isumadd - Intuitionistic Logic Explorer

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Theorem isumadd 11950

Description: Addition of infinite sums. (Contributed by Mario Carneiro, 18-Aug-2013.) (Revised by Mario Carneiro, 23-Apr-2014.)

**Hypotheses**
Ref	Expression
isumadd.1	⊢ 𝑍 = (ℤ_≥‘𝑀)
isumadd.2	⊢ (𝜑 → 𝑀 ∈ ℤ)
isumadd.3	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐴)
isumadd.4	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ)
isumadd.5	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = 𝐵)
isumadd.6	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℂ)
isumadd.7	⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )
isumadd.8	⊢ (𝜑 → seq𝑀( + , 𝐺) ∈ dom ⇝ )

**Assertion**
Ref	Expression
isumadd	⊢ (𝜑 → Σ𝑘 ∈ 𝑍 (𝐴 + 𝐵) = (Σ𝑘 ∈ 𝑍 𝐴 + Σ𝑘 ∈ 𝑍 𝐵))

Distinct variable groups: 𝑘,𝐹 𝑘,𝐺 𝑘,𝑀 𝜑,𝑘 𝑘,𝑍 Allowed substitution hints: 𝐴(𝑘) 𝐵(𝑘)

Proof of Theorem isumadd Dummy variables 𝑗 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isumadd.1	. 2 ⊢ 𝑍 = (ℤ_≥‘𝑀)
2		isumadd.2	. 2 ⊢ (𝜑 → 𝑀 ∈ ℤ)
3		simpr 110	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ 𝑍)
4		isumadd.3	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐴)
5		isumadd.4	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ)
6	4, 5	eqeltrd 2306	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)
7		isumadd.5	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = 𝐵)
8		isumadd.6	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℂ)
9	7, 8	eqeltrd 2306	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ ℂ)
10	6, 9	addcld 8174	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝐹‘𝑘) + (𝐺‘𝑘)) ∈ ℂ)
11		fveq2 5629	. . . . . 6 ⊢ (𝑚 = 𝑘 → (𝐹‘𝑚) = (𝐹‘𝑘))
12		fveq2 5629	. . . . . 6 ⊢ (𝑚 = 𝑘 → (𝐺‘𝑚) = (𝐺‘𝑘))
13	11, 12	oveq12d 6025	. . . . 5 ⊢ (𝑚 = 𝑘 → ((𝐹‘𝑚) + (𝐺‘𝑚)) = ((𝐹‘𝑘) + (𝐺‘𝑘)))
14		eqid 2229	. . . . 5 ⊢ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚) + (𝐺‘𝑚))) = (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚) + (𝐺‘𝑚)))
15	13, 14	fvmptg 5712	. . . 4 ⊢ ((𝑘 ∈ 𝑍 ∧ ((𝐹‘𝑘) + (𝐺‘𝑘)) ∈ ℂ) → ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚) + (𝐺‘𝑚)))‘𝑘) = ((𝐹‘𝑘) + (𝐺‘𝑘)))
16	3, 10, 15	syl2anc 411	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚) + (𝐺‘𝑚)))‘𝑘) = ((𝐹‘𝑘) + (𝐺‘𝑘)))
17	4, 7	oveq12d 6025	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝐹‘𝑘) + (𝐺‘𝑘)) = (𝐴 + 𝐵))
18	16, 17	eqtrd 2262	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚) + (𝐺‘𝑚)))‘𝑘) = (𝐴 + 𝐵))
19	5, 8	addcld 8174	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐴 + 𝐵) ∈ ℂ)
20		isumadd.7	. . . 4 ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )
21	1, 2, 4, 5, 20	isumclim2 11941	. . 3 ⊢ (𝜑 → seq𝑀( + , 𝐹) ⇝ Σ𝑘 ∈ 𝑍 𝐴)
22		seqex 10679	. . . 4 ⊢ seq𝑀( + , (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚) + (𝐺‘𝑚)))) ∈ V
23	22	a1i 9	. . 3 ⊢ (𝜑 → seq𝑀( + , (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚) + (𝐺‘𝑚)))) ∈ V)
24		isumadd.8	. . . 4 ⊢ (𝜑 → seq𝑀( + , 𝐺) ∈ dom ⇝ )
25	1, 2, 7, 8, 24	isumclim2 11941	. . 3 ⊢ (𝜑 → seq𝑀( + , 𝐺) ⇝ Σ𝑘 ∈ 𝑍 𝐵)
26	1, 2, 6	serf 10713	. . . 4 ⊢ (𝜑 → seq𝑀( + , 𝐹):𝑍⟶ℂ)
27	26	ffvelcdmda 5772	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (seq𝑀( + , 𝐹)‘𝑗) ∈ ℂ)
28	1, 2, 9	serf 10713	. . . 4 ⊢ (𝜑 → seq𝑀( + , 𝐺):𝑍⟶ℂ)
29	28	ffvelcdmda 5772	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (seq𝑀( + , 𝐺)‘𝑗) ∈ ℂ)
30		simpr 110	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ 𝑍)
31	30, 1	eleqtrdi 2322	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ (ℤ_≥‘𝑀))
32		simpll 527	. . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → 𝜑)
33	1	eleq2i 2296	. . . . . . 7 ⊢ (𝑘 ∈ 𝑍 ↔ 𝑘 ∈ (ℤ_≥‘𝑀))
34	33	biimpri 133	. . . . . 6 ⊢ (𝑘 ∈ (ℤ_≥‘𝑀) → 𝑘 ∈ 𝑍)
35	34	adantl 277	. . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → 𝑘 ∈ 𝑍)
36	32, 35, 6	syl2anc 411	. . . 4 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (𝐹‘𝑘) ∈ ℂ)
37	32, 35, 9	syl2anc 411	. . . 4 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (𝐺‘𝑘) ∈ ℂ)
38	32, 35, 10	syl2anc 411	. . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((𝐹‘𝑘) + (𝐺‘𝑘)) ∈ ℂ)
39	35, 38, 15	syl2anc 411	. . . 4 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚) + (𝐺‘𝑚)))‘𝑘) = ((𝐹‘𝑘) + (𝐺‘𝑘)))
40	31, 36, 37, 39	ser3add 10752	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (seq𝑀( + , (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚) + (𝐺‘𝑚))))‘𝑗) = ((seq𝑀( + , 𝐹)‘𝑗) + (seq𝑀( + , 𝐺)‘𝑗)))
41	1, 2, 21, 23, 25, 27, 29, 40	climadd 11845	. 2 ⊢ (𝜑 → seq𝑀( + , (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚) + (𝐺‘𝑚)))) ⇝ (Σ𝑘 ∈ 𝑍 𝐴 + Σ𝑘 ∈ 𝑍 𝐵))
42	1, 2, 18, 19, 41	isumclim 11940	1 ⊢ (𝜑 → Σ𝑘 ∈ 𝑍 (𝐴 + 𝐵) = (Σ𝑘 ∈ 𝑍 𝐴 + Σ𝑘 ∈ 𝑍 𝐵))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 = wceq 1395 ∈ wcel 2200 Vcvv 2799 ↦ cmpt 4145 dom cdm 4719 ‘cfv 5318 (class class class)co 6007 ℂcc 8005 + caddc 8010 ℤcz 9454 ℤ_≥cuz 9730 seqcseq 10677 ⇝ cli 11797 Σcsu 11872

