infleinflem1 - Mathbox for Glauco Siliprandi

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Theorem infleinflem1 46207

Description: Lemma for infleinf 46209, 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 13390	. . . 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 3935	. 2 ⊢ (𝜑 → 𝑍 ∈ ℝ^*)
8		infleinflem1.b	. . . 4 ⊢ (𝜑 → 𝐵 ⊆ ℝ^*)
9		infxrcl 13390	. . . 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 13056	. . . 4 ⊢ (𝑊 ∈ ℝ⁺ → 𝑊 ∈ ℝ^*)
13	11, 12	syl 18	. . 3 ⊢ (𝜑 → 𝑊 ∈ ℝ^*)
14	10, 13	xaddcld 13357	. 2 ⊢ (𝜑 → (inf(𝐵, ℝ^, < ) +_𝑒 𝑊) ∈ ℝ^)
15		infxrlb 13391	. . 3 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝑍 ∈ 𝐴) → inf(𝐴, ℝ^*, < ) ≤ 𝑍)
16	1, 6, 15	syl2anc 596	. 2 ⊢ (𝜑 → inf(𝐴, ℝ^*, < ) ≤ 𝑍)
17		infleinflem1.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
18	8	sselda 3934	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ ℝ^*)
19	17, 18	mpdan 700	. . . 4 ⊢ (𝜑 → 𝑋 ∈ ℝ^*)
20	11	rpred 13090	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ ℝ)
21	20	rehalfcld 12519	. . . . 5 ⊢ (𝜑 → (𝑊 / 2) ∈ ℝ)
22	21	rexrd 11287	. . . 4 ⊢ (𝜑 → (𝑊 / 2) ∈ ℝ^*)
23	19, 22	xaddcld 13357	. . 3 ⊢ (𝜑 → (𝑋 +_𝑒 (𝑊 / 2)) ∈ ℝ^*)
24		infleinflem1.l	. . 3 ⊢ (𝜑 → 𝑍 ≤ (𝑋 +_𝑒 (𝑊 / 2)))
25		pnfge 13185	. . . . . . 7 ⊢ ((𝑋 +_𝑒 (𝑊 / 2)) ∈ ℝ^* → (𝑋 +_𝑒 (𝑊 / 2)) ≤ +∞)
26	23, 25	syl 18	. . . . . 6 ⊢ (𝜑 → (𝑋 +_𝑒 (𝑊 / 2)) ≤ +∞)
27	26	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ inf(𝐵, ℝ^*, < ) = +∞) → (𝑋 +_𝑒 (𝑊 / 2)) ≤ +∞)
28		oveq1 7424	. . . . . . 7 ⊢ (inf(𝐵, ℝ^, < ) = +∞ → (inf(𝐵, ℝ^, < ) +_𝑒 𝑊) = (+∞ +_𝑒 𝑊))
29	28	adantl 487	. . . . . 6 ⊢ ((𝜑 ∧ inf(𝐵, ℝ^, < ) = +∞) → (inf(𝐵, ℝ^, < ) +_𝑒 𝑊) = (+∞ +_𝑒 𝑊))
30		rpre 13055	. . . . . . . . . 10 ⊢ (𝑊 ∈ ℝ⁺ → 𝑊 ∈ ℝ)
31		renemnf 11286	. . . . . . . . . 10 ⊢ (𝑊 ∈ ℝ → 𝑊 ≠ -∞)
32	30, 31	syl 18	. . . . . . . . 9 ⊢ (𝑊 ∈ ℝ⁺ → 𝑊 ≠ -∞)
33		xaddpnf2 13283	. . . . . . . . 9 ⊢ ((𝑊 ∈ ℝ^* ∧ 𝑊 ≠ -∞) → (+∞ +_𝑒 𝑊) = +∞)
34	12, 32, 33	syl2anc 596	. . . . . . . 8 ⊢ (𝑊 ∈ ℝ⁺ → (+∞ +_𝑒 𝑊) = +∞)
35	11, 34	syl 18	. . . . . . 7 ⊢ (𝜑 → (+∞ +_𝑒 𝑊) = +∞)
36	35	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ inf(𝐵, ℝ^*, < ) = +∞) → (+∞ +_𝑒 𝑊) = +∞)
37	29, 36	eqtr2d 2798	. . . . 5 ⊢ ((𝜑 ∧ inf(𝐵, ℝ^, < ) = +∞) → +∞ = (inf(𝐵, ℝ^, < ) +_𝑒 𝑊))
38	27, 37	breqtrd 5135	. . . 4 ⊢ ((𝜑 ∧ inf(𝐵, ℝ^, < ) = +∞) → (𝑋 +_𝑒 (𝑊 / 2)) ≤ (inf(𝐵, ℝ^, < ) +_𝑒 𝑊))
39	8, 17	sseldd 3935	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ ℝ^*)
40	10, 22	xaddcld 13357	. . . . . . 7 ⊢ (𝜑 → (inf(𝐵, ℝ^, < ) +_𝑒 (𝑊 / 2)) ∈ ℝ^)
41		rphalfcl 13075	. . . . . . . . 9 ⊢ (𝑊 ∈ ℝ⁺ → (𝑊 / 2) ∈ ℝ⁺)
42	11, 41	syl 18	. . . . . . . 8 ⊢ (𝜑 → (𝑊 / 2) ∈ ℝ⁺)
43	42	rpxrd 13091	. . . . . . 7 ⊢ (𝜑 → (𝑊 / 2) ∈ ℝ^*)
44		infleinflem1.i	. . . . . . 7 ⊢ (𝜑 → 𝑋 ≤ (inf(𝐵, ℝ^*, < ) +_𝑒 (𝑊 / 2)))
45	39, 40, 43, 44	xleadd1d 46167	. . . . . 6 ⊢ (𝜑 → (𝑋 +_𝑒 (𝑊 / 2)) ≤ ((inf(𝐵, ℝ^*, < ) +_𝑒 (𝑊 / 2)) +_𝑒 (𝑊 / 2)))
46	45	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ ¬ inf(𝐵, ℝ^, < ) = +∞) → (𝑋 +_𝑒 (𝑊 / 2)) ≤ ((inf(𝐵, ℝ^, < ) +_𝑒 (𝑊 / 2)) +_𝑒 (𝑊 / 2)))
