readvcot - Mathbox for Steven Nguyen

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Theorem readvcot 42848

Description: Real antiderivative of cotangent. (Contributed by SN, 7-Oct-2025.)

**Hypothesis**
Ref	Expression
readvcot.d	⊢ 𝐷 = {𝑦 ∈ ℝ ∣ (sin‘𝑦) ≠ 0}

**Assertion**
Ref	Expression
readvcot	⊢ (ℝ D (𝑥 ∈ 𝐷 ↦ (log‘(abs‘(sin‘𝑥))))) = (𝑥 ∈ 𝐷 ↦ ((cos‘𝑥) / (sin‘𝑥)))

Distinct variable groups: 𝑥,𝐷 𝑥,𝑦 Allowed substitution hint: 𝐷(𝑦)

Proof of Theorem readvcot Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		reelprrecn 11128	. . . . 5 ⊢ ℝ ∈ {ℝ, ℂ}
2	1	a1i 11	. . . 4 ⊢ (⊤ → ℝ ∈ {ℝ, ℂ})
3		fveq2 6834	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (sin‘𝑦) = (sin‘𝑥))
4	3	neeq1d 2994	. . . . . . 7 ⊢ (𝑦 = 𝑥 → ((sin‘𝑦) ≠ 0 ↔ (sin‘𝑥) ≠ 0))
5		readvcot.d	. . . . . . 7 ⊢ 𝐷 = {𝑦 ∈ ℝ ∣ (sin‘𝑦) ≠ 0}
6	4, 5	elrab2 3639	. . . . . 6 ⊢ (𝑥 ∈ 𝐷 ↔ (𝑥 ∈ ℝ ∧ (sin‘𝑥) ≠ 0))
7		resincl 16105	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ → (sin‘𝑥) ∈ ℝ)
8	7	adantr 481	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ ∧ (sin‘𝑥) ≠ 0) → (sin‘𝑥) ∈ ℝ)
9		simpr 485	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ ∧ (sin‘𝑥) ≠ 0) → (sin‘𝑥) ≠ 0)
10	8, 9	eldifsnd 4727	. . . . . 6 ⊢ ((𝑥 ∈ ℝ ∧ (sin‘𝑥) ≠ 0) → (sin‘𝑥) ∈ (ℝ ∖ {0}))
11	6, 10	sylbi 218	. . . . 5 ⊢ (𝑥 ∈ 𝐷 → (sin‘𝑥) ∈ (ℝ ∖ {0}))
12	11	adantl 482	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐷) → (sin‘𝑥) ∈ (ℝ ∖ {0}))
13		fvexd 6849	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐷) → (cos‘𝑥) ∈ V)
14		eldifi 4068	. . . . . . . . 9 ⊢ (𝑧 ∈ (ℝ ∖ {0}) → 𝑧 ∈ ℝ)
15	14	adantl 482	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑧 ∈ (ℝ ∖ {0})) → 𝑧 ∈ ℝ)
16	15	recnd 11171	. . . . . . 7 ⊢ ((⊤ ∧ 𝑧 ∈ (ℝ ∖ {0})) → 𝑧 ∈ ℂ)
17	16	abscld 15399	. . . . . 6 ⊢ ((⊤ ∧ 𝑧 ∈ (ℝ ∖ {0})) → (abs‘𝑧) ∈ ℝ)
