Theorem maxabslemval 11185

Description: Lemma for maxabs 11186. 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 7912	. . . 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 7960	. . . . . 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 7960	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> B e. CC )
6	3, 5	subcld 8242	. . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( A - B ) e. CC )
7	6	abscld 11158	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( abs ` ( A - B ) ) e. RR )
8	1, 7	readdcld 7961	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( ( A + B ) + ( abs ` ( A - B ) ) ) e. RR )
9	8	rehalfcld 9138	. 2 |- ( ( A e. RR /\ B e. RR ) -> ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) e. RR )
10		vex 2738	. . . . 5 |- x e. _V
11	10	elpr 3610	. . . 4 |- ( x e. { A , B } <-> ( x = A \/ x = B ) )
12		maxabsle 11181	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> A <_ ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) )
13	2, 9, 12	lensymd 8053	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> -. ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) < A )
14		breq2 4002	. . . . . . 7 |- ( x = A -> ( ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) < x <-> ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) < A ) )
15	14	notbid 667	. . . . . 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 11181	. . . . . . . . 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 8082	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR ) -> ( B + A ) = ( A + B ) )
20	5, 3	abssubd 11170	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR ) -> ( abs ` ( B - A ) ) = ( abs ` ( A - B ) ) )
21	19, 20	oveq12d 5883	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR ) -> ( ( B + A ) + ( abs ` ( B - A ) ) ) = ( ( A + B ) + ( abs ` ( A - B ) ) ) )
22	21	oveq1d 5880	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR ) -> ( ( ( B + A ) + ( abs ` ( B - A ) ) ) / 2 ) = ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) )
23	18, 22	breqtrd 4024	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> B <_ ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) )
24	4, 9, 23	lensymd 8053	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> -. ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) < B )
25		breq2 4002	. . . . . . 7 |- ( x = B -> ( ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) < x <-> ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) < B ) )
26	25	notbid 667	. . . . . 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 717	. . . 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 2547	. 2 |- ( ( A e. RR /\ B e. RR ) -> A. x e. { A , B } -. ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) < x )
31		prid1g 3693	. . . . . . 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 4002	. . . . . . 7 |- ( z = A -> ( x < z <-> x < A ) )
34	33	rspcev 2839	. . . . . 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 3694	. . . . . . 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 4002	. . . . . . 7 |- ( z = B -> ( x < z <-> x < B ) )
39	38	rspcev 2839	. . . . . 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 11184	. . . . 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 798	. . . 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 2548	. 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 1177	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 708   /\ w3a 978   = wceq 1353   e. wcel 2146   A.wral 2453   E.wrex 2454   {cpr 3590   class class class wbr 3998   ` cfv 5208  (class class class)co 5865   RRcr 7785   + caddc 7789   < clt 7966   <_ cle 7967   - cmin 8102   / cdiv 8602   2c2 8943   abscabs 10974

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 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-iinf 4581  ax-cnex 7877  ax-resscn 7878  ax-1cn 7879  ax-1re 7880  ax-icn 7881  ax-addcl 7882  ax-addrcl 7883  ax-mulcl 7884  ax-mulrcl 7885  ax-addcom 7886  ax-mulcom 7887  ax-addass 7888  ax-mulass 7889  ax-distr 7890  ax-i2m1 7891  ax-0lt1 7892  ax-1rid 7893  ax-0id 7894  ax-rnegex 7895  ax-precex 7896  ax-cnre 7897  ax-pre-ltirr 7898  ax-pre-ltwlin 7899  ax-pre-lttrn 7900  ax-pre-apti 7901  ax-pre-ltadd 7902  ax-pre-mulgt0 7903  ax-pre-mulext 7904  ax-arch 7905  ax-caucvg 7906

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-nel 2441  df-ral 2458  df-rex 2459  df-reu 2460  df-rmo 2461  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-if 3533  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-id 4287  df-po 4290  df-iso 4291  df-iord 4360  df-on 4362  df-ilim 4363  df-suc 4365  df-iom 4584  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-riota 5821  df-ov 5868  df-oprab 5869  df-mpo 5870  df-1st 6131  df-2nd 6132  df-recs 6296  df-frec 6382  df-pnf 7968  df-mnf 7969  df-xr 7970  df-ltxr 7971  df-le 7972  df-sub 8104  df-neg 8105  df-reap 8506  df-ap 8513  df-div 8603  df-inn 8893  df-2 8951  df-3 8952  df-4 8953  df-n0 9150  df-z 9227  df-uz 9502  df-rp 9625  df-seqfrec 10416  df-exp 10490  df-cj 10819  df-re 10820  df-im 10821  df-rsqrt 10975  df-abs 10976

This theorem is referenced by:  maxabs  11186  maxleast  11190