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Theorem maxabslemlub 11389

Description: Lemma for maxabs 11391. A least upper bound for {𝐴, 𝐵}. (Contributed by Jim Kingdon, 20-Dec-2021.)

**Hypotheses**
Ref	Expression
maxabslemlub.a	⊢ (𝜑 → 𝐴 ∈ ℝ)
maxabslemlub.b	⊢ (𝜑 → 𝐵 ∈ ℝ)
maxabslemlub.c	⊢ (𝜑 → 𝐶 ∈ ℝ)
maxabslemlub.clt	⊢ (𝜑 → 𝐶 < (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2))

**Assertion**
Ref	Expression
maxabslemlub	⊢ (𝜑 → (𝐶 < 𝐴 ∨ 𝐶 < 𝐵))

**Proof of Theorem maxabslemlub**
Step	Hyp	Ref	Expression
1		maxabslemlub.clt	. . 3 ⊢ (𝜑 → 𝐶 < (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2))
2		maxabslemlub.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ ℝ)
3		maxabslemlub.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ)
4		maxabslemlub.b	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ)
5	3, 4	readdcld 8073	. . . . . 6 ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℝ)
6	3	recnd 8072	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℂ)
7	4	recnd 8072	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℂ)
8	6, 7	subcld 8354	. . . . . . 7 ⊢ (𝜑 → (𝐴 − 𝐵) ∈ ℂ)
9	8	abscld 11363	. . . . . 6 ⊢ (𝜑 → (abs‘(𝐴 − 𝐵)) ∈ ℝ)
10	5, 9	readdcld 8073	. . . . 5 ⊢ (𝜑 → ((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) ∈ ℝ)
11	10	rehalfcld 9255	. . . 4 ⊢ (𝜑 → (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2) ∈ ℝ)
12		axltwlin 8111	. . . 4 ⊢ ((𝐶 ∈ ℝ ∧ (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2) ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝐶 < (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2) → (𝐶 < 𝐴 ∨ 𝐴 < (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2))))
13	2, 11, 3, 12	syl3anc 1249	. . 3 ⊢ (𝜑 → (𝐶 < (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2) → (𝐶 < 𝐴 ∨ 𝐴 < (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2))))
14	1, 13	mpd 13	. 2 ⊢ (𝜑 → (𝐶 < 𝐴 ∨ 𝐴 < (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2)))
15	1	adantr 276	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 < (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2)) → 𝐶 < (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2))
16	3	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 < (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2)) → 𝐴 ∈ ℝ)
17	4	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 < (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2)) → 𝐵 ∈ ℝ)
18	16, 17	resubcld 8424	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 < (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2)) → (𝐴 − 𝐵) ∈ ℝ)
19		2re 9077	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℝ
20	19	a1i 9	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐴 < (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2)) → 2 ∈ ℝ)
21	20, 16	remulcld 8074	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐴 < (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2)) → (2 · 𝐴) ∈ ℝ)
22	21	recnd 8072	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐴 < (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2)) → (2 · 𝐴) ∈ ℂ)
23	6	adantr 276	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐴 < (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2)) → 𝐴 ∈ ℂ)
24	7	adantr 276	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐴 < (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2)) → 𝐵 ∈ ℂ)
25	22, 23, 24	subsub4d 8385	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 < (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2)) → (((2 · 𝐴) − 𝐴) − 𝐵) = ((2 · 𝐴) − (𝐴 + 𝐵)))
26		2cnd 9080	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐴 < (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2)) → 2 ∈ ℂ)
27	26, 23	mulsubfacd 8462	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐴 < (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2)) → ((2 · 𝐴) − 𝐴) = ((2 − 1) · 𝐴))
28		2m1e1 9125	. . . . . . . . . . . . . 14 ⊢ (2 − 1) = 1
29	28	oveq1i 5935	. . . . . . . . . . . . 13 ⊢ ((2 − 1) · 𝐴) = (1 · 𝐴)
30	27, 29	eqtrdi 2245	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐴 < (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2)) → ((2 · 𝐴) − 𝐴) = (1 · 𝐴))
