e 1 ) )
121118renegcld 9998 . . . . . . . 8  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  -u ( B +e  -e 1 )  e.  RR )
122120, 121eqeltrd 2555 . . . . . . 7  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  -e
( B +e  -e 1 )  e.  RR )
123122renemnfd 9657 . . . . . 6  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  -e
( B +e  -e 1 )  =/= -oo )
124118renemnfd 9657 . . . . . 6  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  ( B +e  -e 1 )  =/= -oo )
125 xaddass 11453 . . . . . 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 ) ) ) )
12697, 103, 98, 123, 96, 124, 125syl222anc 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 ) ) ) )
127 xaddcom 11449 . . . . . . . . 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 ) ) )
12898, 96, 127syl2anc 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 ) ) )
129 xnegid 11447 . . . . . . . . 9  |-  ( ( B +e  -e 1 )  e. 
RR*  ->  ( ( B +e  -e 1 ) +e  -e ( B +e  -e 1 ) )  =  0 )
13096, 129syl 16 . . . . . . . 8  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  (
( B +e  -e 1 ) +e  -e ( B +e  -e 1 ) )  =  0 )
131128, 130eqtrd 2508 . . . . . . 7  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  (  -e ( B +e  -e 1 ) +e ( B +e  -e 1 ) )  =  0 )
132131oveq2d 6311 . . . . . 6  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  ( A +e (  -e ( B +e  -e 1 ) +e ( B +e  -e 1 ) ) )  =  ( A +e 0 ) )
13397, 6syl 16 . . . . . 6  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  ( A +e 0 )  =  A )
134132, 133eqtrd 2508 . . . . 5  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  ( A +e (  -e ( B +e  -e 1 ) +e ( B +e  -e 1 ) ) )  =  A )
135101, 126, 1343eqtrrd 2513 . . . 4  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  A  =  ( ( B +e  -e 1 ) +e
( A +e  -e ( B +e  -e 1 ) ) ) )
136113ltm1d 10490 . . . . 5  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  ( B  -  1 )  <  B )
137115, 136eqbrtrd 4473 . . . 4  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  ( B +e  -e 1 )  <  B
)
138102, 117resubcld 9999 . . . . . 6  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  ( A  -  ( B  -  1 ) )  e.  RR )
139 ltpnf 11343 . . . . . 6  |-  ( ( A  -  ( B  -  1 ) )  e.  RR  ->  ( A  -  ( B  -  1 ) )  < +oo )
140138, 139syl 16 . . . . 5  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  ( A  -  ( B  -  1 ) )  < +oo )
141 rexsub 11444 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( B +e  -e 1 )  e.  RR )  ->  ( A +e  -e
( B +e  -e 1 ) )  =  ( A  -  ( B +e  -e 1 ) ) )
142102, 118, 141syl2anc 661 . . . . . 6  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  ( A +e  -e
( B +e  -e 1 ) )  =  ( A  -  ( B +e  -e 1 ) ) )
143115oveq2d 6311 . . . . . 6  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  ( A  -  ( B +e  -e 1 ) )  =  ( A  -  ( B  -  1 ) ) )
144142, 143eqtrd 2508 . . . . 5  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  ( A +e  -e
( B +e  -e 1 ) )  =  ( A  -  ( B  -  1
) ) )
145 simpr 461 . . . . 5  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  C  = +oo )
146140, 144, 1453brtr4d 4483 . . . 4  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  ( A +e  -e
( B +e  -e 1 ) )  <  C )
147 oveq1 6302 . . . . . . 7  |-  ( b  =  ( B +e  -e 1 )  ->  ( b +e c )  =  ( ( B +e  -e 1 ) +e c ) )
148147eqeq2d 2481 . . . . . 6  |-  ( b  =  ( B +e  -e 1 )  ->  ( A  =  ( b +e
c )  <->  A  =  ( ( B +e  -e 1 ) +e c ) ) )
149 breq1 4456 . . . . . 6  |-  ( b  =  ( B +e  -e 1 )  ->  ( b  < 
B  <->  ( B +e  -e 1 )  <  B ) )
150148, 1493anbi12d 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 ) ) )
151 oveq2 6303 . . . . . . 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 ) ) ) )
152151eqeq2d 2481 . . . . . 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 ) ) ) ) )
153 breq1 4456 . . . . . 6  |-  ( c  =  ( A +e  -e ( B +e  -e 1 ) )  -> 
( c  <  C  <->  ( A +e  -e ( B +e  -e 1 ) )  <  C ) )
154152, 1533anbi13d 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 ) ) )
155150, 154rspc2ev 3230 . . . 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 ) )
15696, 99, 135, 137, 146, 155syl113anc 1240 . . 3  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  E. b  e.  RR*  E. c  e. 
