Theorem isumclim3 15777

Description: The sequence of partial finite sums of a converging infinite series converges to the infinite sum of the series. Note that 𝑗 must not occur in 𝐴. (Contributed by NM, 9-Jan-2006.) (Revised by Mario Carneiro, 23-Apr-2014.)

Hypotheses

Ref	Expression
isumclim3.1	⊢ 𝑍 = (ℤ≥‘𝑀)
isumclim3.2	⊢ (𝜑 → 𝑀 ∈ ℤ)
isumclim3.3	⊢ (𝜑 → 𝐹 ∈ dom ⇝ )
isumclim3.4	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ)
isumclim3.5	⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) = Σ𝑘 ∈ (𝑀...𝑗)𝐴)

Assertion

Ref	Expression
isumclim3	⊢ (𝜑 → 𝐹 ⇝ Σ𝑘 ∈ 𝑍 𝐴)

Distinct variable groups:   𝐴,𝑗   𝑗,𝑘,𝑀   𝜑,𝑗,𝑘   𝑗,𝑍,𝑘   𝑗,𝐹

Allowed substitution hints:   𝐴(𝑘)   𝐹(𝑘)

Proof of Theorem isumclim3

Dummy variables 𝑚 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isumclim3.3	. . 3 ⊢ (𝜑 → 𝐹 ∈ dom ⇝ )
2		climdm 15572	. . 3 ⊢ (𝐹 ∈ dom ⇝ ↔ 𝐹 ⇝ ( ⇝ ‘𝐹))
3	1, 2	sylib 220	. 2 ⊢ (𝜑 → 𝐹 ⇝ ( ⇝ ‘𝐹))
4		sumfc 15727	. . . 4 ⊢ Σ𝑚 ∈ 𝑍 ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) = Σ𝑘 ∈ 𝑍 𝐴
5		isumclim3.1	. . . . 5 ⊢ 𝑍 = (ℤ≥‘𝑀)
6		isumclim3.2	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ)
7		eqidd 2762	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) = ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚))
8		isumclim3.4	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ)
9	8	fmpttd 7091	. . . . . 6 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ 𝐴):𝑍⟶ℂ)
10	9	ffvelcdmda 7060	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) ∈ ℂ)
11	5, 6, 7, 10	isum 15737	. . . 4 ⊢ (𝜑 → Σ𝑚 ∈ 𝑍 ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) = ( ⇝ ‘seq𝑀( + , (𝑘 ∈ 𝑍 ↦ 𝐴))))
12	4, 11	eqtr3id 2810	. . 3 ⊢ (𝜑 → Σ𝑘 ∈ 𝑍 𝐴 = ( ⇝ ‘seq𝑀( + , (𝑘 ∈ 𝑍 ↦ 𝐴))))
13		seqex 14010	. . . . . . 7 ⊢ seq𝑀( + , (𝑘 ∈ 𝑍 ↦ 𝐴)) ∈ V
14	13	a1i 11	. . . . . 6 ⊢ (𝜑 → seq𝑀( + , (𝑘 ∈ 𝑍 ↦ 𝐴)) ∈ V)
15		isumclim3.5	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) = Σ𝑘 ∈ (𝑀...𝑗)𝐴)
16		fvres 6881	. . . . . . . . . . 11 ⊢ (𝑚 ∈ (𝑀...𝑗) → (((𝑘 ∈ 𝑍 ↦ 𝐴) ↾ (𝑀...𝑗))‘𝑚) = ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚))
17		fzssuz 13564	. . . . . . . . . . . . . 14 ⊢ (𝑀...𝑗) ⊆ (ℤ≥‘𝑀)
18	17, 5	sseqtrri 3983	. . . . . . . . . . . . 13 ⊢ (𝑀...𝑗) ⊆ 𝑍
19		resmpt 6022	. . . . . . . . . . . . 13 ⊢ ((𝑀...𝑗) ⊆ 𝑍 → ((𝑘 ∈ 𝑍 ↦ 𝐴) ↾ (𝑀...𝑗)) = (𝑘 ∈ (𝑀...𝑗) ↦ 𝐴))
20	18, 19	ax-mp 5	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ 𝑍 ↦ 𝐴) ↾ (𝑀...𝑗)) = (𝑘 ∈ (𝑀...𝑗) ↦ 𝐴)
21	20	fveq1i 6863	. . . . . . . . . . 11 ⊢ (((𝑘 ∈ 𝑍 ↦ 𝐴) ↾ (𝑀...𝑗))‘𝑚) = ((𝑘 ∈ (𝑀...𝑗) ↦ 𝐴)‘𝑚)
22	16, 21	eqtr3di 2811	. . . . . . . . . 10 ⊢ (𝑚 ∈ (𝑀...𝑗) → ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) = ((𝑘 ∈ (𝑀...𝑗) ↦ 𝐴)‘𝑚))
23	22	sumeq2i 15716	. . . . . . . . 9 ⊢ Σ𝑚 ∈ (𝑀...𝑗)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) = Σ𝑚 ∈ (𝑀...𝑗)((𝑘 ∈ (𝑀...𝑗) ↦ 𝐴)‘𝑚)
24		sumfc 15727	. . . . . . . . 9 ⊢ Σ𝑚 ∈ (𝑀...𝑗)((𝑘 ∈ (𝑀...𝑗) ↦ 𝐴)‘𝑚) = Σ𝑘 ∈ (𝑀...𝑗)𝐴
25	23, 24	eqtri 2784	. . . . . . . 8 ⊢ Σ𝑚 ∈ (𝑀...𝑗)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) = Σ𝑘 ∈ (𝑀...𝑗)𝐴
26		eqidd 2762	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑚 ∈ (𝑀...𝑗)) → ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) = ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚))
27		simpr 488	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ 𝑍)
