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Theorem dvmulxx 14950

Description: The product rule for derivatives at a point. For the (more general) relation version, see dvmulxxbr 14948. (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 8005	. . . . . 6 ⊢ ℂ ∈ V
3	2	a1i 9	. . . . 5 ⊢ (𝜑 → ℂ ∈ V)
4		mulcl 8008	. . . . . . 7 ⊢ ((𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ) → (𝑢 · 𝑣) ∈ ℂ)
5	4	adantl 277	. . . . . 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 4174	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ V)
10		inidm 3373	. . . . . 6 ⊢ (𝑋 ∩ 𝑋) = 𝑋
11	5, 6, 7, 9, 9, 10	off 6149	. . . . 5 ⊢ (𝜑 → (𝐹 ∘_𝑓 · 𝐺):𝑋⟶ℂ)
12		elpm2r 6726	. . . . 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 1250	. . . 4 ⊢ (𝜑 → (𝐹 ∘_𝑓 · 𝐺) ∈ (ℂ ↑_pm 𝑆))
14		dvfgg 14934	. . . 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 411	. . 3 ⊢ (𝜑 → (𝑆 D (𝐹 ∘_𝑓 · 𝐺)):dom (𝑆 D (𝐹 ∘_𝑓 · 𝐺))⟶ℂ)
16	15	ffund 5412	. 2 ⊢ (𝜑 → Fun (𝑆 D (𝐹 ∘_𝑓 · 𝐺)))
17		recnprss 14933	. . . 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 6726	. . . . . . 7 ⊢ (((ℂ ∈ V ∧ 𝑆 ∈ {ℝ, ℂ}) ∧ (𝐹:𝑋⟶ℂ ∧ 𝑋 ⊆ 𝑆)) → 𝐹 ∈ (ℂ ↑_pm 𝑆))
21	3, 1, 6, 8, 20	syl22anc 1250	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (ℂ ↑_pm 𝑆))
22		dvfgg 14934	. . . . . 6 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) → (𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ)
23	1, 21, 22	syl2anc 411	. . . . 5 ⊢ (𝜑 → (𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ)
24		ffun 5411	. . . . 5 ⊢ ((𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ → Fun (𝑆 D 𝐹))
25		funfvbrb 5676	. . . . 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 147	. . 3 ⊢ (𝜑 → 𝐶(𝑆 D 𝐹)((𝑆 D 𝐹)‘𝐶))
28		dvadd.dg	. . . 4 ⊢ (𝜑 → 𝐶 ∈ dom (𝑆 D 𝐺))
29		elpm2r 6726	. . . . . . 7 ⊢ (((ℂ ∈ V ∧ 𝑆 ∈ {ℝ, ℂ}) ∧ (𝐺:𝑋⟶ℂ ∧ 𝑋 ⊆ 𝑆)) → 𝐺 ∈ (ℂ ↑_pm 𝑆))
30	3, 1, 7, 8, 29	syl22anc 1250	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ (ℂ ↑_pm 𝑆))
31		dvfgg 14934	. . . . . 6 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐺 ∈ (ℂ ↑_pm 𝑆)) → (𝑆 D 𝐺):dom (𝑆 D 𝐺)⟶ℂ)
32	1, 30, 31	syl2anc 411	. . . . 5 ⊢ (𝜑 → (𝑆 D 𝐺):dom (𝑆 D 𝐺)⟶ℂ)
33		ffun 5411	. . . . 5 ⊢ ((𝑆 D 𝐺):dom (𝑆 D 𝐺)⟶ℂ → Fun (𝑆 D 𝐺))
34		funfvbrb 5676	. . . . 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 147	. . 3 ⊢ (𝜑 → 𝐶(𝑆 D 𝐺)((𝑆 D 𝐺)‘𝐶))
37		eqid 2196	. . 3 ⊢ (MetOpen‘(abs ∘ − )) = (MetOpen‘(abs ∘ − ))
38	6, 8, 7, 18, 27, 36, 37	dvmulxxbr 14948	. 2 ⊢ (𝜑 → 𝐶(𝑆 D (𝐹 ∘_𝑓 · 𝐺))((((𝑆 D 𝐹)‘𝐶) · (𝐺‘𝐶)) + (((𝑆 D 𝐺)‘𝐶) · (𝐹‘𝐶))))
39		funbrfv 5600	. 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 104 ↔ wb 105 = wceq 1364 ∈ wcel 2167 Vcvv 2763 ⊆ wss 3157 {cpr 3624 class class class wbr 4034 dom cdm 4664 ∘ ccom 4668 Fun wfun 5253 ⟶wf 5255 ‘cfv 5259 (class class class)co 5923 ∘_𝑓 cof 6134 ↑_pm cpm 6709 ℂcc 7879 ℝcr 7880 + caddc 7884 · cmul 7886 − cmin 8199 abscabs 11164 MetOpencmopn 14107 D cdv 14901

