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Theorem xlt2addrd 28344
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 9697 . . . . 5  |-  ( ph  ->  A  e.  RR* )
32ad2antrr 730 . . . 4  |-  ( ( ( ph  /\  B  = +oo )  /\  C  = +oo )  ->  A  e.  RR* )
4 0xr 9694 . . . . 5  |-  0  e.  RR*
54a1i 11 . . . 4  |-  ( ( ( ph  /\  B  = +oo )  /\  C  = +oo )  ->  0  e.  RR* )
6 xaddid1 11539 . . . . . 6  |-  ( A  e.  RR*  ->  ( A +e 0 )  =  A )
76eqcomd 2430 . . . . 5  |-  ( A  e.  RR*  ->  A  =  ( A +e 0 ) )
83, 7syl 17 . . . 4  |-  ( ( ( ph  /\  B  = +oo )  /\  C  = +oo )  ->  A  =  ( A +e 0 ) )
91ad2antrr 730 . . . . . 6  |-  ( ( ( ph  /\  B  = +oo )  /\  C  = +oo )  ->  A  e.  RR )
10 ltpnf 11429 . . . . . 6  |-  ( A  e.  RR  ->  A  < +oo )
119, 10syl 17 . . . . 5  |-  ( ( ( ph  /\  B  = +oo )  /\  C  = +oo )  ->  A  < +oo )
12 simplr 760 . . . . 5  |-  ( ( ( ph  /\  B  = +oo )  /\  C  = +oo )  ->  B  = +oo )
1311, 12breqtrrd 4450 . . . 4  |-  ( ( ( ph  /\  B  = +oo )  /\  C  = +oo )  ->  A  <  B )
14 0ltpnf 11431 . . . . 5  |-  0  < +oo
15 simpr 462 . . . . 5  |-  ( ( ( ph  /\  B  = +oo )  /\  C  = +oo )  ->  C  = +oo )
1614, 15syl5breqr 4460 . . . 4  |-  ( ( ( ph  /\  B  = +oo )  /\  C  = +oo )  ->  0  <  C )
17 oveq1 6312 . . . . . . 7  |-  ( b  =  A  ->  (
b +e c )  =  ( A +e c ) )
1817eqeq2d 2436 . . . . . 6  |-  ( b  =  A  ->  ( A  =  ( b +e c )  <-> 
A  =  ( A +e c ) ) )
19 breq1 4426 . . . . . 6  |-  ( b  =  A  ->  (
b  <  B  <->  A  <  B ) )
2018, 193anbi12d 1336 . . . . 5  |-  ( b  =  A  ->  (
( A  =  ( b +e c )  /\  b  < 
B  /\  c  <  C )  <->  ( A  =  ( A +e
c )  /\  A  <  B  /\  c  < 
C ) ) )
21 oveq2 6313 . . . . . . 7  |-  ( c  =  0  ->  ( A +e c )  =  ( A +e 0 ) )
2221eqeq2d 2436 . . . . . 6  |-  ( c  =  0  ->  ( A  =  ( A +e c )  <-> 
A  =  ( A +e 0 ) ) )
23 breq1 4426 . . . . . 6  |-  ( c  =  0  ->  (
c  <  C  <->  0  <  C ) )
2422, 233anbi13d 1337 . . . . 5  |-  ( c  =  0  ->  (
( A  =  ( A +e c )  /\  A  < 
B  /\  c  <  C )  <->  ( A  =  ( A +e 0 )  /\  A  <  B  /\  0  < 
C ) ) )
2520, 24rspc2ev 3193 . . . 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 1276 . . 3  |-  ( ( ( ph  /\  B  = +oo )  /\  C  = +oo )  ->  E. b  e.  RR*  E. c  e. 
