isummulc2 - Intuitionistic Logic Explorer

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Theorem isummulc2 11569

Description: An infinite sum multiplied by a constant. (Contributed by NM, 12-Nov-2005.) (Revised by Mario Carneiro, 23-Apr-2014.)

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

**Assertion**
Ref	Expression
isummulc2	⊢ (𝜑 → (𝐵 · Σ𝑘 ∈ 𝑍 𝐴) = Σ𝑘 ∈ 𝑍 (𝐵 · 𝐴))

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

Proof of Theorem isummulc2 Dummy variable 𝑚 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		isumcl.1	. . 3 ⊢ 𝑍 = (ℤ_≥‘𝑀)
2		isumcl.2	. . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ)
3		eqidd 2194	. . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ (𝐵 · 𝐴))‘𝑚) = ((𝑘 ∈ 𝑍 ↦ (𝐵 · 𝐴))‘𝑚))
4		summulc.6	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℂ)
5	4	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℂ)
6		isumcl.4	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ)
7	5, 6	mulcld 8040	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐵 · 𝐴) ∈ ℂ)
8	7	fmpttd 5713	. . . 4 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ (𝐵 · 𝐴)):𝑍⟶ℂ)
9	8	ffvelcdmda 5693	. . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ (𝐵 · 𝐴))‘𝑚) ∈ ℂ)
10		isumcl.3	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐴)
11		isumcl.5	. . . . 5 ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )
12	1, 2, 10, 6, 11	isumclim2 11565	. . . 4 ⊢ (𝜑 → seq𝑀( + , 𝐹) ⇝ Σ𝑘 ∈ 𝑍 𝐴)
13	10, 6	eqeltrd 2270	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)
14	13	ralrimiva 2567	. . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ)
15		fveq2 5554	. . . . . . 7 ⊢ (𝑘 = 𝑚 → (𝐹‘𝑘) = (𝐹‘𝑚))
16	15	eleq1d 2262	. . . . . 6 ⊢ (𝑘 = 𝑚 → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘𝑚) ∈ ℂ))
17	16	rspccva 2863	. . . . 5 ⊢ ((∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚) ∈ ℂ)
18	14, 17	sylan 283	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚) ∈ ℂ)
19		simpr 110	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ 𝑍)
20		eqid 2193	. . . . . . . . 9 ⊢ (𝑘 ∈ 𝑍 ↦ (𝐵 · 𝐴)) = (𝑘 ∈ 𝑍 ↦ (𝐵 · 𝐴))
21	20	fvmpt2 5641	. . . . . . . 8 ⊢ ((𝑘 ∈ 𝑍 ∧ (𝐵 · 𝐴) ∈ ℂ) → ((𝑘 ∈ 𝑍 ↦ (𝐵 · 𝐴))‘𝑘) = (𝐵 · 𝐴))
22	19, 7, 21	syl2anc 411	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ (𝐵 · 𝐴))‘𝑘) = (𝐵 · 𝐴))
23	10	oveq2d 5934	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐵 · (𝐹‘𝑘)) = (𝐵 · 𝐴))
24	22, 23	eqtr4d 2229	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ (𝐵 · 𝐴))‘𝑘) = (𝐵 · (𝐹‘𝑘)))
25	24	ralrimiva 2567	. . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ 𝑍 ((𝑘 ∈ 𝑍 ↦ (𝐵 · 𝐴))‘𝑘) = (𝐵 · (𝐹‘𝑘)))
26		nffvmpt1 5565	. . . . . . 7 ⊢ Ⅎ𝑘((𝑘 ∈ 𝑍 ↦ (𝐵 · 𝐴))‘𝑚)
27	26	nfeq1 2346	. . . . . 6 ⊢ Ⅎ𝑘((𝑘 ∈ 𝑍 ↦ (𝐵 · 𝐴))‘𝑚) = (𝐵 · (𝐹‘𝑚))
28		fveq2 5554	. . . . . . 7 ⊢ (𝑘 = 𝑚 → ((𝑘 ∈ 𝑍 ↦ (𝐵 · 𝐴))‘𝑘) = ((𝑘 ∈ 𝑍 ↦ (𝐵 · 𝐴))‘𝑚))
29	15	oveq2d 5934	. . . . . . 7 ⊢ (𝑘 = 𝑚 → (𝐵 · (𝐹‘𝑘)) = (𝐵 · (𝐹‘𝑚)))
30	28, 29	eqeq12d 2208	. . . . . 6 ⊢ (𝑘 = 𝑚 → (((𝑘 ∈ 𝑍 ↦ (𝐵 · 𝐴))‘𝑘) = (𝐵 · (𝐹‘𝑘)) ↔ ((𝑘 ∈ 𝑍 ↦ (𝐵 · 𝐴))‘𝑚) = (𝐵 · (𝐹‘𝑚))))
31	27, 30	rspc 2858	. . . . 5 ⊢ (𝑚 ∈ 𝑍 → (∀𝑘 ∈ 𝑍 ((𝑘 ∈ 𝑍 ↦ (𝐵 · 𝐴))‘𝑘) = (𝐵 · (𝐹‘𝑘)) → ((𝑘 ∈ 𝑍 ↦ (𝐵 · 𝐴))‘𝑚) = (𝐵 · (𝐹‘𝑚))))
32	25, 31	mpan9 281	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ (𝐵 · 𝐴))‘𝑚) = (𝐵 · (𝐹‘𝑚)))
33	1, 2, 4, 12, 18, 32	isermulc2 11483	. . 3 ⊢ (𝜑 → seq𝑀( + , (𝑘 ∈ 𝑍 ↦ (𝐵 · 𝐴))) ⇝ (𝐵 · Σ𝑘 ∈ 𝑍 𝐴))
34	1, 2, 3, 9, 33	isumclim 11564	. 2 ⊢ (𝜑 → Σ𝑚 ∈ 𝑍 ((𝑘 ∈ 𝑍 ↦ (𝐵 · 𝐴))‘𝑚) = (𝐵 · Σ𝑘 ∈ 𝑍 𝐴))
35	7	ralrimiva 2567	. . 3 ⊢ (𝜑 → ∀𝑘 ∈ 𝑍 (𝐵 · 𝐴) ∈ ℂ)
36		sumfct 11517	. . 3 ⊢ (∀𝑘 ∈ 𝑍 (𝐵 · 𝐴) ∈ ℂ → Σ𝑚 ∈ 𝑍 ((𝑘 ∈ 𝑍 ↦ (𝐵 · 𝐴))‘𝑚) = Σ𝑘 ∈ 𝑍 (𝐵 · 𝐴))
37	35, 36	syl 14	. 2 ⊢ (𝜑 → Σ𝑚 ∈ 𝑍 ((𝑘 ∈ 𝑍 ↦ (𝐵 · 𝐴))‘𝑚) = Σ𝑘 ∈ 𝑍 (𝐵 · 𝐴))
38	34, 37	eqtr3d 2228	1 ⊢ (𝜑 → (𝐵 · Σ𝑘 ∈ 𝑍 𝐴) = Σ𝑘 ∈ 𝑍 (𝐵 · 𝐴))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 = wceq 1364 ∈ wcel 2164 ∀wral 2472 ↦ cmpt 4090 dom cdm 4659 ‘cfv 5254 (class class class)co 5918 ℂcc 7870 + caddc 7875 · cmul 7877 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: isummulc1 11570 trirecip 11644 geoisum1c 11663

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