dvrecg - Metamath Proof Explorer

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Theorem dvrecg 25133

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 10963	. . . 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 481	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {0})) → 𝐴 ∈ ℂ)
8		eldifi 4066	. . . . 5 ⊢ (𝑦 ∈ (ℂ ∖ {0}) → 𝑦 ∈ ℂ)
9	8	adantl 482	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {0})) → 𝑦 ∈ ℂ)
10		eldifsni 4729	. . . . 5 ⊢ (𝑦 ∈ (ℂ ∖ {0}) → 𝑦 ≠ 0)
11	10	adantl 482	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {0})) → 𝑦 ≠ 0)
12	7, 9, 11	divcld 11749	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {0})) → (𝐴 / 𝑦) ∈ ℂ)
13	9	sqcld 13858	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {0})) → (𝑦↑2) ∈ ℂ)
14		2z 12350	. . . . . . 7 ⊢ 2 ∈ ℤ
15	14	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {0})) → 2 ∈ ℤ)
16	9, 11, 15	expne0d 13866	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {0})) → (𝑦↑2) ≠ 0)
17	7, 13, 16	divcld 11749	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {0})) → (𝐴 / (𝑦↑2)) ∈ ℂ)
18	17	negcld 11317	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {0})) → -(𝐴 / (𝑦↑2)) ∈ ℂ)
19		dvrecg.db	. . 3 ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐵)) = (𝑥 ∈ 𝑋 ↦ 𝐶))
20		dvrec 25115	. . . 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 7277	. . 3 ⊢ (𝑦 = 𝐵 → (𝐴 / 𝑦) = (𝐴 / 𝐵))
23		oveq1 7276	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦↑2) = (𝐵↑2))
24	23	oveq2d 7285	. . . 4 ⊢ (𝑦 = 𝐵 → (𝐴 / (𝑦↑2)) = (𝐴 / (𝐵↑2)))
25	24	negeqd 11213	. . 3 ⊢ (𝑦 = 𝐵 → -(𝐴 / (𝑦↑2)) = -(𝐴 / (𝐵↑2)))
26	1, 3, 4, 5, 12, 18, 19, 21, 22, 25	dvmptco 25132	. 2 ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐵))) = (𝑥 ∈ 𝑋 ↦ (-(𝐴 / (𝐵↑2)) · 𝐶)))
27	6	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ)
28		eldifi 4066	. . . . . . . 8 ⊢ (𝐵 ∈ (ℂ ∖ {0}) → 𝐵 ∈ ℂ)
29	4, 28	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ ℂ)
30	29	sqcld 13858	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐵↑2) ∈ ℂ)
31		eldifsn 4726	. . . . . . . . 9 ⊢ (𝐵 ∈ (ℂ ∖ {0}) ↔ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0))
32	4, 31	sylib 217	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0))
33	32	simprd 496	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ≠ 0)
34	14	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 2 ∈ ℤ)
35	29, 33, 34	expne0d 13866	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐵↑2) ≠ 0)
36	27, 30, 35	divcld 11749	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐴 / (𝐵↑2)) ∈ ℂ)
37	1, 29, 5, 19	dvmptcl 25119	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐶 ∈ ℂ)
38	36, 37	mulneg1d 11426	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (-(𝐴 / (𝐵↑2)) · 𝐶) = -((𝐴 / (𝐵↑2)) · 𝐶))
39	27, 37, 30, 35	div23d 11786	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝐴 · 𝐶) / (𝐵↑2)) = ((𝐴 / (𝐵↑2)) · 𝐶))
40	39	eqcomd 2746	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝐴 / (𝐵↑2)) · 𝐶) = ((𝐴 · 𝐶) / (𝐵↑2)))
41	40	negeqd 11213	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → -((𝐴 / (𝐵↑2)) · 𝐶) = -((𝐴 · 𝐶) / (𝐵↑2)))
42	38, 41	eqtrd 2780	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (-(𝐴 / (𝐵↑2)) · 𝐶) = -((𝐴 · 𝐶) / (𝐵↑2)))
43	42	mpteq2dva 5179	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (-(𝐴 / (𝐵↑2)) · 𝐶)) = (𝑥 ∈ 𝑋 ↦ -((𝐴 · 𝐶) / (𝐵↑2))))
44	26, 43	eqtrd 2780	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 396 = wceq 1542 ∈ wcel 2110 ≠ wne 2945 ∖ cdif 3889 {csn 4567 {cpr 4569 ↦ cmpt 5162 (class class class)co 7269 ℂcc 10868 ℝcr 10869 0cc0 10870 · cmul 10875 -cneg 11204 / cdiv 11630 2c2 12026 ℤcz 12317 ↑cexp 13778 D cdv 25023

