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Theorem xlt2addrd 28020
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 9673 . . . . 5  |-  ( ph  ->  A  e.  RR* )
32ad2antrr 724 . . . 4  |-  ( ( ( ph  /\  B  = +oo )  /\  C  = +oo )  ->  A  e.  RR* )
4 0xr 9670 . . . . 5  |-  0  e.  RR*
54a1i 11 . . . 4  |-  ( ( ( ph  /\  B  = +oo )  /\  C  = +oo )  ->  0  e.  RR* )
6 xaddid1 11491 . . . . . 6  |-  ( A  e.  RR*  ->  ( A +e 0 )  =  A )
76eqcomd 2410 . . . . 5  |-  ( A  e.  RR*  ->  A  =  ( A +e 0 ) )
83, 7syl 17 . . . 4  |-  ( ( ( ph  /\  B  = +oo )  /\  C  = +oo )  ->  A  =  ( A +e 0 ) )
91ad2antrr 724 . . . . . 6  |-  ( ( ( ph  /\  B  = +oo )  /\  C  = +oo )  ->  A  e.  RR )
10 ltpnf 11384 . . . . . 6  |-  ( A  e.  RR  ->  A  < +oo )
119, 10syl 17 . . . . 5  |-  ( ( ( ph  /\  B  = +oo )  /\  C  = +oo )  ->  A  < +oo )
12 simplr 754 . . . . 5  |-  ( ( ( ph  /\  B  = +oo )  /\  C  = +oo )  ->  B  = +oo )
1311, 12breqtrrd 4421 . . . 4  |-  ( ( ( ph  /\  B  = +oo )  /\  C  = +oo )  ->  A  <  B )
14 0ltpnf 11385 . . . . 5  |-  0  < +oo
15 simpr 459 . . . . 5  |-  ( ( ( ph  /\  B  = +oo )  /\  C  = +oo )  ->  C  = +oo )
1614, 15syl5breqr 4431 . . . 4  |-  ( ( ( ph  /\  B  = +oo )  /\  C  = +oo )  ->  0  <  C )
17 oveq1 6285 . . . . . . 7  |-  ( b  =  A  ->  (
b +e c )  =  ( A +e c ) )
1817eqeq2d 2416 . . . . . 6  |-  ( b  =  A  ->  ( A  =  ( b +e c )  <-> 
A  =  ( A +e c ) ) )
19 breq1 4398 . . . . . 6  |-  ( b  =  A  ->  (
b  <  B  <->  A  <  B ) )
2018, 193anbi12d 1302 . . . . 5  |-  ( b  =  A  ->  (
( A  =  ( b +e c )  /\  b  < 
B  /\  c  <  C )  <->  ( A  =  ( A +e
c )  /\  A  <  B  /\  c  < 
C ) ) )
21 oveq2 6286 . . . . . . 7  |-  ( c  =  0  ->  ( A +e c )  =  ( A +e 0 ) )
2221eqeq2d 2416 . . . . . 6  |-  ( c  =  0  ->  ( A  =  ( A +e c )  <-> 
A  =  ( A +e 0 ) ) )
23 breq1 4398 . . . . . 6  |-  ( c  =  0  ->  (
c  <  C  <->  0  <  C ) )
2422, 233anbi13d 1303 . . . . 5  |-  ( c  =  0  ->  (
( A  =  ( A +e c )  /\  A  < 
B  /\  c  <  C )  <->  ( A  =  ( A +e 0 )  /\  A  <  B  /\  0  < 
C ) ) )
2520, 24rspc2ev 3171 . . . 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 1242 . . 3  |-  ( ( ( ph  /\  B  = +oo )  /\  C  = +oo )  ->  E. b  e.  RR*  E. c  e. 
