dvmptdiv - Metamath Proof Explorer

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Theorem dvmptdiv 26024

Description: Function-builder for derivative, quotient rule. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

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

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

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

**Proof of Theorem dvmptdiv**
Step	Hyp	Ref	Expression
1		dvmptdiv.a	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ)
2		dvmptdiv.c	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐶 ∈ (ℂ ∖ {0}))
3	2	eldifad 3914	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐶 ∈ ℂ)
4		eldifsn 4743	. . . . . . 7 ⊢ (𝐶 ∈ (ℂ ∖ {0}) ↔ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0))
5	2, 4	sylib 220	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0))
6	5	simprd 499	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐶 ≠ 0)
7	1, 3, 6	divrecd 11964	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐴 / 𝐶) = (𝐴 · (1 / 𝐶)))
8	7	mpteq2dva 5190	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)) = (𝑥 ∈ 𝑋 ↦ (𝐴 · (1 / 𝐶))))
9	8	oveq2d 7407	. 2 ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶))) = (𝑆 D (𝑥 ∈ 𝑋 ↦ (𝐴 · (1 / 𝐶)))))
10		dvmptdiv.s	. . 3 ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})
11		dvmptdiv.b	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑉)
12		dvmptdiv.da	. . 3 ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐵))
13	3, 6	reccld 11954	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (1 / 𝐶) ∈ ℂ)
14		1cnd 11169	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 1 ∈ ℂ)
15		dvmptdiv.d	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐷 ∈ ℂ)
16	14, 15	mulcld 11196	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (1 · 𝐷) ∈ ℂ)
17	3	sqcld 14151	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐶↑2) ∈ ℂ)
18	6	neneqd 2961	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ¬ 𝐶 = 0)
19		sqeq0 14127	. . . . . . . 8 ⊢ (𝐶 ∈ ℂ → ((𝐶↑2) = 0 ↔ 𝐶 = 0))
20	3, 19	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝐶↑2) = 0 ↔ 𝐶 = 0))
21	18, 20	mtbird 327	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ¬ (𝐶↑2) = 0)
22	21	neqned 2963	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐶↑2) ≠ 0)
23	16, 17, 22	divcld 11961	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((1 · 𝐷) / (𝐶↑2)) ∈ ℂ)
24	23	negcld 11523	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → -((1 · 𝐷) / (𝐶↑2)) ∈ ℂ)
25		1cnd 11169	. . . 4 ⊢ (𝜑 → 1 ∈ ℂ)
26		dvmptdiv.dc	. . . 4 ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐶)) = (𝑥 ∈ 𝑋 ↦ 𝐷))
27	10, 25, 2, 15, 26	dvrecg 26023	. . 3 ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ (1 / 𝐶))) = (𝑥 ∈ 𝑋 ↦ -((1 · 𝐷) / (𝐶↑2))))
28	10, 1, 11, 12, 13, 24, 27	dvmptmul 26011	. 2 ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ (𝐴 · (1 / 𝐶)))) = (𝑥 ∈ 𝑋 ↦ ((𝐵 · (1 / 𝐶)) + (-((1 · 𝐷) / (𝐶↑2)) · 𝐴))))
29	10, 1, 11, 12	dvmptcl 26009	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ ℂ)
30	29, 3	mulcld 11196	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐵 · 𝐶) ∈ ℂ)
31	30, 17, 22	divcld 11961	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝐵 · 𝐶) / (𝐶↑2)) ∈ ℂ)
32	15, 1	mulcld 11196	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐷 · 𝐴) ∈ ℂ)
