readvcot - Mathbox for Steven Nguyen

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Theorem readvcot 42937

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 11162	. . . . 5 ⊢ ℝ ∈ {ℝ, ℂ}
2	1	a1i 11	. . . 4 ⊢ (⊤ → ℝ ∈ {ℝ, ℂ})
3		fveq2 6863	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (sin‘𝑦) = (sin‘𝑥))
4	3	neeq1d 3015	. . . . . . 7 ⊢ (𝑦 = 𝑥 → ((sin‘𝑦) ≠ 0 ↔ (sin‘𝑥) ≠ 0))
5		readvcot.d	. . . . . . 7 ⊢ 𝐷 = {𝑦 ∈ ℝ ∣ (sin‘𝑦) ≠ 0}
6	4, 5	elrab2 3653	. . . . . 6 ⊢ (𝑥 ∈ 𝐷 ↔ (𝑥 ∈ ℝ ∧ (sin‘𝑥) ≠ 0))
7		resincl 16155	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ → (sin‘𝑥) ∈ ℝ)
8	7	adantr 484	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ ∧ (sin‘𝑥) ≠ 0) → (sin‘𝑥) ∈ ℝ)
9		simpr 488	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ ∧ (sin‘𝑥) ≠ 0) → (sin‘𝑥) ≠ 0)
10	8, 9	eldifsnd 4746	. . . . . 6 ⊢ ((𝑥 ∈ ℝ ∧ (sin‘𝑥) ≠ 0) → (sin‘𝑥) ∈ (ℝ ∖ {0}))
11	6, 10	sylbi 219	. . . . 5 ⊢ (𝑥 ∈ 𝐷 → (sin‘𝑥) ∈ (ℝ ∖ {0}))
12	11	adantl 485	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐷) → (sin‘𝑥) ∈ (ℝ ∖ {0}))
13		fvexd 6878	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐷) → (cos‘𝑥) ∈ V)
14		eldifi 4084	. . . . . . . . 9 ⊢ (𝑧 ∈ (ℝ ∖ {0}) → 𝑧 ∈ ℝ)
15	14	adantl 485	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑧 ∈ (ℝ ∖ {0})) → 𝑧 ∈ ℝ)
16	15	recnd 11207	. . . . . . 7 ⊢ ((⊤ ∧ 𝑧 ∈ (ℝ ∖ {0})) → 𝑧 ∈ ℂ)
17	16	abscld 15449	. . . . . 6 ⊢ ((⊤ ∧ 𝑧 ∈ (ℝ ∖ {0})) → (abs‘𝑧) ∈ ℝ)
18	17	recnd 11207	. . . . 5 ⊢ ((⊤ ∧ 𝑧 ∈ (ℝ ∖ {0})) → (abs‘𝑧) ∈ ℂ)
19		eldifsni 4749	. . . . . . 7 ⊢ (𝑧 ∈ (ℝ ∖ {0}) → 𝑧 ≠ 0)
20	19	adantl 485	. . . . . 6 ⊢ ((⊤ ∧ 𝑧 ∈ (ℝ ∖ {0})) → 𝑧 ≠ 0)
21	16, 20	absne0d 15460	. . . . 5 ⊢ ((⊤ ∧ 𝑧 ∈ (ℝ ∖ {0})) → (abs‘𝑧) ≠ 0)
22	18, 21	logcld 26612	. . . 4 ⊢ ((⊤ ∧ 𝑧 ∈ (ℝ ∖ {0})) → (log‘(abs‘𝑧)) ∈ ℂ)
23		ovexd 7427	. . . 4 ⊢ ((⊤ ∧ 𝑧 ∈ (ℝ ∖ {0})) → (1 / 𝑧) ∈ V)
24	7	recnd 11207	. . . . . 6 ⊢ (𝑥 ∈ ℝ → (sin‘𝑥) ∈ ℂ)
25	24	adantl 485	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ) → (sin‘𝑥) ∈ ℂ)
26		fvexd 6878	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ) → (cos‘𝑥) ∈ V)
27		eqid 2761	. . . . . 6 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
28		cnopn 24826	. . . . . . 7 ⊢ ℂ ∈ (TopOpen‘ℂ_fld)
29	28	a1i 11	. . . . . 6 ⊢ (⊤ → ℂ ∈ (TopOpen‘ℂ_fld))
30		ax-resscn 11127	. . . . . . . 8 ⊢ ℝ ⊆ ℂ
31		dfss2 3922	. . . . . . . 8 ⊢ (ℝ ⊆ ℂ ↔ (ℝ ∩ ℂ) = ℝ)
32	30, 31	mpbi 232	. . . . . . 7 ⊢ (ℝ ∩ ℂ) = ℝ
33	32	a1i 11	. . . . . 6 ⊢ (⊤ → (ℝ ∩ ℂ) = ℝ)
