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Theorem dvmulxx 13308

Description: The product rule for derivatives at a point. For the (more general) relation version, see dvmulxxbr 13306. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Jim Kingdon, 2-Dec-2023.)

**Hypotheses**
Ref	Expression
dvadd.f	⊢ (𝜑 → 𝐹:𝑋⟶ℂ)
dvadd.x	⊢ (𝜑 → 𝑋 ⊆ 𝑆)
dvaddxx.g	⊢ (𝜑 → 𝐺:𝑋⟶ℂ)
dvadd.s	⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})
dvadd.df	⊢ (𝜑 → 𝐶 ∈ dom (𝑆 D 𝐹))
dvadd.dg	⊢ (𝜑 → 𝐶 ∈ dom (𝑆 D 𝐺))

**Assertion**
Ref	Expression
dvmulxx	⊢ (𝜑 → ((𝑆 D (𝐹 ∘_𝑓 · 𝐺))‘𝐶) = ((((𝑆 D 𝐹)‘𝐶) · (𝐺‘𝐶)) + (((𝑆 D 𝐺)‘𝐶) · (𝐹‘𝐶))))

Proof of Theorem dvmulxx Dummy variables 𝑢 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dvadd.s	. . . 4 ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})
2		cnex 7877	. . . . . 6 ⊢ ℂ ∈ V
3	2	a1i 9	. . . . 5 ⊢ (𝜑 → ℂ ∈ V)
4		mulcl 7880	. . . . . . 7 ⊢ ((𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ) → (𝑢 · 𝑣) ∈ ℂ)
5	4	adantl 275	. . . . . 6 ⊢ ((𝜑 ∧ (𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ)) → (𝑢 · 𝑣) ∈ ℂ)
6		dvadd.f	. . . . . 6 ⊢ (𝜑 → 𝐹:𝑋⟶ℂ)
7		dvaddxx.g	. . . . . 6 ⊢ (𝜑 → 𝐺:𝑋⟶ℂ)
8		dvadd.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ⊆ 𝑆)
9	1, 8	ssexd 4122	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ V)
10		inidm 3331	. . . . . 6 ⊢ (𝑋 ∩ 𝑋) = 𝑋
11	5, 6, 7, 9, 9, 10	off 6062	. . . . 5 ⊢ (𝜑 → (𝐹 ∘_𝑓 · 𝐺):𝑋⟶ℂ)
12		elpm2r 6632	. . . . 5 ⊢ (((ℂ ∈ V ∧ 𝑆 ∈ {ℝ, ℂ}) ∧ ((𝐹 ∘_𝑓 · 𝐺):𝑋⟶ℂ ∧ 𝑋 ⊆ 𝑆)) → (𝐹 ∘_𝑓 · 𝐺) ∈ (ℂ ↑_pm 𝑆))
13	3, 1, 11, 8, 12	syl22anc 1229	. . . 4 ⊢ (𝜑 → (𝐹 ∘_𝑓 · 𝐺) ∈ (ℂ ↑_pm 𝑆))
14		dvfgg 13297	. . . 4 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ (𝐹 ∘_𝑓 · 𝐺) ∈ (ℂ ↑_pm 𝑆)) → (𝑆 D (𝐹 ∘_𝑓 · 𝐺)):dom (𝑆 D (𝐹 ∘_𝑓 · 𝐺))⟶ℂ)
15	1, 13, 14	syl2anc 409	. . 3 ⊢ (𝜑 → (𝑆 D (𝐹 ∘_𝑓 · 𝐺)):dom (𝑆 D (𝐹 ∘_𝑓 · 𝐺))⟶ℂ)
16	15	ffund 5341	. 2 ⊢ (𝜑 → Fun (𝑆 D (𝐹 ∘_𝑓 · 𝐺)))
17		recnprss 13296	. . . 4 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ)
18	1, 17	syl 14	. . 3 ⊢ (𝜑 → 𝑆 ⊆ ℂ)
19		dvadd.df	. . . 4 ⊢ (𝜑 → 𝐶 ∈ dom (𝑆 D 𝐹))
20		elpm2r 6632	. . . . . . 7 ⊢ (((ℂ ∈ V ∧ 𝑆 ∈ {ℝ, ℂ}) ∧ (𝐹:𝑋⟶ℂ ∧ 𝑋 ⊆ 𝑆)) → 𝐹 ∈ (ℂ ↑_pm 𝑆))
