infleinflem1 - Mathbox for Glauco Siliprandi

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Theorem infleinflem1 46105

Description: Lemma for infleinf 46107, 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 13355	. . . 4 ⊢ (𝐴 ⊆ ℝ^* → inf(𝐴, ℝ^, < ) ∈ ℝ^)
3	1, 2	syl 18	. . 3 ⊢ (𝜑 → inf(𝐴, ℝ^, < ) ∈ ℝ^)
4		id 23	. . 3 ⊢ (inf(𝐴, ℝ^, < ) ∈ ℝ^ → inf(𝐴, ℝ^, < ) ∈ ℝ^)
5	3, 4	syl 18	. 2 ⊢ (𝜑 → inf(𝐴, ℝ^, < ) ∈ ℝ^)
6		infleinflem1.z	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝐴)
7	1, 6	sseldd 3938	. 2 ⊢ (𝜑 → 𝑍 ∈ ℝ^*)
8		infleinflem1.b	. . . 4 ⊢ (𝜑 → 𝐵 ⊆ ℝ^*)
9		infxrcl 13355	. . . 4 ⊢ (𝐵 ⊆ ℝ^* → inf(𝐵, ℝ^, < ) ∈ ℝ^)
10	8, 9	syl 18	. . 3 ⊢ (𝜑 → inf(𝐵, ℝ^, < ) ∈ ℝ^)
11		infleinflem1.w	. . . 4 ⊢ (𝜑 → 𝑊 ∈ ℝ⁺)
12		rpxr 13021	. . . 4 ⊢ (𝑊 ∈ ℝ⁺ → 𝑊 ∈ ℝ^*)
13	11, 12	syl 18	. . 3 ⊢ (𝜑 → 𝑊 ∈ ℝ^*)
14	10, 13	xaddcld 13322	. 2 ⊢ (𝜑 → (inf(𝐵, ℝ^, < ) +_𝑒 𝑊) ∈ ℝ^)
15		infxrlb 13356	. . 3 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝑍 ∈ 𝐴) → inf(𝐴, ℝ^*, < ) ≤ 𝑍)
16	1, 6, 15	syl2anc 595	. 2 ⊢ (𝜑 → inf(𝐴, ℝ^*, < ) ≤ 𝑍)
17		infleinflem1.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
18	8	sselda 3937	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ ℝ^*)
19	17, 18	mpdan 699	. . . 4 ⊢ (𝜑 → 𝑋 ∈ ℝ^*)
20	11	rpred 13055	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ ℝ)
21	20	rehalfcld 12486	. . . . 5 ⊢ (𝜑 → (𝑊 / 2) ∈ ℝ)
22	21	rexrd 11254	. . . 4 ⊢ (𝜑 → (𝑊 / 2) ∈ ℝ^*)
23	19, 22	xaddcld 13322	. . 3 ⊢ (𝜑 → (𝑋 +_𝑒 (𝑊 / 2)) ∈ ℝ^*)
24		infleinflem1.l	. . 3 ⊢ (𝜑 → 𝑍 ≤ (𝑋 +_𝑒 (𝑊 / 2)))
25		pnfge 13150	. . . . . . 7 ⊢ ((𝑋 +_𝑒 (𝑊 / 2)) ∈ ℝ^* → (𝑋 +_𝑒 (𝑊 / 2)) ≤ +∞)
26	23, 25	syl 18	. . . . . 6 ⊢ (𝜑 → (𝑋 +_𝑒 (𝑊 / 2)) ≤ +∞)
27	26	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ inf(𝐵, ℝ^*, < ) = +∞) → (𝑋 +_𝑒 (𝑊 / 2)) ≤ +∞)
28		oveq1 7417	. . . . . . 7 ⊢ (inf(𝐵, ℝ^, < ) = +∞ → (inf(𝐵, ℝ^, < ) +_𝑒 𝑊) = (+∞ +_𝑒 𝑊))
29	28	adantl 486	. . . . . 6 ⊢ ((𝜑 ∧ inf(𝐵, ℝ^, < ) = +∞) → (inf(𝐵, ℝ^, < ) +_𝑒 𝑊) = (+∞ +_𝑒 𝑊))
30		rpre 13020	. . . . . . . . . 10 ⊢ (𝑊 ∈ ℝ⁺ → 𝑊 ∈ ℝ)
31		renemnf 11253	. . . . . . . . . 10 ⊢ (𝑊 ∈ ℝ → 𝑊 ≠ -∞)
32	30, 31	syl 18	. . . . . . . . 9 ⊢ (𝑊 ∈ ℝ⁺ → 𝑊 ≠ -∞)
33		xaddpnf2 13248	. . . . . . . . 9 ⊢ ((𝑊 ∈ ℝ^* ∧ 𝑊 ≠ -∞) → (+∞ +_𝑒 𝑊) = +∞)
