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Theorem xlt2addrd 26056
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 9438 . . . . 5  |-  ( ph  ->  A  e.  RR* )
32ad2antrr 725 . . . 4  |-  ( ( ( ph  /\  B  = +oo )  /\  C  = +oo )  ->  A  e.  RR* )
4 0xr 9435 . . . . 5  |-  0  e.  RR*
54a1i 11 . . . 4  |-  ( ( ( ph  /\  B  = +oo )  /\  C  = +oo )  ->  0  e.  RR* )
6 xaddid1 11214 . . . . . 6  |-  ( A  e.  RR*  ->  ( A +e 0 )  =  A )
76eqcomd 2448 . . . . 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 11107 . . . . . 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 4323 . . . 4  |-  ( ( ( ph  /\  B  = +oo )  /\  C  = +oo )  ->  A  <  B )
14 0ltpnf 11108 . . . . . 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 4323 . . . 4  |-  ( ( ( ph  /\  B  = +oo )  /\  C  = +oo )  ->  0  <  C )
18 oveq1 6103 . . . . . . 7  |-  ( b  =  A  ->  (
b +e c )  =  ( A +e c ) )
1918eqeq2d 2454 . . . . . 6  |-  ( b  =  A  ->  ( A  =  ( b +e c )  <-> 
A  =  ( A +e c ) ) )
20 breq1 4300 . . . . . 6  |-  ( b  =  A  ->  (
b  <  B  <->  A  <  B ) )
2119, 203anbi12d 1290 . . . . 5  |-  ( b  =  A  ->  (
( A  =  ( b +e c )  /\  b  < 
B  /\  c  <  C )  <->  ( A  =  ( A +e
c )  /\  A  <  B  /\  c  < 
C ) ) )
22 oveq2 6104 . . . . . . 7  |-  ( c  =  0  ->  ( A +e c )  =  ( A +e 0 ) )
2322eqeq2d 2454 . . . . . 6  |-  ( c  =  0  ->  ( A  =  ( A +e c )  <-> 
A  =  ( A +e 0 ) ) )
24 breq1 4300 . . . . . 6  |-  ( c  =  0  ->  (
c  <  C  <->  0  <  C ) )
2523, 243anbi13d 1291 . . . . 5  |-  ( c  =  0  ->  (
( A  =  ( A +e c )  /\  A  < 
B  /\  c  <  C )  <->  ( A  =  ( A +e 0 )  /\  A  <  B  /\  0  < 
C ) ) )
2621, 25rspc2ev 3086 . . . 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 1230 . . 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 9390 . . . . . . . . . 10  |-  1  e.  RR
3231rexri 9441 . . . . . . . . 9  |-  1  e.  RR*
3332a1i 11 . . . . . . . 8  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  1  e.  RR* )
3433xnegcld 11268 . . . . . . 7  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  -e 1  e.  RR* )
3530, 34xaddcld 11269 . . . . . 6  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  ( C +e  -e 1 )  e.  RR* )
3635xnegcld 11268 . . . . 5  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  -e
( C +e  -e 1 )  e. 
