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Theorem dvrecg 25875

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 11102	. . . 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 480	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {0})) → 𝐴 ∈ ℂ)
8		eldifi 4082	. . . . 5 ⊢ (𝑦 ∈ (ℂ ∖ {0}) → 𝑦 ∈ ℂ)
9	8	adantl 481	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {0})) → 𝑦 ∈ ℂ)
10		eldifsni 4741	. . . . 5 ⊢ (𝑦 ∈ (ℂ ∖ {0}) → 𝑦 ≠ 0)
11	10	adantl 481	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {0})) → 𝑦 ≠ 0)
12	7, 9, 11	divcld 11900	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {0})) → (𝐴 / 𝑦) ∈ ℂ)
13	9	sqcld 14051	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {0})) → (𝑦↑2) ∈ ℂ)
14		2z 12507	. . . . . . 7 ⊢ 2 ∈ ℤ
15	14	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {0})) → 2 ∈ ℤ)
16	9, 11, 15	expne0d 14059	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {0})) → (𝑦↑2) ≠ 0)
17	7, 13, 16	divcld 11900	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {0})) → (𝐴 / (𝑦↑2)) ∈ ℂ)
18	17	negcld 11462	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {0})) → -(𝐴 / (𝑦↑2)) ∈ ℂ)
19		dvrecg.db	. . 3 ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐵)) = (𝑥 ∈ 𝑋 ↦ 𝐶))
20		dvrec 25857	. . . 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 7357	. . 3 ⊢ (𝑦 = 𝐵 → (𝐴 / 𝑦) = (𝐴 / 𝐵))
23		oveq1 7356	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦↑2) = (𝐵↑2))
24	23	oveq2d 7365	. . . 4 ⊢ (𝑦 = 𝐵 → (𝐴 / (𝑦↑2)) = (𝐴 / (𝐵↑2)))
25	24	negeqd 11357	. . 3 ⊢ (𝑦 = 𝐵 → -(𝐴 / (𝑦↑2)) = -(𝐴 / (𝐵↑2)))
26	1, 3, 4, 5, 12, 18, 19, 21, 22, 25	dvmptco 25874	. 2 ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐵))) = (𝑥 ∈ 𝑋 ↦ (-(𝐴 / (𝐵↑2)) · 𝐶)))
27	6	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ)
28		eldifi 4082	. . . . . . . 8 ⊢ (𝐵 ∈ (ℂ ∖ {0}) → 𝐵 ∈ ℂ)
29	4, 28	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ ℂ)
30	29	sqcld 14051	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐵↑2) ∈ ℂ)
31		eldifsn 4737	. . . . . . . . 9 ⊢ (𝐵 ∈ (ℂ ∖ {0}) ↔ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0))
32	4, 31	sylib 218	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0))
33	32	simprd 495	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ≠ 0)
34	14	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 2 ∈ ℤ)
35	29, 33, 34	expne0d 14059	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐵↑2) ≠ 0)
36	27, 30, 35	divcld 11900	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐴 / (𝐵↑2)) ∈ ℂ)
37	1, 29, 5, 19	dvmptcl 25861	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐶 ∈ ℂ)
38	36, 37	mulneg1d 11573	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (-(𝐴 / (𝐵↑2)) · 𝐶) = -((𝐴 / (𝐵↑2)) · 𝐶))
39	27, 37, 30, 35	div23d 11937	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝐴 · 𝐶) / (𝐵↑2)) = ((𝐴 / (𝐵↑2)) · 𝐶))
40	39	eqcomd 2735	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝐴 / (𝐵↑2)) · 𝐶) = ((𝐴 · 𝐶) / (𝐵↑2)))
41	40	negeqd 11357	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → -((𝐴 / (𝐵↑2)) · 𝐶) = -((𝐴 · 𝐶) / (𝐵↑2)))
42	38, 41	eqtrd 2764	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (-(𝐴 / (𝐵↑2)) · 𝐶) = -((𝐴 · 𝐶) / (𝐵↑2)))
43	42	mpteq2dva 5185	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (-(𝐴 / (𝐵↑2)) · 𝐶)) = (𝑥 ∈ 𝑋 ↦ -((𝐴 · 𝐶) / (𝐵↑2))))
44	26, 43	eqtrd 2764	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 395 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 ∖ cdif 3900 {csn 4577 {cpr 4579 ↦ cmpt 5173 (class class class)co 7349 ℂcc 11007 ℝcr 11008 0cc0 11009 · cmul 11014 -cneg 11348 / cdiv 11777 2c2 12183 ℤcz 12471 ↑cexp 13968 D cdv 25762

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5218 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-pre-sup 11087 ax-addf 11088

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3343 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-uni 4859 df-int 4897 df-iun 4943 df-iin 4944 df-br 5093 df-opab 5155 df-mpt 5174 df-tr 5200 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-se 5573 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6249 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-isom 6491 df-riota 7306 df-ov 7352 df-oprab 7353 df-mpo 7354 df-of 7613 df-om 7800 df-1st 7924 df-2nd 7925 df-supp 8094 df-frecs 8214 df-wrecs 8245 df-recs 8294 df-rdg 8332 df-1o 8388 df-2o 8389 df-er 8625 df-map 8755 df-pm 8756 df-ixp 8825 df-en 8873 df-dom 8874 df-sdom 8875 df-fin 8876 df-fsupp 9252 df-fi 9301 df-sup 9332 df-inf 9333 df-oi 9402 df-card 9835 df-pnf 11151 df-mnf 11152 df-xr 11153 df-ltxr 11154 df-le 11155 df-sub 11349 df-neg 11350 df-div 11778 df-nn 12129 df-2 12191 df-3 12192 df-4 12193 df-5 12194 df-6 12195 df-7 12196 df-8 12197 df-9 12198 df-n0 12385 df-z 12472 df-dec 12592 df-uz 12736 df-q 12850 df-rp 12894 df-xneg 13014 df-xadd 13015 df-xmul 13016 df-icc 13255 df-fz 13411 df-fzo 13558 df-seq 13909 df-exp 13969 df-hash 14238 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-starv 17176 df-sca 17177 df-vsca 17178 df-ip 17179 df-tset 17180 df-ple 17181 df-ds 17183 df-unif 17184 df-hom 17185 df-cco 17186 df-rest 17326 df-topn 17327 df-0g 17345 df-gsum 17346 df-topgen 17347 df-pt 17348 df-prds 17351 df-xrs 17406 df-qtop 17411 df-imas 17412 df-xps 17414 df-mre 17488 df-mrc 17489 df-acs 17491 df-mgm 18514 df-sgrp 18593 df-mnd 18609 df-submnd 18658 df-mulg 18947 df-cntz 19196 df-cmn 19661 df-psmet 21253 df-xmet 21254 df-met 21255 df-bl 21256 df-mopn 21257 df-fbas 21258 df-fg 21259 df-cnfld 21262 df-top 22779 df-topon 22796 df-topsp 22818 df-bases 22831 df-cld 22904 df-ntr 22905 df-cls 22906 df-nei 22983 df-lp 23021 df-perf 23022 df-cn 23112 df-cnp 23113 df-t1 23199 df-haus 23200 df-tx 23447 df-hmeo 23640 df-fil 23731 df-fm 23823 df-flim 23824 df-flf 23825 df-xms 24206 df-ms 24207 df-tms 24208 df-cncf 24769 df-limc 25765 df-dv 25766

This theorem is referenced by: dvmptdiv 25876

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