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Theorem lt2addrd 27432
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
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 9623 . . . . 5  |-  ( ph  ->  ( B  +  C
)  e.  RR )
4 lt2addrd.1 . . . . 5  |-  ( ph  ->  A  e.  RR )
53, 4resubcld 9990 . . . 4  |-  ( ph  ->  ( ( B  +  C )  -  A
)  e.  RR )
65rehalfcld 10788 . . 3  |-  ( ph  ->  ( ( ( B  +  C )  -  A )  /  2
)  e.  RR )
71, 6resubcld 9990 . 2  |-  ( ph  ->  ( B  -  (
( ( B  +  C )  -  A
)  /  2 ) )  e.  RR )
82, 6resubcld 9990 . 2  |-  ( ph  ->  ( C  -  (
( ( B  +  C )  -  A
)  /  2 ) )  e.  RR )
92recnd 9622 . . . . . 6  |-  ( ph  ->  C  e.  CC )
101recnd 9622 . . . . . . . . 9  |-  ( ph  ->  B  e.  CC )
1110, 9addcld 9615 . . . . . . . 8  |-  ( ph  ->  ( B  +  C
)  e.  CC )
124recnd 9622 . . . . . . . 8  |-  ( ph  ->  A  e.  CC )
1311, 12subcld 9933 . . . . . . 7  |-  ( ph  ->  ( ( B  +  C )  -  A
)  e.  CC )
1413halfcld 10786 . . . . . 6  |-  ( ph  ->  ( ( ( B  +  C )  -  A )  /  2
)  e.  CC )
159, 14, 14subsub4d 9964 . . . . 5  |-  ( ph  ->  ( ( C  -  ( ( ( B  +  C )  -  A )  /  2
) )  -  (
( ( B  +  C )  -  A
)  /  2 ) )  =  ( C  -  ( ( ( ( B  +  C
)  -  A )  /  2 )  +  ( ( ( B  +  C )  -  A )  /  2
) ) ) )
1615oveq2d 6294 . . . 4  |-  ( ph  ->  ( B  +  ( ( C  -  (
( ( B  +  C )  -  A
)  /  2 ) )  -  ( ( ( B  +  C
)  -  A )  /  2 ) ) )  =  ( B  +  ( C  -  ( ( ( ( B  +  C )  -  A )  / 
2 )  +  ( ( ( B  +  C )  -  A
)  /  2 ) ) ) ) )
179, 14subcld 9933 . . . . 5  |-  ( ph  ->  ( C  -  (
( ( B  +  C )  -  A
)  /  2 ) )  e.  CC )
1810, 14, 17subadd23d 9955 . . . 4  |-  ( ph  ->  ( ( B  -  ( ( ( B  +  C )  -  A )  /  2
) )  +  ( C  -  ( ( ( B  +  C
)  -  A )  /  2 ) ) )  =  ( B  +  ( ( C  -  ( ( ( B  +  C )  -  A )  / 
2 ) )  -  ( ( ( B  +  C )  -  A )  /  2
) ) ) )
19132halvesd 10787 . . . . . 6  |-  ( ph  ->  ( ( ( ( B  +  C )  -  A )  / 
2 )  +  ( ( ( B  +  C )  -  A
)  /  2 ) )  =  ( ( B  +  C )  -  A ) )
2019, 13eqeltrd 2529 . . . . 5  |-  ( ph  ->  ( ( ( ( B  +  C )  -  A )  / 
2 )  +  ( ( ( B  +  C )  -  A
)  /  2 ) )  e.  CC )
2110, 9, 20addsubassd 9953 . . . 4  |-  ( ph  ->  ( ( B  +  C )  -  (
( ( ( B  +  C )  -  A )  /  2
)  +  ( ( ( B  +  C
)  -  A )  /  2 ) ) )  =  ( B  +  ( C  -  ( ( ( ( B  +  C )  -  A )  / 
2 )  +  ( ( ( B  +  C )  -  A
)  /  2 ) ) ) ) )
2216, 18, 213eqtr4d 2492 . . 3  |-  ( ph  ->  ( ( B  -  ( ( ( B  +  C )  -  A )  /  2
) )  +  ( C  -  ( ( ( B  +  C
)  -  A )  /  2 ) ) )  =  ( ( B  +  C )  -  ( ( ( ( B  +  C
)  -  A )  /  2 )  +  ( ( ( B  +  C )  -  A )  /  2
) ) ) )
2319oveq2d 6294 . . 3  |-  ( ph  ->  ( ( B  +  C )  -  (
( ( ( B  +  C )  -  A )  /  2
)  +  ( ( ( B  +  C
)  -  A )  /  2 ) ) )  =  ( ( B  +  C )  -  ( ( B  +  C )  -  A ) ) )
2411, 12nncand 9938 . . 3  |-  ( ph  ->  ( ( B  +  C )  -  (
( B  +  C
)  -  A ) )  =  A )
2522, 23, 243eqtrrd 2487 . 2  |-  ( ph  ->  A  =  ( ( B  -  ( ( ( B  +  C
)  -  A )  /  2 ) )  +  ( C  -  ( ( ( B  +  C )  -  A )  /  2
) ) ) )
26 lt2addrd.4 . . . . 5  |-  ( ph  ->  A  <  ( B  +  C ) )