34	12, 32, 33	syl2anc 595	. . . . . . . 8 ⊢ (𝑊 ∈ ℝ⁺ → (+∞ +_𝑒 𝑊) = +∞)
35	11, 34	syl 18	. . . . . . 7 ⊢ (𝜑 → (+∞ +_𝑒 𝑊) = +∞)
36	35	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ inf(𝐵, ℝ^*, < ) = +∞) → (+∞ +_𝑒 𝑊) = +∞)
37	29, 36	eqtr2d 2799	. . . . 5 ⊢ ((𝜑 ∧ inf(𝐵, ℝ^, < ) = +∞) → +∞ = (inf(𝐵, ℝ^, < ) +_𝑒 𝑊))
38	27, 37	breqtrd 5137	. . . 4 ⊢ ((𝜑 ∧ inf(𝐵, ℝ^, < ) = +∞) → (𝑋 +_𝑒 (𝑊 / 2)) ≤ (inf(𝐵, ℝ^, < ) +_𝑒 𝑊))
39	8, 17	sseldd 3938	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ ℝ^*)
40	10, 22	xaddcld 13322	. . . . . . 7 ⊢ (𝜑 → (inf(𝐵, ℝ^, < ) +_𝑒 (𝑊 / 2)) ∈ ℝ^)
41		rphalfcl 13040	. . . . . . . . 9 ⊢ (𝑊 ∈ ℝ⁺ → (𝑊 / 2) ∈ ℝ⁺)
42	11, 41	syl 18	. . . . . . . 8 ⊢ (𝜑 → (𝑊 / 2) ∈ ℝ⁺)
43	42	rpxrd 13056	. . . . . . 7 ⊢ (𝜑 → (𝑊 / 2) ∈ ℝ^*)
44		infleinflem1.i	. . . . . . 7 ⊢ (𝜑 → 𝑋 ≤ (inf(𝐵, ℝ^*, < ) +_𝑒 (𝑊 / 2)))
45	39, 40, 43, 44	xleadd1d 46065	. . . . . 6 ⊢ (𝜑 → (𝑋 +_𝑒 (𝑊 / 2)) ≤ ((inf(𝐵, ℝ^*, < ) +_𝑒 (𝑊 / 2)) +_𝑒 (𝑊 / 2)))
46	45	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ ¬ inf(𝐵, ℝ^, < ) = +∞) → (𝑋 +_𝑒 (𝑊 / 2)) ≤ ((inf(𝐵, ℝ^, < ) +_𝑒 (𝑊 / 2)) +_𝑒 (𝑊 / 2)))
47	10	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ inf(𝐵, ℝ^, < ) = +∞) → inf(𝐵, ℝ^, < ) ∈ ℝ^*)
48		neqne 2966	. . . . . . . 8 ⊢ (¬ inf(𝐵, ℝ^, < ) = +∞ → inf(𝐵, ℝ^, < ) ≠ +∞)
49	48	adantl 486	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ inf(𝐵, ℝ^, < ) = +∞) → inf(𝐵, ℝ^, < ) ≠ +∞)
50	43	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ inf(𝐵, ℝ^, < ) = +∞) → (𝑊 / 2) ∈ ℝ^)
51	11	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ inf(𝐵, ℝ^*, < ) = +∞) → 𝑊 ∈ ℝ⁺)
52		rpre 13020	. . . . . . . 8 ⊢ ((𝑊 / 2) ∈ ℝ⁺ → (𝑊 / 2) ∈ ℝ)
53		renepnf 11252	. . . . . . . 8 ⊢ ((𝑊 / 2) ∈ ℝ → (𝑊 / 2) ≠ +∞)
54	51, 41, 52, 53	4syl 20	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ inf(𝐵, ℝ^*, < ) = +∞) → (𝑊 / 2) ≠ +∞)
55		xaddass2 13271	. . . . . . 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 1413	. . . . . 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 12466	. . . . . . . . . 10 ⊢ (𝑊 ∈ ℝ → (𝑊 / 2) ∈ ℝ)
58	57, 57	rexaddd 13255	. . . . . . . . 9 ⊢ (𝑊 ∈ ℝ → ((𝑊 / 2) +_𝑒 (𝑊 / 2)) = ((𝑊 / 2) + (𝑊 / 2)))
59		recn 11185	. . . . . . . . . 10 ⊢ (𝑊 ∈ ℝ → 𝑊 ∈ ℂ)