21	3, 1, 6, 8, 20	syl22anc 1229	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (ℂ ↑_pm 𝑆))
22		dvfgg 13297	. . . . . 6 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) → (𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ)
23	1, 21, 22	syl2anc 409	. . . . 5 ⊢ (𝜑 → (𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ)
24		ffun 5340	. . . . 5 ⊢ ((𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ → Fun (𝑆 D 𝐹))
25		funfvbrb 5598	. . . . 5 ⊢ (Fun (𝑆 D 𝐹) → (𝐶 ∈ dom (𝑆 D 𝐹) ↔ 𝐶(𝑆 D 𝐹)((𝑆 D 𝐹)‘𝐶)))
26	23, 24, 25	3syl 17	. . . 4 ⊢ (𝜑 → (𝐶 ∈ dom (𝑆 D 𝐹) ↔ 𝐶(𝑆 D 𝐹)((𝑆 D 𝐹)‘𝐶)))
27	19, 26	mpbid 146	. . 3 ⊢ (𝜑 → 𝐶(𝑆 D 𝐹)((𝑆 D 𝐹)‘𝐶))
28		dvadd.dg	. . . 4 ⊢ (𝜑 → 𝐶 ∈ dom (𝑆 D 𝐺))
29		elpm2r 6632	. . . . . . 7 ⊢ (((ℂ ∈ V ∧ 𝑆 ∈ {ℝ, ℂ}) ∧ (𝐺:𝑋⟶ℂ ∧ 𝑋 ⊆ 𝑆)) → 𝐺 ∈ (ℂ ↑_pm 𝑆))
30	3, 1, 7, 8, 29	syl22anc 1229	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ (ℂ ↑_pm 𝑆))
31		dvfgg 13297	. . . . . 6 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐺 ∈ (ℂ ↑_pm 𝑆)) → (𝑆 D 𝐺):dom (𝑆 D 𝐺)⟶ℂ)
32	1, 30, 31	syl2anc 409	. . . . 5 ⊢ (𝜑 → (𝑆 D 𝐺):dom (𝑆 D 𝐺)⟶ℂ)
33		ffun 5340	. . . . 5 ⊢ ((𝑆 D 𝐺):dom (𝑆 D 𝐺)⟶ℂ → Fun (𝑆 D 𝐺))
34		funfvbrb 5598	. . . . 5 ⊢ (Fun (𝑆 D 𝐺) → (𝐶 ∈ dom (𝑆 D 𝐺) ↔ 𝐶(𝑆 D 𝐺)((𝑆 D 𝐺)‘𝐶)))
35	32, 33, 34	3syl 17	. . . 4 ⊢ (𝜑 → (𝐶 ∈ dom (𝑆 D 𝐺) ↔ 𝐶(𝑆 D 𝐺)((𝑆 D 𝐺)‘𝐶)))
36	28, 35	mpbid 146	. . 3 ⊢ (𝜑 → 𝐶(𝑆 D 𝐺)((𝑆 D 𝐺)‘𝐶))
37		eqid 2165	. . 3 ⊢ (MetOpen‘(abs ∘ − )) = (MetOpen‘(abs ∘ − ))
38	6, 8, 7, 18, 27, 36, 37	dvmulxxbr 13306	. 2 ⊢ (𝜑 → 𝐶(𝑆 D (𝐹 ∘_𝑓 · 𝐺))((((𝑆 D 𝐹)‘𝐶) · (𝐺‘𝐶)) + (((𝑆 D 𝐺)‘𝐶) · (𝐹‘𝐶))))
39		funbrfv 5525	. 2 ⊢ (Fun (𝑆 D (𝐹 ∘_𝑓 · 𝐺)) → (𝐶(𝑆 D (𝐹 ∘_𝑓 · 𝐺))((((𝑆 D 𝐹)‘𝐶) · (𝐺‘𝐶)) + (((𝑆 D 𝐺)‘𝐶) · (𝐹‘𝐶))) → ((𝑆 D (𝐹 ∘_𝑓 · 𝐺))‘𝐶) = ((((𝑆 D 𝐹)‘𝐶) · (𝐺‘𝐶)) + (((𝑆 D 𝐺)‘𝐶) · (𝐹‘𝐶)))))
40	16, 38, 39	sylc 62	1 ⊢ (𝜑 → ((𝑆 D (𝐹 ∘_𝑓 · 𝐺))‘𝐶) = ((((𝑆 D 𝐹)‘𝐶) · (𝐺‘𝐶)) + (((𝑆 D 𝐺)‘𝐶) · (𝐹‘𝐶))))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1343 ∈ wcel 2136 Vcvv 2726 ⊆ wss 3116 {cpr 3577 class class class wbr 3982 dom cdm 4604 ∘ ccom 4608 Fun wfun 5182 ⟶wf 5184 ‘cfv 5188 (class class class)co 5842 ∘_𝑓 cof 6048 ↑_pm cpm 6615 ℂcc 7751 ℝcr 7752 + caddc 7756 · cmul 7758 − cmin 8069 abscabs 10939 MetOpencmopn 12625 D cdv 13264