18	17	recnd 11171	. . . . 5 ⊢ ((⊤ ∧ 𝑧 ∈ (ℝ ∖ {0})) → (abs‘𝑧) ∈ ℂ)
19		eldifsni 4730	. . . . . . 7 ⊢ (𝑧 ∈ (ℝ ∖ {0}) → 𝑧 ≠ 0)
20	19	adantl 482	. . . . . 6 ⊢ ((⊤ ∧ 𝑧 ∈ (ℝ ∖ {0})) → 𝑧 ≠ 0)
21	16, 20	absne0d 15410	. . . . 5 ⊢ ((⊤ ∧ 𝑧 ∈ (ℝ ∖ {0})) → (abs‘𝑧) ≠ 0)
22	18, 21	logcld 26559	. . . 4 ⊢ ((⊤ ∧ 𝑧 ∈ (ℝ ∖ {0})) → (log‘(abs‘𝑧)) ∈ ℂ)
23		ovexd 7398	. . . 4 ⊢ ((⊤ ∧ 𝑧 ∈ (ℝ ∖ {0})) → (1 / 𝑧) ∈ V)
24	7	recnd 11171	. . . . . 6 ⊢ (𝑥 ∈ ℝ → (sin‘𝑥) ∈ ℂ)
25	24	adantl 482	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ) → (sin‘𝑥) ∈ ℂ)
26		fvexd 6849	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ) → (cos‘𝑥) ∈ V)
27		eqid 2740	. . . . . 6 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
28		cnopn 24776	. . . . . . 7 ⊢ ℂ ∈ (TopOpen‘ℂ_fld)
29	28	a1i 11	. . . . . 6 ⊢ (⊤ → ℂ ∈ (TopOpen‘ℂ_fld))
30		ax-resscn 11093	. . . . . . . 8 ⊢ ℝ ⊆ ℂ
31		dfss2 3908	. . . . . . . 8 ⊢ (ℝ ⊆ ℂ ↔ (ℝ ∩ ℂ) = ℝ)
32	30, 31	mpbi 231	. . . . . . 7 ⊢ (ℝ ∩ ℂ) = ℝ
33	32	a1i 11	. . . . . 6 ⊢ (⊤ → (ℝ ∩ ℂ) = ℝ)
34		sincl 16091	. . . . . . 7 ⊢ (𝑥 ∈ ℂ → (sin‘𝑥) ∈ ℂ)
35	34	adantl 482	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → (sin‘𝑥) ∈ ℂ)
36		fvexd 6849	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → (cos‘𝑥) ∈ V)
37		dvsin 25974	. . . . . . 7 ⊢ (ℂ D sin) = cos
38		sinf 16089	. . . . . . . . . 10 ⊢ sin:ℂ⟶ℂ
39	38	a1i 11	. . . . . . . . 9 ⊢ (⊤ → sin:ℂ⟶ℂ)
40	39	feqmptd 6902	. . . . . . . 8 ⊢ (⊤ → sin = (𝑥 ∈ ℂ ↦ (sin‘𝑥)))
41	40	oveq2d 7379	. . . . . . 7 ⊢ (⊤ → (ℂ D sin) = (ℂ D (𝑥 ∈ ℂ ↦ (sin‘𝑥))))
42		cosf 16090	. . . . . . . . 9 ⊢ cos:ℂ⟶ℂ
43	42	a1i 11	. . . . . . . 8 ⊢ (⊤ → cos:ℂ⟶ℂ)
44	43	feqmptd 6902	. . . . . . 7 ⊢ (⊤ → cos = (𝑥 ∈ ℂ ↦ (cos‘𝑥)))
45	37, 41, 44	3eqtr3a 2799	. . . . . 6 ⊢ (⊤ → (ℂ D (𝑥 ∈ ℂ ↦ (sin‘𝑥))) = (𝑥 ∈ ℂ ↦ (cos‘𝑥)))
