dvmptmulx - Intuitionistic Logic Explorer

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Theorem dvmptmulx 15379

Description: Function-builder for derivative, product rule. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)

**Hypotheses**
Ref	Expression
dvmptadd.s	⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})
dvmptadd.a	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ)
dvmptadd.b	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑉)
dvmptadd.da	⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐵))
dvmptclx.ss	⊢ (𝜑 → 𝑋 ⊆ 𝑆)
dvmptadd.c	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐶 ∈ ℂ)
dvmptadd.d	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐷 ∈ 𝑊)
dvmptadd.dc	⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐶)) = (𝑥 ∈ 𝑋 ↦ 𝐷))

**Assertion**
Ref	Expression
dvmptmulx	⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐶))) = (𝑥 ∈ 𝑋 ↦ ((𝐵 · 𝐶) + (𝐷 · 𝐴))))

Distinct variable groups: 𝜑,𝑥 𝑥,𝑆 𝑥,𝑉 𝑥,𝑊 𝑥,𝑋 Allowed substitution hints: 𝐴(𝑥) 𝐵(𝑥) 𝐶(𝑥) 𝐷(𝑥)

**Proof of Theorem dvmptmulx**
Step	Hyp	Ref	Expression
1		dvmptadd.s	. . 3 ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})
2		dvmptclx.ss	. . 3 ⊢ (𝜑 → 𝑋 ⊆ 𝑆)
3		dvmptadd.a	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ)
4	3	fmpttd 5783	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶ℂ)
5		dvmptadd.c	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐶 ∈ ℂ)
6	5	fmpttd 5783	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐶):𝑋⟶ℂ)
7		dvmptadd.da	. . . . 5 ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐵))
8	7	dmeqd 4922	. . . 4 ⊢ (𝜑 → dom (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) = dom (𝑥 ∈ 𝑋 ↦ 𝐵))
9		dvmptadd.b	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑉)
10	9	ralrimiva 2603	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 𝐵 ∈ 𝑉)
11		dmmptg 5222	. . . . 5 ⊢ (∀𝑥 ∈ 𝑋 𝐵 ∈ 𝑉 → dom (𝑥 ∈ 𝑋 ↦ 𝐵) = 𝑋)
12	10, 11	syl 14	. . . 4 ⊢ (𝜑 → dom (𝑥 ∈ 𝑋 ↦ 𝐵) = 𝑋)
13	8, 12	eqtrd 2262	. . 3 ⊢ (𝜑 → dom (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) = 𝑋)
14		dvmptadd.dc	. . . . 5 ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐶)) = (𝑥 ∈ 𝑋 ↦ 𝐷))
15	14	dmeqd 4922	. . . 4 ⊢ (𝜑 → dom (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐶)) = dom (𝑥 ∈ 𝑋 ↦ 𝐷))
16		dvmptadd.d	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐷 ∈ 𝑊)
17	16	ralrimiva 2603	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 𝐷 ∈ 𝑊)
18		dmmptg 5222	. . . . 5 ⊢ (∀𝑥 ∈ 𝑋 𝐷 ∈ 𝑊 → dom (𝑥 ∈ 𝑋 ↦ 𝐷) = 𝑋)
19	17, 18	syl 14	. . . 4 ⊢ (𝜑 → dom (𝑥 ∈ 𝑋 ↦ 𝐷) = 𝑋)
20	15, 19	eqtrd 2262	. . 3 ⊢ (𝜑 → dom (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐶)) = 𝑋)
21	1, 2, 4, 6, 13, 20	dvimulf 15365	. 2 ⊢ (𝜑 → (𝑆 D ((𝑥 ∈ 𝑋 ↦ 𝐴) ∘_𝑓 · (𝑥 ∈ 𝑋 ↦ 𝐶))) = (((𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) ∘_𝑓 · (𝑥 ∈ 𝑋 ↦ 𝐶)) ∘_𝑓 + ((𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐶)) ∘_𝑓 · (𝑥 ∈ 𝑋 ↦ 𝐴))))
22	1, 2	ssexd 4223	. . . 4 ⊢ (𝜑 → 𝑋 ∈ V)
23		eqidd 2230	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) = (𝑥 ∈ 𝑋 ↦ 𝐴))
24		eqidd 2230	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐶) = (𝑥 ∈ 𝑋 ↦ 𝐶))
25	22, 3, 5, 23, 24	offval2 6224	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ↦ 𝐴) ∘_𝑓 · (𝑥 ∈ 𝑋 ↦ 𝐶)) = (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐶)))
26	25	oveq2d 6010	. 2 ⊢ (𝜑 → (𝑆 D ((𝑥 ∈ 𝑋 ↦ 𝐴) ∘_𝑓 · (𝑥 ∈ 𝑋 ↦ 𝐶))) = (𝑆 D (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐶))))
27	1, 3, 9, 7, 2	dvmptclx 15377	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ ℂ)
28	27, 5	mulcld 8155	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐵 · 𝐶) ∈ ℂ)
29	1, 5, 16, 14, 2	dvmptclx 15377	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐷 ∈ ℂ)
30	29, 3	mulcld 8155	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐷 · 𝐴) ∈ ℂ)
31	22, 9, 5, 7, 24	offval2 6224	. . 3 ⊢ (𝜑 → ((𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) ∘_𝑓 · (𝑥 ∈ 𝑋 ↦ 𝐶)) = (𝑥 ∈ 𝑋 ↦ (𝐵 · 𝐶)))
32	22, 16, 3, 14, 23	offval2 6224	. . 3 ⊢ (𝜑 → ((𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐶)) ∘_𝑓 · (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ (𝐷 · 𝐴)))
33	22, 28, 30, 31, 32	offval2 6224	. 2 ⊢ (𝜑 → (((𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) ∘_𝑓 · (𝑥 ∈ 𝑋 ↦ 𝐶)) ∘_𝑓 + ((𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐶)) ∘_𝑓 · (𝑥 ∈ 𝑋 ↦ 𝐴))) = (𝑥 ∈ 𝑋 ↦ ((𝐵 · 𝐶) + (𝐷 · 𝐴))))
34	21, 26, 33	3eqtr3d 2270	1 ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐶))) = (𝑥 ∈ 𝑋 ↦ ((𝐵 · 𝐶) + (𝐷 · 𝐴))))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 = wceq 1395 ∈ wcel 2200 ∀wral 2508 Vcvv 2799 ⊆ wss 3197 {cpr 3667 ↦ cmpt 4144 dom cdm 4716 (class class class)co 5994 ∘_𝑓 cof 6206 ℂcc 7985 ℝcr 7986 + caddc 7990 · cmul 7992 D cdv 15314

