![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 10516 | . . . 4 ⊢ (𝑘 ∈ 𝑍 → (seq𝑀( + , (𝑍 × {0}))‘𝑘) = 0) |
3 | c0ex 7953 | . . . . 5 ⊢ 0 ∈ V | |
4 | 3 | fvconst2 5734 | . . . 4 ⊢ (𝑘 ∈ 𝑍 → ((𝑍 × {0})‘𝑘) = 0) |
5 | 2, 4 | eqtr4d 2213 | . . 3 ⊢ (𝑘 ∈ 𝑍 → (seq𝑀( + , (𝑍 × {0}))‘𝑘) = ((𝑍 × {0})‘𝑘)) |
6 | 5 | rgen 2530 | . 2 ⊢ ∀𝑘 ∈ 𝑍 (seq𝑀( + , (𝑍 × {0}))‘𝑘) = ((𝑍 × {0})‘𝑘) |
7 | eqid 2177 | . . . . . 6 ⊢ (ℤ≥‘𝑀) = (ℤ≥‘𝑀) | |
8 | id 19 | . . . . . 6 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℤ) | |
9 | 1 | eleq2i 2244 | . . . . . . . 8 ⊢ (𝑘 ∈ 𝑍 ↔ 𝑘 ∈ (ℤ≥‘𝑀)) |
10 | 0cnd 7952 | . . . . . . . . 9 ⊢ (𝑘 ∈ 𝑍 → 0 ∈ ℂ) | |
11 | 4, 10 | eqeltrd 2254 | . . . . . . . 8 ⊢ (𝑘 ∈ 𝑍 → ((𝑍 × {0})‘𝑘) ∈ ℂ) |
12 | 9, 11 | sylbir 135 | . . . . . . 7 ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → ((𝑍 × {0})‘𝑘) ∈ ℂ) |
13 | 12 | adantl 277 | . . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → ((𝑍 × {0})‘𝑘) ∈ ℂ) |
14 | 7, 8, 13 | serf 10476 | . . . . 5 ⊢ (𝑀 ∈ ℤ → seq𝑀( + , (𝑍 × {0})):(ℤ≥‘𝑀)⟶ℂ) |
15 | 14 | ffnd 5368 | . . . 4 ⊢ (𝑀 ∈ ℤ → seq𝑀( + , (𝑍 × {0})) Fn (ℤ≥‘𝑀)) |
16 | 1 | fneq2i 5313 | . . . 4 ⊢ (seq𝑀( + , (𝑍 × {0})) Fn 𝑍 ↔ seq𝑀( + , (𝑍 × {0})) Fn (ℤ≥‘𝑀)) |
17 | 15, 16 | sylibr 134 | . . 3 ⊢ (𝑀 ∈ ℤ → seq𝑀( + , (𝑍 × {0})) Fn 𝑍) |
18 | 3 | fconst 5413 | . . . 4 ⊢ (𝑍 × {0}):𝑍⟶{0} |
19 | ffn 5367 | . . . 4 ⊢ ((𝑍 × {0}):𝑍⟶{0} → (𝑍 × {0}) Fn 𝑍) | |
20 | 18, 19 | ax-mp 5 | . . 3 ⊢ (𝑍 × {0}) Fn 𝑍 |
21 | eqfnfv 5615 | . . 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 413 | . 2 ⊢ (𝑀 ∈ ℤ → (seq𝑀( + , (𝑍 × {0})) = (𝑍 × {0}) ↔ ∀𝑘 ∈ 𝑍 (seq𝑀( + , (𝑍 × {0}))‘𝑘) = ((𝑍 × {0})‘𝑘))) |
23 | 6, 22 | mpbiri 168 | 1 ⊢ (𝑀 ∈ ℤ → seq𝑀( + , (𝑍 × {0})) = (𝑍 × {0})) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 = wceq 1353 ∈ wcel 2148 ∀wral 2455 {csn 3594 × cxp 4626 Fn wfn 5213 ⟶wf 5214 ‘cfv 5218 ℂcc 7811 0cc0 7813 + caddc 7816 ℤcz 9255 ℤ≥cuz 9530 seqcseq 10447 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4120 ax-sep 4123 ax-nul 4131 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-iinf 4589 ax-cnex 7904 ax-resscn 7905 ax-1cn 7906 ax-1re 7907 ax-icn 7908 ax-addcl 7909 ax-addrcl 7910 ax-mulcl 7911 ax-addcom 7913 ax-addass 7915 ax-distr 7917 ax-i2m1 7918 ax-0lt1 7919 ax-0id 7921 ax-rnegex 7922 ax-cnre 7924 ax-pre-ltirr 7925 ax-pre-ltwlin 7926 ax-pre-lttrn 7927 ax-pre-ltadd 7929 |
This theorem depends on definitions: df-bi 117 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2741 df-sbc 2965 df-csb 3060 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-nul 3425 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-iun 3890 df-br 4006 df-opab 4067 df-mpt 4068 df-tr 4104 df-id 4295 df-iord 4368 df-on 4370 df-ilim 4371 df-suc 4373 df-iom 4592 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-f1 5223 df-fo 5224 df-f1o 5225 df-fv 5226 df-riota 5833 df-ov 5880 df-oprab 5881 df-mpo 5882 df-1st 6143 df-2nd 6144 df-recs 6308 df-frec 6394 df-pnf 7996 df-mnf 7997 df-xr 7998 df-ltxr 7999 df-le 8000 df-sub 8132 df-neg 8133 df-inn 8922 df-n0 9179 df-z 9256 df-uz 9531 df-fz 10011 df-fzo 10145 df-seqfrec 10448 |
This theorem is referenced by: serclim0 11315 |
Copyright terms: Public domain | W3C validator |