Theorem lt2addrd 24068

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

Step	Hyp	Ref	Expression
1		lt2addrd.2	. . 3 |- ( ph -> B e. RR )
2		lt2addrd.3	. . . . . 6 |- ( ph -> C e. RR )
3	1, 2	readdcld 9071	. . . . 5 |- ( ph -> ( B + C ) e. RR )
4		lt2addrd.1	. . . . 5 |- ( ph -> A e. RR )
5	3, 4	resubcld 9421	. . . 4 |- ( ph -> ( ( B + C ) - A ) e. RR )
6	5	rehalfcld 10170	. . 3 |- ( ph -> ( ( ( B + C ) - A ) / 2 ) e. RR )
7	1, 6	resubcld 9421	. 2 |- ( ph -> ( B - ( ( ( B + C ) - A ) / 2 ) ) e. RR )
8	2, 6	resubcld 9421	. 2 |- ( ph -> ( C - ( ( ( B + C ) - A ) / 2 ) ) e. RR )
9	2	recnd 9070	. . . . . 6 |- ( ph -> C e. CC )
10	1	recnd 9070	. . . . . . . . 9 |- ( ph -> B e. CC )
11	10, 9	addcld 9063	. . . . . . . 8 |- ( ph -> ( B + C ) e. CC )
12	4	recnd 9070	. . . . . . . 8 |- ( ph -> A e. CC )
13	11, 12	subcld 9367	. . . . . . 7 |- ( ph -> ( ( B + C ) - A ) e. CC )
14	13	halfcld 10168	. . . . . 6 |- ( ph -> ( ( ( B + C ) - A ) / 2 ) e. CC )
15	9, 14, 14	subsub4d 9398	. . . . 5 |- ( ph -> ( ( C - ( ( ( B + C ) - A ) / 2 ) ) - ( ( ( B + C ) - A ) / 2 ) ) = ( C - ( ( ( ( B + C ) - A ) / 2 ) + ( ( ( B + C ) - A ) / 2 ) ) ) )
16	15	oveq2d 6056	. . . 4 |- ( ph -> ( B + ( ( C - ( ( ( B + C ) - A ) / 2 ) ) - ( ( ( B + C ) - A ) / 2 ) ) ) = ( B + ( C - ( ( ( ( B + C ) - A ) / 2 ) + ( ( ( B + C ) - A ) / 2 ) ) ) ) )
17	9, 14	subcld 9367	. . . . 5 |- ( ph -> ( C - ( ( ( B + C ) - A ) / 2 ) ) e. CC )
18	10, 14, 17	subadd23d 9389	. . . 4 |- ( ph -> ( ( B - ( ( ( B + C ) - A ) / 2 ) ) + ( C - ( ( ( B + C ) - A ) / 2 ) ) ) = ( B + ( ( C - ( ( ( B + C ) - A ) / 2 ) ) - ( ( ( B + C ) - A ) / 2 ) ) ) )
19	13	2halvesd 10169	. . . . . 6 |- ( ph -> ( ( ( ( B + C ) - A ) / 2 ) + ( ( ( B + C ) - A ) / 2 ) ) = ( ( B + C ) - A ) )
20	19, 13	eqeltrd 2478	. . . . 5 |- ( ph -> ( ( ( ( B + C ) - A ) / 2 ) + ( ( ( B + C ) - A ) / 2 ) ) e. CC )
21	10, 9, 20	addsubassd 9387	. . . 4 |- ( ph -> ( ( B + C ) - ( ( ( ( B + C ) - A ) / 2 ) + ( ( ( B + C ) - A ) / 2 ) ) ) = ( B + ( C - ( ( ( ( B + C ) - A ) / 2 ) + ( ( ( B + C ) - A ) / 2 ) ) ) ) )
22	16, 18, 21	3eqtr4d 2446	. . 3 |- ( ph -> ( ( B - ( ( ( B + C ) - A ) / 2 ) ) + ( C - ( ( ( B + C ) - A ) / 2 ) ) ) = ( ( B + C ) - ( ( ( ( B + C ) - A ) / 2 ) + ( ( ( B + C ) - A ) / 2 ) ) ) )
23	19	oveq2d 6056	. . 3 |- ( ph -> ( ( B + C ) - ( ( ( ( B + C ) - A ) / 2 ) + ( ( ( B + C ) - A ) / 2 ) ) ) = ( ( B + C ) - ( ( B + C ) - A ) ) )