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-coll 4149 ax-sep 4152 ax-nul 4160 ax-pow 4208 ax-pr 4243 ax-un 4469 ax-setind 4574 ax-iinf 4625 ax-cnex 7972 ax-resscn 7973 ax-1cn 7974 ax-1re 7975 ax-icn 7976 ax-addcl 7977 ax-addrcl 7978 ax-mulcl 7979 ax-mulrcl 7980 ax-addcom 7981 ax-mulcom 7982 ax-addass 7983 ax-mulass 7984 ax-distr 7985 ax-i2m1 7986 ax-0lt1 7987 ax-1rid 7988 ax-0id 7989 ax-rnegex 7990 ax-precex 7991 ax-cnre 7992 ax-pre-ltirr 7993 ax-pre-ltwlin 7994 ax-pre-lttrn 7995 ax-pre-apti 7996 ax-pre-ltadd 7997 ax-pre-mulgt0 7998 ax-pre-mulext 7999 ax-arch 8000 ax-caucvg 8001 ax-addf 8003 ax-mulf 8004

This theorem depends on definitions: df-bi 117 df-stab 832 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-nel 2463 df-ral 2480 df-rex 2481 df-reu 2482 df-rmo 2483 df-rab 2484 df-v 2765 df-sbc 2990 df-csb 3085 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-nul 3452 df-if 3563 df-pw 3608 df-sn 3629 df-pr 3630 df-op 3632 df-uni 3841 df-int 3876 df-iun 3919 df-br 4035 df-opab 4096 df-mpt 4097 df-tr 4133 df-id 4329 df-po 4332 df-iso 4333 df-iord 4402 df-on 4404 df-ilim 4405 df-suc 4407 df-iom 4628 df-xp 4670 df-rel 4671 df-cnv 4672 df-co 4673 df-dm 4674 df-rn 4675 df-res 4676 df-ima 4677 df-iota 5220 df-fun 5261 df-fn 5262 df-f 5263 df-f1 5264 df-fo 5265 df-f1o 5266 df-fv 5267 df-isom 5268 df-riota 5878 df-ov 5926 df-oprab 5927 df-mpo 5928 df-of 6136 df-1st 6199 df-2nd 6200 df-recs 6364 df-frec 6450 df-map 6710 df-pm 6711 df-sup 7051 df-inf 7052 df-pnf 8065 df-mnf 8066 df-xr 8067 df-ltxr 8068 df-le 8069 df-sub 8201 df-neg 8202 df-reap 8604 df-ap 8611 df-div 8702 df-inn 8993 df-2 9051 df-3 9052 df-4 9053 df-n0 9252 df-z 9329 df-uz 9604 df-q 9696 df-rp 9731 df-xneg 9849 df-xadd 9850 df-seqfrec 10542 df-exp 10633 df-cj 11009 df-re 11010 df-im 11011 df-rsqrt 11165 df-abs 11166 df-rest 12922 df-topgen 12941 df-psmet 14109 df-xmet 14110 df-met 14111 df-bl 14112 df-mopn 14113 df-top 14244 df-topon 14257 df-bases 14289 df-ntr 14342 df-cn 14434 df-cnp 14435 df-tx 14499 df-cncf 14817 df-limced 14902 df-dvap 14903

This theorem is referenced by: dvimulf 14952

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