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Theorem lt2addrd 27713
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 9534 . . . . 5  |-  ( ph  ->  ( B  +  C
)  e.  RR )
4 lt2addrd.1 . . . . 5  |-  ( ph  ->  A  e.  RR )
53, 4resubcld 9905 . . . 4  |-  ( ph  ->  ( ( B  +  C )  -  A
)  e.  RR )
65rehalfcld 10702 . . 3  |-  ( ph  ->  ( ( ( B  +  C )  -  A )  /  2
)  e.  RR )
71, 6resubcld 9905 . 2  |-  ( ph  ->  ( B  -  (
( ( B  +  C )  -  A
)  /  2 ) )  e.  RR )
82, 6resubcld 9905 . 2  |-  ( ph  ->  ( C  -  (
( ( B  +  C )  -  A
)  /  2 ) )  e.  RR )
92recnd 9533 . . . . . 6  |-  ( ph  ->  C  e.  CC )
101recnd 9533 . . . . . . . . 9  |-  ( ph  ->  B  e.  CC )
1110, 9addcld 9526 . . . . . . . 8  |-  ( ph  ->  ( B  +  C
)  e.  CC )
124recnd 9533 . . . . . . . 8  |-  ( ph  ->  A  e.  CC )
1311, 12subcld 9844 . . . . . . 7  |-  ( ph  ->  ( ( B  +  C )  -  A
)  e.  CC )
1413halfcld 10700 . . . . . 6  |-  ( ph  ->  ( ( ( B  +  C )  -  A )  /  2
)  e.  CC )
159, 14, 14subsub4d 9875 . . . . 5  |-  ( ph  ->  ( ( C  -  ( ( ( B  +  C )  -  A )  /  2
) )  -  (
( ( B  +  C )  -  A
)  /  2 ) )  =  ( C  -  ( ( ( ( B  +  C
)  -  A )  /  2 )  +  ( ( ( B  +  C )  -  A )  /  2
) ) ) )
1615oveq2d 6212 . . . 4  |-  ( ph  ->  ( B  +  ( ( C  -  (
( ( B  +  C )  -  A
)  /  2 ) )  -  ( ( ( B  +  C
)  -  A )  /  2 ) ) )  =  ( B  +  ( C  -  ( ( ( ( B  +  C )  -  A )  / 
2 )  +  ( ( ( B  +  C )  -  A
)  /  2 ) ) ) ) )
179, 14subcld 9844 . . . . 5  |-  ( ph  ->  ( C  -  (
( ( B  +  C )  -  A
)  /  2 ) )  e.  CC )
1810, 14, 17subadd23d 9866 . . . 4  |-  ( ph  ->  ( ( B  -  ( ( ( B  +  C )  -  A )  /  2
) )  +  ( C  -  ( ( ( B  +  C
)  -  A )  /  2 ) ) )  =  ( B  +  ( ( C  -  ( ( ( B  +  C )  -  A )  / 
2 ) )  -  ( ( ( B  +  C )  -  A )  /  2
) ) ) )
19132halvesd 10701 . . . . . 6  |-  ( ph  ->  ( ( ( ( B  +  C )  -  A )  / 
2 )  +  ( ( ( B  +  C )  -  A
)  /  2 ) )  =  ( ( B  +  C )  -  A ) )
2019, 13eqeltrd 2470 . . . . 5  |-  ( ph  ->  ( ( ( ( B  +  C )  -  A )  / 
2 )  +  ( ( ( B  +  C )  -  A
)  /  2 ) )  e.  CC )
2110, 9, 20addsubassd 9864 . . . 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 2433 . . 3  |-  ( ph  ->  ( ( B  -  ( ( ( B  +  C )  -  A )  /  2
) )  +  ( C  -  ( ( ( B  +  C
)  -  A )  /  2 ) ) )  =  ( ( B  +  C )  -  ( ( ( ( B  +  C
)  -  A )  /  2 )  +  ( ( ( B  +  C )  -  A )  /  2
) ) ) )
2319oveq2d 6212 . . 3  |-  ( ph  ->  ( ( B  +  C )  -  (
( ( ( B  +  C )  -  A )  /  2
)  +  ( ( ( B  +  C
)  -  A )  /  2 ) ) )  =  ( ( B  +  C )  -  ( ( B  +  C )  -  A ) ) )
2411, 12nncand 9849 . . 3  |-  ( ph  ->  ( ( B  +  C )  -  (
( B  +  C
