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Mirrors > Home > MPE Home > Th. List > seqid3 | Structured version Visualization version GIF version |
Description: A sequence that consists entirely of "zeroes" sums to "zero". More precisely, a constant sequence with value an element which is a + -idempotent sums (or "+'s") to that element. (Contributed by Mario Carneiro, 15-Dec-2014.) |
Ref | Expression |
---|---|
seqid3.1 | ⊢ (𝜑 → (𝑍 + 𝑍) = 𝑍) |
seqid3.2 | ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) |
seqid3.3 | ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹‘𝑥) = 𝑍) |
Ref | Expression |
---|---|
seqid3 | ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = 𝑍) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | seqid3.2 | . . 3 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) | |
2 | seqid3.3 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹‘𝑥) = 𝑍) | |
3 | fvex 6682 | . . . . 5 ⊢ (𝐹‘𝑥) ∈ V | |
4 | 3 | elsn 4581 | . . . 4 ⊢ ((𝐹‘𝑥) ∈ {𝑍} ↔ (𝐹‘𝑥) = 𝑍) |
5 | 2, 4 | sylibr 236 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹‘𝑥) ∈ {𝑍}) |
6 | seqid3.1 | . . . . . 6 ⊢ (𝜑 → (𝑍 + 𝑍) = 𝑍) | |
7 | ovex 7188 | . . . . . . 7 ⊢ (𝑍 + 𝑍) ∈ V | |
8 | 7 | elsn 4581 | . . . . . 6 ⊢ ((𝑍 + 𝑍) ∈ {𝑍} ↔ (𝑍 + 𝑍) = 𝑍) |
9 | 6, 8 | sylibr 236 | . . . . 5 ⊢ (𝜑 → (𝑍 + 𝑍) ∈ {𝑍}) |
10 | elsni 4583 | . . . . . . 7 ⊢ (𝑥 ∈ {𝑍} → 𝑥 = 𝑍) | |
11 | elsni 4583 | . . . . . . 7 ⊢ (𝑦 ∈ {𝑍} → 𝑦 = 𝑍) | |
12 | 10, 11 | oveqan12d 7174 | . . . . . 6 ⊢ ((𝑥 ∈ {𝑍} ∧ 𝑦 ∈ {𝑍}) → (𝑥 + 𝑦) = (𝑍 + 𝑍)) |
13 | 12 | eleq1d 2897 | . . . . 5 ⊢ ((𝑥 ∈ {𝑍} ∧ 𝑦 ∈ {𝑍}) → ((𝑥 + 𝑦) ∈ {𝑍} ↔ (𝑍 + 𝑍) ∈ {𝑍})) |
14 | 9, 13 | syl5ibrcom 249 | . . . 4 ⊢ (𝜑 → ((𝑥 ∈ {𝑍} ∧ 𝑦 ∈ {𝑍}) → (𝑥 + 𝑦) ∈ {𝑍})) |
15 | 14 | imp 409 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ {𝑍} ∧ 𝑦 ∈ {𝑍})) → (𝑥 + 𝑦) ∈ {𝑍}) |
16 | 1, 5, 15 | seqcl 13389 | . 2 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) ∈ {𝑍}) |
17 | elsni 4583 | . 2 ⊢ ((seq𝑀( + , 𝐹)‘𝑁) ∈ {𝑍} → (seq𝑀( + , 𝐹)‘𝑁) = 𝑍) | |
18 | 16, 17 | syl 17 | 1 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 {csn 4566 ‘cfv 6354 (class class class)co 7155 ℤ≥cuz 12242 ...cfz 12891 seqcseq 13368 |
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 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 |
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 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-1st 7688 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-nn 11638 df-n0 11897 df-z 11981 df-uz 12243 df-fz 12892 df-seq 13369 |
This theorem is referenced by: seqid 13414 ser0 13421 prodf1 15246 gsumval2 17895 mulgnn0z 18253 gsumval3 19026 lgsval2lem 25882 |
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