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Theorem dvrecg 25993

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 11242	. . . 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 479	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {0})) → 𝐴 ∈ ℂ)
8		eldifi 4123	. . . . 5 ⊢ (𝑦 ∈ (ℂ ∖ {0}) → 𝑦 ∈ ℂ)
9	8	adantl 480	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {0})) → 𝑦 ∈ ℂ)
10		eldifsni 4789	. . . . 5 ⊢ (𝑦 ∈ (ℂ ∖ {0}) → 𝑦 ≠ 0)
11	10	adantl 480	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {0})) → 𝑦 ≠ 0)
12	7, 9, 11	divcld 12035	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {0})) → (𝐴 / 𝑦) ∈ ℂ)
13	9	sqcld 14157	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {0})) → (𝑦↑2) ∈ ℂ)
14		2z 12640	. . . . . . 7 ⊢ 2 ∈ ℤ
15	14	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {0})) → 2 ∈ ℤ)
16	9, 11, 15	expne0d 14165	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {0})) → (𝑦↑2) ≠ 0)
17	7, 13, 16	divcld 12035	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {0})) → (𝐴 / (𝑦↑2)) ∈ ℂ)
18	17	negcld 11599	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ {0})) → -(𝐴 / (𝑦↑2)) ∈ ℂ)
19		dvrecg.db	. . 3 ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐵)) = (𝑥 ∈ 𝑋 ↦ 𝐶))
20		dvrec 25975	. . . 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 7424	. . 3 ⊢ (𝑦 = 𝐵 → (𝐴 / 𝑦) = (𝐴 / 𝐵))
23		oveq1 7423	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦↑2) = (𝐵↑2))
24	23	oveq2d 7432	. . . 4 ⊢ (𝑦 = 𝐵 → (𝐴 / (𝑦↑2)) = (𝐴 / (𝐵↑2)))
25	24	negeqd 11495	. . 3 ⊢ (𝑦 = 𝐵 → -(𝐴 / (𝑦↑2)) = -(𝐴 / (𝐵↑2)))
26	1, 3, 4, 5, 12, 18, 19, 21, 22, 25	dvmptco 25992	. 2 ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐵))) = (𝑥 ∈ 𝑋 ↦ (-(𝐴 / (𝐵↑2)) · 𝐶)))
27	6	adantr 479	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ)
28		eldifi 4123	. . . . . . . 8 ⊢ (𝐵 ∈ (ℂ ∖ {0}) → 𝐵 ∈ ℂ)
29	4, 28	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ ℂ)
30	29	sqcld 14157	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐵↑2) ∈ ℂ)
31		eldifsn 4785	. . . . . . . . 9 ⊢ (𝐵 ∈ (ℂ ∖ {0}) ↔ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0))
32	4, 31	sylib 217	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0))
33	32	simprd 494	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ≠ 0)
34	14	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 2 ∈ ℤ)
35	29, 33, 34	expne0d 14165	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐵↑2) ≠ 0)
36	27, 30, 35	divcld 12035	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐴 / (𝐵↑2)) ∈ ℂ)
37	1, 29, 5, 19	dvmptcl 25979	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐶 ∈ ℂ)
38	36, 37	mulneg1d 11708	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (-(𝐴 / (𝐵↑2)) · 𝐶) = -((𝐴 / (𝐵↑2)) · 𝐶))
39	27, 37, 30, 35	div23d 12072	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝐴 · 𝐶) / (𝐵↑2)) = ((𝐴 / (𝐵↑2)) · 𝐶))
40	39	eqcomd 2732	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝐴 / (𝐵↑2)) · 𝐶) = ((𝐴 · 𝐶) / (𝐵↑2)))
41	40	negeqd 11495	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → -((𝐴 / (𝐵↑2)) · 𝐶) = -((𝐴 · 𝐶) / (𝐵↑2)))
42	38, 41	eqtrd 2766	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (-(𝐴 / (𝐵↑2)) · 𝐶) = -((𝐴 · 𝐶) / (𝐵↑2)))
43	42	mpteq2dva 5245	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (-(𝐴 / (𝐵↑2)) · 𝐶)) = (𝑥 ∈ 𝑋 ↦ -((𝐴 · 𝐶) / (𝐵↑2))))
44	26, 43	eqtrd 2766	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 394 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 ∖ cdif 3943 {csn 4623 {cpr 4625 ↦ cmpt 5228 (class class class)co 7416 ℂcc 11147 ℝcr 11148 0cc0 11149 · cmul 11154 -cneg 11486 / cdiv 11912 2c2 12313 ℤcz 12604 ↑cexp 14075 D cdv 25880