24	11, 12	nncand 9372	. . 3 |- ( ph -> ( ( B + C ) - ( ( B + C ) - A ) ) = A )
25	22, 23, 24	3eqtrrd 2441	. 2 |- ( ph -> A = ( ( B - ( ( ( B + C ) - A ) / 2 ) ) + ( C - ( ( ( B + C ) - A ) / 2 ) ) ) )
26		lt2addrd.4	. . . . 5 |- ( ph -> A < ( B + C ) )
27		difrp 10601	. . . . . 6 |- ( ( A e. RR /\ ( B + C ) e. RR ) -> ( A < ( B + C ) <-> ( ( B + C ) - A ) e. RR+ ) )
28	4, 3, 27	syl2anc 643	. . . . 5 |- ( ph -> ( A < ( B + C ) <-> ( ( B + C ) - A ) e. RR+ ) )
29	26, 28	mpbid 202	. . . 4 |- ( ph -> ( ( B + C ) - A ) e. RR+ )
30	29	rphalfcld 10616	. . 3 |- ( ph -> ( ( ( B + C ) - A ) / 2 ) e. RR+ )
31	1, 30	ltsubrpd 10632	. 2 |- ( ph -> ( B - ( ( ( B + C ) - A ) / 2 ) ) < B )
32	2, 30	ltsubrpd 10632	. 2 |- ( ph -> ( C - ( ( ( B + C ) - A ) / 2 ) ) < C )
33		oveq1 6047	. . . . 5 |- ( b = ( B - ( ( ( B + C ) - A ) / 2 ) ) -> ( b + c ) = ( ( B - ( ( ( B + C ) - A ) / 2 ) ) + c ) )
34	33	eqeq2d 2415	. . . 4 |- ( b = ( B - ( ( ( B + C ) - A ) / 2 ) ) -> ( A = ( b + c ) <-> A = ( ( B - ( ( ( B + C ) - A ) / 2 ) ) + c ) ) )
35		breq1 4175	. . . 4 |- ( b = ( B - ( ( ( B + C ) - A ) / 2 ) ) -> ( b < B <-> ( B - ( ( ( B + C ) - A ) / 2 ) ) < B ) )
36	34, 35	3anbi12d 1255	. . 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 6048	. . . . 5 |- ( c = ( C - ( ( ( B + C ) - A ) / 2 ) ) -> ( ( B - ( ( ( B + C ) - A ) / 2 ) ) + c ) = ( ( B - ( ( ( B + C ) - A ) / 2 ) ) + ( C - ( ( ( B + C ) - A ) / 2 ) ) ) )
38	37	eqeq2d 2415	. . . 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 4175	. . . 4 |- ( c = ( C - ( ( ( B + C ) - A ) / 2 ) ) -> ( c < C <-> ( C - ( ( ( B + C ) - A ) / 2 ) ) < C ) )
40	38, 39	3anbi13d 1256	. . 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 ) ) )
41	36, 40	rspc2ev 3020	. 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 ) )
42	7, 8, 25, 31, 32, 41	syl113anc 1196	1 |- ( ph -> E. b e. RR E. c e. RR ( A = ( b + c ) /\ b < B /\ c < C ) )

Syntax hints:   -> wi 4   <-> wb 177   /\ w3a 936   = wceq 1649   e. wcel 1721   E.wrex 2667   class class class wbr 4172  (class class class)co 6040   CCcc 8944   RRcr 8945   + caddc 8949   < clt 9076   - cmin 9247   / cdiv 9633   2c2 10005   RR+crp 10568

This theorem is referenced by:  xlt2addrd  24077

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-po 4463  df-so 4464  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-riota 6508  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-2 10014  df-rp 10569