infleinflem1 - Mathbox for Glauco Siliprandi

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Theorem infleinflem1 45055

Description: Lemma for infleinf 45057, case 𝐵 ≠ ∅ ∧ -∞ < inf(𝐵, ℝ^*, < ). (Contributed by Glauco Siliprandi, 3-Mar-2021.)

**Hypotheses**
Ref	Expression
infleinflem1.a	⊢ (𝜑 → 𝐴 ⊆ ℝ^*)
infleinflem1.b	⊢ (𝜑 → 𝐵 ⊆ ℝ^*)
infleinflem1.w	⊢ (𝜑 → 𝑊 ∈ ℝ⁺)
infleinflem1.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
infleinflem1.i	⊢ (𝜑 → 𝑋 ≤ (inf(𝐵, ℝ^*, < ) +_𝑒 (𝑊 / 2)))
infleinflem1.z	⊢ (𝜑 → 𝑍 ∈ 𝐴)
infleinflem1.l	⊢ (𝜑 → 𝑍 ≤ (𝑋 +_𝑒 (𝑊 / 2)))

**Assertion**
Ref	Expression
infleinflem1	⊢ (𝜑 → inf(𝐴, ℝ^, < ) ≤ (inf(𝐵, ℝ^, < ) +_𝑒 𝑊))

**Proof of Theorem infleinflem1**
Step	Hyp	Ref	Expression
1		infleinflem1.a	. . . 4 ⊢ (𝜑 → 𝐴 ⊆ ℝ^*)
2		infxrcl 13376	. . . 4 ⊢ (𝐴 ⊆ ℝ^* → inf(𝐴, ℝ^, < ) ∈ ℝ^)
3	1, 2	syl 17	. . 3 ⊢ (𝜑 → inf(𝐴, ℝ^, < ) ∈ ℝ^)
4		id 22	. . 3 ⊢ (inf(𝐴, ℝ^, < ) ∈ ℝ^ → inf(𝐴, ℝ^, < ) ∈ ℝ^)
5	3, 4	syl 17	. 2 ⊢ (𝜑 → inf(𝐴, ℝ^, < ) ∈ ℝ^)
6		infleinflem1.z	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝐴)
7	1, 6	sseldd 3992	. 2 ⊢ (𝜑 → 𝑍 ∈ ℝ^*)
8		infleinflem1.b	. . . 4 ⊢ (𝜑 → 𝐵 ⊆ ℝ^*)
9		infxrcl 13376	. . . 4 ⊢ (𝐵 ⊆ ℝ^* → inf(𝐵, ℝ^, < ) ∈ ℝ^)
10	8, 9	syl 17	. . 3 ⊢ (𝜑 → inf(𝐵, ℝ^, < ) ∈ ℝ^)
11		infleinflem1.w	. . . 4 ⊢ (𝜑 → 𝑊 ∈ ℝ⁺)
12		rpxr 13047	. . . 4 ⊢ (𝑊 ∈ ℝ⁺ → 𝑊 ∈ ℝ^*)
13	11, 12	syl 17	. . 3 ⊢ (𝜑 → 𝑊 ∈ ℝ^*)
14	10, 13	xaddcld 13344	. 2 ⊢ (𝜑 → (inf(𝐵, ℝ^, < ) +_𝑒 𝑊) ∈ ℝ^)
15		infxrlb 13377	. . 3 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝑍 ∈ 𝐴) → inf(𝐴, ℝ^*, < ) ≤ 𝑍)
16	1, 6, 15	syl2anc 582	. 2 ⊢ (𝜑 → inf(𝐴, ℝ^*, < ) ≤ 𝑍)
17		infleinflem1.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
18	8	sselda 3991	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ ℝ^*)
19	17, 18	mpdan 685	. . . 4 ⊢ (𝜑 → 𝑋 ∈ ℝ^*)
20	11	rpred 13080	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ ℝ)
21	20	rehalfcld 12521	. . . . 5 ⊢ (𝜑 → (𝑊 / 2) ∈ ℝ)
22	21	rexrd 11321	. . . 4 ⊢ (𝜑 → (𝑊 / 2) ∈ ℝ^*)
23	19, 22	xaddcld 13344	. . 3 ⊢ (𝜑 → (𝑋 +_𝑒 (𝑊 / 2)) ∈ ℝ^*)
24		infleinflem1.l	. . 3 ⊢ (𝜑 → 𝑍 ≤ (𝑋 +_𝑒 (𝑊 / 2)))
25		pnfge 13174	. . . . . . 7 ⊢ ((𝑋 +_𝑒 (𝑊 / 2)) ∈ ℝ^* → (𝑋 +_𝑒 (𝑊 / 2)) ≤ +∞)