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5282 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 ax-cnex 11205 ax-resscn 11206 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-addrcl 11210 ax-mulcl 11211 ax-mulrcl 11212 ax-mulcom 11213 ax-addass 11214 ax-mulass 11215 ax-distr 11216 ax-i2m1 11217 ax-1ne0 11218 ax-1rid 11219 ax-rnegex 11220 ax-rrecex 11221 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 ax-pre-ltadd 11225 ax-pre-mulgt0 11226 ax-pre-sup 11227 ax-addf 11228

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4906 df-int 4947 df-iun 4995 df-iin 4996 df-br 5146 df-opab 5208 df-mpt 5229 df-tr 5263 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-se 5630 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6304 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-isom 6555 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7682 df-om 7869 df-1st 7995 df-2nd 7996 df-supp 8167 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-1o 8488 df-2o 8489 df-er 8726 df-map 8849 df-pm 8850 df-ixp 8919 df-en 8967 df-dom 8968 df-sdom 8969 df-fin 8970 df-fsupp 9399 df-fi 9447 df-sup 9478 df-inf 9479 df-oi 9546 df-card 9975 df-pnf 11291 df-mnf 11292 df-xr 11293 df-ltxr 11294 df-le 11295 df-sub 11487 df-neg 11488 df-div 11913 df-nn 12259 df-2 12321 df-3 12322 df-4 12323 df-5 12324 df-6 12325 df-7 12326 df-8 12327 df-9 12328 df-n0 12519 df-z 12605 df-dec 12724 df-uz 12869 df-q 12979 df-rp 13023 df-xneg 13140 df-xadd 13141 df-xmul 13142 df-icc 13379 df-fz 13533 df-fzo 13676 df-seq 14016 df-exp 14076 df-hash 14343 df-cj 15099 df-re 15100 df-im 15101 df-sqrt 15235 df-abs 15236 df-struct 17144 df-sets 17161 df-slot 17179 df-ndx 17191 df-base 17209 df-ress 17238 df-plusg 17274 df-mulr 17275 df-starv 17276 df-sca 17277 df-vsca 17278 df-ip 17279 df-tset 17280 df-ple 17281 df-ds 17283 df-unif 17284 df-hom 17285 df-cco 17286 df-rest 17432 df-topn 17433 df-0g 17451 df-gsum 17452 df-topgen 17453 df-pt 17454 df-prds 17457 df-xrs 17512 df-qtop 17517 df-imas 17518 df-xps 17520 df-mre 17594 df-mrc 17595 df-acs 17597 df-mgm 18628 df-sgrp 18707 df-mnd 18723 df-submnd 18769 df-mulg 19058 df-cntz 19307 df-cmn 19776 df-psmet 21331 df-xmet 21332 df-met 21333 df-bl 21334 df-mopn 21335 df-fbas 21336 df-fg 21337 df-cnfld 21340 df-top 22884 df-topon 22901 df-topsp 22923 df-bases 22937 df-cld 23011 df-ntr 23012 df-cls 23013 df-nei 23090 df-lp 23128 df-perf 23129 df-cn 23219 df-cnp 23220 df-t1 23306 df-haus 23307 df-tx 23554 df-hmeo 23747 df-fil 23838 df-fm 23930 df-flim 23931 df-flf 23932 df-xms 24314 df-ms 24315 df-tms 24316 df-cncf 24886 df-limc 25883 df-dv 25884

This theorem is referenced by: dvmptdiv 25994

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