readvcot - Mathbox for Steven Nguyen

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Theorem readvcot 42796

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 11130	. . . . 5 ⊢ ℝ ∈ {ℝ, ℂ}
2	1	a1i 11	. . . 4 ⊢ (⊤ → ℝ ∈ {ℝ, ℂ})
3		fveq2 6840	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (sin‘𝑦) = (sin‘𝑥))
4	3	neeq1d 2991	. . . . . . 7 ⊢ (𝑦 = 𝑥 → ((sin‘𝑦) ≠ 0 ↔ (sin‘𝑥) ≠ 0))
5		readvcot.d	. . . . . . 7 ⊢ 𝐷 = {𝑦 ∈ ℝ ∣ (sin‘𝑦) ≠ 0}
6	4, 5	elrab2 3637	. . . . . 6 ⊢ (𝑥 ∈ 𝐷 ↔ (𝑥 ∈ ℝ ∧ (sin‘𝑥) ≠ 0))
7		resincl 16107	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ → (sin‘𝑥) ∈ ℝ)
8	7	adantr 480	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ ∧ (sin‘𝑥) ≠ 0) → (sin‘𝑥) ∈ ℝ)
9		simpr 484	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ ∧ (sin‘𝑥) ≠ 0) → (sin‘𝑥) ≠ 0)
10	8, 9	eldifsnd 4732	. . . . . 6 ⊢ ((𝑥 ∈ ℝ ∧ (sin‘𝑥) ≠ 0) → (sin‘𝑥) ∈ (ℝ ∖ {0}))
11	6, 10	sylbi 217	. . . . 5 ⊢ (𝑥 ∈ 𝐷 → (sin‘𝑥) ∈ (ℝ ∖ {0}))
12	11	adantl 481	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐷) → (sin‘𝑥) ∈ (ℝ ∖ {0}))
13		fvexd 6855	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐷) → (cos‘𝑥) ∈ V)
14		eldifi 4071	. . . . . . . . 9 ⊢ (𝑧 ∈ (ℝ ∖ {0}) → 𝑧 ∈ ℝ)
15	14	adantl 481	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑧 ∈ (ℝ ∖ {0})) → 𝑧 ∈ ℝ)
16	15	recnd 11173	. . . . . . 7 ⊢ ((⊤ ∧ 𝑧 ∈ (ℝ ∖ {0})) → 𝑧 ∈ ℂ)
17	16	abscld 15401	. . . . . 6 ⊢ ((⊤ ∧ 𝑧 ∈ (ℝ ∖ {0})) → (abs‘𝑧) ∈ ℝ)
18	17	recnd 11173	. . . . 5 ⊢ ((⊤ ∧ 𝑧 ∈ (ℝ ∖ {0})) → (abs‘𝑧) ∈ ℂ)
19		eldifsni 4735	. . . . . . 7 ⊢ (𝑧 ∈ (ℝ ∖ {0}) → 𝑧 ≠ 0)
20	19	adantl 481	. . . . . 6 ⊢ ((⊤ ∧ 𝑧 ∈ (ℝ ∖ {0})) → 𝑧 ≠ 0)
21	16, 20	absne0d 15412	. . . . 5 ⊢ ((⊤ ∧ 𝑧 ∈ (ℝ ∖ {0})) → (abs‘𝑧) ≠ 0)
22	18, 21	logcld 26534	. . . 4 ⊢ ((⊤ ∧ 𝑧 ∈ (ℝ ∖ {0})) → (log‘(abs‘𝑧)) ∈ ℂ)
23		ovexd 7402	. . . 4 ⊢ ((⊤ ∧ 𝑧 ∈ (ℝ ∖ {0})) → (1 / 𝑧) ∈ V)
24	7	recnd 11173	. . . . . 6 ⊢ (𝑥 ∈ ℝ → (sin‘𝑥) ∈ ℂ)
25	24	adantl 481	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ) → (sin‘𝑥) ∈ ℂ)
26		fvexd 6855	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ) → (cos‘𝑥) ∈ V)
27		eqid 2736	. . . . . 6 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
28		cnopn 24751	. . . . . . 7 ⊢ ℂ ∈ (TopOpen‘ℂ_fld)
29	28	a1i 11	. . . . . 6 ⊢ (⊤ → ℂ ∈ (TopOpen‘ℂ_fld))
30		ax-resscn 11095	. . . . . . . 8 ⊢ ℝ ⊆ ℂ
31		dfss2 3907	. . . . . . . 8 ⊢ (ℝ ⊆ ℂ ↔ (ℝ ∩ ℂ) = ℝ)
32	30, 31	mpbi 230	. . . . . . 7 ⊢ (ℝ ∩ ℂ) = ℝ
33	32	a1i 11	. . . . . 6 ⊢ (⊤ → (ℝ ∩ ℂ) = ℝ)