60		2halves 12457	. . . . . . . . . 10 ⊢ (𝑊 ∈ ℂ → ((𝑊 / 2) + (𝑊 / 2)) = 𝑊)
61	59, 60	syl 18	. . . . . . . . 9 ⊢ (𝑊 ∈ ℝ → ((𝑊 / 2) + (𝑊 / 2)) = 𝑊)
62	58, 61	eqtrd 2798	. . . . . . . 8 ⊢ (𝑊 ∈ ℝ → ((𝑊 / 2) +_𝑒 (𝑊 / 2)) = 𝑊)
63	62	oveq2d 7426	. . . . . . 7 ⊢ (𝑊 ∈ ℝ → (inf(𝐵, ℝ^, < ) +_𝑒 ((𝑊 / 2) +_𝑒 (𝑊 / 2))) = (inf(𝐵, ℝ^, < ) +_𝑒 𝑊))
64	51, 30, 63	3syl 19	. . . . . 6 ⊢ ((𝜑 ∧ ¬ inf(𝐵, ℝ^, < ) = +∞) → (inf(𝐵, ℝ^, < ) +_𝑒 ((𝑊 / 2) +_𝑒 (𝑊 / 2))) = (inf(𝐵, ℝ^*, < ) +_𝑒 𝑊))
65	56, 64	eqtrd 2798	. . . . 5 ⊢ ((𝜑 ∧ ¬ inf(𝐵, ℝ^, < ) = +∞) → ((inf(𝐵, ℝ^, < ) +_𝑒 (𝑊 / 2)) +_𝑒 (𝑊 / 2)) = (inf(𝐵, ℝ^*, < ) +_𝑒 𝑊))
66	46, 65	breqtrd 5137	. . . 4 ⊢ ((𝜑 ∧ ¬ inf(𝐵, ℝ^, < ) = +∞) → (𝑋 +_𝑒 (𝑊 / 2)) ≤ (inf(𝐵, ℝ^, < ) +_𝑒 𝑊))
67	38, 66	pm2.61dan 824	. . 3 ⊢ (𝜑 → (𝑋 +_𝑒 (𝑊 / 2)) ≤ (inf(𝐵, ℝ^*, < ) +_𝑒 𝑊))
68	7, 23, 14, 24, 67	xrletrd 13182	. 2 ⊢ (𝜑 → 𝑍 ≤ (inf(𝐵, ℝ^*, < ) +_𝑒 𝑊))
69	5, 7, 14, 16, 68	xrletrd 13182	1 ⊢ (𝜑 → inf(𝐴, ℝ^, < ) ≤ (inf(𝐵, ℝ^, < ) +_𝑒 𝑊))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 400 = wceq 1570 ∈ wcel 2143 ≠ wne 2958 ⊆ wss 3905 class class class wbr 5109 (class class class)co 7410 infcinf 9397 ℂcc 11093 ℝcr 11094 + caddc 11098 +∞cpnf 11235 -∞cmnf 11236 ℝ^*cxr 11237 < clt 11238 ≤ cle 11239 / cdiv 11866 2c2 12290 ℝ⁺crp 13011 +_𝑒 cxad 13130

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-cnex 11151 ax-resscn 11152 ax-1cn 11153 ax-icn 11154 ax-addcl 11155 ax-addrcl 11156 ax-mulcl 11157 ax-mulrcl 11158 ax-mulcom 11159 ax-addass 11160 ax-mulass 11161 ax-distr 11162 ax-i2m1 11163 ax-1ne0 11164 ax-1rid 11165 ax-rnegex 11166 ax-rrecex 11167 ax-cnre 11168 ax-pre-lttri 11169 ax-pre-lttrn 11170 ax-pre-ltadd 11171 ax-pre-mulgt0 11172 ax-pre-sup 11173

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4873 df-iun 4958 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7859 df-1st 7982 df-2nd 7983 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-er 8690 df-en 8940 df-dom 8941 df-sdom 8942 df-sup 9398 df-inf 9399 df-pnf 11240 df-mnf 11241 df-xr 11242 df-ltxr 11243 df-le 11244 df-sub 11438 df-neg 11439 df-div 11867 df-nn 12229 df-2 12298 df-rp 13012 df-xneg 13132 df-xadd 13133

This theorem is referenced by: infleinf 46107

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