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Theorem maxabslemlub 11997

Description: Lemma for maxabs 11999. 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 8362	. . . . . 6 ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℝ)
6	3	recnd 8361	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℂ)
7	4	recnd 8361	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℂ)
8	6, 7	subcld 8645	. . . . . . 7 ⊢ (𝜑 → (𝐴 − 𝐵) ∈ ℂ)
9	8	abscld 11971	. . . . . 6 ⊢ (𝜑 → (abs‘(𝐴 − 𝐵)) ∈ ℝ)
10	5, 9	readdcld 8362	. . . . 5 ⊢ (𝜑 → ((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) ∈ ℝ)
11	10	rehalfcld 9564	. . . 4 ⊢ (𝜑 → (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2) ∈ ℝ)
12		axltwlin 8400	. . . 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 1278	. . 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 8716	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 < (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2)) → (𝐴 − 𝐵) ∈ ℝ)
19		2re 9384	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℝ
20	19	a1i 9	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐴 < (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2)) → 2 ∈ ℝ)
21	20, 16	remulcld 8363	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐴 < (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2)) → (2 · 𝐴) ∈ ℝ)
22	21	recnd 8361	. . . . . . . . . . 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 8676	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 < (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2)) → (((2 · 𝐴) − 𝐴) − 𝐵) = ((2 · 𝐴) − (𝐴 + 𝐵)))
26		2cnd 9387	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐴 < (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2)) → 2 ∈ ℂ)
27	26, 23	mulsubfacd 8754	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐴 < (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2)) → ((2 · 𝐴) − 𝐴) = ((2 − 1) · 𝐴))
28		2m1e1 9432	. . . . . . . . . . . . . 14 ⊢ (2 − 1) = 1
29	28	oveq1i 6098	. . . . . . . . . . . . 13 ⊢ ((2 − 1) · 𝐴) = (1 · 𝐴)
30	27, 29	eqtrdi 2287	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐴 < (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2)) → ((2 · 𝐴) − 𝐴) = (1 · 𝐴))
31	23	mullidd 8351	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐴 < (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2)) → (1 · 𝐴) = 𝐴)
32	30, 31	eqtrd 2271	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐴 < (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2)) → ((2 · 𝐴) − 𝐴) = 𝐴)
33	32	oveq1d 6103	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 < (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2)) → (((2 · 𝐴) − 𝐴) − 𝐵) = (𝐴 − 𝐵))
34	25, 33	eqtr3d 2273	. . . . . . . . 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 10077	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℝ⁺
38	37	a1i 9	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐴 < (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2)) → 2 ∈ ℝ⁺)
39	16, 36, 38	ltmuldiv2d 10164	. . . . . . . . . . 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 8879	. . . . . . . . . 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 4155	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 < (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2)) → (𝐴 − 𝐵) < (abs‘(𝐴 − 𝐵)))
46		ltabs 11877	. . . . . . . 8 ⊢ (((𝐴 − 𝐵) ∈ ℝ ∧ (𝐴 − 𝐵) < (abs‘(𝐴 − 𝐵))) → (𝐴 − 𝐵) < 0)
47	18, 45, 46	syl2anc 415	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 < (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2)) → (𝐴 − 𝐵) < 0)
48	16, 17	sublt0d 8907	. . . . . . 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 11996	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 < (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2)) → (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2) = 𝐵)
51	15, 50	breqtrd 4157	. . . 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 800	. 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

This proof depends on syntax axioms: → wi 4 ∧ wa 104 ∨ wo 720 ∈ wcel 2209 class class class wbr 4131 ‘cfv 5380 (class class class)co 6088 ℂcc 8184 ℝcr 8185 0cc0 8186 1c1 8187 + caddc 8189 · cmul 8191 < clt 8367 − cmin 8505 / cdiv 9011 2c2 9365 ℝ⁺crp 10072 abscabs 11786

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 4248 ax-sep 4251 ax-nul 4261 ax-pow 4313 ax-pr 4348 ax-un 4580 ax-setind 4686 ax-iinf 4737 ax-cnex 8277 ax-resscn 8278 ax-1cn 8279 ax-1re 8280 ax-icn 8281 ax-addcl 8282 ax-addrcl 8283 ax-mulcl 8284 ax-mulrcl 8285 ax-addcom 8286 ax-mulcom 8287 ax-addass 8288 ax-mulass 8289 ax-distr 8290 ax-i2m1 8291 ax-0lt1 8292 ax-1rid 8293 ax-0id 8294 ax-rnegex 8295 ax-precex 8296 ax-cnre 8297 ax-pre-ltirr 8298 ax-pre-ltwlin 8299 ax-pre-lttrn 8300 ax-pre-apti 8301 ax-pre-ltadd 8302 ax-pre-mulgt0 8303 ax-pre-mulext 8304 ax-arch 8305 ax-caucvg 8306

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 4132 df-opab 4194 df-mpt 4195 df-tr 4232 df-id 4440 df-po 4443 df-iso 4444 df-iord 4513 df-on 4515 df-ilim 4516 df-suc 4518 df-iom 4740 df-xp 4782 df-rel 4783 df-cnv 4784 df-co 4785 df-dm 4786 df-rn 4787 df-res 4788 df-ima 4789 df-iota 5340 df-fun 5382 df-fn 5383 df-f 5384 df-f1 5385 df-fo 5386 df-f1o 5387 df-fv 5388 df-riota 6041 df-ov 6091 df-oprab 6092 df-mpo 6093 df-1st 6378 df-2nd 6379 df-recs 6580 df-frec 6666 df-pnf 8369 df-mnf 8370 df-xr 8371 df-ltxr 8372 df-le 8373 df-sub 8507 df-neg 8508 df-reap 8912 df-ap 8919 df-div 9012 df-inn 9315 df-2 9373 df-3 9374 df-4 9375 df-n0 9576 df-z 9657 df-uz 9939 df-rp 10073 df-seqfrec 10907 df-exp 10998 df-cj 11630 df-re 11631 df-im 11632 df-rsqrt 11787 df-abs 11788

This theorem is used by: maxabslemval 11998 maxleastlt 12005

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