33	32, 17, 22	divcld 11961	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝐷 · 𝐴) / (𝐶↑2)) ∈ ℂ)
34	31, 33	negsubd 11542	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (((𝐵 · 𝐶) / (𝐶↑2)) + -((𝐷 · 𝐴) / (𝐶↑2))) = (((𝐵 · 𝐶) / (𝐶↑2)) − ((𝐷 · 𝐴) / (𝐶↑2))))
35	29, 14, 3, 6	div12d 11997	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐵 · (1 / 𝐶)) = (1 · (𝐵 / 𝐶)))
36	29, 3, 6	divcld 11961	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐵 / 𝐶) ∈ ℂ)
37	36	mullidd 11194	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (1 · (𝐵 / 𝐶)) = (𝐵 / 𝐶))
38	3	sqvald 14150	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐶↑2) = (𝐶 · 𝐶))
39	38	oveq2d 7407	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝐵 · 𝐶) / (𝐶↑2)) = ((𝐵 · 𝐶) / (𝐶 · 𝐶)))
40	29, 3, 3, 6, 6	divcan5rd 11988	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝐵 · 𝐶) / (𝐶 · 𝐶)) = (𝐵 / 𝐶))
41	39, 40	eqtr2d 2797	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐵 / 𝐶) = ((𝐵 · 𝐶) / (𝐶↑2)))
42	35, 37, 41	3eqtrd 2800	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐵 · (1 / 𝐶)) = ((𝐵 · 𝐶) / (𝐶↑2)))
43	15	mullidd 11194	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (1 · 𝐷) = 𝐷)
44	43	oveq1d 7406	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((1 · 𝐷) / (𝐶↑2)) = (𝐷 / (𝐶↑2)))
45	44	negeqd 11418	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → -((1 · 𝐷) / (𝐶↑2)) = -(𝐷 / (𝐶↑2)))
46	45	oveq1d 7406	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (-((1 · 𝐷) / (𝐶↑2)) · 𝐴) = (-(𝐷 / (𝐶↑2)) · 𝐴))
47	15, 17, 22	divcld 11961	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐷 / (𝐶↑2)) ∈ ℂ)
48	47, 1	mulneg1d 11634	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (-(𝐷 / (𝐶↑2)) · 𝐴) = -((𝐷 / (𝐶↑2)) · 𝐴))
49	15, 1, 17, 22	div23d 11998	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝐷 · 𝐴) / (𝐶↑2)) = ((𝐷 / (𝐶↑2)) · 𝐴))
50	49	eqcomd 2767	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝐷 / (𝐶↑2)) · 𝐴) = ((𝐷 · 𝐴) / (𝐶↑2)))
51	50	negeqd 11418	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → -((𝐷 / (𝐶↑2)) · 𝐴) = -((𝐷 · 𝐴) / (𝐶↑2)))
52	46, 48, 51	3eqtrd 2800	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (-((1 · 𝐷) / (𝐶↑2)) · 𝐴) = -((𝐷 · 𝐴) / (𝐶↑2)))
53	42, 52	oveq12d 7409	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝐵 · (1 / 𝐶)) + (-((1 · 𝐷) / (𝐶↑2)) · 𝐴)) = (((𝐵 · 𝐶) / (𝐶↑2)) + -((𝐷 · 𝐴) / (𝐶↑2))))
54	30, 32, 17, 22	divsubdird 12000	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (((𝐵 · 𝐶) − (𝐷 · 𝐴)) / (𝐶↑2)) = (((𝐵 · 𝐶) / (𝐶↑2)) − ((𝐷 · 𝐴) / (𝐶↑2))))
55	34, 53, 54	3eqtr4d 2806	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝐵 · (1 / 𝐶)) + (-((1 · 𝐷) / (𝐶↑2)) · 𝐴)) = (((𝐵 · 𝐶) − (𝐷 · 𝐴)) / (𝐶↑2)))
56	55	mpteq2dva 5190	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ((𝐵 · (1 / 𝐶)) + (-((1 · 𝐷) / (𝐶↑2)) · 𝐴))) = (𝑥 ∈ 𝑋 ↦ (((𝐵 · 𝐶) − (𝐷 · 𝐴)) / (𝐶↑2))))
57	9, 28, 56	3eqtrd 2800	1 ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶))) = (𝑥 ∈ 𝑋 ↦ (((𝐵 · 𝐶) − (𝐷 · 𝐴)) / (𝐶↑2))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 399 = wceq 1559 ∈ wcel 2141 ≠ wne 2956 ∖ cdif 3899 {csn 4579 {cpr 4581 ↦ cmpt 5178 (class class class)co 7391 ℂcc 11065 ℝcr 11066 0cc0 11067 1c1 11068 + caddc 11070 · cmul 11072 − cmin 11408 -cneg 11409 / cdiv 11838 2c2 12266 ↑cexp 14068 D cdv 25913