RR*  ( A  =  ( b +e
c )  /\  b  <  B  /\  c  < 
C ) )
272ad2antrr 730 . . . . 5  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  A  e.  RR* )
28 xlt2addrd.3 . . . . . . . 8  |-  ( ph  ->  C  e.  RR* )
2928ad2antrr 730 . . . . . . 7  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  C  e.  RR* )
30 1re 9649 . . . . . . . . . 10  |-  1  e.  RR
3130rexri 9700 . . . . . . . . 9  |-  1  e.  RR*
3231a1i 11 . . . . . . . 8  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  1  e.  RR* )
3332xnegcld 11593 . . . . . . 7  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  -e 1  e.  RR* )
3429, 33xaddcld 11594 . . . . . 6  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  ( C +e  -e 1 )  e.  RR* )
3534xnegcld 11593 . . . . 5  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  -e
( C +e  -e 1 )  e. 
RR* )
3627, 35xaddcld 11594 . . . 4  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  ( A +e  -e
( C +e  -e 1 ) )  e.  RR* )
371ad2antrr 730 . . . . . . 7  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  A  e.  RR )
3837renemnfd 9699 . . . . . 6  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  A  =/= -oo )
39 xrnepnf 11427 . . . . . . . . . . . . . . 15  |-  ( ( C  e.  RR*  /\  C  =/= +oo )  <->  ( C  e.  RR  \/  C  = -oo ) )
4039biimpi 197 . . . . . . . . . . . . . 14  |-  ( ( C  e.  RR*  /\  C  =/= +oo )  ->  ( C  e.  RR  \/  C  = -oo )
)
4129, 40sylancom 671 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  ( C  e.  RR  \/  C  = -oo )
)
4241orcomd 389 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  ( C  = -oo  \/  C  e.  RR ) )
43 xlt2addrd.5 . . . . . . . . . . . . . 14  |-  ( ph  ->  C  =/= -oo )
4443ad2antrr 730 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  C  =/= -oo )
4544neneqd 2621 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  -.  C  = -oo )
46 pm2.53 374 . . . . . . . . . . . 12  |-  ( ( C  = -oo  \/  C  e.  RR )  ->  ( -.  C  = -oo  ->  C  e.  RR ) )
4742, 45, 46sylc 62 . . . . . . . . . . 11  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  C  e.  RR )
48 rexsub 11533 . . . . . . . . . . 11  |-  ( ( C  e.  RR  /\  1  e.  RR )  ->  ( C +e  -e 1 )  =  ( C  -  1 ) )
4947, 30, 48sylancl 666 . . . . . . . . . 10  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  ( C +e  -e 1 )  =  ( C  -  1 ) )
50 resubcl 9945 . . . . . . . . . . 11  |-  ( ( C  e.  RR  /\  1  e.  RR )  ->  ( C  -  1 )  e.  RR )
5147, 30, 50sylancl 666 . . . . . . . . . 10  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  ( C  -  1 )  e.  RR )
5249, 51eqeltrd 2507 . . . . . . . . 9  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  ( C +e  -e 1 )  e.  RR )
53 rexneg 11511 . . . . . . . . 9  |-  ( ( C +e  -e 1 )  e.  RR  ->  -e ( C +e  -e 1 )  = 
-u ( C +e  -e 1 ) )
5452, 53syl 17 . . . . . . . 8  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  -e
( C +e  -e 1 )  = 
-u ( C +e  -e 1 ) )
5552renegcld 10053 . . . . . . . 8  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  -u ( C +e  -e 1 )  e.  RR )
5654, 55eqeltrd 2507 . . . . . . 7  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  -e
( C +e  -e 1 )  e.  RR )
5756renemnfd 9699 . . . . . 6  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  -e
( C +e  -e 1 )  =/= -oo )
5852renemnfd 9699 . . . . . 6  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  ( C +e  -e 1 )  =/= -oo )
59 xaddass 11542 . . . . . 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 1280 . . . . 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 11538 . . . . . . . 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 665 . . . . . . 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 11536 . . . . . . . 8  |-  ( ( C +e  -e 1 )  e. 