34		sincl 16141	. . . . . . 7 ⊢ (𝑥 ∈ ℂ → (sin‘𝑥) ∈ ℂ)
35	34	adantl 485	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → (sin‘𝑥) ∈ ℂ)
36		fvexd 6878	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → (cos‘𝑥) ∈ V)
37		dvsin 26024	. . . . . . 7 ⊢ (ℂ D sin) = cos
38		sinf 16139	. . . . . . . . . 10 ⊢ sin:ℂ⟶ℂ
39	38	a1i 11	. . . . . . . . 9 ⊢ (⊤ → sin:ℂ⟶ℂ)
40	39	feqmptd 6931	. . . . . . . 8 ⊢ (⊤ → sin = (𝑥 ∈ ℂ ↦ (sin‘𝑥)))
41	40	oveq2d 7408	. . . . . . 7 ⊢ (⊤ → (ℂ D sin) = (ℂ D (𝑥 ∈ ℂ ↦ (sin‘𝑥))))
42		cosf 16140	. . . . . . . . 9 ⊢ cos:ℂ⟶ℂ
43	42	a1i 11	. . . . . . . 8 ⊢ (⊤ → cos:ℂ⟶ℂ)
44	43	feqmptd 6931	. . . . . . 7 ⊢ (⊤ → cos = (𝑥 ∈ ℂ ↦ (cos‘𝑥)))
45	37, 41, 44	3eqtr3a 2820	. . . . . 6 ⊢ (⊤ → (ℂ D (𝑥 ∈ ℂ ↦ (sin‘𝑥))) = (𝑥 ∈ ℂ ↦ (cos‘𝑥)))
46	27, 2, 29, 33, 35, 36, 45	dvmptres3 25998	. . . . 5 ⊢ (⊤ → (ℝ D (𝑥 ∈ ℝ ↦ (sin‘𝑥))) = (𝑥 ∈ ℝ ↦ (cos‘𝑥)))
47	5	ssrab3 4035	. . . . . 6 ⊢ 𝐷 ⊆ ℝ
48	47	a1i 11	. . . . 5 ⊢ (⊤ → 𝐷 ⊆ ℝ)
49		tgioo4 24845	. . . . 5 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
50	5	resuppsinopn 42936	. . . . . 6 ⊢ 𝐷 ∈ (topGen‘ran (,))
51	50	a1i 11	. . . . 5 ⊢ (⊤ → 𝐷 ∈ (topGen‘ran (,)))
52	2, 25, 26, 46, 48, 49, 27, 51	dvmptres 26005	. . . 4 ⊢ (⊤ → (ℝ D (𝑥 ∈ 𝐷 ↦ (sin‘𝑥))) = (𝑥 ∈ 𝐷 ↦ (cos‘𝑥)))
53		eqid 2761	. . . . . 6 ⊢ (ℝ ∖ {0}) = (ℝ ∖ {0})
54	53	readvrec 42935	. . . . 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 6868	. . . 4 ⊢ (𝑧 = (sin‘𝑥) → (log‘(abs‘𝑧)) = (log‘(abs‘(sin‘𝑥))))
57		oveq2 7400	. . . 4 ⊢ (𝑧 = (sin‘𝑥) → (1 / 𝑧) = (1 / (sin‘𝑥)))
58	2, 2, 12, 13, 22, 23, 52, 55, 56, 57	dvmptco 26014	. . 3 ⊢ (⊤ → (ℝ D (𝑥 ∈ 𝐷 ↦ (log‘(abs‘(sin‘𝑥))))) = (𝑥 ∈ 𝐷 ↦ ((1 / (sin‘𝑥)) · (cos‘𝑥))))
59	58	mptru 1566	. 2 ⊢ (ℝ D (𝑥 ∈ 𝐷 ↦ (log‘(abs‘(sin‘𝑥))))) = (𝑥 ∈ 𝐷 ↦ ((1 / (sin‘𝑥)) · (cos‘𝑥)))
60	6	simplbi 500	. . . . . 6 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ∈ ℝ)
61	60	recoscld 16159	. . . . 5 ⊢ (𝑥 ∈ 𝐷 → (cos‘𝑥) ∈ ℝ)
62	61	recnd 11207	. . . 4 ⊢ (𝑥 ∈ 𝐷 → (cos‘𝑥) ∈ ℂ)
63	6, 8	sylbi 219	. . . . 5 ⊢ (𝑥 ∈ 𝐷 → (sin‘𝑥) ∈ ℝ)
64	63	recnd 11207	. . . 4 ⊢ (𝑥 ∈ 𝐷 → (sin‘𝑥) ∈ ℂ)
65	6, 9	sylbi 219	. . . 4 ⊢ (𝑥 ∈ 𝐷 → (sin‘𝑥) ≠ 0)
66	62, 64, 65	divrec2d 11968	. . 3 ⊢ (𝑥 ∈ 𝐷 → ((cos‘𝑥) / (sin‘𝑥)) = ((1 / (sin‘𝑥)) · (cos‘𝑥)))
67	66	mpteq2ia 5194	. 2 ⊢ (𝑥 ∈ 𝐷 ↦ ((cos‘𝑥) / (sin‘𝑥))) = (𝑥 ∈ 𝐷 ↦ ((1 / (sin‘𝑥)) · (cos‘𝑥)))
68	59, 67	eqtr4i 2787	1 ⊢ (ℝ D (𝑥 ∈ 𝐷 ↦ (log‘(abs‘(sin‘𝑥))))) = (𝑥 ∈ 𝐷 ↦ ((cos‘𝑥) / (sin‘𝑥)))

