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Mirrors > Home > ILE Home > Th. List > ser0f | GIF version |
Description: A zero-valued infinite series is equal to the constant zero function. (Contributed by Mario Carneiro, 8-Feb-2014.) |
Ref | Expression |
---|---|
ser0.1 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
Ref | Expression |
---|---|
ser0f | ⊢ (𝑀 ∈ ℤ → seq𝑀( + , (𝑍 × {0})) = (𝑍 × {0})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ser0.1 | . . . . 5 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
2 | 1 | ser0 10064 | . . . 4 ⊢ (𝑘 ∈ 𝑍 → (seq𝑀( + , (𝑍 × {0}))‘𝑘) = 0) |
3 | c0ex 7579 | . . . . 5 ⊢ 0 ∈ V | |
4 | 3 | fvconst2 5552 | . . . 4 ⊢ (𝑘 ∈ 𝑍 → ((𝑍 × {0})‘𝑘) = 0) |
5 | 2, 4 | eqtr4d 2130 | . . 3 ⊢ (𝑘 ∈ 𝑍 → (seq𝑀( + , (𝑍 × {0}))‘𝑘) = ((𝑍 × {0})‘𝑘)) |
6 | 5 | rgen 2439 | . 2 ⊢ ∀𝑘 ∈ 𝑍 (seq𝑀( + , (𝑍 × {0}))‘𝑘) = ((𝑍 × {0})‘𝑘) |
7 | eqid 2095 | . . . . . 6 ⊢ (ℤ≥‘𝑀) = (ℤ≥‘𝑀) | |
8 | id 19 | . . . . . 6 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℤ) | |
9 | 1 | eleq2i 2161 | . . . . . . . 8 ⊢ (𝑘 ∈ 𝑍 ↔ 𝑘 ∈ (ℤ≥‘𝑀)) |
10 | 0cnd 7578 | . . . . . . . . 9 ⊢ (𝑘 ∈ 𝑍 → 0 ∈ ℂ) | |
11 | 4, 10 | eqeltrd 2171 | . . . . . . . 8 ⊢ (𝑘 ∈ 𝑍 → ((𝑍 × {0})‘𝑘) ∈ ℂ) |
12 | 9, 11 | sylbir 134 | . . . . . . 7 ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → ((𝑍 × {0})‘𝑘) ∈ ℂ) |
13 | 12 | adantl 272 | . . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → ((𝑍 × {0})‘𝑘) ∈ ℂ) |
14 | 7, 8, 13 | serf 10024 | . . . . 5 ⊢ (𝑀 ∈ ℤ → seq𝑀( + , (𝑍 × {0})):(ℤ≥‘𝑀)⟶ℂ) |
15 | 14 | ffnd 5196 | . . . 4 ⊢ (𝑀 ∈ ℤ → seq𝑀( + , (𝑍 × {0})) Fn (ℤ≥‘𝑀)) |
16 | 1 | fneq2i 5143 | . . . 4 ⊢ (seq𝑀( + , (𝑍 × {0})) Fn 𝑍 ↔ seq𝑀( + , (𝑍 × {0})) Fn (ℤ≥‘𝑀)) |
17 | 15, 16 | sylibr 133 | . . 3 ⊢ (𝑀 ∈ ℤ → seq𝑀( + , (𝑍 × {0})) Fn 𝑍) |
18 | 3 | fconst 5241 | . . . 4 ⊢ (𝑍 × {0}):𝑍⟶{0} |
19 | ffn 5195 | . . . 4 ⊢ ((𝑍 × {0}):𝑍⟶{0} → (𝑍 × {0}) Fn 𝑍) | |
20 | 18, 19 | ax-mp 7 | . . 3 ⊢ (𝑍 × {0}) Fn 𝑍 |
21 | eqfnfv 5436 | . . 3 ⊢ ((seq𝑀( + , (𝑍 × {0})) Fn 𝑍 ∧ (𝑍 × {0}) Fn 𝑍) → (seq𝑀( + , (𝑍 × {0})) = (𝑍 × {0}) ↔ ∀𝑘 ∈ 𝑍 (seq𝑀( + , (𝑍 × {0}))‘𝑘) = ((𝑍 × {0})‘𝑘))) | |
22 | 17, 20, 21 | sylancl 405 | . 2 ⊢ (𝑀 ∈ ℤ → (seq𝑀( + , (𝑍 × {0})) = (𝑍 × {0}) ↔ ∀𝑘 ∈ 𝑍 (seq𝑀( + , (𝑍 × {0}))‘𝑘) = ((𝑍 × {0})‘𝑘))) |
23 | 6, 22 | mpbiri 167 | 1 ⊢ (𝑀 ∈ ℤ → seq𝑀( + , (𝑍 × {0})) = (𝑍 × {0})) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1296 ∈ wcel 1445 ∀wral 2370 {csn 3466 × cxp 4465 Fn wfn 5044 ⟶wf 5045 ‘cfv 5049 ℂcc 7445 0cc0 7447 + caddc 7450 ℤcz 8848 ℤ≥cuz 9118 seqcseq 10000 |
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 582 ax-in2 583 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-13 1456 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-coll 3975 ax-sep 3978 ax-nul 3986 ax-pow 4030 ax-pr 4060 ax-un 4284 ax-setind 4381 ax-iinf 4431 ax-cnex 7533 ax-resscn 7534 ax-1cn 7535 ax-1re 7536 ax-icn 7537 ax-addcl 7538 ax-addrcl 7539 ax-mulcl 7540 ax-addcom 7542 ax-addass 7544 ax-distr 7546 ax-i2m1 7547 ax-0lt1 7548 ax-0id 7550 ax-rnegex 7551 ax-cnre 7553 ax-pre-ltirr 7554 ax-pre-ltwlin 7555 ax-pre-lttrn 7556 ax-pre-ltadd 7558 |
This theorem depends on definitions: df-bi 116 df-3or 928 df-3an 929 df-tru 1299 df-fal 1302 df-nf 1402 df-sb 1700 df-eu 1958 df-mo 1959 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ne 2263 df-nel 2358 df-ral 2375 df-rex 2376 df-reu 2377 df-rab 2379 df-v 2635 df-sbc 2855 df-csb 2948 df-dif 3015 df-un 3017 df-in 3019 df-ss 3026 df-nul 3303 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-uni 3676 df-int 3711 df-iun 3754 df-br 3868 df-opab 3922 df-mpt 3923 df-tr 3959 df-id 4144 df-iord 4217 df-on 4219 df-ilim 4220 df-suc 4222 df-iom 4434 df-xp 4473 df-rel 4474 df-cnv 4475 df-co 4476 df-dm 4477 df-rn 4478 df-res 4479 df-ima 4480 df-iota 5014 df-fun 5051 df-fn 5052 df-f 5053 df-f1 5054 df-fo 5055 df-f1o 5056 df-fv 5057 df-riota 5646 df-ov 5693 df-oprab 5694 df-mpt2 5695 df-1st 5949 df-2nd 5950 df-recs 6108 df-frec 6194 df-pnf 7621 df-mnf 7622 df-xr 7623 df-ltxr 7624 df-le 7625 df-sub 7752 df-neg 7753 df-inn 8521 df-n0 8772 df-z 8849 df-uz 9119 df-fz 9574 df-fzo 9703 df-seqfrec 10001 |
This theorem is referenced by: serclim0 10848 |
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