46	27, 2, 29, 33, 35, 36, 45	dvmptres3 25948	. . . . 5 ⊢ (⊤ → (ℝ D (𝑥 ∈ ℝ ↦ (sin‘𝑥))) = (𝑥 ∈ ℝ ↦ (cos‘𝑥)))
47	5	ssrab3 4020	. . . . . 6 ⊢ 𝐷 ⊆ ℝ
48	47	a1i 11	. . . . 5 ⊢ (⊤ → 𝐷 ⊆ ℝ)
49		tgioo4 24795	. . . . 5 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
50	5	resuppsinopn 42847	. . . . . 6 ⊢ 𝐷 ∈ (topGen‘ran (,))
51	50	a1i 11	. . . . 5 ⊢ (⊤ → 𝐷 ∈ (topGen‘ran (,)))
52	2, 25, 26, 46, 48, 49, 27, 51	dvmptres 25955	. . . 4 ⊢ (⊤ → (ℝ D (𝑥 ∈ 𝐷 ↦ (sin‘𝑥))) = (𝑥 ∈ 𝐷 ↦ (cos‘𝑥)))
53		eqid 2740	. . . . . 6 ⊢ (ℝ ∖ {0}) = (ℝ ∖ {0})
54	53	readvrec 42846	. . . . 5 ⊢ (ℝ D (𝑧 ∈ (ℝ ∖ {0}) ↦ (log‘(abs‘𝑧)))) = (𝑧 ∈ (ℝ ∖ {0}) ↦ (1 / 𝑧))
55	54	a1i 11	. . . 4 ⊢ (⊤ → (ℝ D (𝑧 ∈ (ℝ ∖ {0}) ↦ (log‘(abs‘𝑧)))) = (𝑧 ∈ (ℝ ∖ {0}) ↦ (1 / 𝑧)))
56		2fveq3 6839	. . . 4 ⊢ (𝑧 = (sin‘𝑥) → (log‘(abs‘𝑧)) = (log‘(abs‘(sin‘𝑥))))
57		oveq2 7371	. . . 4 ⊢ (𝑧 = (sin‘𝑥) → (1 / 𝑧) = (1 / (sin‘𝑥)))
58	2, 2, 12, 13, 22, 23, 52, 55, 56, 57	dvmptco 25964	. . 3 ⊢ (⊤ → (ℝ D (𝑥 ∈ 𝐷 ↦ (log‘(abs‘(sin‘𝑥))))) = (𝑥 ∈ 𝐷 ↦ ((1 / (sin‘𝑥)) · (cos‘𝑥))))
59	58	mptru 1554	. 2 ⊢ (ℝ D (𝑥 ∈ 𝐷 ↦ (log‘(abs‘(sin‘𝑥))))) = (𝑥 ∈ 𝐷 ↦ ((1 / (sin‘𝑥)) · (cos‘𝑥)))
60	6	simplbi 497	. . . . . 6 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ∈ ℝ)
61	60	recoscld 16109	. . . . 5 ⊢ (𝑥 ∈ 𝐷 → (cos‘𝑥) ∈ ℝ)
62	61	recnd 11171	. . . 4 ⊢ (𝑥 ∈ 𝐷 → (cos‘𝑥) ∈ ℂ)
63	6, 8	sylbi 218	. . . . 5 ⊢ (𝑥 ∈ 𝐷 → (sin‘𝑥) ∈ ℝ)
64	63	recnd 11171	. . . 4 ⊢ (𝑥 ∈ 𝐷 → (sin‘𝑥) ∈ ℂ)
65	6, 9	sylbi 218	. . . 4 ⊢ (𝑥 ∈ 𝐷 → (sin‘𝑥) ≠ 0)
66	62, 64, 65	divrec2d 11933	. . 3 ⊢ (𝑥 ∈ 𝐷 → ((cos‘𝑥) / (sin‘𝑥)) = ((1 / (sin‘𝑥)) · (cos‘𝑥)))
67	66	mpteq2ia 5174	. 2 ⊢ (𝑥 ∈ 𝐷 ↦ ((cos‘𝑥) / (sin‘𝑥))) = (𝑥 ∈ 𝐷 ↦ ((1 / (sin‘𝑥)) · (cos‘𝑥)))
68	59, 67	eqtr4i 2766	1 ⊢ (ℝ D (𝑥 ∈ 𝐷 ↦ (log‘(abs‘(sin‘𝑥))))) = (𝑥 ∈ 𝐷 ↦ ((cos‘𝑥) / (sin‘𝑥)))

