dvrecg - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > dvrecg		Structured version Visualization version GIF version

Theorem dvrecg 25137

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 10964	. . . 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 4061	. . . . 5 ⊢ (𝑦 ∈ (ℂ ∖ {0}) → 𝑦 ∈ ℂ)
9	8	adantl 482	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {0})) → 𝑦 ∈ ℂ)
10		eldifsni 4723	. . . . 5 ⊢ (𝑦 ∈ (ℂ ∖ {0}) → 𝑦 ≠ 0)
11	10	adantl 482	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {0})) → 𝑦 ≠ 0)
12	7, 9, 11	divcld 11751	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {0})) → (𝐴 / 𝑦) ∈ ℂ)
13	9	sqcld 13862	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {0})) → (𝑦↑2) ∈ ℂ)
14		2z 12352	. . . . . . 7 ⊢ 2 ∈ ℤ
15	14	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {0})) → 2 ∈ ℤ)
16	9, 11, 15	expne0d 13870	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {0})) → (𝑦↑2) ≠ 0)
17	7, 13, 16	divcld 11751	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {0})) → (𝐴 / (𝑦↑2)) ∈ ℂ)
18	17	negcld 11319	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {0})) → -(𝐴 / (𝑦↑2)) ∈ ℂ)
19		dvrecg.db	. . 3 ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐵)) = (𝑥 ∈ 𝑋 ↦ 𝐶))
20		dvrec 25119	. . . 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 7283	. . 3 ⊢ (𝑦 = 𝐵 → (𝐴 / 𝑦) = (𝐴 / 𝐵))
23		oveq1 7282	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦↑2) = (𝐵↑2))
24	23	oveq2d 7291	. . . 4 ⊢ (𝑦 = 𝐵 → (𝐴 / (𝑦↑2)) = (𝐴 / (𝐵↑2)))
25	24	negeqd 11215	. . 3 ⊢ (𝑦 = 𝐵 → -(𝐴 / (𝑦↑2)) = -(𝐴 / (𝐵↑2)))
26	1, 3, 4, 5, 12, 18, 19, 21, 22, 25	dvmptco 25136	. 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 4061	. . . . . . . 8 ⊢ (𝐵 ∈ (ℂ ∖ {0}) → 𝐵 ∈ ℂ)
29	4, 28	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ ℂ)
30	29	sqcld 13862	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐵↑2) ∈ ℂ)
31		eldifsn 4720	. . . . . . . . 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 13870	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐵↑2) ≠ 0)
36	27, 30, 35	divcld 11751	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐴 / (𝐵↑2)) ∈ ℂ)
37	1, 29, 5, 19	dvmptcl 25123	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐶 ∈ ℂ)
38	36, 37	mulneg1d 11428	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (-(𝐴 / (𝐵↑2)) · 𝐶) = -((𝐴 / (𝐵↑2)) · 𝐶))
39	27, 37, 30, 35	div23d 11788	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝐴 · 𝐶) / (𝐵↑2)) = ((𝐴 / (𝐵↑2)) · 𝐶))
40	39	eqcomd 2744	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝐴 / (𝐵↑2)) · 𝐶) = ((𝐴 · 𝐶) / (𝐵↑2)))
41	40	negeqd 11215	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → -((𝐴 / (𝐵↑2)) · 𝐶) = -((𝐴 · 𝐶) / (𝐵↑2)))
42	38, 41	eqtrd 2778	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (-(𝐴 / (𝐵↑2)) · 𝐶) = -((𝐴 · 𝐶) / (𝐵↑2)))
43	42	mpteq2dva 5174	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (-(𝐴 / (𝐵↑2)) · 𝐶)) = (𝑥 ∈ 𝑋 ↦ -((𝐴 · 𝐶) / (𝐵↑2))))
44	26, 43	eqtrd 2778	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 1539 ∈ wcel 2106 ≠ wne 2943 ∖ cdif 3884 {csn 4561 {cpr 4563 ↦ cmpt 5157 (class class class)co 7275 ℂcc 10869 ℝcr 10870 0cc0 10871 · cmul 10876 -cneg 11206 / cdiv 11632 2c2 12028 ℤcz 12319 ↑cexp 13782 D cdv 25027

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949 ax-addf 10950 ax-mulf 10951

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 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 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-om 7713 df-1st 7831 df-2nd 7832 df-supp 7978 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-2o 8298 df-er 8498 df-map 8617 df-pm 8618 df-ixp 8686 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fsupp 9129 df-fi 9170 df-sup 9201 df-inf 9202 df-oi 9269 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-q 12689 df-rp 12731 df-xneg 12848 df-xadd 12849 df-xmul 12850 df-icc 13086 df-fz 13240 df-fzo 13383 df-seq 13722 df-exp 13783 df-hash 14045 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-starv 16977 df-sca 16978 df-vsca 16979 df-ip 16980 df-tset 16981 df-ple 16982 df-ds 16984 df-unif 16985 df-hom 16986 df-cco 16987 df-rest 17133 df-topn 17134 df-0g 17152 df-gsum 17153 df-topgen 17154 df-pt 17155 df-prds 17158 df-xrs 17213 df-qtop 17218 df-imas 17219 df-xps 17221 df-mre 17295 df-mrc 17296 df-acs 17298 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-submnd 18431 df-mulg 18701 df-cntz 18923 df-cmn 19388 df-psmet 20589 df-xmet 20590 df-met 20591 df-bl 20592 df-mopn 20593 df-fbas 20594 df-fg 20595 df-cnfld 20598 df-top 22043 df-topon 22060 df-topsp 22082 df-bases 22096 df-cld 22170 df-ntr 22171 df-cls 22172 df-nei 22249 df-lp 22287 df-perf 22288 df-cn 22378 df-cnp 22379 df-t1 22465 df-haus 22466 df-tx 22713 df-hmeo 22906 df-fil 22997 df-fm 23089 df-flim 23090 df-flf 23091 df-xms 23473 df-ms 23474 df-tms 23475 df-cncf 24041 df-limc 25030 df-dv 25031

This theorem is referenced by: dvmptdiv 25138

W3C validator