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5224 ax-sep 5243 ax-nul 5253 ax-pow 5319 ax-pr 5387 ax-un 7713 ax-cnex 11123 ax-resscn 11124 ax-1cn 11125 ax-icn 11126 ax-addcl 11127 ax-addrcl 11128 ax-mulcl 11129 ax-mulrcl 11130 ax-mulcom 11131 ax-addass 11132 ax-mulass 11133 ax-distr 11134 ax-i2m1 11135 ax-1ne0 11136 ax-1rid 11137 ax-rnegex 11138 ax-rrecex 11139 ax-cnre 11140 ax-pre-lttri 11141 ax-pre-lttrn 11142 ax-pre-ltadd 11143 ax-pre-mulgt0 11144 ax-pre-sup 11145 ax-addf 11146

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-uni 4863 df-int 4903 df-iun 4948 df-iin 4949 df-br 5098 df-opab 5160 df-mpt 5179 df-tr 5205 df-id 5538 df-eprel 5543 df-po 5551 df-so 5552 df-fr 5596 df-se 5597 df-we 5598 df-xp 5649 df-rel 5650 df-cnv 5651 df-co 5652 df-dm 5653 df-rn 5654 df-res 5655 df-ima 5656 df-pred 6283 df-ord 6344 df-on 6345 df-lim 6346 df-suc 6347 df-iota 6472 df-fun 6518 df-fn 6519 df-f 6520 df-f1 6521 df-fo 6522 df-f1o 6523 df-fv 6524 df-isom 6525 df-riota 7348 df-ov 7394 df-oprab 7395 df-mpo 7396 df-of 7655 df-om 7842 df-1st 7965 df-2nd 7966 df-supp 8135 df-frecs 8256 df-wrecs 8287 df-recs 8336 df-rdg 8375 df-1o 8431 df-2o 8432 df-er 8672 df-map 8804 df-pm 8805 df-ixp 8874 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-fsupp 9302 df-fi 9351 df-sup 9382 df-inf 9383 df-oi 9452 df-card 9891 df-pnf 11212 df-mnf 11213 df-xr 11214 df-ltxr 11215 df-le 11216 df-sub 11410 df-neg 11411 df-div 11839 df-nn 12205 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12476 df-z 12563 df-dec 12683 df-uz 12834 df-q 12944 df-rp 12988 df-xneg 13108 df-xadd 13109 df-xmul 13110 df-icc 13350 df-fz 13507 df-fzo 13654 df-seq 14009 df-exp 14069 df-hash 14338 df-cj 15117 df-re 15118 df-im 15119 df-sqrt 15253 df-abs 15254 df-struct 17174 df-sets 17191 df-slot 17209 df-ndx 17221 df-base 17237 df-ress 17258 df-plusg 17290 df-mulr 17291 df-starv 17292 df-sca 17293 df-vsca 17294 df-ip 17295 df-tset 17296 df-ple 17297 df-ds 17299 df-unif 17300 df-hom 17301 df-cco 17302 df-rest 17442 df-topn 17443 df-0g 17461 df-gsum 17462 df-topgen 17463 df-pt 17464 df-prds 17467 df-xrs 17523 df-qtop 17528 df-imas 17529 df-xps 17531 df-mre 17605 df-mrc 17606 df-acs 17608 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-submnd 18809 df-mulg 19101 df-cntz 19348 df-cmn 19813 df-psmet 21404 df-xmet 21405 df-met 21406 df-bl 21407 df-mopn 21408 df-fbas 21409 df-fg 21410 df-cnfld 21413 df-top 22942 df-topon 22959 df-topsp 22981 df-bases 22994 df-cld 23067 df-ntr 23068 df-cls 23069 df-nei 23146 df-lp 23184 df-perf 23185 df-cn 23275 df-cnp 23276 df-t1 23362 df-haus 23363 df-tx 23610 df-hmeo 23803 df-fil 23894 df-fm 23986 df-flim 23987 df-flf 23988 df-xms 24368 df-ms 24369 df-tms 24370 df-cncf 24928 df-limc 25916 df-dv 25917

This theorem is referenced by: dvdivf 46457 dvdivbd 46458 fourierdlem56 46697 fourierdlem57 46698

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