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-13 2138 ax-14 2139 ax-ext 2147 ax-coll 4097 ax-sep 4100 ax-nul 4108 ax-pow 4153 ax-pr 4187 ax-un 4411 ax-setind 4514 ax-iinf 4565 ax-cnex 7844 ax-resscn 7845 ax-1cn 7846 ax-1re 7847 ax-icn 7848 ax-addcl 7849 ax-addrcl 7850 ax-mulcl 7851 ax-mulrcl 7852 ax-addcom 7853 ax-mulcom 7854 ax-addass 7855 ax-mulass 7856 ax-distr 7857 ax-i2m1 7858 ax-0lt1 7859 ax-1rid 7860 ax-0id 7861 ax-rnegex 7862 ax-precex 7863 ax-cnre 7864 ax-pre-ltirr 7865 ax-pre-ltwlin 7866 ax-pre-lttrn 7867 ax-pre-apti 7868 ax-pre-ltadd 7869 ax-pre-mulgt0 7870 ax-pre-mulext 7871 ax-arch 7872 ax-caucvg 7873 ax-addf 7875 ax-mulf 7876

This theorem depends on definitions: df-bi 116 df-stab 821 df-dc 825 df-3or 969 df-3an 970 df-tru 1346 df-fal 1349 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ne 2337 df-nel 2432 df-ral 2449 df-rex 2450 df-reu 2451 df-rmo 2452 df-rab 2453 df-v 2728 df-sbc 2952 df-csb 3046 df-dif 3118 df-un 3120 df-in 3122 df-ss 3129 df-nul 3410 df-if 3521 df-pw 3561 df-sn 3582 df-pr 3583 df-op 3585 df-uni 3790 df-int 3825 df-iun 3868 df-br 3983 df-opab 4044 df-mpt 4045 df-tr 4081 df-id 4271 df-po 4274 df-iso 4275 df-iord 4344 df-on 4346 df-ilim 4347 df-suc 4349 df-iom 4568 df-xp 4610 df-rel 4611 df-cnv 4612 df-co 4613 df-dm 4614 df-rn 4615 df-res 4616 df-ima 4617 df-iota 5153 df-fun 5190 df-fn 5191 df-f 5192 df-f1 5193 df-fo 5194 df-f1o 5195 df-fv 5196 df-isom 5197 df-riota 5798 df-ov 5845 df-oprab 5846 df-mpo 5847 df-of 6050 df-1st 6108 df-2nd 6109 df-recs 6273 df-frec 6359 df-map 6616 df-pm 6617 df-sup 6949 df-inf 6950 df-pnf 7935 df-mnf 7936 df-xr 7937 df-ltxr 7938 df-le 7939 df-sub 8071 df-neg 8072 df-reap 8473 df-ap 8480 df-div 8569 df-inn 8858 df-2 8916 df-3 8917 df-4 8918 df-n0 9115 df-z 9192 df-uz 9467 df-q 9558 df-rp 9590 df-xneg 9708 df-xadd 9709 df-seqfrec 10381 df-exp 10455 df-cj 10784 df-re 10785 df-im 10786 df-rsqrt 10940 df-abs 10941 df-rest 12558 df-topgen 12577 df-psmet 12627 df-xmet 12628 df-met 12629 df-bl 12630 df-mopn 12631 df-top 12636 df-topon 12649 df-bases 12681 df-ntr 12736 df-cn 12828 df-cnp 12829 df-tx 12893 df-cncf 13198 df-limced 13265 df-dvap 13266

This theorem is referenced by: dvimulf 13310

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