Theorem maxabslemval 11390

Description: Lemma for maxabs 11391. Value of the supremum. (Contributed by Jim Kingdon, 22-Dec-2021.)

Assertion

Ref	Expression
maxabslemval	|- ( ( A e. RR /\ B e. RR ) -> ( ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) e. RR /\ A. x e. { A , B } -. ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) < x /\ A. x e. RR ( x < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) -> E. z e. { A , B } x < z ) ) )

Distinct variable groups:   x, A, z   x, B, z

Proof of Theorem maxabslemval

Step	Hyp	Ref	Expression
1		readdcl 8022	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( A + B ) e. RR )
2		simpl 109	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> A e. RR )
3	2	recnd 8072	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> A e. CC )
4		simpr 110	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> B e. RR )
5	4	recnd 8072	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> B e. CC )
6	3, 5	subcld 8354	. . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( A - B ) e. CC )
7	6	abscld 11363	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( abs ` ( A - B ) ) e. RR )
8	1, 7	readdcld 8073	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( ( A + B ) + ( abs ` ( A - B ) ) ) e. RR )
9	8	rehalfcld 9255	. 2 |- ( ( A e. RR /\ B e. RR ) -> ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) e. RR )
10		vex 2766	. . . . 5 |- x e. _V
11	10	elpr 3644	. . . 4 |- ( x e. { A , B } <-> ( x = A \/ x = B ) )
12		maxabsle 11386	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> A <_ ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) )
13	2, 9, 12	lensymd 8165	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> -. ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) < A )
14		breq2 4038	. . . . . . 7 |- ( x = A -> ( ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) < x <-> ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) < A ) )
15	14	notbid 668	. . . . . 6 |- ( x = A -> ( -. ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) < x <-> -. ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) < A ) )
16	13, 15	syl5ibrcom 157	. . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( x = A -> -. ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) < x ) )
17		maxabsle 11386	. . . . . . . . 9 |- ( ( B e. RR /\ A e. RR ) -> B <_ ( ( ( B + A ) + ( abs ` ( B - A ) ) ) / 2 ) )
18	17	ancoms 268	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR ) -> B <_ ( ( ( B + A ) + ( abs ` ( B - A ) ) ) / 2 ) )
19	5, 3	addcomd 8194	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR ) -> ( B + A ) = ( A + B ) )
20	5, 3	abssubd 11375	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR ) -> ( abs ` ( B - A ) ) = ( abs ` ( A - B ) ) )
21	19, 20	oveq12d 5943	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR ) -> ( ( B + A ) + ( abs ` ( B - A ) ) ) = ( ( A + B ) + ( abs ` ( A - B ) ) ) )
22	21	oveq1d 5940	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR ) -> ( ( ( B + A ) + ( abs ` ( B - A ) ) ) / 2 ) = ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) )
23	18, 22	breqtrd 4060	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> B <_ ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) )
24	4, 9, 23	lensymd 8165	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> -. ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) < B )
25		breq2 4038	. . . . . . 7 |- ( x = B -> ( ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) < x <-> ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) < B ) )
26	25	notbid 668	. . . . . 6 |- ( x = B -> ( -. ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) < x <-> -. ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) < B ) )
27	24, 26	syl5ibrcom 157	. . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( x = B -> -. ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) < x ) )
28	16, 27	jaod 718	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( ( x = A \/ x = B ) -> -. ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) < x ) )