Colors of variables: wff setvar class

Syntax hints: ∧ wa 396 = wceq 1547 ⊤wtru 1548 ∈ wcel 2119 ≠ wne 2935 {crab 3392 Vcvv 3432 ∖ cdif 3887 ∩ cin 3889 ⊆ wss 3890 {csn 4562 {cpr 4564 ↦ cmpt 5160 ran crn 5626 ⟶wf 6488 ‘cfv 6492 (class class class)co 7363 ℂcc 11034 ℝcr 11035 0cc0 11036 1c1 11037 · cmul 11041 / cdiv 11805 (,)cioo 13296 abscabs 15194 sincsin 16026 cosccos 16027 TopOpenctopn 17382 topGenctg 17398 ℂ_fldccnfld 21354 D cdv 25855 logclog 26543

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2712 ax-rep 5206 ax-sep 5225 ax-nul 5235 ax-pow 5301 ax-pr 5369 ax-un 7685 ax-inf2 9560 ax-cnex 11092 ax-resscn 11093 ax-1cn 11094 ax-icn 11095 ax-addcl 11096 ax-addrcl 11097 ax-mulcl 11098 ax-mulrcl 11099 ax-mulcom 11100 ax-addass 11101 ax-mulass 11102 ax-distr 11103 ax-i2m1 11104 ax-1ne0 11105 ax-1rid 11106 ax-rnegex 11107 ax-rrecex 11108 ax-cnre 11109 ax-pre-lttri 11110 ax-pre-lttrn 11111 ax-pre-ltadd 11112 ax-pre-mulgt0 11113 ax-pre-sup 11114 ax-addf 11115

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2719 df-cleq 2732 df-clel 2815 df-nfc 2889 df-ne 2936 df-nel 3040 df-ral 3055 df-rex 3065 df-rmo 3345 df-reu 3346 df-rab 3393 df-v 3434 df-sbc 3731 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4269 df-if 4462 df-pw 4538 df-sn 4563 df-pr 4565 df-tp 4567 df-op 4569 df-uni 4846 df-int 4885 df-iun 4930 df-iin 4931 df-br 5080 df-opab 5142 df-mpt 5161 df-tr 5187 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-se 5579 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7320 df-ov 7366 df-oprab 7367 df-mpo 7368 df-of 7627 df-om 7814 df-1st 7938 df-2nd 7939 df-supp 8108 df-frecs 8228 df-wrecs 8259 df-recs 8308 df-rdg 8346 df-1o 8402 df-2o 8403 df-er 8640 df-map 8772 df-pm 8773 df-ixp 8843 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-fsupp 9272 df-fi 9321 df-sup 9352 df-inf 9353 df-oi 9422 df-card 9861 df-pnf 11179 df-mnf 11180 df-xr 11181 df-ltxr 11182 df-le 11183 df-sub 11377 df-neg 11378 df-div 11806 df-nn 12173 df-2 12242 df-3 12243 df-4 12244 df-5 12245 df-6 12246 df-7 12247 df-8 12248 df-9 12249 df-n0 12436 df-z 12523 df-dec 12643 df-uz 12787 df-q 12897 df-rp 12941 df-xneg 13061 df-xadd 13062 df-xmul 13063 df-ioo 13300 df-ioc 13301 df-ico 13302 df-icc 13303 df-fz 13460 df-fzo 13607 df-fl 13749 df-mod 13827 df-seq 13962 df-exp 14022 df-fac 14234 df-bc 14263 df-hash 14291 df-shft 15027 df-cj 15059 df-re 15060 df-im 15061 df-sqrt 15195 df-abs 15196 df-limsup 15431 df-clim 15448 df-rlim 15449 df-sum 15647 df-ef 16030 df-sin 16032 df-cos 16033 df-tan 16034 df-pi 16035 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17178 df-ress 17199 df-plusg 17231 df-mulr 17232 df-starv 17233 df-sca 17234 df-vsca 17235 df-ip 17236 df-tset 17237 df-ple 17238 df-ds 17240 df-unif 17241 df-hom 17242 df-cco 17243 df-rest 17383 df-topn 17384 df-0g 17402 df-gsum 17403 df-topgen 17404 df-pt 17405 df-prds 17408 df-xrs 17464 df-qtop 17469 df-imas 17470 df-xps 17472 df-mre 17546 df-mrc 17547 df-acs 17549 df-mgm 18606 df-sgrp 18685 df-mnd 18701 df-submnd 18750 df-mulg 19042 df-cntz 19290 df-cmn 19755 df-psmet 21346 df-xmet 21347 df-met 21348 df-bl 21349 df-mopn 21350 df-fbas 21351 df-fg 21352 df-cnfld 21355 df-top 22884 df-topon 22901 df-topsp 22923 df-bases 22936 df-cld 23009 df-ntr 23010 df-cls 23011 df-nei 23088 df-lp 23126 df-perf 23127 df-cn 23217 df-cnp 23218 df-t1 23304 df-haus 23305 df-cmp 23377 df-tx 23552 df-hmeo 23745 df-fil 23836 df-fm 23928 df-flim 23929 df-flf 23930 df-xms 24310 df-ms 24311 df-tms 24312 df-cncf 24870 df-limc 25858 df-dv 25859 df-log 26545

This theorem is referenced by: (None)

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