47	10	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ inf(𝐵, ℝ^, < ) = +∞) → inf(𝐵, ℝ^, < ) ∈ ℝ^*)
48		neqne 2965	. . . . . . . 8 ⊢ (¬ inf(𝐵, ℝ^, < ) = +∞ → inf(𝐵, ℝ^, < ) ≠ +∞)
49	48	adantl 487	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ inf(𝐵, ℝ^, < ) = +∞) → inf(𝐵, ℝ^, < ) ≠ +∞)
50	43	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ inf(𝐵, ℝ^, < ) = +∞) → (𝑊 / 2) ∈ ℝ^)
51	11	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ inf(𝐵, ℝ^*, < ) = +∞) → 𝑊 ∈ ℝ⁺)
52		rpre 13055	. . . . . . . 8 ⊢ ((𝑊 / 2) ∈ ℝ⁺ → (𝑊 / 2) ∈ ℝ)
53		renepnf 11285	. . . . . . . 8 ⊢ ((𝑊 / 2) ∈ ℝ → (𝑊 / 2) ≠ +∞)
54	51, 41, 52, 53	4syl 20	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ inf(𝐵, ℝ^*, < ) = +∞) → (𝑊 / 2) ≠ +∞)
55		xaddass2 13306	. . . . . . 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 12499	. . . . . . . . . 10 ⊢ (𝑊 ∈ ℝ → (𝑊 / 2) ∈ ℝ)
58	57, 57	rexaddd 13290	. . . . . . . . 9 ⊢ (𝑊 ∈ ℝ → ((𝑊 / 2) +_𝑒 (𝑊 / 2)) = ((𝑊 / 2) + (𝑊 / 2)))
59		recn 11218	. . . . . . . . . 10 ⊢ (𝑊 ∈ ℝ → 𝑊 ∈ ℂ)
60		2halves 12490	. . . . . . . . . 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 2797	. . . . . . . 8 ⊢ (𝑊 ∈ ℝ → ((𝑊 / 2) +_𝑒 (𝑊 / 2)) = 𝑊)
63	62	oveq2d 7433	. . . . . . 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 2797	. . . . 5 ⊢ ((𝜑 ∧ ¬ inf(𝐵, ℝ^, < ) = +∞) → ((inf(𝐵, ℝ^, < ) +_𝑒 (𝑊 / 2)) +_𝑒 (𝑊 / 2)) = (inf(𝐵, ℝ^*, < ) +_𝑒 𝑊))
66	46, 65	breqtrd 5135	. . . 4 ⊢ ((𝜑 ∧ ¬ inf(𝐵, ℝ^, < ) = +∞) → (𝑋 +_𝑒 (𝑊 / 2)) ≤ (inf(𝐵, ℝ^, < ) +_𝑒 𝑊))
67	38, 66	pm2.61dan 825	. . 3 ⊢ (𝜑 → (𝑋 +_𝑒 (𝑊 / 2)) ≤ (inf(𝐵, ℝ^*, < ) +_𝑒 𝑊))
68	7, 23, 14, 24, 67	xrletrd 13217	. 2 ⊢ (𝜑 → 𝑍 ≤ (inf(𝐵, ℝ^*, < ) +_𝑒 𝑊))
69	5, 7, 14, 16, 68	xrletrd 13217	1 ⊢ (𝜑 → inf(𝐴, ℝ^, < ) ≤ (inf(𝐵, ℝ^, < ) +_𝑒 𝑊))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ≠ wne 2957 ⊆ wss 3902 class class class wbr 5107 (class class class)co 7417 infcinf 9415 ℂcc 11126 ℝcr 11127 + caddc 11131 +∞cpnf 11268 -∞cmnf 11269 ℝ^*cxr 11270 < clt 11271 ≤ cle 11272 / cdiv 11899 2c2 12323 ℝ⁺crp 13046 +_𝑒 cxad 13165

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7740 ax-cnex 11184 ax-resscn 11185 ax-1cn 11186 ax-icn 11187 ax-addcl 11188 ax-addrcl 11189 ax-mulcl 11190 ax-mulrcl 11191 ax-mulcom 11192 ax-addass 11193 ax-mulass 11194 ax-distr 11195 ax-i2m1 11196 ax-1ne0 11197 ax-1rid 11198 ax-rnegex 11199 ax-rrecex 11200 ax-cnre 11201 ax-pre-lttri 11202 ax-pre-lttrn 11203 ax-pre-ltadd 11204 ax-pre-mulgt0 11205 ax-pre-sup 11206

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4871 df-iun 4956 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7374 df-ov 7420 df-oprab 7421 df-mpo 7422 df-om 7867 df-1st 7990 df-2nd 7991 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-er 8700 df-en 8957 df-dom 8958 df-sdom 8959 df-sup 9416 df-inf 9417 df-pnf 11273 df-mnf 11274 df-xr 11275 df-ltxr 11276 df-le 11277 df-sub 11471 df-neg 11472 df-div 11900 df-nn 12262 df-2 12331 df-rp 13047 df-xneg 13167 df-xadd 13168

This theorem is used by: infleinf 46209

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