34		sincl 16093	. . . . . . 7 ⊢ (𝑥 ∈ ℂ → (sin‘𝑥) ∈ ℂ)
35	34	adantl 481	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → (sin‘𝑥) ∈ ℂ)
36		fvexd 6855	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → (cos‘𝑥) ∈ V)
37		dvsin 25949	. . . . . . 7 ⊢ (ℂ D sin) = cos
38		sinf 16091	. . . . . . . . . 10 ⊢ sin:ℂ⟶ℂ
39	38	a1i 11	. . . . . . . . 9 ⊢ (⊤ → sin:ℂ⟶ℂ)
40	39	feqmptd 6908	. . . . . . . 8 ⊢ (⊤ → sin = (𝑥 ∈ ℂ ↦ (sin‘𝑥)))
41	40	oveq2d 7383	. . . . . . 7 ⊢ (⊤ → (ℂ D sin) = (ℂ D (𝑥 ∈ ℂ ↦ (sin‘𝑥))))
42		cosf 16092	. . . . . . . . 9 ⊢ cos:ℂ⟶ℂ
43	42	a1i 11	. . . . . . . 8 ⊢ (⊤ → cos:ℂ⟶ℂ)
44	43	feqmptd 6908	. . . . . . 7 ⊢ (⊤ → cos = (𝑥 ∈ ℂ ↦ (cos‘𝑥)))
45	37, 41, 44	3eqtr3a 2795	. . . . . 6 ⊢ (⊤ → (ℂ D (𝑥 ∈ ℂ ↦ (sin‘𝑥))) = (𝑥 ∈ ℂ ↦ (cos‘𝑥)))
46	27, 2, 29, 33, 35, 36, 45	dvmptres3 25923	. . . . 5 ⊢ (⊤ → (ℝ D (𝑥 ∈ ℝ ↦ (sin‘𝑥))) = (𝑥 ∈ ℝ ↦ (cos‘𝑥)))
47	5	ssrab3 4022	. . . . . 6 ⊢ 𝐷 ⊆ ℝ
48	47	a1i 11	. . . . 5 ⊢ (⊤ → 𝐷 ⊆ ℝ)
49		tgioo4 24770	. . . . 5 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
50	5	resuppsinopn 42795	. . . . . 6 ⊢ 𝐷 ∈ (topGen‘ran (,))
51	50	a1i 11	. . . . 5 ⊢ (⊤ → 𝐷 ∈ (topGen‘ran (,)))
52	2, 25, 26, 46, 48, 49, 27, 51	dvmptres 25930	. . . 4 ⊢ (⊤ → (ℝ D (𝑥 ∈ 𝐷 ↦ (sin‘𝑥))) = (𝑥 ∈ 𝐷 ↦ (cos‘𝑥)))
53		eqid 2736	. . . . . 6 ⊢ (ℝ ∖ {0}) = (ℝ ∖ {0})
54	53	readvrec 42794	. . . . 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 6845	. . . 4 ⊢ (𝑧 = (sin‘𝑥) → (log‘(abs‘𝑧)) = (log‘(abs‘(sin‘𝑥))))
57		oveq2 7375	. . . 4 ⊢ (𝑧 = (sin‘𝑥) → (1 / 𝑧) = (1 / (sin‘𝑥)))
58	2, 2, 12, 13, 22, 23, 52, 55, 56, 57	dvmptco 25939	. . 3 ⊢ (⊤ → (ℝ D (𝑥 ∈ 𝐷 ↦ (log‘(abs‘(sin‘𝑥))))) = (𝑥 ∈ 𝐷 ↦ ((1 / (sin‘𝑥)) · (cos‘𝑥))))
59	58	mptru 1549	. 2 ⊢ (ℝ D (𝑥 ∈ 𝐷 ↦ (log‘(abs‘(sin‘𝑥))))) = (𝑥 ∈ 𝐷 ↦ ((1 / (sin‘𝑥)) · (cos‘𝑥)))
60	6	simplbi 496	. . . . . 6 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ∈ ℝ)
61	60	recoscld 16111	. . . . 5 ⊢ (𝑥 ∈ 𝐷 → (cos‘𝑥) ∈ ℝ)
62	61	recnd 11173	. . . 4 ⊢ (𝑥 ∈ 𝐷 → (cos‘𝑥) ∈ ℂ)
63	6, 8	sylbi 217	. . . . 5 ⊢ (𝑥 ∈ 𝐷 → (sin‘𝑥) ∈ ℝ)
64	63	recnd 11173	. . . 4 ⊢ (𝑥 ∈ 𝐷 → (sin‘𝑥) ∈ ℂ)
65	6, 9	sylbi 217	. . . 4 ⊢ (𝑥 ∈ 𝐷 → (sin‘𝑥) ≠ 0)
66	62, 64, 65	divrec2d 11935	. . 3 ⊢ (𝑥 ∈ 𝐷 → ((cos‘𝑥) / (sin‘𝑥)) = ((1 / (sin‘𝑥)) · (cos‘𝑥)))
67	66	mpteq2ia 5180	. 2 ⊢ (𝑥 ∈ 𝐷 ↦ ((cos‘𝑥) / (sin‘𝑥))) = (𝑥 ∈ 𝐷 ↦ ((1 / (sin‘𝑥)) · (cos‘𝑥)))
68	59, 67	eqtr4i 2762	1 ⊢ (ℝ D (𝑥 ∈ 𝐷 ↦ (log‘(abs‘(sin‘𝑥))))) = (𝑥 ∈ 𝐷 ↦ ((cos‘𝑥) / (sin‘𝑥)))