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 4199 ax-sep 4202 ax-nul 4210 ax-pow 4258 ax-pr 4293 ax-un 4524 ax-setind 4629 ax-iinf 4680 ax-cnex 8098 ax-resscn 8099 ax-1cn 8100 ax-1re 8101 ax-icn 8102 ax-addcl 8103 ax-addrcl 8104 ax-mulcl 8105 ax-mulrcl 8106 ax-addcom 8107 ax-mulcom 8108 ax-addass 8109 ax-mulass 8110 ax-distr 8111 ax-i2m1 8112 ax-0lt1 8113 ax-1rid 8114 ax-0id 8115 ax-rnegex 8116 ax-precex 8117 ax-cnre 8118 ax-pre-ltirr 8119 ax-pre-ltwlin 8120 ax-pre-lttrn 8121 ax-pre-apti 8122 ax-pre-ltadd 8123 ax-pre-mulgt0 8124 ax-pre-mulext 8125 ax-arch 8126 ax-caucvg 8127

This theorem depends on definitions: df-bi 117 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 3889 df-int 3924 df-iun 3967 df-br 4084 df-opab 4146 df-mpt 4147 df-tr 4183 df-id 4384 df-po 4387 df-iso 4388 df-iord 4457 df-on 4459 df-ilim 4460 df-suc 4462 df-iom 4683 df-xp 4725 df-rel 4726 df-cnv 4727 df-co 4728 df-dm 4729 df-rn 4730 df-res 4731 df-ima 4732 df-iota 5278 df-fun 5320 df-fn 5321 df-f 5322 df-f1 5323 df-fo 5324 df-f1o 5325 df-fv 5326 df-isom 5327 df-riota 5960 df-ov 6010 df-oprab 6011 df-mpo 6012 df-1st 6292 df-2nd 6293 df-recs 6457 df-irdg 6522 df-frec 6543 df-1o 6568 df-oadd 6572 df-er 6688 df-en 6896 df-dom 6897 df-fin 6898 df-pnf 8191 df-mnf 8192 df-xr 8193 df-ltxr 8194 df-le 8195 df-sub 8327 df-neg 8328 df-reap 8730 df-ap 8737 df-div 8828 df-inn 9119 df-2 9177 df-3 9178 df-4 9179 df-n0 9378 df-z 9455 df-uz 9731 df-q 9823 df-rp 9858 df-fz 10213 df-fzo 10347 df-seqfrec 10678 df-exp 10769 df-ihash 11006 df-cj 11361 df-re 11362 df-im 11363 df-rsqrt 11517 df-abs 11518 df-clim 11798 df-sumdc 11873

This theorem is referenced by: sumsplitdc 11951

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