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4198 ax-sep 4201 ax-nul 4209 ax-pow 4257 ax-pr 4292 ax-un 4521 ax-setind 4626 ax-iinf 4677 ax-cnex 8078 ax-resscn 8079 ax-1cn 8080 ax-1re 8081 ax-icn 8082 ax-addcl 8083 ax-addrcl 8084 ax-mulcl 8085 ax-mulrcl 8086 ax-addcom 8087 ax-mulcom 8088 ax-addass 8089 ax-mulass 8090 ax-distr 8091 ax-i2m1 8092 ax-0lt1 8093 ax-1rid 8094 ax-0id 8095 ax-rnegex 8096 ax-precex 8097 ax-cnre 8098 ax-pre-ltirr 8099 ax-pre-ltwlin 8100 ax-pre-lttrn 8101 ax-pre-apti 8102 ax-pre-ltadd 8103 ax-pre-mulgt0 8104 ax-pre-mulext 8105 ax-arch 8106 ax-caucvg 8107 ax-addf 8109 ax-mulf 8110

This theorem depends on definitions: df-bi 117 df-stab 836 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rmo 2516 df-rab 2517 df-v 2801 df-sbc 3029 df-csb 3125 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-nul 3492 df-if 3603 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3888 df-int 3923 df-iun 3966 df-br 4083 df-opab 4145 df-mpt 4146 df-tr 4182 df-id 4381 df-po 4384 df-iso 4385 df-iord 4454 df-on 4456 df-ilim 4457 df-suc 4459 df-iom 4680 df-xp 4722 df-rel 4723 df-cnv 4724 df-co 4725 df-dm 4726 df-rn 4727 df-res 4728 df-ima 4729 df-iota 5274 df-fun 5316 df-fn 5317 df-f 5318 df-f1 5319 df-fo 5320 df-f1o 5321 df-fv 5322 df-isom 5323 df-riota 5947 df-ov 5997 df-oprab 5998 df-mpo 5999 df-of 6208 df-1st 6276 df-2nd 6277 df-recs 6441 df-frec 6527 df-map 6787 df-pm 6788 df-sup 7139 df-inf 7140 df-pnf 8171 df-mnf 8172 df-xr 8173 df-ltxr 8174 df-le 8175 df-sub 8307 df-neg 8308 df-reap 8710 df-ap 8717 df-div 8808 df-inn 9099 df-2 9157 df-3 9158 df-4 9159 df-n0 9358 df-z 9435 df-uz 9711 df-q 9803 df-rp 9838 df-xneg 9956 df-xadd 9957 df-seqfrec 10657 df-exp 10748 df-cj 11339 df-re 11340 df-im 11341 df-rsqrt 11495 df-abs 11496 df-rest 13260 df-topgen 13279 df-psmet 14492 df-xmet 14493 df-met 14494 df-bl 14495 df-mopn 14496 df-top 14657 df-topon 14670 df-bases 14702 df-ntr 14755 df-cn 14847 df-cnp 14848 df-tx 14912 df-cncf 15230 df-limced 15315 df-dvap 15316

This theorem is referenced by: dvmptcmulcn 15380

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