Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > prodfclim1 | GIF version |
Description: The constant one product converges to one. (Contributed by Scott Fenton, 5-Dec-2017.) |
Ref | Expression |
---|---|
prodf1.1 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
Ref | Expression |
---|---|
prodfclim1 | ⊢ (𝑀 ∈ ℤ → seq𝑀( · , (𝑍 × {1})) ⇝ 1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prodf1.1 | . . 3 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
2 | 1 | prodf1f 11315 | . 2 ⊢ (𝑀 ∈ ℤ → seq𝑀( · , (𝑍 × {1})) = (𝑍 × {1})) |
3 | ax-1cn 7716 | . . 3 ⊢ 1 ∈ ℂ | |
4 | 1 | eqimss2i 3154 | . . . 4 ⊢ (ℤ≥‘𝑀) ⊆ 𝑍 |
5 | zex 9066 | . . . . . 6 ⊢ ℤ ∈ V | |
6 | uzssz 9348 | . . . . . 6 ⊢ (ℤ≥‘𝑀) ⊆ ℤ | |
7 | 5, 6 | ssexi 4066 | . . . . 5 ⊢ (ℤ≥‘𝑀) ∈ V |
8 | 1, 7 | eqeltri 2212 | . . . 4 ⊢ 𝑍 ∈ V |
9 | 4, 8 | climconst2 11063 | . . 3 ⊢ ((1 ∈ ℂ ∧ 𝑀 ∈ ℤ) → (𝑍 × {1}) ⇝ 1) |
10 | 3, 9 | mpan 420 | . 2 ⊢ (𝑀 ∈ ℤ → (𝑍 × {1}) ⇝ 1) |
11 | 2, 10 | eqbrtrd 3950 | 1 ⊢ (𝑀 ∈ ℤ → seq𝑀( · , (𝑍 × {1})) ⇝ 1) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 ∈ wcel 1480 Vcvv 2686 {csn 3527 class class class wbr 3929 × cxp 4537 ‘cfv 5123 ℂcc 7621 1c1 7624 · cmul 7628 ℤcz 9057 ℤ≥cuz 9329 seqcseq 10221 ⇝ cli 11050 |
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 7714 ax-resscn 7715 ax-1cn 7716 ax-1re 7717 ax-icn 7718 ax-addcl 7719 ax-addrcl 7720 ax-mulcl 7721 ax-mulrcl 7722 ax-addcom 7723 ax-mulcom 7724 ax-addass 7725 ax-mulass 7726 ax-distr 7727 ax-i2m1 7728 ax-0lt1 7729 ax-1rid 7730 ax-0id 7731 ax-rnegex 7732 ax-precex 7733 ax-cnre 7734 ax-pre-ltirr 7735 ax-pre-ltwlin 7736 ax-pre-lttrn 7737 ax-pre-apti 7738 ax-pre-ltadd 7739 ax-pre-mulgt0 7740 ax-pre-mulext 7741 |
This theorem depends on definitions: df-bi 116 df-dc 820 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-rmo 2424 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-if 3475 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-po 4218 df-iso 4219 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 7805 df-mnf 7806 df-xr 7807 df-ltxr 7808 df-le 7809 df-sub 7938 df-neg 7939 df-reap 8340 df-ap 8347 df-div 8436 df-inn 8724 df-2 8782 df-n0 8981 df-z 9058 df-uz 9330 df-rp 9445 df-fz 9794 df-fzo 9923 df-seqfrec 10222 df-exp 10296 df-cj 10617 df-rsqrt 10773 df-abs 10774 df-clim 11051 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |