isumadd - Intuitionistic Logic Explorer

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Theorem isumadd 11574

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 2270	. . . . 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 2270	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ ℂ)
10	6, 9	addcld 8039	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝐹‘𝑘) + (𝐺‘𝑘)) ∈ ℂ)
11		fveq2 5554	. . . . . 6 ⊢ (𝑚 = 𝑘 → (𝐹‘𝑚) = (𝐹‘𝑘))
12		fveq2 5554	. . . . . 6 ⊢ (𝑚 = 𝑘 → (𝐺‘𝑚) = (𝐺‘𝑘))
13	11, 12	oveq12d 5936	. . . . 5 ⊢ (𝑚 = 𝑘 → ((𝐹‘𝑚) + (𝐺‘𝑚)) = ((𝐹‘𝑘) + (𝐺‘𝑘)))
14		eqid 2193	. . . . 5 ⊢ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚) + (𝐺‘𝑚))) = (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚) + (𝐺‘𝑚)))
15	13, 14	fvmptg 5633	. . . 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 5936	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝐹‘𝑘) + (𝐺‘𝑘)) = (𝐴 + 𝐵))
18	16, 17	eqtrd 2226	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚) + (𝐺‘𝑚)))‘𝑘) = (𝐴 + 𝐵))
19	5, 8	addcld 8039	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐴 + 𝐵) ∈ ℂ)
20		isumadd.7	. . . 4 ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )
21	1, 2, 4, 5, 20	isumclim2 11565	. . 3 ⊢ (𝜑 → seq𝑀( + , 𝐹) ⇝ Σ𝑘 ∈ 𝑍 𝐴)
22		seqex 10520	. . . 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 11565	. . 3 ⊢ (𝜑 → seq𝑀( + , 𝐺) ⇝ Σ𝑘 ∈ 𝑍 𝐵)
26	1, 2, 6	serf 10554	. . . 4 ⊢ (𝜑 → seq𝑀( + , 𝐹):𝑍⟶ℂ)
27	26	ffvelcdmda 5693	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (seq𝑀( + , 𝐹)‘𝑗) ∈ ℂ)
28	1, 2, 9	serf 10554	. . . 4 ⊢ (𝜑 → seq𝑀( + , 𝐺):𝑍⟶ℂ)
29	28	ffvelcdmda 5693	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (seq𝑀( + , 𝐺)‘𝑗) ∈ ℂ)
30		simpr 110	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ 𝑍)
31	30, 1	eleqtrdi 2286	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ (ℤ_≥‘𝑀))
32		simpll 527	. . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → 𝜑)
33	1	eleq2i 2260	. . . . . . 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 10593	. . 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 11469	. 2 ⊢ (𝜑 → seq𝑀( + , (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚) + (𝐺‘𝑚)))) ⇝ (Σ𝑘 ∈ 𝑍 𝐴 + Σ𝑘 ∈ 𝑍 𝐵))
42	1, 2, 18, 19, 41	isumclim 11564	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 1364 ∈ wcel 2164 Vcvv 2760 ↦ cmpt 4090 dom cdm 4659 ‘cfv 5254 (class class class)co 5918 ℂcc 7870 + caddc 7875 ℤcz 9317 ℤ_≥cuz 9592 seqcseq 10518 ⇝ cli 11421 Σcsu 11496

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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-coll 4144 ax-sep 4147 ax-nul 4155 ax-pow 4203 ax-pr 4238 ax-un 4464 ax-setind 4569 ax-iinf 4620 ax-cnex 7963 ax-resscn 7964 ax-1cn 7965 ax-1re 7966 ax-icn 7967 ax-addcl 7968 ax-addrcl 7969 ax-mulcl 7970 ax-mulrcl 7971 ax-addcom 7972 ax-mulcom 7973 ax-addass 7974 ax-mulass 7975 ax-distr 7976 ax-i2m1 7977 ax-0lt1 7978 ax-1rid 7979 ax-0id 7980 ax-rnegex 7981 ax-precex 7982 ax-cnre 7983 ax-pre-ltirr 7984 ax-pre-ltwlin 7985 ax-pre-lttrn 7986 ax-pre-apti 7987 ax-pre-ltadd 7988 ax-pre-mulgt0 7989 ax-pre-mulext 7990 ax-arch 7991 ax-caucvg 7992

This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rmo 2480 df-rab 2481 df-v 2762 df-sbc 2986 df-csb 3081 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-nul 3447 df-if 3558 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-int 3871 df-iun 3914 df-br 4030 df-opab 4091 df-mpt 4092 df-tr 4128 df-id 4324 df-po 4327 df-iso 4328 df-iord 4397 df-on 4399 df-ilim 4400 df-suc 4402 df-iom 4623 df-xp 4665 df-rel 4666 df-cnv 4667 df-co 4668 df-dm 4669 df-rn 4670 df-res 4671 df-ima 4672 df-iota 5215 df-fun 5256 df-fn 5257 df-f 5258 df-f1 5259 df-fo 5260 df-f1o 5261 df-fv 5262 df-isom 5263 df-riota 5873 df-ov 5921 df-oprab 5922 df-mpo 5923 df-1st 6193 df-2nd 6194 df-recs 6358 df-irdg 6423 df-frec 6444 df-1o 6469 df-oadd 6473 df-er 6587 df-en 6795 df-dom 6796 df-fin 6797 df-pnf 8056 df-mnf 8057 df-xr 8058 df-ltxr 8059 df-le 8060 df-sub 8192 df-neg 8193 df-reap 8594 df-ap 8601 df-div 8692 df-inn 8983 df-2 9041 df-3 9042 df-4 9043 df-n0 9241 df-z 9318 df-uz 9593 df-q 9685 df-rp 9720 df-fz 10075 df-fzo 10209 df-seqfrec 10519 df-exp 10610 df-ihash 10847 df-cj 10986 df-re 10987 df-im 10988 df-rsqrt 11142 df-abs 11143 df-clim 11422 df-sumdc 11497

This theorem is referenced by: sumsplitdc 11575

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