RR*  ( A  =  ( b +e
c )  /\  b  <  B  /\  c  < 
C ) )
272ad2antrr 724 . . . . 5  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  A  e.  RR* )
28 xlt2addrd.3 . . . . . . . 8  |-  ( ph  ->  C  e.  RR* )
2928ad2antrr 724 . . . . . . 7  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  C  e.  RR* )
30 1re 9625 . . . . . . . . . 10  |-  1  e.  RR
3130rexri 9676 . . . . . . . . 9  |-  1  e.  RR*
3231a1i 11 . . . . . . . 8  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  1  e.  RR* )
3332xnegcld 11545 . . . . . . 7  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  -e 1  e.  RR* )
3429, 33xaddcld 11546 . . . . . 6  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  ( C +e  -e 1 )  e.  RR* )
3534xnegcld 11545 . . . . 5  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  -e
( C +e  -e 1 )  e. 
RR* )
3627, 35xaddcld 11546 . . . 4  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  ( A +e  -e
( C +e  -e 1 ) )  e.  RR* )
371ad2antrr 724 . . . . . . 7  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  A  e.  RR )
3837renemnfd 9675 . . . . . 6  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  A  =/= -oo )
39 xrnepnf 11382 . . . . . . . . . . . . . . 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 665 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  ( C  e.  RR  \/  C  = -oo )
)
4241orcomd 386 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  ( C  = -oo  \/  C  e.  RR ) )
43 xlt2addrd.5 . . . . . . . . . . . . . 14  |-  ( ph  ->  C  =/= -oo )
4443ad2antrr 724 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  C  =/= -oo )
4544neneqd 2605 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  -.  C  = -oo )
46 pm2.53 371 . . . . . . . . . . . 12  |-  ( ( C  = -oo  \/  C  e.  RR )  ->  ( -.  C  = -oo  ->  C  e.  RR ) )
4742, 45, 46sylc 59 . . . . . . . . . . 11  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  C  e.  RR )
48 rexsub 11485 . . . . . . . . . . 11  |-  ( ( C  e.  RR  /\  1  e.  RR )  ->  ( C +e  -e 1 )  =  ( C  -  1 ) )
4947, 30, 48sylancl 660 . . . . . . . . . 10  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  ( C +e  -e 1 )  =  ( C  -  1 ) )
50 resubcl 9919 . . . . . . . . . . 11  |-  ( ( C  e.  RR  /\  1  e.  RR )  ->  ( C  -  1 )  e.  RR )
5147, 30, 50sylancl 660 . . . . . . . . . 10  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  ( C  -  1 )  e.  RR )
5249, 51eqeltrd 2490 . . . . . . . . 9  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  ( C +e  -e 1 )  e.  RR )
53 rexneg 11463 . . . . . . . . 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 10027 . . . . . . . 8  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  -u ( C +e  -e 1 )  e.  RR )
5654, 55eqeltrd 2490 . . . . . . 7  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  -e
( C +e  -e 1 )  e.  RR )
5756renemnfd 9675 . . . . . 6  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  -e
( C +e  -e 1 )  =/= -oo )
5852renemnfd 9675 . . . . . 6  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  ( C +e  -e 1 )  =/= -oo )
59 xaddass 11494 . . . . . 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 1246 . . . . 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 11490 . . . . . . . 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 659 . . . . . . 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 11488 . . . . . . . 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 2443 . . . . . 6  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  (  -e ( C +e  -e 1 ) +e ( C +e  -e 1 ) )  =  0 )
6665oveq2d 6294 . . . . 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 2448 . . . 4  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  A  =  ( ( A +e  -e
( C +e  -e 1 ) ) +e ( C +e  -e 1 ) ) )
6937, 51resubcld 10028 . . . . . 6  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  ( A  -  ( C  -  1 ) )  e.  RR )
