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Theorem dvrecg 24824

Description: Derivative of the reciprocal of a function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

**Hypotheses**
Ref	Expression
dvrecg.s	⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})
dvrecg.a	⊢ (𝜑 → 𝐴 ∈ ℂ)
dvrecg.b	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ (ℂ ∖ {0}))
dvrecg.c	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐶 ∈ 𝑉)
dvrecg.db	⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐵)) = (𝑥 ∈ 𝑋 ↦ 𝐶))

**Assertion**
Ref	Expression
dvrecg	⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐵))) = (𝑥 ∈ 𝑋 ↦ -((𝐴 · 𝐶) / (𝐵↑2))))

Distinct variable groups: 𝑥,𝐴 𝑥,𝑆 𝑥,𝑉 𝑥,𝑋 𝜑,𝑥 Allowed substitution hints: 𝐵(𝑥) 𝐶(𝑥)

Proof of Theorem dvrecg Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dvrecg.s	. . 3 ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})
2		cnelprrecn 10787	. . . 4 ⊢ ℂ ∈ {ℝ, ℂ}
3	2	a1i 11	. . 3 ⊢ (𝜑 → ℂ ∈ {ℝ, ℂ})
4		dvrecg.b	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ (ℂ ∖ {0}))
5		dvrecg.c	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐶 ∈ 𝑉)
6		dvrecg.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℂ)
7	6	adantr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {0})) → 𝐴 ∈ ℂ)
8		eldifi 4027	. . . . 5 ⊢ (𝑦 ∈ (ℂ ∖ {0}) → 𝑦 ∈ ℂ)
9	8	adantl 485	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {0})) → 𝑦 ∈ ℂ)
10		eldifsni 4689	. . . . 5 ⊢ (𝑦 ∈ (ℂ ∖ {0}) → 𝑦 ≠ 0)
11	10	adantl 485	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {0})) → 𝑦 ≠ 0)
12	7, 9, 11	divcld 11573	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {0})) → (𝐴 / 𝑦) ∈ ℂ)
13	9	sqcld 13679	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {0})) → (𝑦↑2) ∈ ℂ)
14		2z 12174	. . . . . . 7 ⊢ 2 ∈ ℤ
15	14	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {0})) → 2 ∈ ℤ)
16	9, 11, 15	expne0d 13687	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {0})) → (𝑦↑2) ≠ 0)
17	7, 13, 16	divcld 11573	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {0})) → (𝐴 / (𝑦↑2)) ∈ ℂ)
18	17	negcld 11141	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {0})) → -(𝐴 / (𝑦↑2)) ∈ ℂ)
19		dvrecg.db	. . 3 ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐵)) = (𝑥 ∈ 𝑋 ↦ 𝐶))
20		dvrec 24806	. . . 4 ⊢ (𝐴 ∈ ℂ → (ℂ D (𝑦 ∈ (ℂ ∖ {0}) ↦ (𝐴 / 𝑦))) = (𝑦 ∈ (ℂ ∖ {0}) ↦ -(𝐴 / (𝑦↑2))))
21	6, 20	syl 17	. . 3 ⊢ (𝜑 → (ℂ D (𝑦 ∈ (ℂ ∖ {0}) ↦ (𝐴 / 𝑦))) = (𝑦 ∈ (ℂ ∖ {0}) ↦ -(𝐴 / (𝑦↑2))))
22		oveq2 7199	. . 3 ⊢ (𝑦 = 𝐵 → (𝐴 / 𝑦) = (𝐴 / 𝐵))
23		oveq1 7198	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦↑2) = (𝐵↑2))
24	23	oveq2d 7207	. . . 4 ⊢ (𝑦 = 𝐵 → (𝐴 / (𝑦↑2)) = (𝐴 / (𝐵↑2)))
25	24	negeqd 11037	. . 3 ⊢ (𝑦 = 𝐵 → -(𝐴 / (𝑦↑2)) = -(𝐴 / (𝐵↑2)))
26	1, 3, 4, 5, 12, 18, 19, 21, 22, 25	dvmptco 24823	. 2 ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐵))) = (𝑥 ∈ 𝑋 ↦ (-(𝐴 / (𝐵↑2)) · 𝐶)))
27	6	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ)
28		eldifi 4027	. . . . . . . 8 ⊢ (𝐵 ∈ (ℂ ∖ {0}) → 𝐵 ∈ ℂ)
29	4, 28	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ ℂ)
30	29	sqcld 13679	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐵↑2) ∈ ℂ)
31		eldifsn 4686	. . . . . . . . 9 ⊢ (𝐵 ∈ (ℂ ∖ {0}) ↔ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0))
32	4, 31	sylib 221	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0))
33	32	simprd 499	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ≠ 0)
34	14	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 2 ∈ ℤ)
35	29, 33, 34	expne0d 13687	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐵↑2) ≠ 0)
36	27, 30, 35	divcld 11573	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐴 / (𝐵↑2)) ∈ ℂ)
37	1, 29, 5, 19	dvmptcl 24810	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐶 ∈ ℂ)
38	36, 37	mulneg1d 11250	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (-(𝐴 / (𝐵↑2)) · 𝐶) = -((𝐴 / (𝐵↑2)) · 𝐶))
39	27, 37, 30, 35	div23d 11610	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝐴 · 𝐶) / (𝐵↑2)) = ((𝐴 / (𝐵↑2)) · 𝐶))
40	39	eqcomd 2742	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝐴 / (𝐵↑2)) · 𝐶) = ((𝐴 · 𝐶) / (𝐵↑2)))
41	40	negeqd 11037	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → -((𝐴 / (𝐵↑2)) · 𝐶) = -((𝐴 · 𝐶) / (𝐵↑2)))
42	38, 41	eqtrd 2771	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (-(𝐴 / (𝐵↑2)) · 𝐶) = -((𝐴 · 𝐶) / (𝐵↑2)))
43	42	mpteq2dva 5135	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (-(𝐴 / (𝐵↑2)) · 𝐶)) = (𝑥 ∈ 𝑋 ↦ -((𝐴 · 𝐶) / (𝐵↑2))))
44	26, 43	eqtrd 2771	1 ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐵))) = (𝑥 ∈ 𝑋 ↦ -((𝐴 · 𝐶) / (𝐵↑2))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ≠ wne 2932 ∖ cdif 3850 {csn 4527 {cpr 4529 ↦ cmpt 5120 (class class class)co 7191 ℂcc 10692 ℝcr 10693 0cc0 10694 · cmul 10699 -cneg 11028 / cdiv 11454 2c2 11850 ℤcz 12141 ↑cexp 13600 D cdv 24714