RR*  ->  ( ( C +e  -e 1 ) +e  -e ( C +e  -e 1 ) )  =  0 )
6434, 63syl 17 . . . . . . 7  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  (
( C +e  -e 1 ) +e  -e ( C +e  -e 1 ) )  =  0 )
6562, 64eqtrd 2463 . . . . . 6  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  (  -e ( C +e  -e 1 ) +e ( C +e  -e 1 ) )  =  0 )
6665oveq2d 6321 . . . . 5  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  ( A +e (  -e ( C +e  -e 1 ) +e ( C +e  -e 1 ) ) )  =  ( A +e 0 ) )
6727, 6syl 17 . . . . 5  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  ( A +e 0 )  =  A )
6860, 66, 673eqtrrd 2468 . . . 4  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  A  =  ( ( A +e  -e
( C +e  -e 1 ) ) +e ( C +e  -e 1 ) ) )
6937, 51resubcld 10054 . . . . . 6  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  ( A  -  ( C  -  1 ) )  e.  RR )
70 ltpnf 11429 . . . . . 6  |-  ( ( A  -  ( C  -  1 ) )  e.  RR  ->  ( A  -  ( C  -  1 ) )  < +oo )
7169, 70syl 17 . . . . 5  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  ( A  -  ( C  -  1 ) )  < +oo )
72 rexsub 11533 . . . . . . 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 665 . . . . . 6  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  ( A +e  -e
( C +e  -e 1 ) )  =  ( A  -  ( C +e  -e 1 ) ) )
7449oveq2d 6321 . . . . . 6  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  ( A  -  ( C +e  -e 1 ) )  =  ( A  -  ( C  -  1 ) ) )
7573, 74eqtrd 2463 . . . . 5  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  ( A +e  -e
( C +e  -e 1 ) )  =  ( A  -  ( C  -  1
) ) )
76 simplr 760 . . . . 5  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  B  = +oo )
7771, 75, 763brtr4d 4454 . . . 4  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  ( A +e  -e
( C +e  -e 1 ) )  <  B )
7847ltm1d 10546 . . . . 5  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  ( C  -  1 )  <  C )
7949, 78eqbrtrd 4444 . . . 4  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  ( C +e  -e 1 )  <  C
)
80 oveq1 6312 . . . . . . 7  |-  ( b  =  ( A +e  -e ( C +e  -e 1 ) )  -> 
( b +e
c )  =  ( ( A +e  -e ( C +e  -e 1 ) ) +e c ) )
8180eqeq2d 2436 . . . . . 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 4426 . . . . . 6  |-  ( b  =  ( A +e  -e ( C +e  -e 1 ) )  -> 
( b  <  B  <->  ( A +e  -e ( C +e  -e 1 ) )  <  B ) )
8381, 823anbi12d 1336 . . . . 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 6313 . . . . . . 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 2436 . . . . . 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 4426 . . . . . 6  |-  ( c  =  ( C +e  -e 1 )  ->  ( c  < 
C  <->  ( C +e  -e 1 )  <  C ) )
8785, 863anbi13d 1337 . . . . 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 3193 . . . 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 1276 . . 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 2738 . 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 730 . . . . 5  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  B  e.  RR* )
9331a1i 11 . . . . . 6  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  1  e.  RR* )
9493xnegcld 11593 . . . . 5  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  -e 1  e.  RR* )
9592, 94xaddcld 11594 . . . 4  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  ( B +e  -e 1 )  e.  RR* )
962ad2antrr 730 . . . . 5  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  A  e.  RR* )
9795xnegcld 11593 . . . . 5  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  -e
( B +e  -e 1 )  e. 
