Theorem lt2addrd 27231

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 9619	. . . . 5 |- ( ph -> ( B + C ) e. RR )
4		lt2addrd.1	. . . . 5 |- ( ph -> A e. RR )
5	3, 4	resubcld 9983	. . . 4 |- ( ph -> ( ( B + C ) - A ) e. RR )
6	5	rehalfcld 10781	. . 3 |- ( ph -> ( ( ( B + C ) - A ) / 2 ) e. RR )
7	1, 6	resubcld 9983	. 2 |- ( ph -> ( B - ( ( ( B + C ) - A ) / 2 ) ) e. RR )
8	2, 6	resubcld 9983	. 2 |- ( ph -> ( C - ( ( ( B + C ) - A ) / 2 ) ) e. RR )
9	2	recnd 9618	. . . . . 6 |- ( ph -> C e. CC )
10	1	recnd 9618	. . . . . . . . 9 |- ( ph -> B e. CC )
11	10, 9	addcld 9611	. . . . . . . 8 |- ( ph -> ( B + C ) e. CC )
12	4	recnd 9618	. . . . . . . 8 |- ( ph -> A e. CC )
13	11, 12	subcld 9926	. . . . . . 7 |- ( ph -> ( ( B + C ) - A ) e. CC )
14	13	halfcld 10779	. . . . . 6 |- ( ph -> ( ( ( B + C ) - A ) / 2 ) e. CC )
15	9, 14, 14	subsub4d 9957	. . . . 5 |- ( ph -> ( ( C - ( ( ( B + C ) - A ) / 2 ) ) - ( ( ( B + C ) - A ) / 2 ) ) = ( C - ( ( ( ( B + C ) - A ) / 2 ) + ( ( ( B + C ) - A ) / 2 ) ) ) )
16	15	oveq2d 6298	. . . 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 9926	. . . . 5 |- ( ph -> ( C - ( ( ( B + C ) - A ) / 2 ) ) e. CC )
18	10, 14, 17	subadd23d 9948	. . . 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 10780	. . . . . 6 |- ( ph -> ( ( ( ( B + C ) - A ) / 2 ) + ( ( ( B + C ) - A ) / 2 ) ) = ( ( B + C ) - A ) )
20	19, 13	eqeltrd 2555	. . . . 5 |- ( ph -> ( ( ( ( B + C ) - A ) / 2 ) + ( ( ( B + C ) - A ) / 2 ) ) e. CC )
21	10, 9, 20	addsubassd 9946	. . . 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 2518	. . 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 6298	. . 3 |- ( ph -> ( ( B + C ) - ( ( ( ( B + C ) - A ) / 2 ) + ( ( ( B + C ) - A ) / 2 ) ) ) = ( ( B + C ) - ( ( B + C ) - A ) ) )
24	11, 12	nncand 9931	. . 3 |- ( ph -> ( ( B + C ) - ( ( B + C ) - A ) ) = A )
25	22, 23, 24	3eqtrrd 2513	. 2 |- ( ph -> A = ( ( B - ( ( ( B + C ) - A ) / 2 ) ) + ( C - ( ( ( B + C ) - A ) / 2 ) ) ) )
26		lt2addrd.4	. . . . 5 |- ( ph -> A < ( B + C ) )
27		difrp 11249	. . . . . 6 |- ( ( A e. RR /\ ( B + C ) e. RR ) -> ( A < ( B + C ) <-> ( ( B + C ) - A ) e. RR+ ) )
28	4, 3, 27	syl2anc 661	. . . . 5 |- ( ph -> ( A < ( B + C ) <-> ( ( B + C ) - A ) e. RR+ ) )
29	26, 28	mpbid 210	. . . 4 |- ( ph -> ( ( B + C ) - A ) e. RR+ )
30	29	rphalfcld 11264	. . 3 |- ( ph -> ( ( ( B + C ) - A ) / 2 ) e. RR+ )
31	1, 30	ltsubrpd 11280	. 2 |- ( ph -> ( B - ( ( ( B + C ) - A ) / 2 ) ) < B )
32	2, 30	ltsubrpd 11280	. 2 |- ( ph -> ( C - ( ( ( B + C ) - A ) / 2 ) ) < C )
33		oveq1 6289	. . . . 5 |- ( b = ( B - ( ( ( B + C ) - A ) / 2 ) ) -> ( b + c ) = ( ( B - ( ( ( B + C ) - A ) / 2 ) ) + c ) )
34	33	eqeq2d 2481	. . . 4 |- ( b = ( B - ( ( ( B + C ) - A ) / 2 ) ) -> ( A = ( b + c ) <-> A = ( ( B - ( ( ( B + C ) - A ) / 2 ) ) + c ) ) )
35		breq1 4450	. . . 4 |- ( b = ( B - ( ( ( B + C ) - A ) / 2 ) ) -> ( b < B <-> ( B - ( ( ( B + C ) - A ) / 2 ) ) < B ) )
36	34, 35	3anbi12d 1300	. . 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 6290	. . . . 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 2481	. . . 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 4450	. . . 4 |- ( c = ( C - ( ( ( B + C ) - A ) / 2 ) ) -> ( c < C <-> ( C - ( ( ( B + C ) - A ) / 2 ) ) < C ) )
40	38, 39	3anbi13d 1301	. . 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 3225	. 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 1240	1 |- ( ph -> E. b e. RR E. c e. RR ( A = ( b + c ) /\ b < B /\ c < C ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ w3a 973   = wceq 1379   e. wcel 1767   E.wrex 2815   class class class wbr 4447  (class class class)co 6282   CCcc 9486   RRcr 9487   + caddc 9491   < clt 9624   - cmin 9801   / cdiv 10202   2c2 10581   RR+crp 11216

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-2 10590  df-rp 11217

This theorem is referenced by:  xlt2addrd  27246