Theorem maxabslemlub 11351

Description: Lemma for maxabs 11353. A least upper bound for { A , B }. (Contributed by Jim Kingdon, 20-Dec-2021.)

Hypotheses

Ref	Expression
maxabslemlub.a	|- ( ph -> A e. RR )
maxabslemlub.b	|- ( ph -> B e. RR )
maxabslemlub.c	|- ( ph -> C e. RR )
maxabslemlub.clt	|- ( ph -> C < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) )

Assertion

Ref	Expression
maxabslemlub	|- ( ph -> ( C < A \/ C < B ) )

Proof of Theorem maxabslemlub

Step	Hyp	Ref	Expression
1		maxabslemlub.clt	. . 3 |- ( ph -> C < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) )
2		maxabslemlub.c	. . . 4 |- ( ph -> C e. RR )
3		maxabslemlub.a	. . . . . . 7 |- ( ph -> A e. RR )
4		maxabslemlub.b	. . . . . . 7 |- ( ph -> B e. RR )
5	3, 4	readdcld 8049	. . . . . 6 |- ( ph -> ( A + B ) e. RR )
6	3	recnd 8048	. . . . . . . 8 |- ( ph -> A e. CC )
7	4	recnd 8048	. . . . . . . 8 |- ( ph -> B e. CC )
8	6, 7	subcld 8330	. . . . . . 7 |- ( ph -> ( A - B ) e. CC )
9	8	abscld 11325	. . . . . 6 |- ( ph -> ( abs ` ( A - B ) ) e. RR )
10	5, 9	readdcld 8049	. . . . 5 |- ( ph -> ( ( A + B ) + ( abs ` ( A - B ) ) ) e. RR )
11	10	rehalfcld 9229	. . . 4 |- ( ph -> ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) e. RR )
12		axltwlin 8087	. . . 4 |- ( ( C e. RR /\ ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) e. RR /\ A e. RR ) -> ( C < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) -> ( C < A \/ A < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) ) )
13	2, 11, 3, 12	syl3anc 1249	. . 3 |- ( ph -> ( C < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) -> ( C < A \/ A < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) ) )
14	1, 13	mpd 13	. 2 |- ( ph -> ( C < A \/ A < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) )
15	1	adantr 276	. . . . 5 |- ( ( ph /\ A < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) -> C < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) )
16	3	adantr 276	. . . . . 6 |- ( ( ph /\ A < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) -> A e. RR )
17	4	adantr 276	. . . . . 6 |- ( ( ph /\ A < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) -> B e. RR )
18	16, 17	resubcld 8400	. . . . . . . 8 |- ( ( ph /\ A < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) -> ( A - B ) e. RR )
19		2re 9052	. . . . . . . . . . . . . 14 |- 2 e. RR
20	19	a1i 9	. . . . . . . . . . . . 13 |- ( ( ph /\ A < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) -> 2 e. RR )
21	20, 16	remulcld 8050	. . . . . . . . . . . 12 |- ( ( ph /\ A < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) -> ( 2 x. A ) e. RR )
22	21	recnd 8048	. . . . . . . . . . 11 |- ( ( ph /\ A < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) -> ( 2 x. A ) e. CC )
23	6	adantr 276	. . . . . . . . . . 11 |- ( ( ph /\ A < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) -> A e. CC )
24	7	adantr 276	. . . . . . . . . . 11 |- ( ( ph /\ A < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) -> B e. CC )
25	22, 23, 24	subsub4d 8361	. . . . . . . . . 10 |- ( ( ph /\ A < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) -> ( ( ( 2 x. A ) - A ) - B ) = ( ( 2 x. A ) - ( A + B ) ) )
26		2cnd 9055	. . . . . . . . . . . . . 14 |- ( ( ph /\ A < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) -> 2 e. CC )
27	26, 23	mulsubfacd 8437	. . . . . . . . . . . . 13 |- ( ( ph /\ A < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) -> ( ( 2 x. A ) - A ) = ( ( 2 - 1 ) x. A ) )
28		2m1e1 9100	. . . . . . . . . . . . . 14 |- ( 2 - 1 ) = 1
29	28	oveq1i 5928	. . . . . . . . . . . . 13 |- ( ( 2 - 1 ) x. A ) = ( 1 x. A )
30	27, 29	eqtrdi 2242	. . . . . . . . . . . 12 |- ( ( ph /\ A < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) -> ( ( 2 x. A ) - A ) = ( 1 x. A ) )
31	23	mulid2d 8038	. . . . . . . . . . . 12 |- ( ( ph /\ A < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) -> ( 1 x. A ) = A )
32	30, 31	eqtrd 2226	. . . . . . . . . . 11 |- ( ( ph /\ A < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) -> ( ( 2 x. A ) - A ) = A )
33	32	oveq1d 5933	. . . . . . . . . 10 |- ( ( ph /\ A < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) -> ( ( ( 2 x. A ) - A ) - B ) = ( A - B ) )
