Theorem maxabslemlub 11460

Description: Lemma for maxabs 11462. 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 8101	. . . . . 6 |- ( ph -> ( A + B ) e. RR )
6	3	recnd 8100	. . . . . . . 8 |- ( ph -> A e. CC )
7	4	recnd 8100	. . . . . . . 8 |- ( ph -> B e. CC )
8	6, 7	subcld 8382	. . . . . . 7 |- ( ph -> ( A - B ) e. CC )
9	8	abscld 11434	. . . . . 6 |- ( ph -> ( abs ` ( A - B ) ) e. RR )
10	5, 9	readdcld 8101	. . . . 5 |- ( ph -> ( ( A + B ) + ( abs ` ( A - B ) ) ) e. RR )
11	10	rehalfcld 9283	. . . 4 |- ( ph -> ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) e. RR )
12		axltwlin 8139	. . . 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 8452	. . . . . . . 8 |- ( ( ph /\ A < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) -> ( A - B ) e. RR )
19		2re 9105	. . . . . . . . . . . . . 14 |- 2 e. RR
20	19	a1i 9	. . . . . . . . . . . . 13 |- ( ( ph /\ A < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) -> 2 e. RR )
21	20, 16	remulcld 8102	. . . . . . . . . . . 12 |- ( ( ph /\ A < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) -> ( 2 x. A ) e. RR )
22	21	recnd 8100	. . . . . . . . . . 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 8413	. . . . . . . . . 10 |- ( ( ph /\ A < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) -> ( ( ( 2 x. A ) - A ) - B ) = ( ( 2 x. A ) - ( A + B ) ) )
26		2cnd 9108	. . . . . . . . . . . . . 14 |- ( ( ph /\ A < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) -> 2 e. CC )
27	26, 23	mulsubfacd 8490	. . . . . . . . . . . . 13 |- ( ( ph /\ A < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) -> ( ( 2 x. A ) - A ) = ( ( 2 - 1 ) x. A ) )
28		2m1e1 9153	. . . . . . . . . . . . . 14 |- ( 2 - 1 ) = 1
29	28	oveq1i 5953	. . . . . . . . . . . . 13 |- ( ( 2 - 1 ) x. A ) = ( 1 x. A )
30	27, 29	eqtrdi 2253	. . . . . . . . . . . 12 |- ( ( ph /\ A < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) -> ( ( 2 x. A ) - A ) = ( 1 x. A ) )
31	23	mulid2d 8090	. . . . . . . . . . . 12 |- ( ( ph /\ A < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) -> ( 1 x. A ) = A )
32	30, 31	eqtrd 2237	. . . . . . . . . . 11 |- ( ( ph /\ A < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) -> ( ( 2 x. A ) - A ) = A )
33	32	oveq1d 5958	. . . . . . . . . 10 |- ( ( ph /\ A < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) -> ( ( ( 2 x. A ) - A ) - B ) = ( A - B ) )
34	25, 33	eqtr3d 2239	. . . . . . . . 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 9779	. . . . . . . . . . . . 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 9866	. . . . . . . . . . 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 8615	. . . . . . . . . 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 4067	. . . . . . . 8 |- ( ( ph /\ A < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) -> ( A - B ) < ( abs ` ( A - B ) ) )
46		ltabs 11340	. . . . . . . 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 8642	. . . . . . 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 11459	. . . . 5 |- ( ( ph /\ A < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) -> ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) = B )
51	15, 50	breqtrd 4069	. . . 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 2175   class class class wbr 4043   ` cfv 5270  (class class class)co 5943   CCcc 7922   RRcr 7923   0cc0 7924   1c1 7925   + caddc 7927   x. cmul 7929   < clt 8106   - cmin 8242   / cdiv 8744   2c2 9086   RR+crp 9774   abscabs 11250

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-nul 4169  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-iinf 4635  ax-cnex 8015  ax-resscn 8016  ax-1cn 8017  ax-1re 8018  ax-icn 8019  ax-addcl 8020  ax-addrcl 8021  ax-mulcl 8022  ax-mulrcl 8023  ax-addcom 8024  ax-mulcom 8025  ax-addass 8026  ax-mulass 8027  ax-distr 8028  ax-i2m1 8029  ax-0lt1 8030  ax-1rid 8031  ax-0id 8032  ax-rnegex 8033  ax-precex 8034  ax-cnre 8035  ax-pre-ltirr 8036  ax-pre-ltwlin 8037  ax-pre-lttrn 8038  ax-pre-apti 8039  ax-pre-ltadd 8040  ax-pre-mulgt0 8041  ax-pre-mulext 8042  ax-arch 8043  ax-caucvg 8044

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-nel 2471  df-ral 2488  df-rex 2489  df-reu 2490  df-rmo 2491  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-if 3571  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-tr 4142  df-id 4339  df-po 4342  df-iso 4343  df-iord 4412  df-on 4414  df-ilim 4415  df-suc 4417  df-iom 4638  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-riota 5898  df-ov 5946  df-oprab 5947  df-mpo 5948  df-1st 6225  df-2nd 6226  df-recs 6390  df-frec 6476  df-pnf 8108  df-mnf 8109  df-xr 8110  df-ltxr 8111  df-le 8112  df-sub 8244  df-neg 8245  df-reap 8647  df-ap 8654  df-div 8745  df-inn 9036  df-2 9094  df-3 9095  df-4 9096  df-n0 9295  df-z 9372  df-uz 9648  df-rp 9775  df-seqfrec 10591  df-exp 10682  df-cj 11095  df-re 11096  df-im 11097  df-rsqrt 11251  df-abs 11252

This theorem is referenced by:  maxabslemval  11461  maxleastlt  11468