infleinflem1 - Mathbox for Glauco Siliprandi

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Theorem infleinflem1 45556

Description: Lemma for infleinf 45558, 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 13247	. . . 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 3932	. 2 ⊢ (𝜑 → 𝑍 ∈ ℝ^*)
8		infleinflem1.b	. . . 4 ⊢ (𝜑 → 𝐵 ⊆ ℝ^*)
9		infxrcl 13247	. . . 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 12913	. . . 4 ⊢ (𝑊 ∈ ℝ⁺ → 𝑊 ∈ ℝ^*)
13	11, 12	syl 17	. . 3 ⊢ (𝜑 → 𝑊 ∈ ℝ^*)
14	10, 13	xaddcld 13214	. 2 ⊢ (𝜑 → (inf(𝐵, ℝ^, < ) +_𝑒 𝑊) ∈ ℝ^)
15		infxrlb 13248	. . 3 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝑍 ∈ 𝐴) → inf(𝐴, ℝ^*, < ) ≤ 𝑍)
16	1, 6, 15	syl2anc 584	. 2 ⊢ (𝜑 → inf(𝐴, ℝ^*, < ) ≤ 𝑍)
17		infleinflem1.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
18	8	sselda 3931	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ ℝ^*)
19	17, 18	mpdan 687	. . . 4 ⊢ (𝜑 → 𝑋 ∈ ℝ^*)
20	11	rpred 12947	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ ℝ)
21	20	rehalfcld 12386	. . . . 5 ⊢ (𝜑 → (𝑊 / 2) ∈ ℝ)
22	21	rexrd 11180	. . . 4 ⊢ (𝜑 → (𝑊 / 2) ∈ ℝ^*)
23	19, 22	xaddcld 13214	. . 3 ⊢ (𝜑 → (𝑋 +_𝑒 (𝑊 / 2)) ∈ ℝ^*)
24		infleinflem1.l	. . 3 ⊢ (𝜑 → 𝑍 ≤ (𝑋 +_𝑒 (𝑊 / 2)))
25		pnfge 13042	. . . . . . 7 ⊢ ((𝑋 +_𝑒 (𝑊 / 2)) ∈ ℝ^* → (𝑋 +_𝑒 (𝑊 / 2)) ≤ +∞)
26	23, 25	syl 17	. . . . . 6 ⊢ (𝜑 → (𝑋 +_𝑒 (𝑊 / 2)) ≤ +∞)
27	26	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ inf(𝐵, ℝ^*, < ) = +∞) → (𝑋 +_𝑒 (𝑊 / 2)) ≤ +∞)
28		oveq1 7363	. . . . . . 7 ⊢ (inf(𝐵, ℝ^, < ) = +∞ → (inf(𝐵, ℝ^, < ) +_𝑒 𝑊) = (+∞ +_𝑒 𝑊))
29	28	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ inf(𝐵, ℝ^, < ) = +∞) → (inf(𝐵, ℝ^, < ) +_𝑒 𝑊) = (+∞ +_𝑒 𝑊))
30		rpre 12912	. . . . . . . . . 10 ⊢ (𝑊 ∈ ℝ⁺ → 𝑊 ∈ ℝ)
31		renemnf 11179	. . . . . . . . . 10 ⊢ (𝑊 ∈ ℝ → 𝑊 ≠ -∞)
32	30, 31	syl 17	. . . . . . . . 9 ⊢ (𝑊 ∈ ℝ⁺ → 𝑊 ≠ -∞)
33		xaddpnf2 13140	. . . . . . . . 9 ⊢ ((𝑊 ∈ ℝ^* ∧ 𝑊 ≠ -∞) → (+∞ +_𝑒 𝑊) = +∞)
34	12, 32, 33	syl2anc 584	. . . . . . . 8 ⊢ (𝑊 ∈ ℝ⁺ → (+∞ +_𝑒 𝑊) = +∞)
35	11, 34	syl 17	. . . . . . 7 ⊢ (𝜑 → (+∞ +_𝑒 𝑊) = +∞)
36	35	adantr 480	. . . . . 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 5122	. . . 4 ⊢ ((𝜑 ∧ inf(𝐵, ℝ^, < ) = +∞) → (𝑋 +_𝑒 (𝑊 / 2)) ≤ (inf(𝐵, ℝ^, < ) +_𝑒 𝑊))
39	8, 17	sseldd 3932	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ ℝ^*)
40	10, 22	xaddcld 13214	. . . . . . 7 ⊢ (𝜑 → (inf(𝐵, ℝ^, < ) +_𝑒 (𝑊 / 2)) ∈ ℝ^)
41		rphalfcl 12932	. . . . . . . . 9 ⊢ (𝑊 ∈ ℝ⁺ → (𝑊 / 2) ∈ ℝ⁺)
42	11, 41	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝑊 / 2) ∈ ℝ⁺)
43	42	rpxrd 12948	. . . . . . 7 ⊢ (𝜑 → (𝑊 / 2) ∈ ℝ^*)
44		infleinflem1.i	. . . . . . 7 ⊢ (𝜑 → 𝑋 ≤ (inf(𝐵, ℝ^*, < ) +_𝑒 (𝑊 / 2)))
45	39, 40, 43, 44	xleadd1d 45516	. . . . . 6 ⊢ (𝜑 → (𝑋 +_𝑒 (𝑊 / 2)) ≤ ((inf(𝐵, ℝ^*, < ) +_𝑒 (𝑊 / 2)) +_𝑒 (𝑊 / 2)))
46	45	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ¬ inf(𝐵, ℝ^, < ) = +∞) → (𝑋 +_𝑒 (𝑊 / 2)) ≤ ((inf(𝐵, ℝ^, < ) +_𝑒 (𝑊 / 2)) +_𝑒 (𝑊 / 2)))
47	10	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ inf(𝐵, ℝ^, < ) = +∞) → inf(𝐵, ℝ^, < ) ∈ ℝ^*)
48		neqne 2938	. . . . . . . 8 ⊢ (¬ inf(𝐵, ℝ^, < ) = +∞ → inf(𝐵, ℝ^, < ) ≠ +∞)
49	48	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ inf(𝐵, ℝ^, < ) = +∞) → inf(𝐵, ℝ^, < ) ≠ +∞)
50	43	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ inf(𝐵, ℝ^, < ) = +∞) → (𝑊 / 2) ∈ ℝ^)
51	11	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ inf(𝐵, ℝ^*, < ) = +∞) → 𝑊 ∈ ℝ⁺)
52		rpre 12912	. . . . . . . 8 ⊢ ((𝑊 / 2) ∈ ℝ⁺ → (𝑊 / 2) ∈ ℝ)
53		renepnf 11178	. . . . . . . 8 ⊢ ((𝑊 / 2) ∈ ℝ → (𝑊 / 2) ≠ +∞)
54	51, 41, 52, 53	4syl 19	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ inf(𝐵, ℝ^*, < ) = +∞) → (𝑊 / 2) ≠ +∞)
55		xaddass2 13163	. . . . . . 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 1388	. . . . . 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 12366	. . . . . . . . . 10 ⊢ (𝑊 ∈ ℝ → (𝑊 / 2) ∈ ℝ)
58	57, 57	rexaddd 13147	. . . . . . . . 9 ⊢ (𝑊 ∈ ℝ → ((𝑊 / 2) +_𝑒 (𝑊 / 2)) = ((𝑊 / 2) + (𝑊 / 2)))
59		recn 11114	. . . . . . . . . 10 ⊢ (𝑊 ∈ ℝ → 𝑊 ∈ ℂ)
60		2halves 12357	. . . . . . . . . 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 7372	. . . . . . 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 5122	. . . 4 ⊢ ((𝜑 ∧ ¬ inf(𝐵, ℝ^, < ) = +∞) → (𝑋 +_𝑒 (𝑊 / 2)) ≤ (inf(𝐵, ℝ^, < ) +_𝑒 𝑊))
67	38, 66	pm2.61dan 812	. . 3 ⊢ (𝜑 → (𝑋 +_𝑒 (𝑊 / 2)) ≤ (inf(𝐵, ℝ^*, < ) +_𝑒 𝑊))
68	7, 23, 14, 24, 67	xrletrd 13074	. 2 ⊢ (𝜑 → 𝑍 ≤ (inf(𝐵, ℝ^*, < ) +_𝑒 𝑊))
69	5, 7, 14, 16, 68	xrletrd 13074	1 ⊢ (𝜑 → inf(𝐴, ℝ^, < ) ≤ (inf(𝐵, ℝ^, < ) +_𝑒 𝑊))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2113 ≠ wne 2930 ⊆ wss 3899 class class class wbr 5096 (class class class)co 7356 infcinf 9342 ℂcc 11022 ℝcr 11023 + caddc 11027 +∞cpnf 11161 -∞cmnf 11162 ℝ^*cxr 11163 < clt 11164 ≤ cle 11165 / cdiv 11792 2c2 12198 ℝ⁺crp 12903 +_𝑒 cxad 13022

