dvmptaddx - Intuitionistic Logic Explorer

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Theorem dvmptaddx 15513

Description: Function-builder for derivative, addition 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
dvmptaddx	⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ (𝐴 + 𝐶))) = (𝑥 ∈ 𝑋 ↦ (𝐵 + 𝐷)))

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 dvmptaddx**
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 5810	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶ℂ)
5		dvmptadd.c	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐶 ∈ ℂ)
6	5	fmpttd 5810	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐶):𝑋⟶ℂ)
7		dvmptadd.da	. . . . 5 ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐵))
8	7	dmeqd 4939	. . . 4 ⊢ (𝜑 → dom (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) = dom (𝑥 ∈ 𝑋 ↦ 𝐵))
9		dvmptadd.b	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑉)
10	9	ralrimiva 2606	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 𝐵 ∈ 𝑉)
11		dmmptg 5241	. . . . 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 2264	. . 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 4939	. . . 4 ⊢ (𝜑 → dom (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐶)) = dom (𝑥 ∈ 𝑋 ↦ 𝐷))
16		dvmptadd.d	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐷 ∈ 𝑊)
17	16	ralrimiva 2606	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 𝐷 ∈ 𝑊)
18		dmmptg 5241	. . . . 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 2264	. . 3 ⊢ (𝜑 → dom (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐶)) = 𝑋)
21	1, 2, 4, 6, 13, 20	dviaddf 15499	. 2 ⊢ (𝜑 → (𝑆 D ((𝑥 ∈ 𝑋 ↦ 𝐴) ∘_𝑓 + (𝑥 ∈ 𝑋 ↦ 𝐶))) = ((𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) ∘_𝑓 + (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐶))))
22	1, 2	ssexd 4234	. . . 4 ⊢ (𝜑 → 𝑋 ∈ V)
23		eqidd 2232	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) = (𝑥 ∈ 𝑋 ↦ 𝐴))
24		eqidd 2232	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐶) = (𝑥 ∈ 𝑋 ↦ 𝐶))
25	22, 3, 5, 23, 24	offval2 6260	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ↦ 𝐴) ∘_𝑓 + (𝑥 ∈ 𝑋 ↦ 𝐶)) = (𝑥 ∈ 𝑋 ↦ (𝐴 + 𝐶)))
26	25	oveq2d 6044	. 2 ⊢ (𝜑 → (𝑆 D ((𝑥 ∈ 𝑋 ↦ 𝐴) ∘_𝑓 + (𝑥 ∈ 𝑋 ↦ 𝐶))) = (𝑆 D (𝑥 ∈ 𝑋 ↦ (𝐴 + 𝐶))))
27	22, 9, 16, 7, 14	offval2 6260	. 2 ⊢ (𝜑 → ((𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) ∘_𝑓 + (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐶))) = (𝑥 ∈ 𝑋 ↦ (𝐵 + 𝐷)))
28	21, 26, 27	3eqtr3d 2272	1 ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ (𝐴 + 𝐶))) = (𝑥 ∈ 𝑋 ↦ (𝐵 + 𝐷)))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 = wceq 1398 ∈ wcel 2202 ∀wral 2511 Vcvv 2803 ⊆ wss 3201 {cpr 3674 ↦ cmpt 4155 dom cdm 4731 (class class class)co 6028 ∘_𝑓 cof 6242 ℂcc 8073 ℝcr 8074 + caddc 8078 D cdv 15449

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 8166 ax-resscn 8167 ax-1cn 8168 ax-1re 8169 ax-icn 8170 ax-addcl 8171 ax-addrcl 8172 ax-mulcl 8173 ax-mulrcl 8174 ax-addcom 8175 ax-mulcom 8176 ax-addass 8177 ax-mulass 8178 ax-distr 8179 ax-i2m1 8180 ax-0lt1 8181 ax-1rid 8182 ax-0id 8183 ax-rnegex 8184 ax-precex 8185 ax-cnre 8186 ax-pre-ltirr 8187 ax-pre-ltwlin 8188 ax-pre-lttrn 8189 ax-pre-apti 8190 ax-pre-ltadd 8191 ax-pre-mulgt0 8192 ax-pre-mulext 8193 ax-arch 8194 ax-caucvg 8195 ax-addf 8197

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 7226 df-inf 7227 df-pnf 8258 df-mnf 8259 df-xr 8260 df-ltxr 8261 df-le 8262 df-sub 8394 df-neg 8395 df-reap 8797 df-ap 8804 df-div 8895 df-inn 9186 df-2 9244 df-3 9245 df-4 9246 df-n0 9445 df-z 9524 df-uz 9800 df-q 9898 df-rp 9933 df-xneg 10051 df-xadd 10052 df-seqfrec 10756 df-exp 10847 df-cj 11465 df-re 11466 df-im 11467 df-rsqrt 11621 df-abs 11622 df-rest 13387 df-topgen 13406 df-psmet 14622 df-xmet 14623 df-met 14624 df-bl 14625 df-mopn 14626 df-top 14792 df-topon 14805 df-bases 14837 df-ntr 14890 df-cn 14982 df-cnp 14983 df-tx 15047 df-limced 15450 df-dvap 15451

This theorem is referenced by: dvmptsubcn 15517 dvmptfsum 15519

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