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Theorem lt2addrd 23264
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
lt2addrd.1  |-  ( ph  ->  A  e.  RR )
lt2addrd.2  |-  ( ph  ->  B  e.  RR )
lt2addrd.3  |-  ( ph  ->  C  e.  RR )
lt2addrd.4  |-  ( ph  ->  A  <  ( B  +  C ) )
Assertion
Ref Expression
lt2addrd  |-  ( ph  ->  E. b  e.  RR  E. c  e.  RR  ( A  =  ( b  +  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 lt2addrd
StepHypRef Expression
1 lt2addrd.2 . . 3  |-  ( ph  ->  B  e.  RR )
2 lt2addrd.3 . . . . . 6  |-  ( ph  ->  C  e.  RR )
31, 2readdcld 8878 . . . . 5  |-  ( ph  ->  ( B  +  C
)  e.  RR )
4 lt2addrd.1 . . . . 5  |-  ( ph  ->  A  e.  RR )
53, 4resubcld 9227 . . . 4  |-  ( ph  ->  ( ( B  +  C )  -  A
)  e.  RR )
65rehalfcld 9974 . . 3  |-  ( ph  ->  ( ( ( B  +  C )  -  A )  /  2
)  e.  RR )
71, 6resubcld 9227 . 2  |-  ( ph  ->  ( B  -  (
( ( B  +  C )  -  A
)  /  2 ) )  e.  RR )
82, 6resubcld 9227 . 2  |-  ( ph  ->  ( C  -  (
( ( B  +  C )  -  A
)  /  2 ) )  e.  RR )
92recnd 8877 . . . . . . 7  |-  ( ph  ->  C  e.  CC )
101recnd 8877 . . . . . . . . . 10  |-  ( ph  ->  B  e.  CC )
1110, 9addcld 8870 . . . . . . . . 9  |-  ( ph  ->  ( B  +  C
)  e.  CC )
124recnd 8877 . . . . . . . . 9  |-  ( ph  ->  A  e.  CC )
1311, 12subcld 9173 . . . . . . . 8  |-  ( ph  ->  ( ( B  +  C )  -  A
)  e.  CC )
1413halfcld 9972 . . . . . . 7  |-  ( ph  ->  ( ( ( B  +  C )  -  A )  /  2
)  e.  CC )
159, 14, 14subsub4d 9204 . . . . . 6  |-  ( ph  ->  ( ( C  -  ( ( ( B  +  C )  -  A )  /  2
) )  -  (
( ( B  +  C )  -  A
)  /  2 ) )  =  ( C  -  ( ( ( ( B  +  C
)  -  A )  /  2 )  +  ( ( ( B  +  C )  -  A )  /  2
) ) ) )
1615oveq2d 5890 . . . . 5  |-  ( ph  ->  ( B  +  ( ( C  -  (
( ( B  +  C )  -  A
)  /  2 ) )  -  ( ( ( B  +  C
)  -  A )  /  2 ) ) )  =  ( B  +  ( C  -  ( ( ( ( B  +  C )  -  A )  / 
2 )  +  ( ( ( B  +  C )  -  A
)  /  2 ) ) ) ) )
179, 14subcld 9173 . . . . . 6  |-  ( ph  ->  ( C  -  (
( ( B  +  C )  -  A
)  /  2 ) )  e.  CC )
1810, 14, 17subadd23d 9195 . . . . 5  |-  ( ph  ->  ( ( B  -  ( ( ( B  +  C )  -  A )  /  2
) )  +  ( C  -  ( ( ( B  +  C
)  -  A )  /  2 ) ) )  =  ( B  +  ( ( C  -  ( ( ( B  +  C )  -  A )  / 
2 ) )  -  ( ( ( B  +  C )  -  A )  /  2
) ) ) )
19132halvesd 9973 . . . . . . 7  |-  ( ph  ->  ( ( ( ( B  +  C )  -  A )  / 
2 )  +  ( ( ( B  +  C )  -  A
)  /  2 ) )  =  ( ( B  +  C )  -  A ) )