Colors of variables: wff setvar class

Syntax hints: ∧ wa 399 = wceq 1559 ⊤wtru 1560 ∈ wcel 2141 ≠ wne 2956 {crab 3413 Vcvv 3453 ∖ cdif 3901 ∩ cin 3903 ⊆ wss 3904 {csn 4581 {cpr 4583 ↦ cmpt 5180 ran crn 5646 ⟶wf 6513 ‘cfv 6517 (class class class)co 7392 ℂcc 11068 ℝcr 11069 0cc0 11070 1c1 11071 · cmul 11075 / cdiv 11841 (,)cioo 13346 abscabs 15244 sincsin 16076 cosccos 16077 TopOpenctopn 17433 topGenctg 17449 ℂ_fldccnfld 21404 D cdv 25905 logclog 26596

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5226 ax-sep 5245 ax-nul 5255 ax-pow 5321 ax-pr 5389 ax-un 7714 ax-inf2 9593 ax-cnex 11126 ax-resscn 11127 ax-1cn 11128 ax-icn 11129 ax-addcl 11130 ax-addrcl 11131 ax-mulcl 11132 ax-mulrcl 11133 ax-mulcom 11134 ax-addass 11135 ax-mulass 11136 ax-distr 11137 ax-i2m1 11138 ax-1ne0 11139 ax-1rid 11140 ax-rnegex 11141 ax-rrecex 11142 ax-cnre 11143 ax-pre-lttri 11144 ax-pre-lttrn 11145 ax-pre-ltadd 11146 ax-pre-mulgt0 11147 ax-pre-sup 11148 ax-addf 11149

