dvmulxx - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > dvmulxx		GIF version

Theorem dvmulxx 15515

Description: The product rule for derivatives at a point. For the (more general) relation version, see dvmulxxbr 15513. (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 8216	. . . . . 6 ⊢ ℂ ∈ V
3	2	a1i 9	. . . . 5 ⊢ (𝜑 → ℂ ∈ V)
4		mulcl 8219	. . . . . . 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 4234	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ V)
10		inidm 3418	. . . . . 6 ⊢ (𝑋 ∩ 𝑋) = 𝑋
11	5, 6, 7, 9, 9, 10	off 6257	. . . . 5 ⊢ (𝜑 → (𝐹 ∘_𝑓 · 𝐺):𝑋⟶ℂ)
12		elpm2r 6878	. . . . 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 1275	. . . 4 ⊢ (𝜑 → (𝐹 ∘_𝑓 · 𝐺) ∈ (ℂ ↑_pm 𝑆))
14		dvfgg 15499	. . . 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 5493	. 2 ⊢ (𝜑 → Fun (𝑆 D (𝐹 ∘_𝑓 · 𝐺)))
17		recnprss 15498	. . . 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 6878	. . . . . . 7 ⊢ (((ℂ ∈ V ∧ 𝑆 ∈ {ℝ, ℂ}) ∧ (𝐹:𝑋⟶ℂ ∧ 𝑋 ⊆ 𝑆)) → 𝐹 ∈ (ℂ ↑_pm 𝑆))
21	3, 1, 6, 8, 20	syl22anc 1275	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (ℂ ↑_pm 𝑆))
22		dvfgg 15499	. . . . . 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 5492	. . . . 5 ⊢ ((𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ → Fun (𝑆 D 𝐹))
25		funfvbrb 5769	. . . . 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 6878	. . . . . . 7 ⊢ (((ℂ ∈ V ∧ 𝑆 ∈ {ℝ, ℂ}) ∧ (𝐺:𝑋⟶ℂ ∧ 𝑋 ⊆ 𝑆)) → 𝐺 ∈ (ℂ ↑_pm 𝑆))
30	3, 1, 7, 8, 29	syl22anc 1275	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ (ℂ ↑_pm 𝑆))
31		dvfgg 15499	. . . . . 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 5492	. . . . 5 ⊢ ((𝑆 D 𝐺):dom (𝑆 D 𝐺)⟶ℂ → Fun (𝑆 D 𝐺))
34		funfvbrb 5769	. . . . 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 2231	. . 3 ⊢ (MetOpen‘(abs ∘ − )) = (MetOpen‘(abs ∘ − ))
38	6, 8, 7, 18, 27, 36, 37	dvmulxxbr 15513	. 2 ⊢ (𝜑 → 𝐶(𝑆 D (𝐹 ∘_𝑓 · 𝐺))((((𝑆 D 𝐹)‘𝐶) · (𝐺‘𝐶)) + (((𝑆 D 𝐺)‘𝐶) · (𝐹‘𝐶))))
39		funbrfv 5691	. 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 1398 ∈ wcel 2202 Vcvv 2803 ⊆ wss 3201 {cpr 3674 class class class wbr 4093 dom cdm 4731 ∘ ccom 4735 Fun wfun 5327 ⟶wf 5329 ‘cfv 5333 (class class class)co 6028 ∘_𝑓 cof 6242 ↑_pm cpm 6861 ℂcc 8090 ℝcr 8091 + caddc 8095 · cmul 8097 − cmin 8409 abscabs 11637 MetOpencmopn 14637 D cdv 15466

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 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2204 ax-14 2205 ax-ext 2213 ax-coll 4209 ax-sep 4212 ax-nul 4220 ax-pow 4270 ax-pr 4305 ax-un 4536 ax-setind 4641 ax-iinf 4692 ax-cnex 8183 ax-resscn 8184 ax-1cn 8185 ax-1re 8186 ax-icn 8187 ax-addcl 8188 ax-addrcl 8189 ax-mulcl 8190 ax-mulrcl 8191 ax-addcom 8192 ax-mulcom 8193 ax-addass 8194 ax-mulass 8195 ax-distr 8196 ax-i2m1 8197 ax-0lt1 8198 ax-1rid 8199 ax-0id 8200 ax-rnegex 8201 ax-precex 8202 ax-cnre 8203 ax-pre-ltirr 8204 ax-pre-ltwlin 8205 ax-pre-lttrn 8206 ax-pre-apti 8207 ax-pre-ltadd 8208 ax-pre-mulgt0 8209 ax-pre-mulext 8210 ax-arch 8211 ax-caucvg 8212 ax-addf 8214 ax-mulf 8215

This theorem depends on definitions: df-bi 117 df-stab 839 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2364 df-ne 2404 df-nel 2499 df-ral 2516 df-rex 2517 df-reu 2518 df-rmo 2519 df-rab 2520 df-v 2805 df-sbc 3033 df-csb 3129 df-dif 3203 df-un 3205 df-in 3207 df-ss 3214 df-nul 3497 df-if 3608 df-pw 3658 df-sn 3679 df-pr 3680 df-op 3682 df-uni 3899 df-int 3934 df-iun 3977 df-br 4094 df-opab 4156 df-mpt 4157 df-tr 4193 df-id 4396 df-po 4399 df-iso 4400 df-iord 4469 df-on 4471 df-ilim 4472 df-suc 4474 df-iom 4695 df-xp 4737 df-rel 4738 df-cnv 4739 df-co 4740 df-dm 4741 df-rn 4742 df-res 4743 df-ima 4744 df-iota 5293 df-fun 5335 df-fn 5336 df-f 5337 df-f1 5338 df-fo 5339 df-f1o 5340 df-fv 5341 df-isom 5342 df-riota 5981 df-ov 6031 df-oprab 6032 df-mpo 6033 df-of 6244 df-1st 6312 df-2nd 6313 df-recs 6514 df-frec 6600 df-map 6862 df-pm 6863 df-sup 7243 df-inf 7244 df-pnf 8275 df-mnf 8276 df-xr 8277 df-ltxr 8278 df-le 8279 df-sub 8411 df-neg 8412 df-reap 8814 df-ap 8821 df-div 8912 df-inn 9203 df-2 9261 df-3 9262 df-4 9263 df-n0 9462 df-z 9541 df-uz 9817 df-q 9915 df-rp 9950 df-xneg 10068 df-xadd 10069 df-seqfrec 10773 df-exp 10864 df-cj 11482 df-re 11483 df-im 11484 df-rsqrt 11638 df-abs 11639 df-rest 13404 df-topgen 13423 df-psmet 14639 df-xmet 14640 df-met 14641 df-bl 14642 df-mopn 14643 df-top 14809 df-topon 14822 df-bases 14854 df-ntr 14907 df-cn 14999 df-cnp 15000 df-tx 15064 df-cncf 15382 df-limced 15467 df-dvap 15468

This theorem is referenced by: dvimulf 15517

W3C validator