26	23, 25	syl 17	. . . . . 6 ⊢ (𝜑 → (𝑋 +_𝑒 (𝑊 / 2)) ≤ +∞)
27	26	adantr 479	. . . . 5 ⊢ ((𝜑 ∧ inf(𝐵, ℝ^*, < ) = +∞) → (𝑋 +_𝑒 (𝑊 / 2)) ≤ +∞)
28		oveq1 7437	. . . . . . 7 ⊢ (inf(𝐵, ℝ^, < ) = +∞ → (inf(𝐵, ℝ^, < ) +_𝑒 𝑊) = (+∞ +_𝑒 𝑊))
29	28	adantl 480	. . . . . 6 ⊢ ((𝜑 ∧ inf(𝐵, ℝ^, < ) = +∞) → (inf(𝐵, ℝ^, < ) +_𝑒 𝑊) = (+∞ +_𝑒 𝑊))
30		rpre 13046	. . . . . . . . . 10 ⊢ (𝑊 ∈ ℝ⁺ → 𝑊 ∈ ℝ)
31		renemnf 11320	. . . . . . . . . 10 ⊢ (𝑊 ∈ ℝ → 𝑊 ≠ -∞)
32	30, 31	syl 17	. . . . . . . . 9 ⊢ (𝑊 ∈ ℝ⁺ → 𝑊 ≠ -∞)
33		xaddpnf2 13270	. . . . . . . . 9 ⊢ ((𝑊 ∈ ℝ^* ∧ 𝑊 ≠ -∞) → (+∞ +_𝑒 𝑊) = +∞)
34	12, 32, 33	syl2anc 582	. . . . . . . 8 ⊢ (𝑊 ∈ ℝ⁺ → (+∞ +_𝑒 𝑊) = +∞)
35	11, 34	syl 17	. . . . . . 7 ⊢ (𝜑 → (+∞ +_𝑒 𝑊) = +∞)
36	35	adantr 479	. . . . . 6 ⊢ ((𝜑 ∧ inf(𝐵, ℝ^*, < ) = +∞) → (+∞ +_𝑒 𝑊) = +∞)
37	29, 36	eqtr2d 2770	. . . . 5 ⊢ ((𝜑 ∧ inf(𝐵, ℝ^, < ) = +∞) → +∞ = (inf(𝐵, ℝ^, < ) +_𝑒 𝑊))
38	27, 37	breqtrd 5182	. . . 4 ⊢ ((𝜑 ∧ inf(𝐵, ℝ^, < ) = +∞) → (𝑋 +_𝑒 (𝑊 / 2)) ≤ (inf(𝐵, ℝ^, < ) +_𝑒 𝑊))
39	8, 17	sseldd 3992	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ ℝ^*)
40	10, 22	xaddcld 13344	. . . . . . 7 ⊢ (𝜑 → (inf(𝐵, ℝ^, < ) +_𝑒 (𝑊 / 2)) ∈ ℝ^)
41		rphalfcl 13065	. . . . . . . . 9 ⊢ (𝑊 ∈ ℝ⁺ → (𝑊 / 2) ∈ ℝ⁺)
42	11, 41	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝑊 / 2) ∈ ℝ⁺)
43	42	rpxrd 13081	. . . . . . 7 ⊢ (𝜑 → (𝑊 / 2) ∈ ℝ^*)
44		infleinflem1.i	. . . . . . 7 ⊢ (𝜑 → 𝑋 ≤ (inf(𝐵, ℝ^*, < ) +_𝑒 (𝑊 / 2)))
45	39, 40, 43, 44	xleadd1d 45014	. . . . . 6 ⊢ (𝜑 → (𝑋 +_𝑒 (𝑊 / 2)) ≤ ((inf(𝐵, ℝ^*, < ) +_𝑒 (𝑊 / 2)) +_𝑒 (𝑊 / 2)))
46	45	adantr 479	. . . . 5 ⊢ ((𝜑 ∧ ¬ inf(𝐵, ℝ^, < ) = +∞) → (𝑋 +_𝑒 (𝑊 / 2)) ≤ ((inf(𝐵, ℝ^, < ) +_𝑒 (𝑊 / 2)) +_𝑒 (𝑊 / 2)))
47	10	adantr 479	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ inf(𝐵, ℝ^, < ) = +∞) → inf(𝐵, ℝ^, < ) ∈ ℝ^*)
48		neqne 2941	. . . . . . . 8 ⊢ (¬ inf(𝐵, ℝ^, < ) = +∞ → inf(𝐵, ℝ^, < ) ≠ +∞)
49	48	adantl 480	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ inf(𝐵, ℝ^, < ) = +∞) → inf(𝐵, ℝ^, < ) ≠ +∞)
50	43	adantr 479	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ inf(𝐵, ℝ^, < ) = +∞) → (𝑊 / 2) ∈ ℝ^)
51	11	adantr 479	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ inf(𝐵, ℝ^*, < ) = +∞) → 𝑊 ∈ ℝ⁺)