2019, 13eqeltrd 2370 . . . . . 6  |-  ( ph  ->  ( ( ( ( B  +  C )  -  A )  / 
2 )  +  ( ( ( B  +  C )  -  A
)  /  2 ) )  e.  CC )
2110, 9, 20addsubassd 9193 . . . . 5  |-  ( ph  ->  ( ( B  +  C )  -  (
( ( ( B  +  C )  -  A )  /  2
)  +  ( ( ( B  +  C
)  -  A )  /  2 ) ) )  =  ( B  +  ( C  -  ( ( ( ( B  +  C )  -  A )  / 
2 )  +  ( ( ( B  +  C )  -  A
)  /  2 ) ) ) ) )
2216, 18, 213eqtr4d 2338 . . . 4  |-  ( ph  ->  ( ( B  -  ( ( ( B  +  C )  -  A )  /  2
) )  +  ( C  -  ( ( ( B  +  C
)  -  A )  /  2 ) ) )  =  ( ( B  +  C )  -  ( ( ( ( B  +  C
)  -  A )  /  2 )  +  ( ( ( B  +  C )  -  A )  /  2
) ) ) )
2319oveq2d 5890 . . . 4  |-  ( ph  ->  ( ( B  +  C )  -  (
( ( ( B  +  C )  -  A )  /  2
)  +  ( ( ( B  +  C
)  -  A )  /  2 ) ) )  =  ( ( B  +  C )  -  ( ( B  +  C )  -  A ) ) )
2411, 12nncand 9178 . . . 4  |-  ( ph  ->  ( ( B  +  C )  -  (
( B  +  C
)  -  A ) )  =  A )
2522, 23, 243eqtrrd 2333 . . 3  |-  ( ph  ->  A  =  ( ( B  -  ( ( ( B  +  C
)  -  A )  /  2 ) )  +  ( C  -  ( ( ( B  +  C )  -  A )  /  2
) ) ) )
26 lt2addrd.4 . . . . . 6  |-  ( ph  ->  A  <  ( B  +  C ) )
27 difrp 10403 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( B  +  C
)  e.  RR )  ->  ( A  < 
( B  +  C
)  <->  ( ( B  +  C )  -  A )  e.  RR+ ) )
284, 3, 27syl2anc 642 . . . . . 6  |-  ( ph  ->  ( A  <  ( B  +  C )  <->  ( ( B  +  C
)  -  A )  e.  RR+ ) )
2926, 28mpbid 201 . . . . 5  |-  ( ph  ->  ( ( B  +  C )  -  A
)  e.  RR+ )
3029rphalfcld 10418 . . . 4  |-  ( ph  ->  ( ( ( B  +  C )  -  A )  /  2
)  e.  RR+ )
311, 30ltsubrpd 10434 . . 3  |-  ( ph  ->  ( B  -  (
( ( B  +  C )  -  A
)  /  2 ) )  <  B )
322, 30ltsubrpd 10434 . . 3  |-  ( ph  ->  ( C  -  (
( ( B  +  C )  -  A
)  /  2 ) )  <  C )
3325, 31, 323jca 1132 . 2  |-  ( ph  ->  ( A  =  ( ( B  -  (
( ( B  +  C )  -  A
)  /  2 ) )  +  ( C  -  ( ( ( B  +  C )  -  A )  / 
2 ) ) )  /\  ( B  -  ( ( ( B  +  C )  -  A )  /  2
) )  <  B  /\  ( C  -  (
( ( B  +  C )  -  A
)  /  2 ) )  <  C ) )
34 oveq1 5881 . . . . 5  |-  ( b  =  ( B  -  ( ( ( B  +  C )  -  A )  /  2
) )  ->  (
b  +  c )  =  ( ( B  -  ( ( ( B  +  C )  -  A )  / 
2 ) )  +  c ) )
3534eqeq2d 2307 . . . 4  |-  ( b  =  ( B  -  ( ( ( B  +  C )  -  A )  /  2
) )  ->  ( A  =  ( b  +  c )  <->  A  =  ( ( B  -  ( ( ( B  +  C )  -  A )  /  2
) )  +  c ) ) )
36 breq1 4042 . . . 4  |-  ( b  =  ( B  -  ( ( ( B  +  C )  -  A )  /  2
) )  ->  (
b  <  B  <->  ( B  -  ( ( ( B  +  C )  -  A )  / 