70 ltpnf 11384 . . . . . 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 11485 . . . . . . 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 659 . . . . . 6  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  ( A +e  -e
( C +e  -e 1 ) )  =  ( A  -  ( C +e  -e 1 ) ) )
7449oveq2d 6294 . . . . . 6  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  ( A  -  ( C +e  -e 1 ) )  =  ( A  -  ( C  -  1 ) ) )
7573, 74eqtrd 2443 . . . . 5  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  ( A +e  -e
( C +e  -e 1 ) )  =  ( A  -  ( C  -  1
) ) )
76 simplr 754 . . . . 5  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  B  = +oo )
7771, 75, 763brtr4d 4425 . . . 4  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  ( A +e  -e
( C +e  -e 1 ) )  <  B )
7847ltm1d 10518 . . . . 5  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  ( C  -  1 )  <  C )
7949, 78eqbrtrd 4415 . . . 4  |-  ( ( ( ph  /\  B  = +oo )  /\  C  =/= +oo )  ->  ( C +e  -e 1 )  <  C
)
80 oveq1 6285 . . . . . . 7  |-  ( b  =  ( A +e  -e ( C +e  -e 1 ) )  -> 
( b +e
c )  =  ( ( A +e  -e ( C +e  -e 1 ) ) +e c ) )
8180eqeq2d 2416 . . . . . 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 4398 . . . . . 6  |-  ( b  =  ( A +e  -e ( C +e  -e 1 ) )  -> 
( b  <  B  <->  ( A +e  -e ( C +e  -e 1 ) )  <  B ) )
8381, 823anbi12d 1302 . . . . 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 6286 . . . . . . 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 2416 . . . . . 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 4398 . . . . . 6  |-  ( c  =  ( C +e  -e 1 )  ->  ( c  < 
C  <->  ( C +e  -e 1 )  <  C ) )
8785, 863anbi13d 1303 . . . . 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 3171 . . . 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 1242 . . 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 2721 . 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 724 . . . . 5  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  B  e.  RR* )
9331a1i 11 . . . . . 6  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  1  e.  RR* )
9493xnegcld 11545 . . . . 5  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  -e 1  e.  RR* )
9592, 94xaddcld 11546 . . . 4  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  ( B +e  -e 1 )  e.  RR* )
962ad2antrr 724 . . . . 5  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  A  e.  RR* )
9795xnegcld 11545 . . . . 5  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  -e
( B +e  -e 1 )  e. 
RR* )
9896, 97xaddcld 11546 . . . 4  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  ( A +e  -e
( B +e  -e 1 ) )  e.  RR* )
99 xaddcom 11490 . . . . . 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 659 . . . . 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 724 . . . . . . 7  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  A  e.  RR )
102101renemnfd 9675 . . . . . 6  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  A  =/= -oo )
103 simplr 754 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  B  =/= +oo )
104 xrnepnf 11382 . . . . . . . . . . . . . . 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 659 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  ( B  e.  RR  \/  B  = -oo )
)
107106orcomd 386 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  ( B  = -oo  \/  B  e.  RR ) )
108 xlt2addrd.4 . . . . . . . . . . . . . 14  |-  ( ph  ->  B  =/= -oo )
109108ad2antrr 724 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  B  =/= -oo )
110109neneqd 2605 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  -.  B  = -oo )
111 pm2.53 371 . . . . . . . . . . . 12  |-  ( ( B  = -oo  \/  B  e.  RR )  ->  ( -.  B  = -oo  ->  B  e.  RR ) )
112107, 110, 111sylc 59 . . . . . . . . . . 11  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  B  e.  RR )
113 rexsub 11485 . . . . . . . . . . 11  |-  ( ( B  e.  RR  /\  1  e.  RR )  ->  ( B +e  -e 1 )  =  ( B  -  1 ) )
