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Theorem dvlog2lem 26159

Description: Lemma for dvlog2 26160. (Contributed by Mario Carneiro, 1-Mar-2015.)

**Hypothesis**
Ref	Expression
dvlog2.s	⊢ 𝑆 = (1(ball‘(abs ∘ − ))1)

**Assertion**
Ref	Expression
dvlog2lem	⊢ 𝑆 ⊆ (ℂ ∖ (-∞(,]0))

Proof of Theorem dvlog2lem Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dvlog2.s	. . . . 5 ⊢ 𝑆 = (1(ball‘(abs ∘ − ))1)
2		cnxmet 24288	. . . . . 6 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
3		ax-1cn 11167	. . . . . 6 ⊢ 1 ∈ ℂ
4		1xr 11272	. . . . . 6 ⊢ 1 ∈ ℝ^*
5		blssm 23923	. . . . . 6 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 1 ∈ ℂ ∧ 1 ∈ ℝ^*) → (1(ball‘(abs ∘ − ))1) ⊆ ℂ)
6	2, 3, 4, 5	mp3an 1461	. . . . 5 ⊢ (1(ball‘(abs ∘ − ))1) ⊆ ℂ
7	1, 6	eqsstri 4016	. . . 4 ⊢ 𝑆 ⊆ ℂ
8	7	sseli 3978	. . 3 ⊢ (𝑥 ∈ 𝑆 → 𝑥 ∈ ℂ)
9		1red 11214	. . . . . . 7 ⊢ (𝑥 ∈ (-∞(,]0) → 1 ∈ ℝ)
10		cnmet 24287	. . . . . . . 8 ⊢ (abs ∘ − ) ∈ (Met‘ℂ)
11		mnfxr 11270	. . . . . . . . . . 11 ⊢ -∞ ∈ ℝ^*
12		0re 11215	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ
13		iocssre 13403	. . . . . . . . . . 11 ⊢ ((-∞ ∈ ℝ^* ∧ 0 ∈ ℝ) → (-∞(,]0) ⊆ ℝ)
14	11, 12, 13	mp2an 690	. . . . . . . . . 10 ⊢ (-∞(,]0) ⊆ ℝ
15		ax-resscn 11166	. . . . . . . . . 10 ⊢ ℝ ⊆ ℂ
16	14, 15	sstri 3991	. . . . . . . . 9 ⊢ (-∞(,]0) ⊆ ℂ
17	16	sseli 3978	. . . . . . . 8 ⊢ (𝑥 ∈ (-∞(,]0) → 𝑥 ∈ ℂ)
18		metcl 23837	. . . . . . . 8 ⊢ (((abs ∘ − ) ∈ (Met‘ℂ) ∧ 1 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (1(abs ∘ − )𝑥) ∈ ℝ)
19	10, 3, 17, 18	mp3an12i 1465	. . . . . . 7 ⊢ (𝑥 ∈ (-∞(,]0) → (1(abs ∘ − )𝑥) ∈ ℝ)
20		1m0e1 12332	. . . . . . . . 9 ⊢ (1 − 0) = 1
21	14	sseli 3978	. . . . . . . . . 10 ⊢ (𝑥 ∈ (-∞(,]0) → 𝑥 ∈ ℝ)
22	12	a1i 11	. . . . . . . . . 10 ⊢ (𝑥 ∈ (-∞(,]0) → 0 ∈ ℝ)
23		elioc2 13386	. . . . . . . . . . . 12 ⊢ ((-∞ ∈ ℝ^* ∧ 0 ∈ ℝ) → (𝑥 ∈ (-∞(,]0) ↔ (𝑥 ∈ ℝ ∧ -∞ < 𝑥 ∧ 𝑥 ≤ 0)))
24	11, 12, 23	mp2an 690	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (-∞(,]0) ↔ (𝑥 ∈ ℝ ∧ -∞ < 𝑥 ∧ 𝑥 ≤ 0))
25	24	simp3bi 1147	. . . . . . . . . 10 ⊢ (𝑥 ∈ (-∞(,]0) → 𝑥 ≤ 0)
26	21, 22, 9, 25	lesub2dd 11830	. . . . . . . . 9 ⊢ (𝑥 ∈ (-∞(,]0) → (1 − 0) ≤ (1 − 𝑥))
27	20, 26	eqbrtrrid 5184	. . . . . . . 8 ⊢ (𝑥 ∈ (-∞(,]0) → 1 ≤ (1 − 𝑥))
28		eqid 2732	. . . . . . . . . . 11 ⊢ (abs ∘ − ) = (abs ∘ − )
29	28	cnmetdval 24286	. . . . . . . . . 10 ⊢ ((1 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (1(abs ∘ − )𝑥) = (abs‘(1 − 𝑥)))