29	11, 28	biimtrid 152	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( x e. { A , B } -> -. ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) < x ) )
30	29	ralrimiv 2569	. 2 |- ( ( A e. RR /\ B e. RR ) -> A. x e. { A , B } -. ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) < x )
31		prid1g 3727	. . . . . . 7 |- ( A e. RR -> A e. { A , B } )
32	31	ad4antr 494	. . . . . 6 |- ( ( ( ( ( A e. RR /\ B e. RR ) /\ x e. RR ) /\ x < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) /\ x < A ) -> A e. { A , B } )
33		breq2 4038	. . . . . . 7 |- ( z = A -> ( x < z <-> x < A ) )
34	33	rspcev 2868	. . . . . 6 |- ( ( A e. { A , B } /\ x < A ) -> E. z e. { A , B } x < z )
35	32, 34	sylancom 420	. . . . 5 |- ( ( ( ( ( A e. RR /\ B e. RR ) /\ x e. RR ) /\ x < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) /\ x < A ) -> E. z e. { A , B } x < z )
36		prid2g 3728	. . . . . . 7 |- ( B e. RR -> B e. { A , B } )
37	36	ad4antlr 495	. . . . . 6 |- ( ( ( ( ( A e. RR /\ B e. RR ) /\ x e. RR ) /\ x < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) /\ x < B ) -> B e. { A , B } )
38		breq2 4038	. . . . . . 7 |- ( z = B -> ( x < z <-> x < B ) )
39	38	rspcev 2868	. . . . . 6 |- ( ( B e. { A , B } /\ x < B ) -> E. z e. { A , B } x < z )
40	37, 39	sylancom 420	. . . . 5 |- ( ( ( ( ( A e. RR /\ B e. RR ) /\ x e. RR ) /\ x < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) /\ x < B ) -> E. z e. { A , B } x < z )
41	2	ad2antrr 488	. . . . . 6 |- ( ( ( ( A e. RR /\ B e. RR ) /\ x e. RR ) /\ x < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) -> A e. RR )
42	4	ad2antrr 488	. . . . . 6 |- ( ( ( ( A e. RR /\ B e. RR ) /\ x e. RR ) /\ x < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) -> B e. RR )
43		simplr 528	. . . . . 6 |- ( ( ( ( A e. RR /\ B e. RR ) /\ x e. RR ) /\ x < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) -> x e. RR )
44		simpr 110	. . . . . 6 |- ( ( ( ( A e. RR /\ B e. RR ) /\ x e. RR ) /\ x < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) -> x < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) )
45	41, 42, 43, 44	maxabslemlub 11389	. . . . 5 |- ( ( ( ( A e. RR /\ B e. RR ) /\ x e. RR ) /\ x < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) -> ( x < A \/ x < B ) )
46	35, 40, 45	mpjaodan 799	. . . 4 |- ( ( ( ( A e. RR /\ B e. RR ) /\ x e. RR ) /\ x < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) -> E. z e. { A , B } x < z )
47	46	ex 115	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ x e. RR ) -> ( x < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) -> E. z e. { A , B } x < z ) )
48	47	ralrimiva 2570	. 2 |- ( ( A e. RR /\ B e. RR ) -> A. x e. RR ( x < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) -> E. z e. { A , B } x < z ) )
49	9, 30, 48	3jca 1179	1 |- ( ( A e. RR /\ B e. RR ) -> ( ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) e. RR /\ A. x e. { A , B } -. ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) < x /\ A. x e. RR ( x < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) -> E. z e. { A , B } x < z ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 709   /\ w3a 980   = wceq 1364   e. wcel 2167   A.wral 2475   E.wrex 2476   {cpr 3624   class class class wbr 4034   ` cfv 5259  (class class class)co 5925   RRcr 7895   + caddc 7899   < clt 8078   <_ cle 8079   - cmin 8214   / cdiv 8716   2c2 9058   abscabs 11179

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-mulrcl 7995  ax-addcom 7996  ax-mulcom 7997  ax-addass 7998  ax-mulass 7999  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-1rid 8003  ax-0id 8004  ax-rnegex 8005  ax-precex 8006  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-apti 8011  ax-pre-ltadd 8012  ax-pre-mulgt0 8013  ax-pre-mulext 8014  ax-arch 8015  ax-caucvg 8016

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-frec 6458  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-reap 8619  df-ap 8626  df-div 8717  df-inn 9008  df-2 9066  df-3 9067  df-4 9068  df-n0 9267  df-z 9344  df-uz 9619  df-rp 9746  df-seqfrec 10557  df-exp 10648  df-cj 11024  df-re 11025  df-im 11026  df-rsqrt 11180  df-abs 11181

This theorem is referenced by:  maxabs  11391  maxleast  11395