telfsumo2 - Intuitionistic Logic Explorer

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Theorem telfsumo2 11468

Description: Sum of a telescoping series. (Contributed by Mario Carneiro, 2-May-2016.)

**Hypotheses**
Ref	Expression
telfsumo.1	⊢ (𝑘 = 𝑗 → 𝐴 = 𝐵)
telfsumo.2	⊢ (𝑘 = (𝑗 + 1) → 𝐴 = 𝐶)
telfsumo.3	⊢ (𝑘 = 𝑀 → 𝐴 = 𝐷)
telfsumo.4	⊢ (𝑘 = 𝑁 → 𝐴 = 𝐸)
telfsumo.5	⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
telfsumo.6	⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ)

**Assertion**
Ref	Expression
telfsumo2	⊢ (𝜑 → Σ𝑗 ∈ (𝑀..^𝑁)(𝐶 − 𝐵) = (𝐸 − 𝐷))

Distinct variable groups: 𝐴,𝑗 𝐵,𝑘 𝐶,𝑘 𝑗,𝑘,𝑀 𝑗,𝑁,𝑘 𝜑,𝑗,𝑘 𝐷,𝑘 𝑘,𝐸 Allowed substitution hints: 𝐴(𝑘) 𝐵(𝑗) 𝐶(𝑗) 𝐷(𝑗) 𝐸(𝑗)

**Proof of Theorem telfsumo2**
Step	Hyp	Ref	Expression
1		telfsumo.1	. . . 4 ⊢ (𝑘 = 𝑗 → 𝐴 = 𝐵)
2	1	negeqd 8148	. . 3 ⊢ (𝑘 = 𝑗 → -𝐴 = -𝐵)
3		telfsumo.2	. . . 4 ⊢ (𝑘 = (𝑗 + 1) → 𝐴 = 𝐶)
4	3	negeqd 8148	. . 3 ⊢ (𝑘 = (𝑗 + 1) → -𝐴 = -𝐶)
5		telfsumo.3	. . . 4 ⊢ (𝑘 = 𝑀 → 𝐴 = 𝐷)
6	5	negeqd 8148	. . 3 ⊢ (𝑘 = 𝑀 → -𝐴 = -𝐷)
7		telfsumo.4	. . . 4 ⊢ (𝑘 = 𝑁 → 𝐴 = 𝐸)
8	7	negeqd 8148	. . 3 ⊢ (𝑘 = 𝑁 → -𝐴 = -𝐸)
9		telfsumo.5	. . 3 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
10		telfsumo.6	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ)
11	10	negcld 8251	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → -𝐴 ∈ ℂ)
12	2, 4, 6, 8, 9, 11	telfsumo 11467	. 2 ⊢ (𝜑 → Σ𝑗 ∈ (𝑀..^𝑁)(-𝐵 − -𝐶) = (-𝐷 − -𝐸))
13	10	ralrimiva 2550	. . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ (𝑀...𝑁)𝐴 ∈ ℂ)
14		elfzofz 10157	. . . . 5 ⊢ (𝑗 ∈ (𝑀..^𝑁) → 𝑗 ∈ (𝑀...𝑁))
15	1	eleq1d 2246	. . . . . 6 ⊢ (𝑘 = 𝑗 → (𝐴 ∈ ℂ ↔ 𝐵 ∈ ℂ))
16	15	rspccva 2840	. . . . 5 ⊢ ((∀𝑘 ∈ (𝑀...𝑁)𝐴 ∈ ℂ ∧ 𝑗 ∈ (𝑀...𝑁)) → 𝐵 ∈ ℂ)
17	13, 14, 16	syl2an 289	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → 𝐵 ∈ ℂ)
18		fzofzp1 10222	. . . . 5 ⊢ (𝑗 ∈ (𝑀..^𝑁) → (𝑗 + 1) ∈ (𝑀...𝑁))
19	3	eleq1d 2246	. . . . . 6 ⊢ (𝑘 = (𝑗 + 1) → (𝐴 ∈ ℂ ↔ 𝐶 ∈ ℂ))
20	19	rspccva 2840	. . . . 5 ⊢ ((∀𝑘 ∈ (𝑀...𝑁)𝐴 ∈ ℂ ∧ (𝑗 + 1) ∈ (𝑀...𝑁)) → 𝐶 ∈ ℂ)
21	13, 18, 20	syl2an 289	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → 𝐶 ∈ ℂ)
22	17, 21	neg2subd 8281	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (-𝐵 − -𝐶) = (𝐶 − 𝐵))
23	22	sumeq2dv 11369	. 2 ⊢ (𝜑 → Σ𝑗 ∈ (𝑀..^𝑁)(-𝐵 − -𝐶) = Σ𝑗 ∈ (𝑀..^𝑁)(𝐶 − 𝐵))
24	5	eleq1d 2246	. . . 4 ⊢ (𝑘 = 𝑀 → (𝐴 ∈ ℂ ↔ 𝐷 ∈ ℂ))
25		eluzfz1 10026	. . . . 5 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑀 ∈ (𝑀...𝑁))
26	9, 25	syl 14	. . . 4 ⊢ (𝜑 → 𝑀 ∈ (𝑀...𝑁))
27	24, 13, 26	rspcdva 2846	. . 3 ⊢ (𝜑 → 𝐷 ∈ ℂ)
28	7	eleq1d 2246	. . . 4 ⊢ (𝑘 = 𝑁 → (𝐴 ∈ ℂ ↔ 𝐸 ∈ ℂ))
29		eluzfz2 10027	. . . . 5 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁))
30	9, 29	syl 14	. . . 4 ⊢ (𝜑 → 𝑁 ∈ (𝑀...𝑁))
31	28, 13, 30	rspcdva 2846	. . 3 ⊢ (𝜑 → 𝐸 ∈ ℂ)
32	27, 31	neg2subd 8281	. 2 ⊢ (𝜑 → (-𝐷 − -𝐸) = (𝐸 − 𝐷))
33	12, 23, 32	3eqtr3d 2218	1 ⊢ (𝜑 → Σ𝑗 ∈ (𝑀..^𝑁)(𝐶 − 𝐵) = (𝐸 − 𝐷))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 = wceq 1353 ∈ wcel 2148 ∀wral 2455 ‘cfv 5215 (class class class)co 5872 ℂcc 7806 1c1 7809 + caddc 7811 − cmin 8124 -cneg 8125 ℤ_≥cuz 9524 ...cfz 10004 ..^cfzo 10137 Σcsu 11354

