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Mirrors > Home > ILE Home > Th. List > fsumgcl | GIF version |
Description: Closure for a function used to describe 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 |
---|---|
fsumgcl | ⊢ (𝜑 → ∀𝑛 ∈ (1...𝑀)(𝐺‘𝑛) ∈ ℂ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fsum.5 | . . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑀)) → (𝐺‘𝑛) = 𝐶) | |
2 | fsum.3 | . . . . . . 7 ⊢ (𝜑 → 𝐹:(1...𝑀)–1-1-onto→𝐴) | |
3 | f1of 5253 | . . . . . . 7 ⊢ (𝐹:(1...𝑀)–1-1-onto→𝐴 → 𝐹:(1...𝑀)⟶𝐴) | |
4 | 2, 3 | syl 14 | . . . . . 6 ⊢ (𝜑 → 𝐹:(1...𝑀)⟶𝐴) |
5 | 4 | ffvelrnda 5434 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑀)) → (𝐹‘𝑛) ∈ 𝐴) |
6 | fsum.1 | . . . . . 6 ⊢ (𝑘 = (𝐹‘𝑛) → 𝐵 = 𝐶) | |
7 | 6 | adantl 271 | . . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...𝑀)) ∧ 𝑘 = (𝐹‘𝑛)) → 𝐵 = 𝐶) |
8 | 5, 7 | csbied 2974 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑀)) → ⦋(𝐹‘𝑛) / 𝑘⦌𝐵 = 𝐶) |
9 | fsum.4 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) | |
10 | 9 | ralrimiva 2446 | . . . . . 6 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ) |
11 | 10 | adantr 270 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑀)) → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ) |
12 | nfcsb1v 2963 | . . . . . . 7 ⊢ Ⅎ𝑘⦋(𝐹‘𝑛) / 𝑘⦌𝐵 | |
13 | 12 | nfel1 2239 | . . . . . 6 ⊢ Ⅎ𝑘⦋(𝐹‘𝑛) / 𝑘⦌𝐵 ∈ ℂ |
14 | csbeq1a 2941 | . . . . . . 7 ⊢ (𝑘 = (𝐹‘𝑛) → 𝐵 = ⦋(𝐹‘𝑛) / 𝑘⦌𝐵) | |
15 | 14 | eleq1d 2156 | . . . . . 6 ⊢ (𝑘 = (𝐹‘𝑛) → (𝐵 ∈ ℂ ↔ ⦋(𝐹‘𝑛) / 𝑘⦌𝐵 ∈ ℂ)) |
16 | 13, 15 | rspc 2716 | . . . . 5 ⊢ ((𝐹‘𝑛) ∈ 𝐴 → (∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ → ⦋(𝐹‘𝑛) / 𝑘⦌𝐵 ∈ ℂ)) |
17 | 5, 11, 16 | sylc 61 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑀)) → ⦋(𝐹‘𝑛) / 𝑘⦌𝐵 ∈ ℂ) |
18 | 8, 17 | eqeltrrd 2165 | . . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑀)) → 𝐶 ∈ ℂ) |
19 | 1, 18 | eqeltrd 2164 | . 2 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑀)) → (𝐺‘𝑛) ∈ ℂ) |
20 | 19 | ralrimiva 2446 | 1 ⊢ (𝜑 → ∀𝑛 ∈ (1...𝑀)(𝐺‘𝑛) ∈ ℂ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 = wceq 1289 ∈ wcel 1438 ∀wral 2359 ⦋csb 2933 ⟶wf 5011 –1-1-onto→wf1o 5014 ‘cfv 5015 (class class class)co 5652 ℂcc 7348 1c1 7351 ℕcn 8422 ...cfz 9424 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-sep 3957 ax-pow 4009 ax-pr 4036 |
This theorem depends on definitions: df-bi 115 df-3an 926 df-tru 1292 df-nf 1395 df-sb 1693 df-eu 1951 df-mo 1952 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ral 2364 df-rex 2365 df-v 2621 df-sbc 2841 df-csb 2934 df-un 3003 df-in 3005 df-ss 3012 df-pw 3431 df-sn 3452 df-pr 3453 df-op 3455 df-uni 3654 df-br 3846 df-opab 3900 df-id 4120 df-xp 4444 df-rel 4445 df-cnv 4446 df-co 4447 df-dm 4448 df-rn 4449 df-iota 4980 df-fun 5017 df-fn 5018 df-f 5019 df-f1 5020 df-f1o 5022 df-fv 5023 |
This theorem is referenced by: fisum 10778 fsum3 10779 |
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