fsum3 - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > fsum3		GIF version

Theorem fsum3 10833

Description: The value of a sum over a nonempty finite set. (Contributed by Jim Kingdon, 10-Oct-2022.)

**Hypotheses**
Ref	Expression
fsum.1	⊢ (𝑘 = (𝐹‘𝑛) → 𝐵 = 𝐶)
fsum.2	⊢ (𝜑 → 𝑀 ∈ ℕ)
fsum.3	⊢ (𝜑 → 𝐹:(1...𝑀)–1-1-onto→𝐴)
fsum.4	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
fsum.5	⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑀)) → (𝐺‘𝑛) = 𝐶)

**Assertion**
Ref	Expression
fsum3	⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑀, (𝐺‘𝑛), 0)))‘𝑀))

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

Proof of Theorem fsum3 Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fsum.1	. . 3 ⊢ (𝑘 = (𝐹‘𝑛) → 𝐵 = 𝐶)
2		fsum.2	. . 3 ⊢ (𝜑 → 𝑀 ∈ ℕ)
3		fsum.3	. . 3 ⊢ (𝜑 → 𝐹:(1...𝑀)–1-1-onto→𝐴)
4		fsum.4	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
5		fsum.5	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑀)) → (𝐺‘𝑛) = 𝐶)
6	1, 2, 3, 4, 5	fisum 10832	. 2 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑀, (𝐺‘𝑛), 0)), ℂ)‘𝑀))
7		1zzd 8831	. . . 4 ⊢ (𝜑 → 1 ∈ ℤ)
8		elnnuz 9109	. . . . . . 7 ⊢ (𝑥 ∈ ℕ ↔ 𝑥 ∈ (ℤ_≥‘1))
9	8	biimpri 132	. . . . . 6 ⊢ (𝑥 ∈ (ℤ_≥‘1) → 𝑥 ∈ ℕ)
10		fveq2 5318	. . . . . . . . 9 ⊢ (𝑛 = 𝑥 → (𝐺‘𝑛) = (𝐺‘𝑥))
11	10	eleq1d 2157	. . . . . . . 8 ⊢ (𝑛 = 𝑥 → ((𝐺‘𝑛) ∈ ℂ ↔ (𝐺‘𝑥) ∈ ℂ))
12	1, 2, 3, 4, 5	fsumgcl 10831	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑛 ∈ (1...𝑀)(𝐺‘𝑛) ∈ ℂ)
13	12	ad2antrr 473	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (ℤ_≥‘1)) ∧ 𝑥 ≤ 𝑀) → ∀𝑛 ∈ (1...𝑀)(𝐺‘𝑛) ∈ ℂ)
14		1zzd 8831	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (ℤ_≥‘1)) ∧ 𝑥 ≤ 𝑀) → 1 ∈ ℤ)
15	2	ad2antrr 473	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (ℤ_≥‘1)) ∧ 𝑥 ≤ 𝑀) → 𝑀 ∈ ℕ)
16	15	nnzd 8921	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (ℤ_≥‘1)) ∧ 𝑥 ≤ 𝑀) → 𝑀 ∈ ℤ)
17		eluzelz 9082	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (ℤ_≥‘1) → 𝑥 ∈ ℤ)
18	17	ad2antlr 474	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (ℤ_≥‘1)) ∧ 𝑥 ≤ 𝑀) → 𝑥 ∈ ℤ)
19	14, 16, 18	3jca 1124	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (ℤ_≥‘1)) ∧ 𝑥 ≤ 𝑀) → (1 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑥 ∈ ℤ))
20		eluzle 9085	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (ℤ_≥‘1) → 1 ≤ 𝑥)
21	20	ad2antlr 474	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (ℤ_≥‘1)) ∧ 𝑥 ≤ 𝑀) → 1 ≤ 𝑥)
22		simpr 109	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (ℤ_≥‘1)) ∧ 𝑥 ≤ 𝑀) → 𝑥 ≤ 𝑀)
23	21, 22	jca 301	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (ℤ_≥‘1)) ∧ 𝑥 ≤ 𝑀) → (1 ≤ 𝑥 ∧ 𝑥 ≤ 𝑀))
24		elfz2 9485	. . . . . . . . 9 ⊢ (𝑥 ∈ (1...𝑀) ↔ ((1 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑥 ∈ ℤ) ∧ (1 ≤ 𝑥 ∧ 𝑥 ≤ 𝑀)))
25	19, 23, 24	sylanbrc 409	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (ℤ_≥‘1)) ∧ 𝑥 ≤ 𝑀) → 𝑥 ∈ (1...𝑀))
26	11, 13, 25	rspcdva 2728	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (ℤ_≥‘1)) ∧ 𝑥 ≤ 𝑀) → (𝐺‘𝑥) ∈ ℂ)
27		0cnd 7535	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (ℤ_≥‘1)) ∧ ¬ 𝑥 ≤ 𝑀) → 0 ∈ ℂ)
28	2	nnzd 8921	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℤ)
29		zdcle 8877	. . . . . . . 8 ⊢ ((𝑥 ∈ ℤ ∧ 𝑀 ∈ ℤ) → DECID 𝑥 ≤ 𝑀)
30	17, 28, 29	syl2anr 285	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ_≥‘1)) → DECID 𝑥 ≤ 𝑀)
31	26, 27, 30	ifcldadc 3424	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ_≥‘1)) → if(𝑥 ≤ 𝑀, (𝐺‘𝑥), 0) ∈ ℂ)
32		breq1 3854	. . . . . . . 8 ⊢ (𝑛 = 𝑥 → (𝑛 ≤ 𝑀 ↔ 𝑥 ≤ 𝑀))
33	32, 10	ifbieq1d 3417	. . . . . . 7 ⊢ (𝑛 = 𝑥 → if(𝑛 ≤ 𝑀, (𝐺‘𝑛), 0) = if(𝑥 ≤ 𝑀, (𝐺‘𝑥), 0))
34		eqid 2089	. . . . . . 7 ⊢ (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑀, (𝐺‘𝑛), 0)) = (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑀, (𝐺‘𝑛), 0))
35	33, 34	fvmptg 5393	. . . . . 6 ⊢ ((𝑥 ∈ ℕ ∧ if(𝑥 ≤ 𝑀, (𝐺‘𝑥), 0) ∈ ℂ) → ((𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑀, (𝐺‘𝑛), 0))‘𝑥) = if(𝑥 ≤ 𝑀, (𝐺‘𝑥), 0))
36	9, 31, 35	syl2an2 562	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ_≥‘1)) → ((𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑀, (𝐺‘𝑛), 0))‘𝑥) = if(𝑥 ≤ 𝑀, (𝐺‘𝑥), 0))
37	36, 31	eqeltrd 2165	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ_≥‘1)) → ((𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑀, (𝐺‘𝑛), 0))‘𝑥) ∈ ℂ)
38	7, 37	iseqseq3 9956	. . 3 ⊢ (𝜑 → seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑀, (𝐺‘𝑛), 0)), ℂ) = seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑀, (𝐺‘𝑛), 0))))
39	38	fveq1d 5320	. 2 ⊢ (𝜑 → (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑀, (𝐺‘𝑛), 0)), ℂ)‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑀, (𝐺‘𝑛), 0)))‘𝑀))
40	6, 39	eqtrd 2121	1 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑀, (𝐺‘𝑛), 0)))‘𝑀))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 DECID wdc 781 ∧ w3a 925 = wceq 1290 ∈ wcel 1439 ∀wral 2360 ifcif 3397 class class class wbr 3851 ↦ cmpt 3905 –1-1-onto→wf1o 5027 ‘cfv 5028 (class class class)co 5666 ℂcc 7402 0cc0 7404 1c1 7405 + caddc 7407 ≤ cle 7577 ℕcn 8476 ℤcz 8804 ℤ_≥cuz 9073 ...cfz 9478 seqcseq4 9905 seqcseq 9906 Σcsu 10796