31	23	mulid2d 8062	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐴 < (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2)) → (1 · 𝐴) = 𝐴)
32	30, 31	eqtrd 2229	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐴 < (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2)) → ((2 · 𝐴) − 𝐴) = 𝐴)
33	32	oveq1d 5940	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 < (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2)) → (((2 · 𝐴) − 𝐴) − 𝐵) = (𝐴 − 𝐵))
34	25, 33	eqtr3d 2231	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 < (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2)) → ((2 · 𝐴) − (𝐴 + 𝐵)) = (𝐴 − 𝐵))
35		simpr 110	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐴 < (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2)) → 𝐴 < (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2))
36	10	adantr 276	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐴 < (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2)) → ((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) ∈ ℝ)
37		2rp 9750	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℝ⁺
38	37	a1i 9	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐴 < (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2)) → 2 ∈ ℝ⁺)
39	16, 36, 38	ltmuldiv2d 9837	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐴 < (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2)) → ((2 · 𝐴) < ((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) ↔ 𝐴 < (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2)))
40	35, 39	mpbird 167	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 < (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2)) → (2 · 𝐴) < ((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))))
41	5	adantr 276	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐴 < (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2)) → (𝐴 + 𝐵) ∈ ℝ)
42	9	adantr 276	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐴 < (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2)) → (abs‘(𝐴 − 𝐵)) ∈ ℝ)
43	21, 41, 42	ltsubadd2d 8587	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 < (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2)) → (((2 · 𝐴) − (𝐴 + 𝐵)) < (abs‘(𝐴 − 𝐵)) ↔ (2 · 𝐴) < ((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵)))))
44	40, 43	mpbird 167	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 < (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2)) → ((2 · 𝐴) − (𝐴 + 𝐵)) < (abs‘(𝐴 − 𝐵)))
45	34, 44	eqbrtrrd 4058	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 < (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2)) → (𝐴 − 𝐵) < (abs‘(𝐴 − 𝐵)))
46		ltabs 11269	. . . . . . . 8 ⊢ (((𝐴 − 𝐵) ∈ ℝ ∧ (𝐴 − 𝐵) < (abs‘(𝐴 − 𝐵))) → (𝐴 − 𝐵) < 0)
47	18, 45, 46	syl2anc 411	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 < (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2)) → (𝐴 − 𝐵) < 0)
48	16, 17	sublt0d 8614	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 < (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2)) → ((𝐴 − 𝐵) < 0 ↔ 𝐴 < 𝐵))
49	47, 48	mpbid 147	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 < (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2)) → 𝐴 < 𝐵)
50	16, 17, 49	maxabslemab 11388	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 < (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2)) → (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2) = 𝐵)
51	15, 50	breqtrd 4060	. . . 4 ⊢ ((𝜑 ∧ 𝐴 < (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2)) → 𝐶 < 𝐵)
52	51	ex 115	. . 3 ⊢ (𝜑 → (𝐴 < (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2) → 𝐶 < 𝐵))
53	52	orim2d 789	. 2 ⊢ (𝜑 → ((𝐶 < 𝐴 ∨ 𝐴 < (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2)) → (𝐶 < 𝐴 ∨ 𝐶 < 𝐵)))
54	14, 53	mpd 13	1 ⊢ (𝜑 → (𝐶 < 𝐴 ∨ 𝐶 < 𝐵))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ∨ wo 709 ∈ wcel 2167 class class class wbr 4034 ‘cfv 5259 (class class class)co 5925 ℂcc 7894 ℝcr 7895 0cc0 7896 1c1 7897 + caddc 7899 · cmul 7901 < clt 8078 − cmin 8214 / cdiv 8716 2c2 9058 ℝ⁺crp 9745 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: maxabslemval 11390 maxleastlt 11397

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