114112, 30, 113sylancl 660 . . . . . . . . . 10  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  ( B +e  -e 1 )  =  ( B  -  1 ) )
115 resubcl 9919 . . . . . . . . . . 11  |-  ( ( B  e.  RR  /\  1  e.  RR )  ->  ( B  -  1 )  e.  RR )
116112, 30, 115sylancl 660 . . . . . . . . . 10  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  ( B  -  1 )  e.  RR )
117114, 116eqeltrd 2490 . . . . . . . . 9  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  ( B +e  -e 1 )  e.  RR )
118 rexneg 11463 . . . . . . . . 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 10027 . . . . . . . 8  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  -u ( B +e  -e 1 )  e.  RR )
121119, 120eqeltrd 2490 . . . . . . 7  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  -e
( B +e  -e 1 )  e.  RR )
122121renemnfd 9675 . . . . . 6  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  -e
( B +e  -e 1 )  =/= -oo )
123117renemnfd 9675 . . . . . 6  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  ( B +e  -e 1 )  =/= -oo )
124 xaddass 11494 . . . . . 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 1246 . . . . 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 11490 . . . . . . . . 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 659 . . . . . . . 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 11488 . . . . . . . . 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 2443 . . . . . . 7  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  (  -e ( B +e  -e 1 ) +e ( B +e  -e 1 ) )  =  0 )
131130oveq2d 6294 . . . . . 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 2443 . . . . 5  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  ( A +e (  -e ( B +e  -e 1 ) +e ( B +e  -e 1 ) ) )  =  A )
134100, 125, 1333eqtrrd 2448 . . . 4  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  A  =  ( ( B +e  -e 1 ) +e
( A +e  -e ( B +e  -e 1 ) ) ) )
135112ltm1d 10518 . . . . 5  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  ( B  -  1 )  <  B )
136114, 135eqbrtrd 4415 . . . 4  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  ( B +e  -e 1 )  <  B
)
137101, 116resubcld 10028 . . . . . 6  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  ( A  -  ( B  -  1 ) )  e.  RR )
138 ltpnf 11384 . . . . . 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 11485 . . . . . . 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 659 . . . . . 6  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  ( A +e  -e
( B +e  -e 1 ) )  =  ( A  -  ( B +e  -e 1 ) ) )
142114oveq2d 6294 . . . . . 6  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  ( A  -  ( B +e  -e 1 ) )  =  ( A  -  ( B  -  1 ) ) )
143141, 142eqtrd 2443 . . . . 5  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  ( A +e  -e
( B +e  -e 1 ) )  =  ( A  -  ( B  -  1
) ) )
144 simpr 459 . . . . 5  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  C  = +oo )
145139, 143, 1443brtr4d 4425 . . . 4  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  ( A +e  -e
( B +e  -e 1 ) )  <  C )
146 oveq1 6285 . . . . . . 7  |-  ( b  =  ( B +e  -e 1 )  ->  ( b +e c )  =  ( ( B +e  -e 1 ) +e c ) )
147146eqeq2d 2416 . . . . . 6  |-  ( b  =  ( B +e  -e 1 )  ->  ( A  =  ( b +e
c )  <->  A  =  ( ( B +e  -e 1 ) +e c ) ) )
148 breq1 4398 . . . . . 6  |-  ( b  =  ( B +e  -e 1 )  ->  ( b  < 
B  <->  ( B +e  -e 1 )  <  B ) )
149147, 1483anbi12d 1302 . . . . 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 6286 . . . . . . 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 2416 . . . . . 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 4398 . . . . . 6  |-  ( c  =  ( A +e  -e ( B +e  -e 1 ) )  -> 
( c  <  C  <->  ( A +e  -e ( B +e  -e 1 ) )  <  C ) )
153151, 1523anbi13d 1303 . . . . 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 3171 . . . 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 1242 . . 3  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  = +oo )  ->  E. b  e.  RR*  E. c  e. 