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 ax-pre-sup 10772 ax-addf 10773 ax-mulf 10774

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-iun 4892 df-iin 4893 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-se 5495 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-of 7447 df-om 7623 df-1st 7739 df-2nd 7740 df-supp 7882 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-2o 8181 df-er 8369 df-map 8488 df-pm 8489 df-ixp 8557 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-fsupp 8964 df-fi 9005 df-sup 9036 df-inf 9037 df-oi 9104 df-card 9520 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-div 11455 df-nn 11796 df-2 11858 df-3 11859 df-4 11860 df-5 11861 df-6 11862 df-7 11863 df-8 11864 df-9 11865 df-n0 12056 df-z 12142 df-dec 12259 df-uz 12404 df-q 12510 df-rp 12552 df-xneg 12669 df-xadd 12670 df-xmul 12671 df-icc 12907 df-fz 13061 df-fzo 13204 df-seq 13540 df-exp 13601 df-hash 13862 df-cj 14627 df-re 14628 df-im 14629 df-sqrt 14763 df-abs 14764 df-struct 16668 df-ndx 16669 df-slot 16670 df-base 16672 df-sets 16673 df-ress 16674 df-plusg 16762 df-mulr 16763 df-starv 16764 df-sca 16765 df-vsca 16766 df-ip 16767 df-tset 16768 df-ple 16769 df-ds 16771 df-unif 16772 df-hom 16773 df-cco 16774 df-rest 16881 df-topn 16882 df-0g 16900 df-gsum 16901 df-topgen 16902 df-pt 16903 df-prds 16906 df-xrs 16961 df-qtop 16966 df-imas 16967 df-xps 16969 df-mre 17043 df-mrc 17044 df-acs 17046 df-mgm 18068 df-sgrp 18117 df-mnd 18128 df-submnd 18173 df-mulg 18443 df-cntz 18665 df-cmn 19126 df-psmet 20309 df-xmet 20310 df-met 20311 df-bl 20312 df-mopn 20313 df-fbas 20314 df-fg 20315 df-cnfld 20318 df-top 21745 df-topon 21762 df-topsp 21784 df-bases 21797 df-cld 21870 df-ntr 21871 df-cls 21872 df-nei 21949 df-lp 21987 df-perf 21988 df-cn 22078 df-cnp 22079 df-t1 22165 df-haus 22166 df-tx 22413 df-hmeo 22606 df-fil 22697 df-fm 22789 df-flim 22790 df-flf 22791 df-xms 23172 df-ms 23173 df-tms 23174 df-cncf 23729 df-limc 24717 df-dv 24718

This theorem is referenced by: dvmptdiv 24825

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