RR*  ( A  =  ( b +e
c )  /\  b  <  B  /\  c  < 
C ) )
1571ad2antrr 725 . . . . . 6  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  A  e.  RR )
15892ad2antrr 725 . . . . . . . . 9  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  B  e.  RR* )
159 simplr 754 . . . . . . . . 9  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  B  =/= +oo )
160158, 159, 106syl2anc 661 . . . . . . . 8  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  ( B  e.  RR  \/  B  = -oo )
)
161160orcomd 388 . . . . . . 7  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  ( B  = -oo  \/  B  e.  RR ) )
162109ad2antrr 725 . . . . . . . 8  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  B  =/= -oo )
163162neneqd 2669 . . . . . . 7  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  -.  B  = -oo )
164161, 163, 112sylc 60 . . . . . 6  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  B  e.  RR )
16529ad2antrr 725 . . . . . . . . 9  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  C  e.  RR* )
166165, 41sylancom 667 . . . . . . . 8  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  ( C  e.  RR  \/  C  = -oo )
)
167166orcomd 388 . . . . . . 7  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  ( C  = -oo  \/  C  e.  RR ) )
16844ad2antrr 725 . . . . . . . 8  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  C  =/= -oo )
169168neneqd 2669 . . . . . . 7  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  -.  C  = -oo )
170167, 169, 47sylc 60 . . . . . 6  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  C  e.  RR )
171 xlt2addrd.6 . . . . . . . 8  |-  ( ph  ->  A  <  ( B +e C ) )
172171ad2antrr 725 . . . . . . 7  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  A  <  ( B +e
C ) )
173 rexadd 11443 . . . . . . . 8  |-  ( ( B  e.  RR  /\  C  e.  RR )  ->  ( B +e
C )  =  ( B  +  C ) )
174164, 170, 173syl2anc 661 . . . . . . 7  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  ( B +e C )  =  ( B  +  C ) )
175172, 174breqtrd 4477 . . . . . 6  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  A  <  ( B  +  C
) )
176157, 164, 170, 175lt2addrd 27377 . . . . 5  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  E. b  e.  RR  E. c  e.  RR  ( A  =  ( b  +  c )  /\  b  < 
B  /\  c  <  C ) )
177 rexadd 11443 . . . . . . . 8  |-  ( ( b  e.  RR  /\  c  e.  RR )  ->  ( b +e
c )  =  ( b  +  c ) )
178177eqeq2d 2481 . . . . . . 7  |-  ( ( b  e.  RR  /\  c  e.  RR )  ->  ( A  =  ( b +e c )  <->  A  =  (
b  +  c ) ) )
1791783anbi1d 1303 . . . . . 6  |-  ( ( b  e.  RR  /\  c  e.  RR )  ->  ( ( A  =  ( b +e
c )  /\  b  <  B  /\  c  < 
C )  <->  ( A  =  ( b  +  c )  /\  b  <  B  /\  c  < 
C ) ) )
1801792rexbiia 2983 . . . . 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 ) )
181176, 180sylibr 212 . . . 4  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  E. b  e.  RR  E. c  e.  RR  ( A  =  ( b +e
c )  /\  b  <  B  /\  c  < 
C ) )
182 ressxr 9649 . . . . . 6  |-  RR  C_  RR*
183 ssrexv 3570 . . . . . 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 ) ) )
184182, 183ax-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 ) )
185184reximi 2935 . . . 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 ) )
186 ssrexv 3570 . . . . 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 ) ) )
187182, 186ax-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 ) )
188181, 185, 1873syl 20 . . 3  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  E. b  e.  RR*  E. c  e. 
RR*  ( A  =  ( b +e
c )  /\  b  <  B  /\  c  < 
C ) )
189156, 188pm2.61dane 2785 . 2  |-  ( (
ph  /\  B  =/= +oo )  ->  E. b  e.  RR*  E. c  e. 
RR*  ( A  =  ( b +e
c )  /\  b  <  B  /\  c  < 
C ) )
19091, 189pm2.61dane 2785 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 1379    e. wcel 1767    =/= wne 2662   E.wrex 2818    C_ wss 3481   class class class wbr 4453  (class class class)co 6295   RRcr 9503   0cc0 9504   1c1 9505    + caddc 9507   +oocpnf 9637   -oocmnf 9638   RR*cxr 9639    < clt 9640    - cmin 9817   -ucneg 9818    -ecxne 11327   +ecxad 11328
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 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
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 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-po 4806  df-so 4807  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  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 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-1st 6795  df-2nd 6796  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-2 10606  df-rp 11233  df-xneg 11330  df-xadd 11331
This theorem is referenced by:  xrofsup  27396
  Copyright terms: Public domain W3C validator