52		rpre 13046	. . . . . . . 8 ⊢ ((𝑊 / 2) ∈ ℝ⁺ → (𝑊 / 2) ∈ ℝ)
53		renepnf 11319	. . . . . . . 8 ⊢ ((𝑊 / 2) ∈ ℝ → (𝑊 / 2) ≠ +∞)
54	51, 41, 52, 53	4syl 19	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ inf(𝐵, ℝ^*, < ) = +∞) → (𝑊 / 2) ≠ +∞)
55		xaddass2 13293	. . . . . . 7 ⊢ (((inf(𝐵, ℝ^, < ) ∈ ℝ^ ∧ inf(𝐵, ℝ^, < ) ≠ +∞) ∧ ((𝑊 / 2) ∈ ℝ^ ∧ (𝑊 / 2) ≠ +∞) ∧ ((𝑊 / 2) ∈ ℝ^* ∧ (𝑊 / 2) ≠ +∞)) → ((inf(𝐵, ℝ^, < ) +_𝑒 (𝑊 / 2)) +_𝑒 (𝑊 / 2)) = (inf(𝐵, ℝ^, < ) +_𝑒 ((𝑊 / 2) +_𝑒 (𝑊 / 2))))
56	47, 49, 50, 54, 50, 54, 55	syl222anc 1383	. . . . . 6 ⊢ ((𝜑 ∧ ¬ inf(𝐵, ℝ^, < ) = +∞) → ((inf(𝐵, ℝ^, < ) +_𝑒 (𝑊 / 2)) +_𝑒 (𝑊 / 2)) = (inf(𝐵, ℝ^*, < ) +_𝑒 ((𝑊 / 2) +_𝑒 (𝑊 / 2))))
57		rehalfcl 12500	. . . . . . . . . 10 ⊢ (𝑊 ∈ ℝ → (𝑊 / 2) ∈ ℝ)
58	57, 57	rexaddd 13277	. . . . . . . . 9 ⊢ (𝑊 ∈ ℝ → ((𝑊 / 2) +_𝑒 (𝑊 / 2)) = ((𝑊 / 2) + (𝑊 / 2)))
59		recn 11255	. . . . . . . . . 10 ⊢ (𝑊 ∈ ℝ → 𝑊 ∈ ℂ)
60		2halves 12502	. . . . . . . . . 10 ⊢ (𝑊 ∈ ℂ → ((𝑊 / 2) + (𝑊 / 2)) = 𝑊)
61	59, 60	syl 17	. . . . . . . . 9 ⊢ (𝑊 ∈ ℝ → ((𝑊 / 2) + (𝑊 / 2)) = 𝑊)
62	58, 61	eqtrd 2769	. . . . . . . 8 ⊢ (𝑊 ∈ ℝ → ((𝑊 / 2) +_𝑒 (𝑊 / 2)) = 𝑊)
63	62	oveq2d 7446	. . . . . . 7 ⊢ (𝑊 ∈ ℝ → (inf(𝐵, ℝ^, < ) +_𝑒 ((𝑊 / 2) +_𝑒 (𝑊 / 2))) = (inf(𝐵, ℝ^, < ) +_𝑒 𝑊))
64	51, 30, 63	3syl 18	. . . . . 6 ⊢ ((𝜑 ∧ ¬ inf(𝐵, ℝ^, < ) = +∞) → (inf(𝐵, ℝ^, < ) +_𝑒 ((𝑊 / 2) +_𝑒 (𝑊 / 2))) = (inf(𝐵, ℝ^*, < ) +_𝑒 𝑊))
65	56, 64	eqtrd 2769	. . . . 5 ⊢ ((𝜑 ∧ ¬ inf(𝐵, ℝ^, < ) = +∞) → ((inf(𝐵, ℝ^, < ) +_𝑒 (𝑊 / 2)) +_𝑒 (𝑊 / 2)) = (inf(𝐵, ℝ^*, < ) +_𝑒 𝑊))
66	46, 65	breqtrd 5182	. . . 4 ⊢ ((𝜑 ∧ ¬ inf(𝐵, ℝ^, < ) = +∞) → (𝑋 +_𝑒 (𝑊 / 2)) ≤ (inf(𝐵, ℝ^, < ) +_𝑒 𝑊))
67	38, 66	pm2.61dan 811	. . 3 ⊢ (𝜑 → (𝑋 +_𝑒 (𝑊 / 2)) ≤ (inf(𝐵, ℝ^*, < ) +_𝑒 𝑊))
68	7, 23, 14, 24, 67	xrletrd 13205	. 2 ⊢ (𝜑 → 𝑍 ≤ (inf(𝐵, ℝ^*, < ) +_𝑒 𝑊))
69	5, 7, 14, 16, 68	xrletrd 13205	1 ⊢ (𝜑 → inf(𝐴, ℝ^, < ) ≤ (inf(𝐵, ℝ^, < ) +_𝑒 𝑊))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2100 ≠ wne 2933 ⊆ wss 3959 class class class wbr 5156 (class class class)co 7430 infcinf 9491 ℂcc 11163 ℝcr 11164 + caddc 11168 +∞cpnf 11302 -∞cmnf 11303 ℝ^*cxr 11304 < clt 11305 ≤ cle 11306 / cdiv 11928 2c2 12329 ℝ⁺crp 13038 +_𝑒 cxad 13154