RR* )
9896, 97xaddcld 11594 . . . 4  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  ( A +e  -e
( B +e  -e 1 ) )  e.  RR* )
99 xaddcom 11538 . . . . . 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 665 . . . . 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 730 . . . . . . 7  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  A  e.  RR )
102101renemnfd 9699 . . . . . 6  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  A  =/= -oo )
103 simplr 760 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  B  =/= +oo )
104 xrnepnf 11427 . . . . . . . . . . . . . . 15  |-  ( ( B  e.  RR*  /\  B  =/= +oo )  <->  ( B  e.  RR  \/  B  = -oo ) )
105104biimpi 197 . . . . . . . . . . . . . 14  |-  ( ( B  e.  RR*  /\  B  =/= +oo )  ->  ( B  e.  RR  \/  B  = -oo )
)
10692, 103, 105syl2anc 665 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  ( B  e.  RR  \/  B  = -oo )
)
107106orcomd 389 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  ( B  = -oo  \/  B  e.  RR ) )
108 xlt2addrd.4 . . . . . . . . . . . . . 14  |-  ( ph  ->  B  =/= -oo )
109108ad2antrr 730 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  B  =/= -oo )
110109neneqd 2621 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  -.  B  = -oo )
111 pm2.53 374 . . . . . . . . . . . 12  |-  ( ( B  = -oo  \/  B  e.  RR )  ->  ( -.  B  = -oo  ->  B  e.  RR ) )
112107, 110, 111sylc 62 . . . . . . . . . . 11  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  B  e.  RR )
113 rexsub 11533 . . . . . . . . . . 11  |-  ( ( B  e.  RR  /\  1  e.  RR )  ->  ( B +e  -e 1 )  =  ( B  -  1 ) )
114112, 30, 113sylancl 666 . . . . . . . . . 10  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  ( B +e  -e 1 )  =  ( B  -  1 ) )
115 resubcl 9945 . . . . . . . . . . 11  |-  ( ( B  e.  RR  /\  1  e.  RR )  ->  ( B  -  1 )  e.  RR )
116112, 30, 115sylancl 666 . . . . . . . . . 10  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  ( B  -  1 )  e.  RR )
117114, 116eqeltrd 2507 . . . . . . . . 9  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  ( B +e  -e 1 )  e.  RR )
118 rexneg 11511 . . . . . . . . 9  |-  ( ( B +e  -e 1 )  e.  RR  ->  -e ( B +e  -e 1 )  = 
-u ( B +e  -e 1 ) )
119117, 118syl 17 . . . . . . . 8  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  -e
( B +e  -e 1 )  = 
-u ( B +e  -e 1 ) )
120117renegcld 10053 . . . . . . . 8  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  -u ( B +e  -e 1 )  e.  RR )
121119, 120eqeltrd 2507 . . . . . . 7  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  -e
( B +e  -e 1 )  e.  RR )
122121renemnfd 9699 . . . . . 6  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  -e
( B +e  -e 1 )  =/= -oo )
123117renemnfd 9699 . . . . . 6  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  ( B +e  -e 1 )  =/= -oo )
124 xaddass 11542 . . . . . 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 1280 . . . . 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 11538 . . . . . . . . 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 665 . . . . . . . 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 11536 . . . . . . . . 9  |-  ( ( B +e  -e 1 )  e. 
