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Theorem maxabslemlub 11920

Description: Lemma for maxabs 11922. 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 8319	. . . . . 6 ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℝ)
6	3	recnd 8318	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℂ)
7	4	recnd 8318	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℂ)
8	6, 7	subcld 8601	. . . . . . 7 ⊢ (𝜑 → (𝐴 − 𝐵) ∈ ℂ)
9	8	abscld 11894	. . . . . 6 ⊢ (𝜑 → (abs‘(𝐴 − 𝐵)) ∈ ℝ)
10	5, 9	readdcld 8319	. . . . 5 ⊢ (𝜑 → ((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) ∈ ℝ)
11	10	rehalfcld 9505	. . . 4 ⊢ (𝜑 → (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2) ∈ ℝ)
12		axltwlin 8357	. . . 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 1274	. . 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 8672	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 < (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2)) → (𝐴 − 𝐵) ∈ ℝ)
19		2re 9327	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℝ
20	19	a1i 9	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐴 < (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2)) → 2 ∈ ℝ)
21	20, 16	remulcld 8320	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐴 < (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2)) → (2 · 𝐴) ∈ ℝ)
22	21	recnd 8318	. . . . . . . . . . 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 8632	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 < (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2)) → (((2 · 𝐴) − 𝐴) − 𝐵) = ((2 · 𝐴) − (𝐴 + 𝐵)))
26		2cnd 9330	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐴 < (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2)) → 2 ∈ ℂ)
27	26, 23	mulsubfacd 8710	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐴 < (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2)) → ((2 · 𝐴) − 𝐴) = ((2 − 1) · 𝐴))
28		2m1e1 9375	. . . . . . . . . . . . . 14 ⊢ (2 − 1) = 1
29	28	oveq1i 6068	. . . . . . . . . . . . 13 ⊢ ((2 − 1) · 𝐴) = (1 · 𝐴)
30	27, 29	eqtrdi 2283	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐴 < (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2)) → ((2 · 𝐴) − 𝐴) = (1 · 𝐴))
31	23	mullidd 8308	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐴 < (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2)) → (1 · 𝐴) = 𝐴)
32	30, 31	eqtrd 2267	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐴 < (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2)) → ((2 · 𝐴) − 𝐴) = 𝐴)
33	32	oveq1d 6073	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 < (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2)) → (((2 · 𝐴) − 𝐴) − 𝐵) = (𝐴 − 𝐵))
34	25, 33	eqtr3d 2269	. . . . . . . . 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 10012	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℝ⁺
38	37	a1i 9	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐴 < (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2)) → 2 ∈ ℝ⁺)
39	16, 36, 38	ltmuldiv2d 10099	. . . . . . . . . . 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 8835	. . . . . . . . . 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 4138	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 < (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2)) → (𝐴 − 𝐵) < (abs‘(𝐴 − 𝐵)))
46		ltabs 11800	. . . . . . . 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 8862	. . . . . . 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 11919	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 < (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2)) → (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2) = 𝐵)
51	15, 50	breqtrd 4140	. . . 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 796	. 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 716 ∈ wcel 2205 class class class wbr 4114 ‘cfv 5357 (class class class)co 6058 ℂcc 8141 ℝcr 8142 0cc0 8143 1c1 8144 + caddc 8146 · cmul 8148 < clt 8324 − cmin 8461 / cdiv 8966 2c2 9308 ℝ⁺crp 10007 abscabs 11710

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 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2207 ax-14 2208 ax-ext 2216 ax-coll 4230 ax-sep 4233 ax-nul 4241 ax-pow 4292 ax-pr 4327 ax-un 4559 ax-setind 4664 ax-iinf 4715 ax-cnex 8234 ax-resscn 8235 ax-1cn 8236 ax-1re 8237 ax-icn 8238 ax-addcl 8239 ax-addrcl 8240 ax-mulcl 8241 ax-mulrcl 8242 ax-addcom 8243 ax-mulcom 8244 ax-addass 8245 ax-mulass 8246 ax-distr 8247 ax-i2m1 8248 ax-0lt1 8249 ax-1rid 8250 ax-0id 8251 ax-rnegex 8252 ax-precex 8253 ax-cnre 8254 ax-pre-ltirr 8255 ax-pre-ltwlin 8256 ax-pre-lttrn 8257 ax-pre-apti 8258 ax-pre-ltadd 8259 ax-pre-mulgt0 8260 ax-pre-mulext 8261 ax-arch 8262 ax-caucvg 8263

This theorem depends on definitions: df-bi 117 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2085 df-mo 2086 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ne 2415 df-nel 2510 df-ral 2527 df-rex 2528 df-reu 2529 df-rmo 2530 df-rab 2531 df-v 2817 df-sbc 3046 df-csb 3142 df-dif 3216 df-un 3218 df-in 3220 df-ss 3227 df-nul 3513 df-if 3625 df-pw 3676 df-sn 3700 df-pr 3701 df-op 3703 df-uni 3920 df-int 3955 df-iun 3998 df-br 4115 df-opab 4177 df-mpt 4178 df-tr 4214 df-id 4419 df-po 4422 df-iso 4423 df-iord 4492 df-on 4494 df-ilim 4495 df-suc 4497 df-iom 4718 df-xp 4760 df-rel 4761 df-cnv 4762 df-co 4763 df-dm 4764 df-rn 4765 df-res 4766 df-ima 4767 df-iota 5317 df-fun 5359 df-fn 5360 df-f 5361 df-f1 5362 df-fo 5363 df-f1o 5364 df-fv 5365 df-riota 6011 df-ov 6061 df-oprab 6062 df-mpo 6063 df-1st 6347 df-2nd 6348 df-recs 6549 df-frec 6635 df-pnf 8326 df-mnf 8327 df-xr 8328 df-ltxr 8329 df-le 8330 df-sub 8463 df-neg 8464 df-reap 8867 df-ap 8874 df-div 8967 df-inn 9258 df-2 9316 df-3 9317 df-4 9318 df-n0 9517 df-z 9598 df-uz 9875 df-rp 10008 df-seqfrec 10837 df-exp 10928 df-cj 11555 df-re 11556 df-im 11557 df-rsqrt 11711 df-abs 11712

This theorem is referenced by: maxabslemval 11921 maxleastlt 11928

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