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-13 1450 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-coll 3960 ax-sep 3963 ax-nul 3971 ax-pow 4015 ax-pr 4045 ax-un 4269 ax-setind 4366 ax-iinf 4416 ax-cnex 7490 ax-resscn 7491 ax-1cn 7492 ax-1re 7493 ax-icn 7494 ax-addcl 7495 ax-addrcl 7496 ax-mulcl 7497 ax-mulrcl 7498 ax-addcom 7499 ax-mulcom 7500 ax-addass 7501 ax-mulass 7502 ax-distr 7503 ax-i2m1 7504 ax-0lt1 7505 ax-1rid 7506 ax-0id 7507 ax-rnegex 7508 ax-precex 7509 ax-cnre 7510 ax-pre-ltirr 7511 ax-pre-ltwlin 7512 ax-pre-lttrn 7513 ax-pre-apti 7514 ax-pre-ltadd 7515 ax-pre-mulgt0 7516 ax-pre-mulext 7517 ax-arch 7518 ax-caucvg 7519

This theorem depends on definitions: df-bi 116 df-dc 782 df-3or 926 df-3an 927 df-tru 1293 df-fal 1296 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ne 2257 df-nel 2352 df-ral 2365 df-rex 2366 df-reu 2367 df-rmo 2368 df-rab 2369 df-v 2622 df-sbc 2842 df-csb 2935 df-dif 3002 df-un 3004 df-in 3006 df-ss 3013 df-nul 3288 df-if 3398 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-uni 3660 df-int 3695 df-iun 3738 df-br 3852 df-opab 3906 df-mpt 3907 df-tr 3943 df-id 4129 df-po 4132 df-iso 4133 df-iord 4202 df-on 4204 df-ilim 4205 df-suc 4207 df-iom 4419 df-xp 4457 df-rel 4458 df-cnv 4459 df-co 4460 df-dm 4461 df-rn 4462 df-res 4463 df-ima 4464 df-iota 4993 df-fun 5030 df-fn 5031 df-f 5032 df-f1 5033 df-fo 5034 df-f1o 5035 df-fv 5036 df-isom 5037 df-riota 5622 df-ov 5669 df-oprab 5670 df-mpt2 5671 df-1st 5925 df-2nd 5926 df-recs 6084 df-irdg 6149 df-frec 6170 df-1o 6195 df-oadd 6199 df-er 6306 df-en 6512 df-dom 6513 df-fin 6514 df-pnf 7578 df-mnf 7579 df-xr 7580 df-ltxr 7581 df-le 7582 df-sub 7709 df-neg 7710 df-reap 8106 df-ap 8113 df-div 8194 df-inn 8477 df-2 8535 df-3 8536 df-4 8537 df-n0 8728 df-z 8805 df-uz 9074 df-q 9159 df-rp 9189 df-fz 9479 df-fzo 9608 df-iseq 9907 df-seq3 9908 df-exp 10009 df-ihash 10238 df-cj 10330 df-re 10331 df-im 10332 df-rsqrt 10485 df-abs 10486 df-clim 10721 df-isum 10797

This theorem is referenced by: fsumcl2lem 10846 fsummulc2 10896

W3C validator