RR*  ->  ( ( B +e  -e 1 ) +e  -e ( B +e  -e 1 ) )  =  0 )
12995, 128syl 17 . . . . . . . 8  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  (
( B +e  -e 1 ) +e  -e ( B +e  -e 1 ) )  =  0 )
130127, 129eqtrd 2463 . . . . . . 7  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  (  -e ( B +e  -e 1 ) +e ( B +e  -e 1 ) )  =  0 )
131130oveq2d 6321 . . . . . 6  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  ( A +e (  -e ( B +e  -e 1 ) +e ( B +e  -e 1 ) ) )  =  ( A +e 0 ) )
13296, 6syl 17 . . . . . 6  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  ( A +e 0 )  =  A )
133131, 132eqtrd 2463 . . . . 5  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  ( A +e (  -e ( B +e  -e 1 ) +e ( B +e  -e 1 ) ) )  =  A )
134100, 125, 1333eqtrrd 2468 . . . 4  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  A  =  ( ( B +e  -e 1 ) +e
( A +e  -e ( B +e  -e 1 ) ) ) )
135112ltm1d 10546 . . . . 5  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  ( B  -  1 )  <  B )
136114, 135eqbrtrd 4444 . . . 4  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  ( B +e  -e 1 )  <  B
)
137101, 116resubcld 10054 . . . . . 6  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  ( A  -  ( B  -  1 ) )  e.  RR )
138 ltpnf 11429 . . . . . 6  |-  ( ( A  -  ( B  -  1 ) )  e.  RR  ->  ( A  -  ( B  -  1 ) )  < +oo )
139137, 138syl 17 . . . . 5  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  ( A  -  ( B  -  1 ) )  < +oo )
140 rexsub 11533 . . . . . . 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 665 . . . . . 6  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  ( A +e  -e
( B +e  -e 1 ) )  =  ( A  -  ( B +e  -e 1 ) ) )
142114oveq2d 6321 . . . . . 6  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  ( A  -  ( B +e  -e 1 ) )  =  ( A  -  ( B  -  1 ) ) )
143141, 142eqtrd 2463 . . . . 5  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  ( A +e  -e
( B +e  -e 1 ) )  =  ( A  -  ( B  -  1
) ) )
144 simpr 462 . . . . 5  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  C  = +oo )
145139, 143, 1443brtr4d 4454 . . . 4  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  ( A +e  -e
( B +e  -e 1 ) )  <  C )
146 oveq1 6312 . . . . . . 7  |-  ( b  =  ( B +e  -e 1 )  ->  ( b +e c )  =  ( ( B +e  -e 1 ) +e c ) )
147146eqeq2d 2436 . . . . . 6  |-  ( b  =  ( B +e  -e 1 )  ->  ( A  =  ( b +e
c )  <->  A  =  ( ( B +e  -e 1 ) +e c ) ) )
148 breq1 4426 . . . . . 6  |-  ( b  =  ( B +e  -e 1 )  ->  ( b  < 
B  <->  ( B +e  -e 1 )  <  B ) )
149147, 1483anbi12d 1336 . . . . 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 6313 . . . . . . 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 2436 . . . . . 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 4426 . . . . . 6  |-  ( c  =  ( A +e  -e ( B +e  -e 1 ) )  -> 
( c  <  C  <->  ( A +e  -e ( B +e  -e 1 ) )  <  C ) )
153151, 1523anbi13d 1337 . . . . 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 3193 . . . 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 1276 . . 3  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  E. b  e.  RR*  E. c  e. 
RR*  ( A  =  ( b +e
c )  /\  b  <  B  /\  c  < 
C ) )
1561ad2antrr 730 . . . . . 6  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  A  e.  RR )
15791ad2antrr 730 . . . . . . . . 9  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  B  e.  RR* )
158 simplr 760 . . . . . . . . 9  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  B  =/= +oo )
159157, 158, 105syl2anc 665 . . . . . . . 8  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  ( B  e.  RR  \/  B  = -oo )
)
160159orcomd 389 . . . . . . 7  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  ( B  = -oo  \/  B  e.  RR ) )
161108ad2antrr 730 . . . . . . . 8  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  B  =/= -oo )
162161neneqd 2621 . . . . . . 7  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  -.  B  = -oo )
