dvaddxx - Intuitionistic Logic Explorer

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Theorem dvaddxx 15446

Description: The sum rule for derivatives at a point. For the (more general) relation version, see dvaddxxbr 15444. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Jim Kingdon, 25-Nov-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
dvaddxx	⊢ (𝜑 → ((𝑆 D (𝐹 ∘_𝑓 + 𝐺))‘𝐶) = (((𝑆 D 𝐹)‘𝐶) + ((𝑆 D 𝐺)‘𝐶)))

Proof of Theorem dvaddxx 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 8156	. . . . . 6 ⊢ ℂ ∈ V
3	2	a1i 9	. . . . 5 ⊢ (𝜑 → ℂ ∈ V)
4		addcl 8157	. . . . . . 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 4229	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ V)
10		inidm 3416	. . . . . 6 ⊢ (𝑋 ∩ 𝑋) = 𝑋
11	5, 6, 7, 9, 9, 10	off 6248	. . . . 5 ⊢ (𝜑 → (𝐹 ∘_𝑓 + 𝐺):𝑋⟶ℂ)
12		elpm2r 6835	. . . . 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 1274	. . . 4 ⊢ (𝜑 → (𝐹 ∘_𝑓 + 𝐺) ∈ (ℂ ↑_pm 𝑆))
14		dvfgg 15431	. . . 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 5486	. 2 ⊢ (𝜑 → Fun (𝑆 D (𝐹 ∘_𝑓 + 𝐺)))
17		recnprss 15430	. . . 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 6835	. . . . . . 7 ⊢ (((ℂ ∈ V ∧ 𝑆 ∈ {ℝ, ℂ}) ∧ (𝐹:𝑋⟶ℂ ∧ 𝑋 ⊆ 𝑆)) → 𝐹 ∈ (ℂ ↑_pm 𝑆))
21	3, 1, 6, 8, 20	syl22anc 1274	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (ℂ ↑_pm 𝑆))
22		dvfgg 15431	. . . . . 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 5485	. . . . 5 ⊢ ((𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ → Fun (𝑆 D 𝐹))
25		funfvbrb 5760	. . . . 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 6835	. . . . . . 7 ⊢ (((ℂ ∈ V ∧ 𝑆 ∈ {ℝ, ℂ}) ∧ (𝐺:𝑋⟶ℂ ∧ 𝑋 ⊆ 𝑆)) → 𝐺 ∈ (ℂ ↑_pm 𝑆))
30	3, 1, 7, 8, 29	syl22anc 1274	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ (ℂ ↑_pm 𝑆))
31		dvfgg 15431	. . . . . 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 5485	. . . . 5 ⊢ ((𝑆 D 𝐺):dom (𝑆 D 𝐺)⟶ℂ → Fun (𝑆 D 𝐺))
34		funfvbrb 5760	. . . . 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	dvaddxxbr 15444	. 2 ⊢ (𝜑 → 𝐶(𝑆 D (𝐹 ∘_𝑓 + 𝐺))(((𝑆 D 𝐹)‘𝐶) + ((𝑆 D 𝐺)‘𝐶)))
39		funbrfv 5682	. 2 ⊢ (Fun (𝑆 D (𝐹 ∘_𝑓 + 𝐺)) → (𝐶(𝑆 D (𝐹 ∘_𝑓 + 𝐺))(((𝑆 D 𝐹)‘𝐶) + ((𝑆 D 𝐺)‘𝐶)) → ((𝑆 D (𝐹 ∘_𝑓 + 𝐺))‘𝐶) = (((𝑆 D 𝐹)‘𝐶) + ((𝑆 D 𝐺)‘𝐶))))
40	16, 38, 39	sylc 62	1 ⊢ (𝜑 → ((𝑆 D (𝐹 ∘_𝑓 + 𝐺))‘𝐶) = (((𝑆 D 𝐹)‘𝐶) + ((𝑆 D 𝐺)‘𝐶)))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1397 ∈ wcel 2202 Vcvv 2802 ⊆ wss 3200 {cpr 3670 class class class wbr 4088 dom cdm 4725 ∘ ccom 4729 Fun wfun 5320 ⟶wf 5322 ‘cfv 5326 (class class class)co 6018 ∘_𝑓 cof 6233 ↑_pm cpm 6818 ℂcc 8030 ℝcr 8031 + caddc 8035 − cmin 8350 abscabs 11575 MetOpencmopn 14574 D cdv 15398

