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Theorem dvmulxx 12876

Description: The product rule for derivatives at a point. For the (more general) relation version, see dvmulxxbr 12874. (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 7768	. . . . . 6 ⊢ ℂ ∈ V
3	2	a1i 9	. . . . 5 ⊢ (𝜑 → ℂ ∈ V)
4		mulcl 7771	. . . . . . 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 4076	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ V)
10		inidm 3290	. . . . . 6 ⊢ (𝑋 ∩ 𝑋) = 𝑋
11	5, 6, 7, 9, 9, 10	off 6002	. . . . 5 ⊢ (𝜑 → (𝐹 ∘_𝑓 · 𝐺):𝑋⟶ℂ)
12		elpm2r 6568	. . . . 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 1218	. . . 4 ⊢ (𝜑 → (𝐹 ∘_𝑓 · 𝐺) ∈ (ℂ ↑_pm 𝑆))
14		dvfgg 12865	. . . 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 5284	. 2 ⊢ (𝜑 → Fun (𝑆 D (𝐹 ∘_𝑓 · 𝐺)))
17		recnprss 12864	. . . 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 6568	. . . . . . 7 ⊢ (((ℂ ∈ V ∧ 𝑆 ∈ {ℝ, ℂ}) ∧ (𝐹:𝑋⟶ℂ ∧ 𝑋 ⊆ 𝑆)) → 𝐹 ∈ (ℂ ↑_pm 𝑆))
21	3, 1, 6, 8, 20	syl22anc 1218	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (ℂ ↑_pm 𝑆))
22		dvfgg 12865	. . . . . 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 5283	. . . . 5 ⊢ ((𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ → Fun (𝑆 D 𝐹))
25		funfvbrb 5541	. . . . 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 6568	. . . . . . 7 ⊢ (((ℂ ∈ V ∧ 𝑆 ∈ {ℝ, ℂ}) ∧ (𝐺:𝑋⟶ℂ ∧ 𝑋 ⊆ 𝑆)) → 𝐺 ∈ (ℂ ↑_pm 𝑆))
30	3, 1, 7, 8, 29	syl22anc 1218	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ (ℂ ↑_pm 𝑆))
31		dvfgg 12865	. . . . . 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 5283	. . . . 5 ⊢ ((𝑆 D 𝐺):dom (𝑆 D 𝐺)⟶ℂ → Fun (𝑆 D 𝐺))
34		funfvbrb 5541	. . . . 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 2140	. . 3 ⊢ (MetOpen‘(abs ∘ − )) = (MetOpen‘(abs ∘ − ))
38	6, 8, 7, 18, 27, 36, 37	dvmulxxbr 12874	. 2 ⊢ (𝜑 → 𝐶(𝑆 D (𝐹 ∘_𝑓 · 𝐺))((((𝑆 D 𝐹)‘𝐶) · (𝐺‘𝐶)) + (((𝑆 D 𝐺)‘𝐶) · (𝐹‘𝐶))))
39		funbrfv 5468	. 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 1332 ∈ wcel 1481 Vcvv 2689 ⊆ wss 3076 {cpr 3533 class class class wbr 3937 dom cdm 4547 ∘ ccom 4551 Fun wfun 5125 ⟶wf 5127 ‘cfv 5131 (class class class)co 5782 ∘_𝑓 cof 5988 ↑_pm cpm 6551 ℂcc 7642 ℝcr 7643 + caddc 7647 · cmul 7649 − cmin 7957 abscabs 10801 MetOpencmopn 12193 D cdv 12832

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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-iinf 4510 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-mulrcl 7743 ax-addcom 7744 ax-mulcom 7745 ax-addass 7746 ax-mulass 7747 ax-distr 7748 ax-i2m1 7749 ax-0lt1 7750 ax-1rid 7751 ax-0id 7752 ax-rnegex 7753 ax-precex 7754 ax-cnre 7755 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 ax-pre-apti 7759 ax-pre-ltadd 7760 ax-pre-mulgt0 7761 ax-pre-mulext 7762 ax-arch 7763 ax-caucvg 7764 ax-addf 7766 ax-mulf 7767

This theorem depends on definitions: df-bi 116 df-stab 817 df-dc 821 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-reu 2424 df-rmo 2425 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-if 3480 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-tr 4035 df-id 4223 df-po 4226 df-iso 4227 df-iord 4296 df-on 4298 df-ilim 4299 df-suc 4301 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-isom 5140 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-of 5990 df-1st 6046 df-2nd 6047 df-recs 6210 df-frec 6296 df-map 6552 df-pm 6553 df-sup 6879 df-inf 6880 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-sub 7959 df-neg 7960 df-reap 8361 df-ap 8368 df-div 8457 df-inn 8745 df-2 8803 df-3 8804 df-4 8805 df-n0 9002 df-z 9079 df-uz 9351 df-q 9439 df-rp 9471 df-xneg 9589 df-xadd 9590 df-seqfrec 10250 df-exp 10324 df-cj 10646 df-re 10647 df-im 10648 df-rsqrt 10802 df-abs 10803 df-rest 12161 df-topgen 12180 df-psmet 12195 df-xmet 12196 df-met 12197 df-bl 12198 df-mopn 12199 df-top 12204 df-topon 12217 df-bases 12249 df-ntr 12304 df-cn 12396 df-cnp 12397 df-tx 12461 df-cncf 12766 df-limced 12833 df-dvap 12834

This theorem is referenced by: dvimulf 12878

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