RR* )
3728, 36xaddcld 11269 . . . 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 9440 . . . . . 6  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  A  =/= -oo )
40 xrnepnf 11105 . . . . . . . . . . . . . . 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 2629 . . . . . . . . . . . 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 11208 . . . . . . . . . . 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 9678 . . . . . . . . . . 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 2517 . . . . . . . . 9  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  ( C +e  -e 1 )  e.  RR )
54 rexneg 11186 . . . . . . . . 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 9780 . . . . . . . 8  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  -u ( C +e  -e 1 )  e.  RR )
5755, 56eqeltrd 2517 . . . . . . 7  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  -e
( C +e  -e 1 )  e.  RR )
5857renemnfd 9440 . . . . . 6  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  -e
( C +e  -e 1 )  =/= -oo )
5953renemnfd 9440 . . . . . 6  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  ( C +e  -e 1 )  =/= -oo )
60 xaddass 11217 . . . . . 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 1234 . . . . 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 11213 . . . . . . . 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 11211 . . . . . . . 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 2475 . . . . . 6  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  (  -e ( C +e  -e 1 ) +e ( C +e  -e 1 ) )  =  0 )
6766oveq2d 6112 . . . . 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 2480 . . . 4  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  A  =  ( ( A +e  -e
( C +e  -e 1 ) ) +e ( C +e  -e 1 ) ) )
7038, 52resubcld 9781 . . . . . 6  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  ( A  -  ( C  -  1 ) )  e.  RR )
71 ltpnf 11107 . . . . . 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 11208 . . . . . . 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 6112 . . . . . 6  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  ( A  -  ( C +e  -e 1 ) )  =  ( A  -  ( C  -  1 ) ) )
7674, 75eqtrd 2475 . . . . 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 4327 . . . 4  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  ( A +e  -e
( C +e  -e 1 ) )  <  B )
7948ltm1d 10270 . . . . 5  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  ( C  -  1 )  <  C )
8050, 79eqbrtrd 4317 . . . 4  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  ( C +e  -e 1 )  <  C
)
81 oveq1 6103 . . . . . . 7  |-  ( b  =  ( A +e  -e ( C +e  -e 1 ) )  -> 
( b +e
c )  =  ( ( A +e  -e ( C +e  -e 1 ) ) +e c ) )
8281eqeq2d 2454 . . . . . 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 4300 . . . . . 6  |-  ( b  =  ( A +e  -e ( C +e  -e 1 ) )  -> 
( b  <  B  <->  ( A +e  -e ( C +e  -e 1 ) )  <  B ) )
8482, 833anbi12d 1290 . . . . 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 6104 . . . . . . 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 2454 . . . . . 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 4300 . . . . . 6  |-  ( c  =  ( C +e  -e 1 )  ->  ( c  < 
C  <->  ( C +e  -e 1 )  <  C ) )
8886, 873anbi13d 1291 . . . . 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 3086 . . . 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 1230 . . 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 2694 . 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 11268 . . . . 5  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  -e 1  e.  RR* )
9693, 95xaddcld 11269 . . . 4  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  ( B +e  -e 1 )  e.  RR* )
972ad2antrr 725 . . . . 5  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  A  e.  RR* )
9896xnegcld 11268 . . . . 5  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  -e
( B +e  -e 1 )  e. 
RR* )
9997, 98xaddcld 11269 . . . 4  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  ( A +e  -e
( B +e  -e 1 ) )  e.  RR* )
100 xaddcom 11213 . . . . . 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 9440 . . . . . 6  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  A  =/= -oo )
104 simplr 754 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  B  =/= +oo )
105 xrnepnf 11105 . . . . . . . . . . . . . . 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 2629 . . . . . . . . . . . 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 11208 . . . . . . . . . . 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 9678 . . . . . . . . . . 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 2517 . . . . . . . . 9  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  ( B +e  -e 1 )  e.  RR )
119 rexneg 11186 . . . . . . . . 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 9780 . . . . . . . 8  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  -u ( B +e  -e 1 )  e.  RR )
122120, 121eqeltrd 2517 . . . . . . 7  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  -e
