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Mirrors > Home > ILE Home > Th. List > fsum3 | GIF version |
Description: The value of a sum over a nonempty finite set. (Contributed by Jim Kingdon, 10-Oct-2022.) |
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) |
Ref | Expression |
---|---|
fsum3 | ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑀, (𝐺‘𝑛), 0)))‘𝑀)) |
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 |
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