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 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7580 ax-cnex 10926 ax-resscn 10927 ax-1cn 10928 ax-icn 10929 ax-addcl 10930 ax-addrcl 10931 ax-mulcl 10932 ax-mulrcl 10933 ax-mulcom 10934 ax-addass 10935 ax-mulass 10936 ax-distr 10937 ax-i2m1 10938 ax-1ne0 10939 ax-1rid 10940 ax-rnegex 10941 ax-rrecex 10942 ax-cnre 10943 ax-pre-lttri 10944 ax-pre-lttrn 10945 ax-pre-ltadd 10946 ax-pre-mulgt0 10947 ax-pre-sup 10948 ax-addf 10949 ax-mulf 10950

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-iin 4933 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6200 df-ord 6267 df-on 6268 df-lim 6269 df-suc 6270 df-iota 6389 df-fun 6433 df-fn 6434 df-f 6435 df-f1 6436 df-fo 6437 df-f1o 6438 df-fv 6439 df-isom 6440 df-riota 7226 df-ov 7272 df-oprab 7273 df-mpo 7274 df-of 7525 df-om 7705 df-1st 7822 df-2nd 7823 df-supp 7967 df-frecs 8086 df-wrecs 8117 df-recs 8191 df-rdg 8230 df-1o 8286 df-2o 8287 df-er 8479 df-map 8598 df-pm 8599 df-ixp 8667 df-en 8715 df-dom 8716 df-sdom 8717 df-fin 8718 df-fsupp 9105 df-fi 9146 df-sup 9177 df-inf 9178 df-oi 9245 df-card 9696 df-pnf 11010 df-mnf 11011 df-xr 11012 df-ltxr 11013 df-le 11014 df-sub 11205 df-neg 11206 df-div 11631 df-nn 11972 df-2 12034 df-3 12035 df-4 12036 df-5 12037 df-6 12038 df-7 12039 df-8 12040 df-9 12041 df-n0 12232 df-z 12318 df-dec 12435 df-uz 12580 df-q 12686 df-rp 12728 df-xneg 12845 df-xadd 12846 df-xmul 12847 df-icc 13083 df-fz 13237 df-fzo 13380 df-seq 13718 df-exp 13779 df-hash 14041 df-cj 14806 df-re 14807 df-im 14808 df-sqrt 14942 df-abs 14943 df-struct 16844 df-sets 16861 df-slot 16879 df-ndx 16891 df-base 16909 df-ress 16938 df-plusg 16971 df-mulr 16972 df-starv 16973 df-sca 16974 df-vsca 16975 df-ip 16976 df-tset 16977 df-ple 16978 df-ds 16980 df-unif 16981 df-hom 16982 df-cco 16983 df-rest 17129 df-topn 17130 df-0g 17148 df-gsum 17149 df-topgen 17150 df-pt 17151 df-prds 17154 df-xrs 17209 df-qtop 17214 df-imas 17215 df-xps 17217 df-mre 17291 df-mrc 17292 df-acs 17294 df-mgm 18322 df-sgrp 18371 df-mnd 18382 df-submnd 18427 df-mulg 18697 df-cntz 18919 df-cmn 19384 df-psmet 20585 df-xmet 20586 df-met 20587 df-bl 20588 df-mopn 20589 df-fbas 20590 df-fg 20591 df-cnfld 20594 df-top 22039 df-topon 22056 df-topsp 22078 df-bases 22092 df-cld 22166 df-ntr 22167 df-cls 22168 df-nei 22245 df-lp 22283 df-perf 22284 df-cn 22374 df-cnp 22375 df-t1 22461 df-haus 22462 df-tx 22709 df-hmeo 22902 df-fil 22993 df-fm 23085 df-flim 23086 df-flf 23087 df-xms 23469 df-ms 23470 df-tms 23471 df-cncf 24037 df-limc 25026 df-dv 25027

This theorem is referenced by: dvmptdiv 25134

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