Colors of variables: wff setvar class

Syntax hints: ∧ wa 395 = wceq 1542 ⊤wtru 1543 ∈ wcel 2114 ≠ wne 2932 {crab 3389 Vcvv 3429 ∖ cdif 3886 ∩ cin 3888 ⊆ wss 3889 {csn 4567 {cpr 4569 ↦ cmpt 5166 ran crn 5632 ⟶wf 6494 ‘cfv 6498 (class class class)co 7367 ℂcc 11036 ℝcr 11037 0cc0 11038 1c1 11039 · cmul 11043 / cdiv 11807 (,)cioo 13298 abscabs 15196 sincsin 16028 cosccos 16029 TopOpenctopn 17384 topGenctg 17400 ℂ_fldccnfld 21352 D cdv 25830 logclog 26518

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-inf2 9562 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116 ax-addf 11117

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-iin 4936 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-isom 6507 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-of 7631 df-om 7818 df-1st 7942 df-2nd 7943 df-supp 8111 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-2o 8406 df-er 8643 df-map 8775 df-pm 8776 df-ixp 8846 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-fsupp 9275 df-fi 9324 df-sup 9355 df-inf 9356 df-oi 9425 df-card 9863 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-4 12246 df-5 12247 df-6 12248 df-7 12249 df-8 12250 df-9 12251 df-n0 12438 df-z 12525 df-dec 12645 df-uz 12789 df-q 12899 df-rp 12943 df-xneg 13063 df-xadd 13064 df-xmul 13065 df-ioo 13302 df-ioc 13303 df-ico 13304 df-icc 13305 df-fz 13462 df-fzo 13609 df-fl 13751 df-mod 13829 df-seq 13964 df-exp 14024 df-fac 14236 df-bc 14265 df-hash 14293 df-shft 15029 df-cj 15061 df-re 15062 df-im 15063 df-sqrt 15197 df-abs 15198 df-limsup 15433 df-clim 15450 df-rlim 15451 df-sum 15649 df-ef 16032 df-sin 16034 df-cos 16035 df-tan 16036 df-pi 16037 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-hom 17244 df-cco 17245 df-rest 17385 df-topn 17386 df-0g 17404 df-gsum 17405 df-topgen 17406 df-pt 17407 df-prds 17410 df-xrs 17466 df-qtop 17471 df-imas 17472 df-xps 17474 df-mre 17548 df-mrc 17549 df-acs 17551 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-submnd 18752 df-mulg 19044 df-cntz 19292 df-cmn 19757 df-psmet 21344 df-xmet 21345 df-met 21346 df-bl 21347 df-mopn 21348 df-fbas 21349 df-fg 21350 df-cnfld 21353 df-top 22859 df-topon 22876 df-topsp 22898 df-bases 22911 df-cld 22984 df-ntr 22985 df-cls 22986 df-nei 23063 df-lp 23101 df-perf 23102 df-cn 23192 df-cnp 23193 df-t1 23279 df-haus 23280 df-cmp 23352 df-tx 23527 df-hmeo 23720 df-fil 23811 df-fm 23903 df-flim 23904 df-flf 23905 df-xms 24285 df-ms 24286 df-tms 24287 df-cncf 24845 df-limc 25833 df-dv 25834 df-log 26520

This theorem is referenced by: (None)

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