Theorem maxabslemlub 11973

Description: Lemma for maxabs 11975. 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 8355	. . . . . 6 |- ( ph -> ( A + B ) e. RR )
6	3	recnd 8354	. . . . . . . 8 |- ( ph -> A e. CC )
7	4	recnd 8354	. . . . . . . 8 |- ( ph -> B e. CC )
8	6, 7	subcld 8637	. . . . . . 7 |- ( ph -> ( A - B ) e. CC )
9	8	abscld 11947	. . . . . 6 |- ( ph -> ( abs ` ( A - B ) ) e. RR )
10	5, 9	readdcld 8355	. . . . 5 |- ( ph -> ( ( A + B ) + ( abs ` ( A - B ) ) ) e. RR )
11	10	rehalfcld 9552	. . . 4 |- ( ph -> ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) e. RR )
12		axltwlin 8393	. . . 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 1278	. . 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 8708	. . . . . . . 8 |- ( ( ph /\ A < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) -> ( A - B ) e. RR )
19		2re 9374	. . . . . . . . . . . . . 14 |- 2 e. RR
20	19	a1i 9	. . . . . . . . . . . . 13 |- ( ( ph /\ A < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) -> 2 e. RR )
21	20, 16	remulcld 8356	. . . . . . . . . . . 12 |- ( ( ph /\ A < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) -> ( 2 x. A ) e. RR )
22	21	recnd 8354	. . . . . . . . . . 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 8668	. . . . . . . . . 10 |- ( ( ph /\ A < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) -> ( ( ( 2 x. A ) - A ) - B ) = ( ( 2 x. A ) - ( A + B ) ) )
26		2cnd 9377	. . . . . . . . . . . . . 14 |- ( ( ph /\ A < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) -> 2 e. CC )
27	26, 23	mulsubfacd 8746	. . . . . . . . . . . . 13 |- ( ( ph /\ A < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) -> ( ( 2 x. A ) - A ) = ( ( 2 - 1 ) x. A ) )
28		2m1e1 9422	. . . . . . . . . . . . . 14 |- ( 2 - 1 ) = 1
29	28	oveq1i 6095	. . . . . . . . . . . . 13 |- ( ( 2 - 1 ) x. A ) = ( 1 x. A )
30	27, 29	eqtrdi 2287	. . . . . . . . . . . 12 |- ( ( ph /\ A < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) -> ( ( 2 x. A ) - A ) = ( 1 x. A ) )
31	23	mullidd 8344	. . . . . . . . . . . 12 |- ( ( ph /\ A < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) -> ( 1 x. A ) = A )
32	30, 31	eqtrd 2271	. . . . . . . . . . 11 |- ( ( ph /\ A < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) -> ( ( 2 x. A ) - A ) = A )
33	32	oveq1d 6100	. . . . . . . . . 10 |- ( ( ph /\ A < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) -> ( ( ( 2 x. A ) - A ) - B ) = ( A - B ) )
34	25, 33	eqtr3d 2273	. . . . . . . . 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 10059	. . . . . . . . . . . . 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 10146	. . . . . . . . . . 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 8871	. . . . . . . . . 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 4154	. . . . . . . 8 |- ( ( ph /\ A < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) -> ( A - B ) < ( abs ` ( A - B ) ) )
46		ltabs 11853	. . . . . . . 8 |- ( ( ( A - B ) e. RR /\ ( A - B ) < ( abs ` ( A - B ) ) ) -> ( A - B ) < 0 )
47	18, 45, 46	syl2anc 415	. . . . . . 7 |- ( ( ph /\ A < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) -> ( A - B ) < 0 )
48	16, 17	sublt0d 8898	. . . . . . 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 11972	. . . . 5 |- ( ( ph /\ A < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) -> ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) = B )
51	15, 50	breqtrd 4156	. . . 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 800	. 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 ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   \/ wo 720   e. wcel 2209   class class class wbr 4130   ` cfv 5377  (class class class)co 6085   CCcc 8177   RRcr 8178   0cc0 8179   1c1 8180   + caddc 8182   x. cmul 8184   < clt 8360   - cmin 8497   / cdiv 9002   2c2 9355   RR+crp 10054   abscabs 11763

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297  ax-arch 8298  ax-caucvg 8299

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8903  df-ap 8910  df-div 9003  df-inn 9305  df-2 9363  df-3 9364  df-4 9365  df-n0 9564  df-z 9645  df-uz 9922  df-rp 10055  df-seqfrec 10885  df-exp 10976  df-cj 11607  df-re 11608  df-im 11609  df-rsqrt 11764  df-abs 11765

This theorem is used by:  maxabslemval  11974  maxleastlt  11981