RR*  ( A  =  ( b +e
c )  /\  b  <  B  /\  c  < 
C ) )
1561ad2antrr 724 . . . . . 6  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  A  e.  RR )
15791ad2antrr 724 . . . . . . . . 9  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  B  e.  RR* )
158 simplr 754 . . . . . . . . 9  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  B  =/= +oo )
159157, 158, 105syl2anc 659 . . . . . . . 8  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  ( B  e.  RR  \/  B  = -oo )
)
160159orcomd 386 . . . . . . 7  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  ( B  = -oo  \/  B  e.  RR ) )
161108ad2antrr 724 . . . . . . . 8  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  B  =/= -oo )
162161neneqd 2605 . . . . . . 7  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  -.  B  = -oo )
163160, 162, 111sylc 59 . . . . . 6  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  B  e.  RR )
16428ad2antrr 724 . . . . . . . . 9  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  C  e.  RR* )
165164, 40sylancom 665 . . . . . . . 8  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  ( C  e.  RR  \/  C  = -oo )
)
166165orcomd 386 . . . . . . 7  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  ( C  = -oo  \/  C  e.  RR ) )
16743ad2antrr 724 . . . . . . . 8  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  C  =/= -oo )
168167neneqd 2605 . . . . . . 7  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  -.  C  = -oo )
169166, 168, 46sylc 59 . . . . . 6  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  C  e.  RR )
170 xlt2addrd.6 . . . . . . . 8  |-  ( ph  ->  A  <  ( B +e C ) )
171170ad2antrr 724 . . . . . . 7  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  A  <  ( B +e
C ) )
172 rexadd 11484 . . . . . . . 8  |-  ( ( B  e.  RR  /\  C  e.  RR )  ->  ( B +e
C )  =  ( B  +  C ) )
173163, 169, 172syl2anc 659 . . . . . . 7  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  ( B +e C )  =  ( B  +  C ) )
174171, 173breqtrd 4419 . . . . . 6  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  A  <  ( B  +  C
) )
175156, 163, 169, 174lt2addrd 28010 . . . . 5  |-  ( ( ( ph  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  E. b  e.  RR  E. c  e.  RR  ( A  =  ( b  +  c )  /\  b  < 
B  /\  c  <  C ) )
176 rexadd 11484 . . . . . . . 8  |-  ( ( b  e.  RR  /\  c  e.  RR )  ->  ( b +e
c )  =  ( b  +  c ) )
177176eqeq2d 2416 . . . . . . 7  |-  ( ( b  e.  RR  /\  c  e.  RR )  ->  ( A  =  ( b +e c )  <->  A  =  (
b  +  c ) ) )
1781773anbi1d 1305 . . . . . 6  |-  ( ( b  e.  RR  /\  c  e.  RR )  ->  ( ( A  =  ( b +e
c )  /\  b  <  B  /\  c  < 
C )  <->  ( A  =  ( b  +  c )  /\  b  <  B  /\  c  < 
C ) ) )
1791782rexbiia 2923 . . . . 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 9667 . . . . . 6  |-  RR  C_  RR*
182 ssrexv 3504 . . . . . 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 2872 . . . 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 3504 . . . . 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 2721 . 2  |-  ( (
ph  /\  B  =/= +oo )  ->  E. b  e.  RR*  E. c  e. 
RR*  ( A  =  ( b +e
c )  /\  b  <  B  /\  c  < 
C ) )
18990, 188pm2.61dane 2721 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 366    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842    =/= wne 2598   E.wrex 2755    C_ wss 3414   class class class wbr 4395  (class class class)co 6278   RRcr 9521   0cc0 9522   1c1 9523    + caddc 9525   +oocpnf 9655   -oocmnf 9656   RR*cxr 9657    < clt 9658    - cmin 9841   -ucneg 9842    -ecxne 11368   +ecxad 11369
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-po 4744  df-so 4745  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-1st 6784  df-2nd 6785  df-er 7348  df-en 7555  df-dom 7556  df-sdom 7557  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-div 10248  df-2 10635  df-rp 11266  df-xneg 11371  df-xadd 11372
This theorem is referenced by:  xrofsup  28030
  Copyright terms: Public domain W3C validator