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4582 df-pr 4584 df-tp 4586 df-op 4588 df-uni 4865 df-int 4905 df-iun 4950 df-iin 4951 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5540 df-eprel 5545 df-po 5553 df-so 5554 df-fr 5598 df-se 5599 df-we 5600 df-xp 5651 df-rel 5652 df-cnv 5653 df-co 5654 df-dm 5655 df-rn 5656 df-res 5657 df-ima 5658 df-pred 6284 df-ord 6345 df-on 6346 df-lim 6347 df-suc 6348 df-iota 6473 df-fun 6519 df-fn 6520 df-f 6521 df-f1 6522 df-fo 6523 df-f1o 6524 df-fv 6525 df-isom 6526 df-riota 7349 df-ov 7395 df-oprab 7396 df-mpo 7397 df-of 7656 df-om 7843 df-1st 7966 df-2nd 7967 df-supp 8136 df-frecs 8257 df-wrecs 8288 df-recs 8337 df-rdg 8376 df-1o 8432 df-2o 8433 df-er 8673 df-map 8805 df-pm 8806 df-ixp 8876 df-en 8924 df-dom 8925 df-sdom 8926 df-fin 8927 df-fsupp 9305 df-fi 9354 df-sup 9385 df-inf 9386 df-oi 9455 df-card 9894 df-pnf 11215 df-mnf 11216 df-xr 11217 df-ltxr 11218 df-le 11219 df-sub 11413 df-neg 11414 df-div 11842 df-nn 12208 df-2 12277 df-3 12278 df-4 12279 df-5 12280 df-6 12281 df-7 12282 df-8 12283 df-9 12284 df-n0 12479 df-z 12566 df-dec 12686 df-uz 12837 df-q 12947 df-rp 12991 df-xneg 13111 df-xadd 13112 df-xmul 13113 df-ioo 13350 df-ioc 13351 df-ico 13352 df-icc 13353 df-fz 13510 df-fzo 13657 df-fl 13799 df-mod 13877 df-seq 14012 df-exp 14072 df-fac 14284 df-bc 14313 df-hash 14341 df-shft 15077 df-cj 15109 df-re 15110 df-im 15111 df-sqrt 15245 df-abs 15246 df-limsup 15481 df-clim 15498 df-rlim 15499 df-sum 15697 df-ef 16080 df-sin 16082 df-cos 16083 df-tan 16084 df-pi 16085 df-struct 17166 df-sets 17183 df-slot 17201 df-ndx 17213 df-base 17229 df-ress 17250 df-plusg 17282 df-mulr 17283 df-starv 17284 df-sca 17285 df-vsca 17286 df-ip 17287 df-tset 17288 df-ple 17289 df-ds 17291 df-unif 17292 df-hom 17293 df-cco 17294 df-rest 17434 df-topn 17435 df-0g 17453 df-gsum 17454 df-topgen 17455 df-pt 17456 df-prds 17459 df-xrs 17515 df-qtop 17520 df-imas 17521 df-xps 17523 df-mre 17597 df-mrc 17598 df-acs 17600 df-mgm 18657 df-sgrp 18736 df-mnd 18752 df-submnd 18801 df-mulg 19093 df-cntz 19340 df-cmn 19805 df-psmet 21396 df-xmet 21397 df-met 21398 df-bl 21399 df-mopn 21400 df-fbas 21401 df-fg 21402 df-cnfld 21405 df-top 22934 df-topon 22951 df-topsp 22973 df-bases 22986 df-cld 23059 df-ntr 23060 df-cls 23061 df-nei 23138 df-lp 23176 df-perf 23177 df-cn 23267 df-cnp 23268 df-t1 23354 df-haus 23355 df-cmp 23427 df-tx 23602 df-hmeo 23795 df-fil 23886 df-fm 23978 df-flim 23979 df-flf 23980 df-xms 24360 df-ms 24361 df-tms 24362 df-cncf 24920 df-limc 25908 df-dv 25909 df-log 26598

This theorem is referenced by: (None)

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