2 ) )  < 
B ) )
3735, 363anbi12d 1253 . . 3  |-  ( b  =  ( B  -  ( ( ( B  +  C )  -  A )  /  2
) )  ->  (
( A  =  ( b  +  c )  /\  b  <  B  /\  c  <  C )  <-> 
( A  =  ( ( B  -  (
( ( B  +  C )  -  A
)  /  2 ) )  +  c )  /\  ( B  -  ( ( ( B  +  C )  -  A )  /  2
) )  <  B  /\  c  <  C ) ) )
38 oveq2 5882 . . . . 5  |-  ( c  =  ( C  -  ( ( ( B  +  C )  -  A )  /  2
) )  ->  (
( B  -  (
( ( B  +  C )  -  A
)  /  2 ) )  +  c )  =  ( ( B  -  ( ( ( B  +  C )  -  A )  / 
2 ) )  +  ( C  -  (
( ( B  +  C )  -  A
)  /  2 ) ) ) )
3938eqeq2d 2307 . . . 4  |-  ( c  =  ( C  -  ( ( ( B  +  C )  -  A )  /  2
) )  ->  ( A  =  ( ( B  -  ( (
( B  +  C
)  -  A )  /  2 ) )  +  c )  <->  A  =  ( ( B  -  ( ( ( B  +  C )  -  A )  /  2
) )  +  ( C  -  ( ( ( B  +  C
)  -  A )  /  2 ) ) ) ) )
40 breq1 4042 . . . 4  |-  ( c  =  ( C  -  ( ( ( B  +  C )  -  A )  /  2
) )  ->  (
c  <  C  <->  ( C  -  ( ( ( B  +  C )  -  A )  / 
2 ) )  < 
C ) )
4139, 403anbi13d 1254 . . 3  |-  ( c  =  ( C  -  ( ( ( B  +  C )  -  A )  /  2
) )  ->  (
( A  =  ( ( B  -  (
( ( B  +  C )  -  A
)  /  2 ) )  +  c )  /\  ( B  -  ( ( ( B  +  C )  -  A )  /  2
) )  <  B  /\  c  <  C )  <-> 
( A  =  ( ( B  -  (
( ( B  +  C )  -  A
)  /  2 ) )  +  ( C  -  ( ( ( B  +  C )  -  A )  / 
2 ) ) )  /\  ( B  -  ( ( ( B  +  C )  -  A )  /  2
) )  <  B  /\  ( C  -  (
( ( B  +  C )  -  A
)  /  2 ) )  <  C ) ) )
4237, 41rspc2ev 2905 . 2  |-  ( ( ( B  -  (
( ( B  +  C )  -  A
)  /  2 ) )  e.  RR  /\  ( C  -  (
( ( B  +  C )  -  A
)  /  2 ) )  e.  RR  /\  ( A  =  (
( B  -  (
( ( B  +  C )  -  A
)  /  2 ) )  +  ( C  -  ( ( ( B  +  C )  -  A )  / 
2 ) ) )  /\  ( B  -  ( ( ( B  +  C )  -  A )  /  2
) )  <  B  /\  ( C  -  (
( ( B  +  C )  -  A
)  /  2 ) )  <  C ) )  ->  E. b  e.  RR  E. c  e.  RR  ( A  =  ( b  +  c )  /\  b  < 
B  /\  c  <  C ) )
437, 8, 33, 42syl3anc 1182 1  |-  ( ph  ->  E. b  e.  RR  E. c  e.  RR  ( A  =  ( b  +  c )  /\  b  <  B  /\  c  <  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ w3a 934    = wceq 1632    e. wcel 1696   E.wrex 2557   class class class wbr 4039  (class class class)co 5874   CCcc 8751   RRcr 8752    + caddc 8756    < clt 8883    - cmin 9053    / cdiv 9439   2c2 9811   RR+crp 10370
This theorem is referenced by:  xlt2addrd  23268
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-2 9820  df-rp 10371
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