( B +e  -e 1 )  e.  RR )
123122renemnfd 9440 . . . . . 6  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  -e
( B +e  -e 1 )  =/= -oo )
124118renemnfd 9440 . . . . . 6  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  ( B +e  -e 1 )  =/= -oo )
125 xaddass 11217 . . . . . 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 1234 . . . . 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 11213 . . . . . . . . 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 11211 . . . . . . . . 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 2475 . . . . . . 7  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  (  -e ( B +e  -e 1 ) +e ( B +e  -e 1 ) )  =  0 )
132131oveq2d 6112 . . . . . 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 2475 . . . . 5  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  ( A +e (  -e ( B +e  -e 1 ) +e ( B +e  -e 1 ) ) )  =  A )
135101, 126, 1343eqtrrd 2480 . . . 4  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  A  =  ( ( B +e  -e 1 ) +e
( A +e  -e ( B +e  -e 1 ) ) ) )
136113ltm1d 10270 . . . . 5  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  ( B  -  1 )  <  B )
137115, 136eqbrtrd 4317 . . . 4  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  ( B +e  -e 1 )  <  B
)
138102, 117resubcld 9781 . . . . . 6  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  ( A  -  ( B  -  1 ) )  e.  RR )
139 ltpnf 11107 . . . . . 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 11208 . . . . . . 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 6112 . . . . . 6  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  ( A  -  ( B +e  -e 1 ) )  =  ( A  -  ( B  -  1 ) ) )
144142, 143eqtrd 2475 . . . . 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 4327 . . . 4  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  ( A +e  -e
( B +e  -e 1 ) )  <  C )
147 oveq1 6103 . . . . . . 7  |-  ( b  =  ( B +e  -e 1 )  ->  ( b +e c )  =  ( ( B +e  -e 1 ) +e c ) )
148147eqeq2d 2454 . . . . . 6  |-  ( b  =  ( B +e  -e 1 )  ->  ( A  =  ( b +e
c )  <->  A  =  ( ( B +e  -e 1 ) +e c ) ) )
149 breq1 4300 . . . . . 6  |-  ( b  =  ( B +e  -e 1 )  ->  ( b  < 
B  <->  ( B +e  -e 1 )  <  B ) )
150148, 1493anbi12d 1290 . . . . 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 6104 . . . . . . 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 2454 . . . . . 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 4300 . . . . . 6  |-  ( c  =  ( A +e  -e ( B +e  -e 1 ) )  -> 
( c  <  C  <->  ( A +e  -e ( B +e  -e 1 ) )  <  C ) )
154152, 1533anbi13d 1291 . . . . 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 3086 . . . 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 1230 . . 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 2629 . . . . . . 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 2629 . . . . . . 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 11207 . . . . . . . 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 4321 . . . . . 6  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  A  <  ( B  +  C
) )
176157, 164, 170, 175lt2addrd 26041 . . . . 5  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  E. b  e.  RR  E. c  e.  RR  ( A  =  ( b  +  c )  /\  b  < 
B  /\  c  <  C ) )
177 rexadd 11207 . . . . . . . 8  |-  ( ( b  e.  RR  /\  c  e.  RR )  ->  ( b +e
c )  =  ( b  +  c ) )
178177eqeq2d 2454 . . . . . . 7  |-  ( ( b  e.  RR  /\  c  e.  RR )  ->  ( A  =  ( b +e c )  <->  A  =  (
b  +  c ) ) )
1791783anbi1d 1293 . . . . . 6  |-  ( ( b  e.  RR  /\  c  e.  RR )  ->  ( ( A  =  ( b +e
c )  /\  b  <  B  /\  c  < 
C )  <->  ( A  =  ( b  +  c )  /\  b  <  B  /\  c  < 
C ) ) )
1801792rexbiia 2754 . . . . 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 9432 . . . . . 6  |-  RR  C_  RR*
183 ssrexv 3422 . . . . . 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 2828 . . . 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 3422 . . . . 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 2694 . 2  |-  ( (
ph  /\  B  =/= +oo )  ->  E. b  e.  RR*  E. c  e. 
RR*  ( A  =  ( b +e
c )  /\  b  <  B  /\  c  < 
C ) )
19091, 189pm2.61dane 2694 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 965    = wceq 1369    e. wcel 1756    =/= wne 2611   E.wrex 2721    C_ wss 3333   class class class wbr 4297  (class class class)co 6096   RRcr 9286   0cc0 9287   1c1 9288    + caddc 9290   +oocpnf 9420   -oocmnf 9421   RR*cxr 9422    < clt 9423    - cmin 9600   -ucneg 9601    -ecxne 11091   +ecxad 11092
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 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
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 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-po 4646  df-so 4647  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-1st 6582  df-2nd 6583  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-div 9999  df-2 10385  df-rp 10997  df-xneg 11094  df-xadd 11095
This theorem is referenced by:  xrofsup  26060
  Copyright terms: Public domain W3C validator