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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4117 ax-sep 4120 ax-nul 4128 ax-pow 4173 ax-pr 4208 ax-un 4432 ax-setind 4535 ax-iinf 4586 ax-cnex 7899 ax-resscn 7900 ax-1cn 7901 ax-1re 7902 ax-icn 7903 ax-addcl 7904 ax-addrcl 7905 ax-mulcl 7906 ax-mulrcl 7907 ax-addcom 7908 ax-mulcom 7909 ax-addass 7910 ax-mulass 7911 ax-distr 7912 ax-i2m1 7913 ax-0lt1 7914 ax-1rid 7915 ax-0id 7916 ax-rnegex 7917 ax-precex 7918 ax-cnre 7919 ax-pre-ltirr 7920 ax-pre-ltwlin 7921 ax-pre-lttrn 7922 ax-pre-apti 7923 ax-pre-ltadd 7924 ax-pre-mulgt0 7925 ax-pre-mulext 7926 ax-arch 7927 ax-caucvg 7928

This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-if 3535 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-iun 3888 df-br 4003 df-opab 4064 df-mpt 4065 df-tr 4101 df-id 4292 df-po 4295 df-iso 4296 df-iord 4365 df-on 4367 df-ilim 4368 df-suc 4370 df-iom 4589 df-xp 4631 df-rel 4632 df-cnv 4633 df-co 4634 df-dm 4635 df-rn 4636 df-res 4637 df-ima 4638 df-iota 5177 df-fun 5217 df-fn 5218 df-f 5219 df-f1 5220 df-fo 5221 df-f1o 5222 df-fv 5223 df-isom 5224 df-riota 5828 df-ov 5875 df-oprab 5876 df-mpo 5877 df-1st 6138 df-2nd 6139 df-recs 6303 df-irdg 6368 df-frec 6389 df-1o 6414 df-oadd 6418 df-er 6532 df-en 6738 df-dom 6739 df-fin 6740 df-pnf 7990 df-mnf 7991 df-xr 7992 df-ltxr 7993 df-le 7994 df-sub 8126 df-neg 8127 df-reap 8528 df-ap 8535 df-div 8626 df-inn 8916 df-2 8974 df-3 8975 df-4 8976 df-n0 9173 df-z 9250 df-uz 9525 df-q 9616 df-rp 9650 df-fz 10005 df-fzo 10138 df-seqfrec 10441 df-exp 10515 df-ihash 10749 df-cj 10844 df-re 10845 df-im 10846 df-rsqrt 11000 df-abs 11001 df-clim 11280 df-sumdc 11355

This theorem is referenced by: telfsum2 11470

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