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 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-13 2204 ax-14 2205 ax-ext 2213 ax-coll 4204 ax-sep 4207 ax-nul 4215 ax-pow 4264 ax-pr 4299 ax-un 4530 ax-setind 4635 ax-iinf 4686 ax-cnex 8123 ax-resscn 8124 ax-1cn 8125 ax-1re 8126 ax-icn 8127 ax-addcl 8128 ax-addrcl 8129 ax-mulcl 8130 ax-mulrcl 8131 ax-addcom 8132 ax-mulcom 8133 ax-addass 8134 ax-mulass 8135 ax-distr 8136 ax-i2m1 8137 ax-0lt1 8138 ax-1rid 8139 ax-0id 8140 ax-rnegex 8141 ax-precex 8142 ax-cnre 8143 ax-pre-ltirr 8144 ax-pre-ltwlin 8145 ax-pre-lttrn 8146 ax-pre-apti 8147 ax-pre-ltadd 8148 ax-pre-mulgt0 8149 ax-pre-mulext 8150 ax-arch 8151 ax-caucvg 8152 ax-addf 8154

This theorem depends on definitions: df-bi 117 df-stab 838 df-dc 842 df-3or 1005 df-3an 1006 df-tru 1400 df-fal 1403 df-nf 1509 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2363 df-ne 2403 df-nel 2498 df-ral 2515 df-rex 2516 df-reu 2517 df-rmo 2518 df-rab 2519 df-v 2804 df-sbc 3032 df-csb 3128 df-dif 3202 df-un 3204 df-in 3206 df-ss 3213 df-nul 3495 df-if 3606 df-pw 3654 df-sn 3675 df-pr 3676 df-op 3678 df-uni 3894 df-int 3929 df-iun 3972 df-br 4089 df-opab 4151 df-mpt 4152 df-tr 4188 df-id 4390 df-po 4393 df-iso 4394 df-iord 4463 df-on 4465 df-ilim 4466 df-suc 4468 df-iom 4689 df-xp 4731 df-rel 4732 df-cnv 4733 df-co 4734 df-dm 4735 df-rn 4736 df-res 4737 df-ima 4738 df-iota 5286 df-fun 5328 df-fn 5329 df-f 5330 df-f1 5331 df-fo 5332 df-f1o 5333 df-fv 5334 df-isom 5335 df-riota 5971 df-ov 6021 df-oprab 6022 df-mpo 6023 df-of 6235 df-1st 6303 df-2nd 6304 df-recs 6471 df-frec 6557 df-map 6819 df-pm 6820 df-sup 7183 df-inf 7184 df-pnf 8216 df-mnf 8217 df-xr 8218 df-ltxr 8219 df-le 8220 df-sub 8352 df-neg 8353 df-reap 8755 df-ap 8762 df-div 8853 df-inn 9144 df-2 9202 df-3 9203 df-4 9204 df-n0 9403 df-z 9480 df-uz 9756 df-q 9854 df-rp 9889 df-xneg 10007 df-xadd 10008 df-seqfrec 10711 df-exp 10802 df-cj 11420 df-re 11421 df-im 11422 df-rsqrt 11576 df-abs 11577 df-rest 13342 df-topgen 13361 df-psmet 14576 df-xmet 14577 df-met 14578 df-bl 14579 df-mopn 14580 df-top 14741 df-topon 14754 df-bases 14786 df-ntr 14839 df-cn 14931 df-cnp 14932 df-tx 14996 df-limced 15399 df-dvap 15400

This theorem is referenced by: dviaddf 15448

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