dvmptaddx - Intuitionistic Logic Explorer

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Theorem dvmptaddx 15109

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 5729	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶ℂ)
5		dvmptadd.c	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐶 ∈ ℂ)
6	5	fmpttd 5729	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐶):𝑋⟶ℂ)
7		dvmptadd.da	. . . . 5 ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐵))
8	7	dmeqd 4878	. . . 4 ⊢ (𝜑 → dom (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) = dom (𝑥 ∈ 𝑋 ↦ 𝐵))
9		dvmptadd.b	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑉)
10	9	ralrimiva 2578	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 𝐵 ∈ 𝑉)
11		dmmptg 5177	. . . . 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 2237	. . 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 4878	. . . 4 ⊢ (𝜑 → dom (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐶)) = dom (𝑥 ∈ 𝑋 ↦ 𝐷))
16		dvmptadd.d	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐷 ∈ 𝑊)
17	16	ralrimiva 2578	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 𝐷 ∈ 𝑊)
18		dmmptg 5177	. . . . 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 2237	. . 3 ⊢ (𝜑 → dom (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐶)) = 𝑋)
21	1, 2, 4, 6, 13, 20	dviaddf 15095	. 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 4183	. . . 4 ⊢ (𝜑 → 𝑋 ∈ V)
23		eqidd 2205	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) = (𝑥 ∈ 𝑋 ↦ 𝐴))
24		eqidd 2205	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐶) = (𝑥 ∈ 𝑋 ↦ 𝐶))
25	22, 3, 5, 23, 24	offval2 6164	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ↦ 𝐴) ∘_𝑓 + (𝑥 ∈ 𝑋 ↦ 𝐶)) = (𝑥 ∈ 𝑋 ↦ (𝐴 + 𝐶)))
26	25	oveq2d 5950	. 2 ⊢ (𝜑 → (𝑆 D ((𝑥 ∈ 𝑋 ↦ 𝐴) ∘_𝑓 + (𝑥 ∈ 𝑋 ↦ 𝐶))) = (𝑆 D (𝑥 ∈ 𝑋 ↦ (𝐴 + 𝐶))))
27	22, 9, 16, 7, 14	offval2 6164	. 2 ⊢ (𝜑 → ((𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) ∘_𝑓 + (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐶))) = (𝑥 ∈ 𝑋 ↦ (𝐵 + 𝐷)))
28	21, 26, 27	3eqtr3d 2245	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 1372 ∈ wcel 2175 ∀wral 2483 Vcvv 2771 ⊆ wss 3165 {cpr 3633 ↦ cmpt 4104 dom cdm 4673 (class class class)co 5934 ∘_𝑓 cof 6146 ℂcc 7905 ℝcr 7906 + caddc 7910 D cdv 15045

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 710 ax-5 1469 ax-7 1470 ax-gen 1471 ax-ie1 1515 ax-ie2 1516 ax-8 1526 ax-10 1527 ax-11 1528 ax-i12 1529 ax-bndl 1531 ax-4 1532 ax-17 1548 ax-i9 1552 ax-ial 1556 ax-i5r 1557 ax-13 2177 ax-14 2178 ax-ext 2186 ax-coll 4158 ax-sep 4161 ax-nul 4169 ax-pow 4217 ax-pr 4252 ax-un 4478 ax-setind 4583 ax-iinf 4634 ax-cnex 7998 ax-resscn 7999 ax-1cn 8000 ax-1re 8001 ax-icn 8002 ax-addcl 8003 ax-addrcl 8004 ax-mulcl 8005 ax-mulrcl 8006 ax-addcom 8007 ax-mulcom 8008 ax-addass 8009 ax-mulass 8010 ax-distr 8011 ax-i2m1 8012 ax-0lt1 8013 ax-1rid 8014 ax-0id 8015 ax-rnegex 8016 ax-precex 8017 ax-cnre 8018 ax-pre-ltirr 8019 ax-pre-ltwlin 8020 ax-pre-lttrn 8021 ax-pre-apti 8022 ax-pre-ltadd 8023 ax-pre-mulgt0 8024 ax-pre-mulext 8025 ax-arch 8026 ax-caucvg 8027 ax-addf 8029

This theorem depends on definitions: df-bi 117 df-stab 832 df-dc 836 df-3or 981 df-3an 982 df-tru 1375 df-fal 1378 df-nf 1483 df-sb 1785 df-eu 2056 df-mo 2057 df-clab 2191 df-cleq 2197 df-clel 2200 df-nfc 2336 df-ne 2376 df-nel 2471 df-ral 2488 df-rex 2489 df-reu 2490 df-rmo 2491 df-rab 2492 df-v 2773 df-sbc 2998 df-csb 3093 df-dif 3167 df-un 3169 df-in 3171 df-ss 3178 df-nul 3460 df-if 3571 df-pw 3617 df-sn 3638 df-pr 3639 df-op 3641 df-uni 3850 df-int 3885 df-iun 3928 df-br 4044 df-opab 4105 df-mpt 4106 df-tr 4142 df-id 4338 df-po 4341 df-iso 4342 df-iord 4411 df-on 4413 df-ilim 4414 df-suc 4416 df-iom 4637 df-xp 4679 df-rel 4680 df-cnv 4681 df-co 4682 df-dm 4683 df-rn 4684 df-res 4685 df-ima 4686 df-iota 5229 df-fun 5270 df-fn 5271 df-f 5272 df-f1 5273 df-fo 5274 df-f1o 5275 df-fv 5276 df-isom 5277 df-riota 5889 df-ov 5937 df-oprab 5938 df-mpo 5939 df-of 6148 df-1st 6216 df-2nd 6217 df-recs 6381 df-frec 6467 df-map 6727 df-pm 6728 df-sup 7068 df-inf 7069 df-pnf 8091 df-mnf 8092 df-xr 8093 df-ltxr 8094 df-le 8095 df-sub 8227 df-neg 8228 df-reap 8630 df-ap 8637 df-div 8728 df-inn 9019 df-2 9077 df-3 9078 df-4 9079 df-n0 9278 df-z 9355 df-uz 9631 df-q 9723 df-rp 9758 df-xneg 9876 df-xadd 9877 df-seqfrec 10574 df-exp 10665 df-cj 11072 df-re 11073 df-im 11074 df-rsqrt 11228 df-abs 11229 df-rest 12991 df-topgen 13010 df-psmet 14223 df-xmet 14224 df-met 14225 df-bl 14226 df-mopn 14227 df-top 14388 df-topon 14401 df-bases 14433 df-ntr 14486 df-cn 14578 df-cnp 14579 df-tx 14643 df-limced 15046 df-dvap 15047

This theorem is referenced by: dvmptsubcn 15113 dvmptfsum 15115

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