163160, 162, 111sylc 62 . . . . . 6  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  B  e.  RR )
16428ad2antrr 730 . . . . . . . . 9  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  C  e.  RR* )
165164, 40sylancom 671 . . . . . . . 8  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  ( C  e.  RR  \/  C  = -oo )
)
166165orcomd 389 . . . . . . 7  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  ( C  = -oo  \/  C  e.  RR ) )
16743ad2antrr 730 . . . . . . . 8  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  C  =/= -oo )
168167neneqd 2621 . . . . . . 7  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  -.  C  = -oo )
169166, 168, 46sylc 62 . . . . . 6  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  C  e.  RR )
170 xlt2addrd.6 . . . . . . . 8  |-  ( ph  ->  A  <  ( B +e C ) )
171170ad2antrr 730 . . . . . . 7  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  A  <  ( B +e
C ) )
172 rexadd 11532 . . . . . . . 8  |-  ( ( B  e.  RR  /\  C  e.  RR )  ->  ( B +e
C )  =  ( B  +  C ) )
173163, 169, 172syl2anc 665 . . . . . . 7  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  ( B +e C )  =  ( B  +  C ) )
174171, 173breqtrd 4448 . . . . . 6  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  A  <  ( B  +  C
) )
175156, 163, 169, 174lt2addrd 28332 . . . . 5  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  E. b  e.  RR  E. c  e.  RR  ( A  =  ( b  +  c )  /\  b  < 
B  /\  c  <  C ) )
176 rexadd 11532 . . . . . . . 8  |-  ( ( b  e.  RR  /\  c  e.  RR )  ->  ( b +e
c )  =  ( b  +  c ) )
177176eqeq2d 2436 . . . . . . 7  |-  ( ( b  e.  RR  /\  c  e.  RR )  ->  ( A  =  ( b +e c )  <->  A  =  (
b  +  c ) ) )
1781773anbi1d 1339 . . . . . 6  |-  ( ( b  e.  RR  /\  c  e.  RR )  ->  ( ( A  =  ( b +e
c )  /\  b  <  B  /\  c  < 
C )  <->  ( A  =  ( b  +  c )  /\  b  <  B  /\  c  < 
C ) ) )
1791782rexbiia 2941 . . . . 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 215 . . . 4  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  E. b  e.  RR  E. c  e.  RR  ( A  =  ( b +e
c )  /\  b  <  B  /\  c  < 
C ) )
181 ressxr 9691 . . . . . 6  |-  RR  C_  RR*
182 ssrexv 3526 . . . . . 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 2890 . . . 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 3526 . . . . 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 18 . . 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 2738 . 2  |-  ( (
ph  /\  B  =/= +oo )  ->  E. b  e.  RR*  E. c  e. 
RR*  ( A  =  ( b +e
c )  /\  b  <  B  /\  c  < 
C ) )
18990, 188pm2.61dane 2738 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 369    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872    =/= wne 2614   E.wrex 2772    C_ wss 3436   class class class wbr 4423  (class class class)co 6305   RRcr 9545   0cc0 9546   1c1 9547    + caddc 9549   +oocpnf 9679   -oocmnf 9680   RR*cxr 9681    < clt 9682    - cmin 9867   -ucneg 9868    -ecxne 11413   +ecxad 11414
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597  ax-cnex 9602  ax-resscn 9603  ax-1cn 9604  ax-icn 9605  ax-addcl 9606  ax-addrcl 9607  ax-mulcl 9608  ax-mulrcl 9609  ax-mulcom 9610  ax-addass 9611  ax-mulass 9612  ax-distr 9613  ax-i2m1 9614  ax-1ne0 9615  ax-1rid 9616  ax-rnegex 9617  ax-rrecex 9618  ax-cnre 9619  ax-pre-lttri 9620  ax-pre-lttrn 9621  ax-pre-ltadd 9622  ax-pre-mulgt0 9623
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rmo 2779  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-id 4768  df-po 4774  df-so 4775  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-1st 6807  df-2nd 6808  df-er 7374  df-en 7581  df-dom 7582  df-sdom 7583  df-pnf 9684  df-mnf 9685  df-xr 9686  df-ltxr 9687  df-le 9688  df-sub 9869  df-neg 9870  df-div 10277  df-2 10675  df-rp 11310  df-xneg 11416  df-xadd 11417
This theorem is referenced by:  xrofsup  28359
  Copyright terms: Public domain W3C validator