)  -  A ) )  =  A )
2522, 23, 243eqtrrd 2428 . 2  |-  ( ph  ->  A  =  ( ( B  -  ( ( ( B  +  C
)  -  A )  /  2 ) )  +  ( C  -  ( ( ( B  +  C )  -  A )  /  2
) ) ) )
26 lt2addrd.4 . . . . 5  |-  ( ph  ->  A  <  ( B  +  C ) )
27 difrp 11173 . . . . . 6  |-  ( ( A  e.  RR  /\  ( B  +  C
)  e.  RR )  ->  ( A  < 
( B  +  C
)  <->  ( ( B  +  C )  -  A )  e.  RR+ ) )
284, 3, 27syl2anc 659 . . . . 5  |-  ( ph  ->  ( A  <  ( B  +  C )  <->  ( ( B  +  C
)  -  A )  e.  RR+ ) )
2926, 28mpbid 210 . . . 4  |-  ( ph  ->  ( ( B  +  C )  -  A
)  e.  RR+ )
3029rphalfcld 11189 . . 3  |-  ( ph  ->  ( ( ( B  +  C )  -  A )  /  2
)  e.  RR+ )
311, 30ltsubrpd 11205 . 2  |-  ( ph  ->  ( B  -  (
( ( B  +  C )  -  A
)  /  2 ) )  <  B )
322, 30ltsubrpd 11205 . 2  |-  ( ph  ->  ( C  -  (
( ( B  +  C )  -  A
)  /  2 ) )  <  C )
33 oveq1 6203 . . . . 5  |-  ( b  =  ( B  -  ( ( ( B  +  C )  -  A )  /  2
) )  ->  (
b  +  c )  =  ( ( B  -  ( ( ( B  +  C )  -  A )  / 
2 ) )  +  c ) )
3433eqeq2d 2396 . . . 4  |-  ( b  =  ( B  -  ( ( ( B  +  C )  -  A )  /  2
) )  ->  ( A  =  ( b  +  c )  <->  A  =  ( ( B  -  ( ( ( B  +  C )  -  A )  /  2
) )  +  c ) ) )
35 breq1 4370 . . . 4  |-  ( b  =  ( B  -  ( ( ( B  +  C )  -  A )  /  2
) )  ->  (
b  <  B  <->  ( B  -  ( ( ( B  +  C )  -  A )  / 
2 ) )  < 
B ) )
3634, 353anbi12d 1298 . . 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 6204 . . . . 5  |-  ( c  =  ( C  -  ( ( ( B  +  C )  -  A )  /  2
) )  ->  (
( B  -  (
( ( B  +  C )  -  A
)  /  2 ) )  +  c )  =  ( ( B  -  ( ( ( B  +  C )  -  A )  / 
2 ) )  +  ( C  -  (
( ( B  +  C )  -  A
)  /  2 ) ) ) )
3837eqeq2d 2396 . . . 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 4370 . . . 4  |-  ( c  =  ( C  -  ( ( ( B  +  C )  -  A )  /  2
) )  ->  (
c  <  C  <->  ( C  -  ( ( ( B  +  C )  -  A )  / 
2 ) )  < 
C ) )
4038, 393anbi13d 1299 . . 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 3146 . 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 1238 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 971    = wceq 1399    e. wcel 1826   E.wrex 2733   class class class wbr 4367  (class class class)co 6196   CCcc 9401   RRcr 9402    + caddc 9406    < clt 9539    - cmin 9718    / cdiv 10123   2c2 10502   RR+crp 11139
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-po 4714  df-so 4715  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-er 7229  df-en 7436  df-dom 7437  df-sdom 7438  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-div 10124  df-2 10511  df-rp 11140
This theorem is referenced by:  xlt2addrd  27728
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