34	25, 33	eqtr3d 2228	. . . . . . . . 9 |- ( ( ph /\ A < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) -> ( ( 2 x. A ) - ( A + B ) ) = ( A - B ) )
35		simpr 110	. . . . . . . . . . 11 |- ( ( ph /\ A < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) -> A < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) )
36	10	adantr 276	. . . . . . . . . . . 12 |- ( ( ph /\ A < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) -> ( ( A + B ) + ( abs ` ( A - B ) ) ) e. RR )
37		2rp 9724	. . . . . . . . . . . . 13 |- 2 e. RR+
38	37	a1i 9	. . . . . . . . . . . 12 |- ( ( ph /\ A < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) -> 2 e. RR+ )
39	16, 36, 38	ltmuldiv2d 9811	. . . . . . . . . . 11 |- ( ( ph /\ A < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) -> ( ( 2 x. A ) < ( ( A + B ) + ( abs ` ( A - B ) ) ) <-> A < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) )
40	35, 39	mpbird 167	. . . . . . . . . 10 |- ( ( ph /\ A < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) -> ( 2 x. A ) < ( ( A + B ) + ( abs ` ( A - B ) ) ) )
41	5	adantr 276	. . . . . . . . . . 11 |- ( ( ph /\ A < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) -> ( A + B ) e. RR )
42	9	adantr 276	. . . . . . . . . . 11 |- ( ( ph /\ A < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) -> ( abs ` ( A - B ) ) e. RR )
43	21, 41, 42	ltsubadd2d 8562	. . . . . . . . . 10 |- ( ( ph /\ A < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) -> ( ( ( 2 x. A ) - ( A + B ) ) < ( abs ` ( A - B ) ) <-> ( 2 x. A ) < ( ( A + B ) + ( abs ` ( A - B ) ) ) ) )
44	40, 43	mpbird 167	. . . . . . . . 9 |- ( ( ph /\ A < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) -> ( ( 2 x. A ) - ( A + B ) ) < ( abs ` ( A - B ) ) )
45	34, 44	eqbrtrrd 4053	. . . . . . . 8 |- ( ( ph /\ A < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) -> ( A - B ) < ( abs ` ( A - B ) ) )
46		ltabs 11231	. . . . . . . 8 |- ( ( ( A - B ) e. RR /\ ( A - B ) < ( abs ` ( A - B ) ) ) -> ( A - B ) < 0 )
47	18, 45, 46	syl2anc 411	. . . . . . 7 |- ( ( ph /\ A < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) -> ( A - B ) < 0 )
48	16, 17	sublt0d 8589	. . . . . . 7 |- ( ( ph /\ A < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) -> ( ( A - B ) < 0 <-> A < B ) )
49	47, 48	mpbid 147	. . . . . 6 |- ( ( ph /\ A < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) -> A < B )
50	16, 17, 49	maxabslemab 11350	. . . . 5 |- ( ( ph /\ A < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) -> ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) = B )
51	15, 50	breqtrd 4055	. . . 4 |- ( ( ph /\ A < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) -> C < B )
52	51	ex 115	. . 3 |- ( ph -> ( A < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) -> C < B ) )
53	52	orim2d 789	. 2 |- ( ph -> ( ( C < A \/ A < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) -> ( C < A \/ C < B ) ) )
54	14, 53	mpd 13	1 |- ( ph -> ( C < A \/ C < B ) )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 709   e. wcel 2164   class class class wbr 4029   ` cfv 5254  (class class class)co 5918   CCcc 7870   RRcr 7871   0cc0 7872   1c1 7873   + caddc 7875   x. cmul 7877   < clt 8054   - cmin 8190   / cdiv 8691   2c2 9033   RR+crp 9719   abscabs 11141

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-mulrcl 7971  ax-addcom 7972  ax-mulcom 7973  ax-addass 7974  ax-mulass 7975  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-1rid 7979  ax-0id 7980  ax-rnegex 7981  ax-precex 7982  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988  ax-pre-mulgt0 7989  ax-pre-mulext 7990  ax-arch 7991  ax-caucvg 7992

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 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-ilim 4400  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-frec 6444  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-reap 8594  df-ap 8601  df-div 8692  df-inn 8983  df-2 9041  df-3 9042  df-4 9043  df-n0 9241  df-z 9318  df-uz 9593  df-rp 9720  df-seqfrec 10519  df-exp 10610  df-cj 10986  df-re 10987  df-im 10988  df-rsqrt 11142  df-abs 11143

This theorem is referenced by:  maxabslemval  11352  maxleastlt  11359