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Mirrors > Home > MPE Home > Th. List > prodf1 | Structured version Visualization version GIF version |
Description: The value of the partial products in a one-valued infinite product. (Contributed by Scott Fenton, 5-Dec-2017.) |
Ref | Expression |
---|---|
prodf1.1 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
Ref | Expression |
---|---|
prodf1 | ⊢ (𝑁 ∈ 𝑍 → (seq𝑀( · , (𝑍 × {1}))‘𝑁) = 1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1t1e1 11793 | . . 3 ⊢ (1 · 1) = 1 | |
2 | 1 | a1i 11 | . 2 ⊢ (𝑁 ∈ 𝑍 → (1 · 1) = 1) |
3 | prodf1.1 | . . . 4 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
4 | 3 | eleq2i 2904 | . . 3 ⊢ (𝑁 ∈ 𝑍 ↔ 𝑁 ∈ (ℤ≥‘𝑀)) |
5 | 4 | biimpi 218 | . 2 ⊢ (𝑁 ∈ 𝑍 → 𝑁 ∈ (ℤ≥‘𝑀)) |
6 | ax-1cn 10589 | . . 3 ⊢ 1 ∈ ℂ | |
7 | elfzuz 12898 | . . . . 5 ⊢ (𝑘 ∈ (𝑀...𝑁) → 𝑘 ∈ (ℤ≥‘𝑀)) | |
8 | 7, 3 | eleqtrrdi 2924 | . . . 4 ⊢ (𝑘 ∈ (𝑀...𝑁) → 𝑘 ∈ 𝑍) |
9 | 8 | adantl 484 | . . 3 ⊢ ((𝑁 ∈ 𝑍 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝑘 ∈ 𝑍) |
10 | fvconst2g 6959 | . . 3 ⊢ ((1 ∈ ℂ ∧ 𝑘 ∈ 𝑍) → ((𝑍 × {1})‘𝑘) = 1) | |
11 | 6, 9, 10 | sylancr 589 | . 2 ⊢ ((𝑁 ∈ 𝑍 ∧ 𝑘 ∈ (𝑀...𝑁)) → ((𝑍 × {1})‘𝑘) = 1) |
12 | 2, 5, 11 | seqid3 13408 | 1 ⊢ (𝑁 ∈ 𝑍 → (seq𝑀( · , (𝑍 × {1}))‘𝑁) = 1) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 {csn 4561 × cxp 5548 ‘cfv 6350 (class class class)co 7150 ℂcc 10529 1c1 10532 · cmul 10536 ℤ≥cuz 12237 ...cfz 12886 seqcseq 13363 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12887 df-seq 13364 |
This theorem is referenced by: prodf1f 15242 fprodntriv 15290 prod1 15292 |
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