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2706 ax-sep 5239 ax-nul 5249 ax-pow 5308 ax-pr 5375 ax-un 7678 ax-cnex 11080 ax-resscn 11081 ax-1cn 11082 ax-icn 11083 ax-addcl 11084 ax-addrcl 11085 ax-mulcl 11086 ax-mulrcl 11087 ax-mulcom 11088 ax-addass 11089 ax-mulass 11090 ax-distr 11091 ax-i2m1 11092 ax-1ne0 11093 ax-1rid 11094 ax-rnegex 11095 ax-rrecex 11096 ax-cnre 11097 ax-pre-lttri 11098 ax-pre-lttrn 11099 ax-pre-ltadd 11100 ax-pre-mulgt0 11101 ax-pre-sup 11102

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3059 df-rmo 3348 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4579 df-pr 4581 df-op 4585 df-uni 4862 df-iun 4946 df-br 5097 df-opab 5159 df-mpt 5178 df-tr 5204 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-er 8633 df-en 8882 df-dom 8883 df-sdom 8884 df-sup 9343 df-inf 9344 df-pnf 11166 df-mnf 11167 df-xr 11168 df-ltxr 11169 df-le 11170 df-sub 11364 df-neg 11365 df-div 11793 df-nn 12144 df-2 12206 df-rp 12904 df-xneg 13024 df-xadd 13025

This theorem is referenced by: infleinf 45558

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