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2102 ax-9 2110 ax-10 2133 ax-11 2150 ax-12 2170 ax-ext 2700 ax-sep 5307 ax-nul 5314 ax-pow 5373 ax-pr 5437 ax-un 7751 ax-cnex 11221 ax-resscn 11222 ax-1cn 11223 ax-icn 11224 ax-addcl 11225 ax-addrcl 11226 ax-mulcl 11227 ax-mulrcl 11228 ax-mulcom 11229 ax-addass 11230 ax-mulass 11231 ax-distr 11232 ax-i2m1 11233 ax-1ne0 11234 ax-1rid 11235 ax-rnegex 11236 ax-rrecex 11237 ax-cnre 11238 ax-pre-lttri 11239 ax-pre-lttrn 11240 ax-pre-ltadd 11241 ax-pre-mulgt0 11242 ax-pre-sup 11243

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2062 df-mo 2532 df-eu 2561 df-clab 2707 df-cleq 2721 df-clel 2806 df-nfc 2881 df-ne 2934 df-nel 3040 df-ral 3055 df-rex 3064 df-rmo 3373 df-reu 3374 df-rab 3429 df-v 3474 df-sbc 3789 df-csb 3905 df-dif 3962 df-un 3964 df-in 3966 df-ss 3976 df-nul 4336 df-if 4537 df-pw 4612 df-sn 4637 df-pr 4639 df-op 4643 df-uni 4919 df-iun 5008 df-br 5157 df-opab 5219 df-mpt 5240 df-id 5584 df-po 5598 df-so 5599 df-xp 5692 df-rel 5693 df-cnv 5694 df-co 5695 df-dm 5696 df-rn 5697 df-res 5698 df-ima 5699 df-iota 6510 df-fun 6560 df-fn 6561 df-f 6562 df-f1 6563 df-fo 6564 df-f1o 6565 df-fv 6566 df-riota 7386 df-ov 7433 df-oprab 7434 df-mpo 7435 df-1st 8011 df-2nd 8012 df-er 8742 df-en 8983 df-dom 8984 df-sdom 8985 df-sup 9492 df-inf 9493 df-pnf 11307 df-mnf 11308 df-xr 11309 df-ltxr 11310 df-le 11311 df-sub 11503 df-neg 11504 df-div 11929 df-2 12337 df-rp 13039 df-xneg 13156 df-xadd 13157

This theorem is referenced by: infleinf 45057

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