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Mirrors > Home > ILE Home > Th. List > prodf1f | GIF version |
Description: A one-valued infinite product is equal to the constant one function. (Contributed by Scott Fenton, 5-Dec-2017.) |
Ref | Expression |
---|---|
prodf1.1 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
Ref | Expression |
---|---|
prodf1f | ⊢ (𝑀 ∈ ℤ → seq𝑀( · , (𝑍 × {1})) = (𝑍 × {1})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prodf1.1 | . . . . 5 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
2 | 1 | prodf1 11316 | . . . 4 ⊢ (𝑘 ∈ 𝑍 → (seq𝑀( · , (𝑍 × {1}))‘𝑘) = 1) |
3 | 1ex 7766 | . . . . 5 ⊢ 1 ∈ V | |
4 | 3 | fvconst2 5636 | . . . 4 ⊢ (𝑘 ∈ 𝑍 → ((𝑍 × {1})‘𝑘) = 1) |
5 | 2, 4 | eqtr4d 2175 | . . 3 ⊢ (𝑘 ∈ 𝑍 → (seq𝑀( · , (𝑍 × {1}))‘𝑘) = ((𝑍 × {1})‘𝑘)) |
6 | 5 | rgen 2485 | . 2 ⊢ ∀𝑘 ∈ 𝑍 (seq𝑀( · , (𝑍 × {1}))‘𝑘) = ((𝑍 × {1})‘𝑘) |
7 | id 19 | . . . . 5 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℤ) | |
8 | 1cnd 7787 | . . . . . . 7 ⊢ (𝑘 ∈ 𝑍 → 1 ∈ ℂ) | |
9 | 4, 8 | eqeltrd 2216 | . . . . . 6 ⊢ (𝑘 ∈ 𝑍 → ((𝑍 × {1})‘𝑘) ∈ ℂ) |
10 | 9 | adantl 275 | . . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑘 ∈ 𝑍) → ((𝑍 × {1})‘𝑘) ∈ ℂ) |
11 | 1, 7, 10 | prodf 11312 | . . . 4 ⊢ (𝑀 ∈ ℤ → seq𝑀( · , (𝑍 × {1})):𝑍⟶ℂ) |
12 | 11 | ffnd 5273 | . . 3 ⊢ (𝑀 ∈ ℤ → seq𝑀( · , (𝑍 × {1})) Fn 𝑍) |
13 | 3 | fconst 5318 | . . . 4 ⊢ (𝑍 × {1}):𝑍⟶{1} |
14 | ffn 5272 | . . . 4 ⊢ ((𝑍 × {1}):𝑍⟶{1} → (𝑍 × {1}) Fn 𝑍) | |
15 | 13, 14 | ax-mp 5 | . . 3 ⊢ (𝑍 × {1}) Fn 𝑍 |
16 | eqfnfv 5518 | . . 3 ⊢ ((seq𝑀( · , (𝑍 × {1})) Fn 𝑍 ∧ (𝑍 × {1}) Fn 𝑍) → (seq𝑀( · , (𝑍 × {1})) = (𝑍 × {1}) ↔ ∀𝑘 ∈ 𝑍 (seq𝑀( · , (𝑍 × {1}))‘𝑘) = ((𝑍 × {1})‘𝑘))) | |
17 | 12, 15, 16 | sylancl 409 | . 2 ⊢ (𝑀 ∈ ℤ → (seq𝑀( · , (𝑍 × {1})) = (𝑍 × {1}) ↔ ∀𝑘 ∈ 𝑍 (seq𝑀( · , (𝑍 × {1}))‘𝑘) = ((𝑍 × {1})‘𝑘))) |
18 | 6, 17 | mpbiri 167 | 1 ⊢ (𝑀 ∈ ℤ → seq𝑀( · , (𝑍 × {1})) = (𝑍 × {1})) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1331 ∈ wcel 1480 ∀wral 2416 {csn 3527 × cxp 4537 Fn wfn 5118 ⟶wf 5119 ‘cfv 5123 ℂcc 7623 1c1 7626 · cmul 7630 ℤcz 9059 ℤ≥cuz 9331 seqcseq 10223 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 ax-cnex 7716 ax-resscn 7717 ax-1cn 7718 ax-1re 7719 ax-icn 7720 ax-addcl 7721 ax-addrcl 7722 ax-mulcl 7723 ax-addcom 7725 ax-mulcom 7726 ax-addass 7727 ax-mulass 7728 ax-distr 7729 ax-i2m1 7730 ax-0lt1 7731 ax-1rid 7732 ax-0id 7733 ax-rnegex 7734 ax-cnre 7736 ax-pre-ltirr 7737 ax-pre-ltwlin 7738 ax-pre-lttrn 7739 ax-pre-ltadd 7741 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-iord 4288 df-on 4290 df-ilim 4291 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-frec 6288 df-pnf 7807 df-mnf 7808 df-xr 7809 df-ltxr 7810 df-le 7811 df-sub 7940 df-neg 7941 df-inn 8726 df-n0 8983 df-z 9060 df-uz 9332 df-fz 9796 df-fzo 9925 df-seqfrec 10224 |
This theorem is referenced by: prodfclim1 11318 |
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