infleinflem1 - Mathbox for Glauco Siliprandi

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Theorem infleinflem1 46520

Description: Lemma for infleinf 46522, 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 13474	. . . 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 3913	. 2 ⊢ (𝜑 → 𝑍 ∈ ℝ^*)
8		infleinflem1.b	. . . 4 ⊢ (𝜑 → 𝐵 ⊆ ℝ^*)
9		infxrcl 13474	. . . 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 13140	. . . 4 ⊢ (𝑊 ∈ ℝ⁺ → 𝑊 ∈ ℝ^*)
13	11, 12	syl 18	. . 3 ⊢ (𝜑 → 𝑊 ∈ ℝ^*)
14	10, 13	xaddcld 13441	. 2 ⊢ (𝜑 → (inf(𝐵, ℝ^, < ) +_e 𝑊) ∈ ℝ^)
15		infxrlb 13475	. . 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 3912	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ ℝ^*)
19	17, 18	mpdan 700	. . . 4 ⊢ (𝜑 → 𝑋 ∈ ℝ^*)
20	11	rpred 13174	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ ℝ)
21	20	rehalfcld 12603	. . . . 5 ⊢ (𝜑 → (𝑊 / 2) ∈ ℝ)
22	21	rexrd 11369	. . . 4 ⊢ (𝜑 → (𝑊 / 2) ∈ ℝ^*)
23	19, 22	xaddcld 13441	. . 3 ⊢ (𝜑 → (𝑋 +_e (𝑊 / 2)) ∈ ℝ^*)
24		infleinflem1.l	. . 3 ⊢ (𝜑 → 𝑍 ≤ (𝑋 +_e (𝑊 / 2)))
25		pnfge 13269	. . . . . . 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 7434	. . . . . . 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 13139	. . . . . . . . . 10 ⊢ (𝑊 ∈ ℝ⁺ → 𝑊 ∈ ℝ)
31		renemnf 11368	. . . . . . . . . 10 ⊢ (𝑊 ∈ ℝ → 𝑊 ≠ -∞)
32	30, 31	syl 18	. . . . . . . . 9 ⊢ (𝑊 ∈ ℝ⁺ → 𝑊 ≠ -∞)
33		xaddpnf2 13367	. . . . . . . . 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 2799	. . . . 5 ⊢ ((𝜑 ∧ inf(𝐵, ℝ^, < ) = +∞) → +∞ = (inf(𝐵, ℝ^, < ) +_e 𝑊))
38	27, 37	breqtrd 5133	. . . 4 ⊢ ((𝜑 ∧ inf(𝐵, ℝ^, < ) = +∞) → (𝑋 +_e (𝑊 / 2)) ≤ (inf(𝐵, ℝ^, < ) +_e 𝑊))
39	8, 17	sseldd 3913	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ ℝ^*)
40	10, 22	xaddcld 13441	. . . . . . 7 ⊢ (𝜑 → (inf(𝐵, ℝ^, < ) +_e (𝑊 / 2)) ∈ ℝ^)
41		rphalfcl 13159	. . . . . . . . 9 ⊢ (𝑊 ∈ ℝ⁺ → (𝑊 / 2) ∈ ℝ⁺)
42	11, 41	syl 18	. . . . . . . 8 ⊢ (𝜑 → (𝑊 / 2) ∈ ℝ⁺)
43	42	rpxrd 13175	. . . . . . 7 ⊢ (𝜑 → (𝑊 / 2) ∈ ℝ^*)
44		infleinflem1.i	. . . . . . 7 ⊢ (𝜑 → 𝑋 ≤ (inf(𝐵, ℝ^*, < ) +_e (𝑊 / 2)))
45	39, 40, 43, 44	xleadd1d 46480	. . . . . 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 2966	. . . . . . . 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 13139	. . . . . . . 8 ⊢ ((𝑊 / 2) ∈ ℝ⁺ → (𝑊 / 2) ∈ ℝ)
53		renepnf 11367	. . . . . . . 8 ⊢ ((𝑊 / 2) ∈ ℝ → (𝑊 / 2) ≠ +∞)
54	51, 41, 52, 53	4syl 20	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ inf(𝐵, ℝ^*, < ) = +∞) → (𝑊 / 2) ≠ +∞)
55		xaddass2 13390	. . . . . . 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 12583	. . . . . . . . . 10 ⊢ (𝑊 ∈ ℝ → (𝑊 / 2) ∈ ℝ)
58	57, 57	rexaddd 13374	. . . . . . . . 9 ⊢ (𝑊 ∈ ℝ → ((𝑊 / 2) +_e (𝑊 / 2)) = ((𝑊 / 2) + (𝑊 / 2)))
59		recn 11300	. . . . . . . . . 10 ⊢ (𝑊 ∈ ℝ → 𝑊 ∈ ℂ)
60		2halves 12574	. . . . . . . . . 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 7443	. . . . . . 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 5133	. . . 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 13301	. 2 ⊢ (𝜑 → 𝑍 ≤ (inf(𝐵, ℝ^*, < ) +_e 𝑊))
69	5, 7, 14, 16, 68	xrletrd 13301	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 1569 ∈ wcel 2146 ≠ wne 2958 ⊆ wss 3899 class class class wbr 5105 (class class class)co 7427 infcinf 9443 ℂcc 11208 ℝcr 11209 + caddc 11213 +∞cpnf 11350 -∞cmnf 11351 ℝ^*cxr 11352 < clt 11353 ≤ cle 11354 / cdiv 11983 2c2 12407 ℝ⁺crp 13130 +_e cxad 13249

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1944 ax-6 2001 ax-7 2042 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2196 ax-12 2215 ax-ext 2735 ax-sep 5251 ax-nul 5262 ax-pow 5330 ax-pr 5394 ax-un 7758 ax-cnex 11266 ax-resscn 11267 ax-1cn 11268 ax-icn 11269 ax-addcl 11270 ax-addrcl 11271 ax-mulcl 11272 ax-mulrcl 11273 ax-mulcom 11274 ax-addass 11275 ax-mulass 11276 ax-distr 11277 ax-i2m1 11278 ax-1ne0 11279 ax-1rid 11280 ax-rnegex 11281 ax-rrecex 11282 ax-cnre 11283 ax-pre-lttri 11284 ax-pre-lttrn 11285 ax-pre-ltadd 11286 ax-pre-mulgt0 11287 ax-pre-sup 11288

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2101 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 3368 df-reu 3369 df-rab 3416 df-v 3455 df-sbc 3742 df-csb 3851 df-ss 3900 df-pss 4016 df-dif 4049 df-in 4093 df-un 4153 df-nul 4281 df-if 4485 df-pw 4561 df-sn 4588 df-pr 4590 df-op 4838 df-uni 4870 df-iun 4955 df-br 5106 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5549 df-eprel 5554 df-po 5562 df-so 5563 df-fr 5607 df-we 5609 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6310 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6500 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-riota 7384 df-ov 7430 df-oprab 7431 df-mpo 7432 df-om 7885 df-1st 8008 df-2nd 8009 df-frecs 8310 df-wrecs 8341 df-recs 8390 df-rdg 8429 df-er 8728 df-en 8985 df-dom 8986 df-sdom 8987 df-sup 9444 df-inf 9445 df-pnf 11355 df-mnf 11356 df-xr 11357 df-ltxr 11358 df-le 11359 df-sub 11553 df-neg 11554 df-div 11984 df-nn 12346 df-2 12415 df-rp 13131 df-xneg 13251 df-xadd 13252

This theorem is used by: infleinf 46522

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