27 difrp 11259 . . . . . 6  |-  ( ( A  e.  RR  /\  ( B  +  C
)  e.  RR )  ->  ( A  < 
( B  +  C
)  <->  ( ( B  +  C )  -  A )  e.  RR+ ) )
284, 3, 27syl2anc 661 . . . . 5  |-  ( ph  ->  ( A  <  ( B  +  C )  <->  ( ( B  +  C
)  -  A )  e.  RR+ ) )
2926, 28mpbid 210 . . . 4  |-  ( ph  ->  ( ( B  +  C )  -  A
)  e.  RR+ )
3029rphalfcld 11274 . . 3  |-  ( ph  ->  ( ( ( B  +  C )  -  A )  /  2
)  e.  RR+ )
311, 30ltsubrpd 11290 . 2  |-  ( ph  ->  ( B  -  (
( ( B  +  C )  -  A
)  /  2 ) )  <  B )
322, 30ltsubrpd 11290 . 2  |-  ( ph  ->  ( C  -  (
( ( B  +  C )  -  A
)  /  2 ) )  <  C )
33 oveq1 6285 . . . . 5  |-  ( b  =  ( B  -  ( ( ( B  +  C )  -  A )  /  2
) )  ->  (
b  +  c )  =  ( ( B  -  ( ( ( B  +  C )  -  A )  / 
2 ) )  +  c ) )
3433eqeq2d 2455 . . . 4  |-  ( b  =  ( B  -  ( ( ( B  +  C )  -  A )  /  2
) )  ->  ( A  =  ( b  +  c )  <->  A  =  ( ( B  -  ( ( ( B  +  C )  -  A )  /  2
) )  +  c ) ) )
35 breq1 4437 . . . 4  |-  ( b  =  ( B  -  ( ( ( B  +  C )  -  A )  /  2
) )  ->  (
b  <  B  <->  ( B  -  ( ( ( B  +  C )  -  A )  / 
2 ) )  < 
B ) )
3634, 353anbi12d 1299 . . 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 ) ) )
37 oveq2 6286 . . . . 5  |-  ( c  =  ( C  -  ( ( ( B  +  C )  -  A )  /  2
) )  ->  (
( B  -  (
( ( B  +  C )  -  A
)  /  2 ) )  +  c )  =  ( ( B  -  ( ( ( B  +  C )  -  A )  / 
2 ) )  +  ( C  -  (
( ( B  +  C )  -  A
)  /  2 ) ) ) )
3837eqeq2d 2455 . . . 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 ) ) ) ) )
39 breq1 4437 . . . 4  |-  ( c  =  ( C  -  ( ( ( B  +  C )  -  A )  /  2
) )  ->  (
c  <  C  <->  ( C  -  ( ( ( B  +  C )  -  A )  / 
2 ) )  < 
C ) )
4038, 393anbi13d 1300 . . 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 ) ) )
4136, 40rspc2ev 3205 . 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 ) )
427, 8, 25, 31, 32, 41syl113anc 1239 1  |-  ( ph  ->  E. b  e.  RR  E. c  e.  RR  ( A  =  ( b  +  c )  /\  b  <  B  /\  c  <  C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ w3a 972    = wceq 1381    e. wcel 1802   E.wrex 2792   class class class wbr 4434  (class class class)co 6278   CCcc 9490   RRcr 9491    + caddc 9495    < clt 9628    - cmin 9807    / cdiv 10209   2c2 10588   RR+crp 11226
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4555  ax-nul 4563  ax-pow 4612  ax-pr 4673  ax-un 6574  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3419  df-dif 3462  df-un 3464  df-in 3466  df-ss 3473  df-nul 3769  df-if 3924  df-pw 3996  df-sn 4012  df-pr 4014  df-op 4018  df-uni 4232  df-br 4435  df-opab 4493  df-mpt 4494  df-id 4782  df-po 4787  df-so 4788  df-xp 4992  df-rel 4993  df-cnv 4994  df-co 4995  df-dm 4996  df-rn 4997  df-res 4998  df-ima 4999  df-iota 5538  df-fun 5577  df-fn 5578  df-f 5579  df-f1 5580  df-fo 5581  df-f1o 5582  df-fv 5583  df-riota 6239  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-er 7310  df-en 7516  df-dom 7517  df-sdom 7518  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9809  df-neg 9810  df-div 10210  df-2 10597  df-rp 11227
This theorem is referenced by:  xlt2addrd  27447
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