dvmptaddx - Intuitionistic Logic Explorer

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Theorem dvmptaddx 15266

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 5748	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶ℂ)
5		dvmptadd.c	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐶 ∈ ℂ)
6	5	fmpttd 5748	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐶):𝑋⟶ℂ)
7		dvmptadd.da	. . . . 5 ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐵))
8	7	dmeqd 4889	. . . 4 ⊢ (𝜑 → dom (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) = dom (𝑥 ∈ 𝑋 ↦ 𝐵))
9		dvmptadd.b	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑉)
10	9	ralrimiva 2580	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 𝐵 ∈ 𝑉)
11		dmmptg 5189	. . . . 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 2239	. . 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 4889	. . . 4 ⊢ (𝜑 → dom (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐶)) = dom (𝑥 ∈ 𝑋 ↦ 𝐷))
16		dvmptadd.d	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐷 ∈ 𝑊)
17	16	ralrimiva 2580	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 𝐷 ∈ 𝑊)
18		dmmptg 5189	. . . . 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 2239	. . 3 ⊢ (𝜑 → dom (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐶)) = 𝑋)
21	1, 2, 4, 6, 13, 20	dviaddf 15252	. 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 4192	. . . 4 ⊢ (𝜑 → 𝑋 ∈ V)
23		eqidd 2207	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) = (𝑥 ∈ 𝑋 ↦ 𝐴))
24		eqidd 2207	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐶) = (𝑥 ∈ 𝑋 ↦ 𝐶))
25	22, 3, 5, 23, 24	offval2 6187	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ↦ 𝐴) ∘_𝑓 + (𝑥 ∈ 𝑋 ↦ 𝐶)) = (𝑥 ∈ 𝑋 ↦ (𝐴 + 𝐶)))
26	25	oveq2d 5973	. 2 ⊢ (𝜑 → (𝑆 D ((𝑥 ∈ 𝑋 ↦ 𝐴) ∘_𝑓 + (𝑥 ∈ 𝑋 ↦ 𝐶))) = (𝑆 D (𝑥 ∈ 𝑋 ↦ (𝐴 + 𝐶))))
27	22, 9, 16, 7, 14	offval2 6187	. 2 ⊢ (𝜑 → ((𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) ∘_𝑓 + (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐶))) = (𝑥 ∈ 𝑋 ↦ (𝐵 + 𝐷)))
28	21, 26, 27	3eqtr3d 2247	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 1373 ∈ wcel 2177 ∀wral 2485 Vcvv 2773 ⊆ wss 3170 {cpr 3639 ↦ cmpt 4113 dom cdm 4683 (class class class)co 5957 ∘_𝑓 cof 6169 ℂcc 7943 ℝcr 7944 + caddc 7948 D cdv 15202

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 711 ax-5 1471 ax-7 1472 ax-gen 1473 ax-ie1 1517 ax-ie2 1518 ax-8 1528 ax-10 1529 ax-11 1530 ax-i12 1531 ax-bndl 1533 ax-4 1534 ax-17 1550 ax-i9 1554 ax-ial 1558 ax-i5r 1559 ax-13 2179 ax-14 2180 ax-ext 2188 ax-coll 4167 ax-sep 4170 ax-nul 4178 ax-pow 4226 ax-pr 4261 ax-un 4488 ax-setind 4593 ax-iinf 4644 ax-cnex 8036 ax-resscn 8037 ax-1cn 8038 ax-1re 8039 ax-icn 8040 ax-addcl 8041 ax-addrcl 8042 ax-mulcl 8043 ax-mulrcl 8044 ax-addcom 8045 ax-mulcom 8046 ax-addass 8047 ax-mulass 8048 ax-distr 8049 ax-i2m1 8050 ax-0lt1 8051 ax-1rid 8052 ax-0id 8053 ax-rnegex 8054 ax-precex 8055 ax-cnre 8056 ax-pre-ltirr 8057 ax-pre-ltwlin 8058 ax-pre-lttrn 8059 ax-pre-apti 8060 ax-pre-ltadd 8061 ax-pre-mulgt0 8062 ax-pre-mulext 8063 ax-arch 8064 ax-caucvg 8065 ax-addf 8067

This theorem depends on definitions: df-bi 117 df-stab 833 df-dc 837 df-3or 982 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1485 df-sb 1787 df-eu 2058 df-mo 2059 df-clab 2193 df-cleq 2199 df-clel 2202 df-nfc 2338 df-ne 2378 df-nel 2473 df-ral 2490 df-rex 2491 df-reu 2492 df-rmo 2493 df-rab 2494 df-v 2775 df-sbc 3003 df-csb 3098 df-dif 3172 df-un 3174 df-in 3176 df-ss 3183 df-nul 3465 df-if 3576 df-pw 3623 df-sn 3644 df-pr 3645 df-op 3647 df-uni 3857 df-int 3892 df-iun 3935 df-br 4052 df-opab 4114 df-mpt 4115 df-tr 4151 df-id 4348 df-po 4351 df-iso 4352 df-iord 4421 df-on 4423 df-ilim 4424 df-suc 4426 df-iom 4647 df-xp 4689 df-rel 4690 df-cnv 4691 df-co 4692 df-dm 4693 df-rn 4694 df-res 4695 df-ima 4696 df-iota 5241 df-fun 5282 df-fn 5283 df-f 5284 df-f1 5285 df-fo 5286 df-f1o 5287 df-fv 5288 df-isom 5289 df-riota 5912 df-ov 5960 df-oprab 5961 df-mpo 5962 df-of 6171 df-1st 6239 df-2nd 6240 df-recs 6404 df-frec 6490 df-map 6750 df-pm 6751 df-sup 7101 df-inf 7102 df-pnf 8129 df-mnf 8130 df-xr 8131 df-ltxr 8132 df-le 8133 df-sub 8265 df-neg 8266 df-reap 8668 df-ap 8675 df-div 8766 df-inn 9057 df-2 9115 df-3 9116 df-4 9117 df-n0 9316 df-z 9393 df-uz 9669 df-q 9761 df-rp 9796 df-xneg 9914 df-xadd 9915 df-seqfrec 10615 df-exp 10706 df-cj 11228 df-re 11229 df-im 11230 df-rsqrt 11384 df-abs 11385 df-rest 13148 df-topgen 13167 df-psmet 14380 df-xmet 14381 df-met 14382 df-bl 14383 df-mopn 14384 df-top 14545 df-topon 14558 df-bases 14590 df-ntr 14643 df-cn 14735 df-cnp 14736 df-tx 14800 df-limced 15203 df-dvap 15204

This theorem is referenced by: dvmptsubcn 15270 dvmptfsum 15272

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