30	3, 17, 29	sylancr 587	. . . . . . . . 9 ⊢ (𝑥 ∈ (-∞(,]0) → (1(abs ∘ − )𝑥) = (abs‘(1 − 𝑥)))
31		0le1 11736	. . . . . . . . . . . 12 ⊢ 0 ≤ 1
32	31	a1i 11	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (-∞(,]0) → 0 ≤ 1)
33	21, 22, 9, 25, 32	letrd 11370	. . . . . . . . . 10 ⊢ (𝑥 ∈ (-∞(,]0) → 𝑥 ≤ 1)
34	21, 9, 33	abssubge0d 15377	. . . . . . . . 9 ⊢ (𝑥 ∈ (-∞(,]0) → (abs‘(1 − 𝑥)) = (1 − 𝑥))
35	30, 34	eqtrd 2772	. . . . . . . 8 ⊢ (𝑥 ∈ (-∞(,]0) → (1(abs ∘ − )𝑥) = (1 − 𝑥))
36	27, 35	breqtrrd 5176	. . . . . . 7 ⊢ (𝑥 ∈ (-∞(,]0) → 1 ≤ (1(abs ∘ − )𝑥))
37	9, 19, 36	lensymd 11364	. . . . . 6 ⊢ (𝑥 ∈ (-∞(,]0) → ¬ (1(abs ∘ − )𝑥) < 1)
38	2	a1i 11	. . . . . . 7 ⊢ (𝑥 ∈ (-∞(,]0) → (abs ∘ − ) ∈ (∞Met‘ℂ))
39	4	a1i 11	. . . . . . 7 ⊢ (𝑥 ∈ (-∞(,]0) → 1 ∈ ℝ^*)
40	3	a1i 11	. . . . . . 7 ⊢ (𝑥 ∈ (-∞(,]0) → 1 ∈ ℂ)
41		elbl2 23895	. . . . . . 7 ⊢ ((((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 1 ∈ ℝ^*) ∧ (1 ∈ ℂ ∧ 𝑥 ∈ ℂ)) → (𝑥 ∈ (1(ball‘(abs ∘ − ))1) ↔ (1(abs ∘ − )𝑥) < 1))
42	38, 39, 40, 17, 41	syl22anc 837	. . . . . 6 ⊢ (𝑥 ∈ (-∞(,]0) → (𝑥 ∈ (1(ball‘(abs ∘ − ))1) ↔ (1(abs ∘ − )𝑥) < 1))
43	37, 42	mtbird 324	. . . . 5 ⊢ (𝑥 ∈ (-∞(,]0) → ¬ 𝑥 ∈ (1(ball‘(abs ∘ − ))1))
44	43	con2i 139	. . . 4 ⊢ (𝑥 ∈ (1(ball‘(abs ∘ − ))1) → ¬ 𝑥 ∈ (-∞(,]0))
45	44, 1	eleq2s 2851	. . 3 ⊢ (𝑥 ∈ 𝑆 → ¬ 𝑥 ∈ (-∞(,]0))
46	8, 45	eldifd 3959	. 2 ⊢ (𝑥 ∈ 𝑆 → 𝑥 ∈ (ℂ ∖ (-∞(,]0)))
47	46	ssriv 3986	1 ⊢ 𝑆 ⊆ (ℂ ∖ (-∞(,]0))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ↔ wb 205 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∖ cdif 3945 ⊆ wss 3948 class class class wbr 5148 ∘ ccom 5680 ‘cfv 6543 (class class class)co 7408 ℂcc 11107 ℝcr 11108 0cc0 11109 1c1 11110 -∞cmnf 11245 ℝ^*cxr 11246 < clt 11247 ≤ cle 11248 − cmin 11443 (,]cioc 13324 abscabs 15180 ∞Metcxmet 20928 Metcmet 20929 ballcbl 20930

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7855 df-1st 7974 df-2nd 7975 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-er 8702 df-map 8821 df-en 8939 df-dom 8940 df-sdom 8941 df-sup 9436 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-div 11871 df-nn 12212 df-2 12274 df-3 12275 df-n0 12472 df-z 12558 df-uz 12822 df-rp 12974 df-xadd 13092 df-ioc 13328 df-seq 13966 df-exp 14027 df-cj 15045 df-re 15046 df-im 15047 df-sqrt 15181 df-abs 15182 df-psmet 20935 df-xmet 20936 df-met 20937 df-bl 20938

This theorem is referenced by: dvlog2 26160 logtayl 26167 logtayl2 26169 efrlim 26471 lgamcvg2 26556

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