28	27, 5	eleqtrdi 2871	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ (ℤ≥‘𝑀))
29		simpl 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝜑)
30		elfzuz 13519	. . . . . . . . . . 11 ⊢ (𝑚 ∈ (𝑀...𝑗) → 𝑚 ∈ (ℤ≥‘𝑀))
31	30, 5	eleqtrrdi 2872	. . . . . . . . . 10 ⊢ (𝑚 ∈ (𝑀...𝑗) → 𝑚 ∈ 𝑍)
32	29, 31, 10	syl2an 605	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑚 ∈ (𝑀...𝑗)) → ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) ∈ ℂ)
33	26, 28, 32	fsumser 15748	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → Σ𝑚 ∈ (𝑀...𝑗)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) = (seq𝑀( + , (𝑘 ∈ 𝑍 ↦ 𝐴))‘𝑗))
34	25, 33	eqtr3id 2810	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → Σ𝑘 ∈ (𝑀...𝑗)𝐴 = (seq𝑀( + , (𝑘 ∈ 𝑍 ↦ 𝐴))‘𝑗))
35	15, 34	eqtr2d 2797	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (seq𝑀( + , (𝑘 ∈ 𝑍 ↦ 𝐴))‘𝑗) = (𝐹‘𝑗))
36	5, 14, 1, 6, 35	climeq 15585	. . . . 5 ⊢ (𝜑 → (seq𝑀( + , (𝑘 ∈ 𝑍 ↦ 𝐴)) ⇝ 𝑥 ↔ 𝐹 ⇝ 𝑥))
37	36	iotabidv 6500	. . . 4 ⊢ (𝜑 → (℩𝑥seq𝑀( + , (𝑘 ∈ 𝑍 ↦ 𝐴)) ⇝ 𝑥) = (℩𝑥𝐹 ⇝ 𝑥))
38		df-fv 6524	. . . 4 ⊢ ( ⇝ ‘seq𝑀( + , (𝑘 ∈ 𝑍 ↦ 𝐴))) = (℩𝑥seq𝑀( + , (𝑘 ∈ 𝑍 ↦ 𝐴)) ⇝ 𝑥)
39		df-fv 6524	. . . 4 ⊢ ( ⇝ ‘𝐹) = (℩𝑥𝐹 ⇝ 𝑥)
40	37, 38, 39	3eqtr4g 2821	. . 3 ⊢ (𝜑 → ( ⇝ ‘seq𝑀( + , (𝑘 ∈ 𝑍 ↦ 𝐴))) = ( ⇝ ‘𝐹))
41	12, 40	eqtrd 2796	. 2 ⊢ (𝜑 → Σ𝑘 ∈ 𝑍 𝐴 = ( ⇝ ‘𝐹))
42	3, 41	breqtrrd 5125	1 ⊢ (𝜑 → 𝐹 ⇝ Σ𝑘 ∈ 𝑍 𝐴)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1559   ∈ wcel 2141  Vcvv 3453   ⊆ wss 3902   class class class wbr 5097   ↦ cmpt 5178  dom cdm 5643   ↾ cres 5645  ℩cio 6470  ‘cfv 6516  (class class class)co 7391  ℂcc 11065   + caddc 11070  ℤcz 12562  ℤ_≥cuz 12833  ...cfz 13506  seqcseq 14008   ⇝ cli 15502  Σcsu 15704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5224  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7713  ax-inf2 9590  ax-cnex 11123  ax-resscn 11124  ax-1cn 11125  ax-icn 11126  ax-addcl 11127  ax-addrcl 11128  ax-mulcl 11129  ax-mulrcl 11130  ax-mulcom 11131  ax-addass 11132  ax-mulass 11133  ax-distr 11134  ax-i2m1 11135  ax-1ne0 11136  ax-1rid 11137  ax-rnegex 11138  ax-rrecex 11139  ax-cnre 11140  ax-pre-lttri 11141  ax-pre-lttrn 11142  ax-pre-ltadd 11143  ax-pre-mulgt0 11144  ax-pre-sup 11145

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3061  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-int 4903  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5538  df-eprel 5543  df-po 5551  df-so 5552  df-fr 5596  df-se 5597  df-we 5598  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-pred 6283  df-ord 6344  df-on 6345  df-lim 6346  df-suc 6347  df-iota 6472  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-isom 6525  df-riota 7348  df-ov 7394  df-oprab 7395  df-mpo 7396  df-om 7842  df-1st 7965  df-2nd 7966  df-frecs 8256  df-wrecs 8287  df-recs 8336  df-rdg 8375  df-1o 8431  df-er 8672  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-sup 9382  df-oi 9452  df-card 9891  df-pnf 11212  df-mnf 11213  df-xr 11214  df-ltxr 11215  df-le 11216  df-sub 11410  df-neg 11411  df-div 11839  df-nn 12205  df-2 12274  df-3 12275  df-n0 12476  df-z 12563  df-uz 12834  df-rp 12988  df-fz 13507  df-fzo 13654  df-seq 14009  df-exp 14069  df-hash 14338  df-cj 15117  df-re 15118  df-im 15119  df-sqrt 15253  df-abs 15254  df-clim 15506  df-sum 15705

This theorem is referenced by:  esumcvg  34344