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Mirrors > Home > MPE Home > Th. List > seqid | Structured version Visualization version GIF version |
Description: Discarding the first few terms of a sequence that starts with all zeroes (or any element which is a left-identity for +) has no effect on its sum. (Contributed by Mario Carneiro, 13-Jul-2013.) (Revised by Mario Carneiro, 27-May-2014.) |
Ref | Expression |
---|---|
seqid.1 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝑍 + 𝑥) = 𝑥) |
seqid.2 | ⊢ (𝜑 → 𝑍 ∈ 𝑆) |
seqid.3 | ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) |
seqid.4 | ⊢ (𝜑 → (𝐹‘𝑁) ∈ 𝑆) |
seqid.5 | ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...(𝑁 − 1))) → (𝐹‘𝑥) = 𝑍) |
Ref | Expression |
---|---|
seqid | ⊢ (𝜑 → (seq𝑀( + , 𝐹) ↾ (ℤ≥‘𝑁)) = seq𝑁( + , 𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | seqid.3 | . 2 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) | |
2 | eluzelz 12071 | . . . . 5 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℤ) | |
3 | seq1 13200 | . . . . 5 ⊢ (𝑁 ∈ ℤ → (seq𝑁( + , 𝐹)‘𝑁) = (𝐹‘𝑁)) | |
4 | 1, 2, 3 | 3syl 18 | . . . 4 ⊢ (𝜑 → (seq𝑁( + , 𝐹)‘𝑁) = (𝐹‘𝑁)) |
5 | seqeq1 13190 | . . . . . 6 ⊢ (𝑁 = 𝑀 → seq𝑁( + , 𝐹) = seq𝑀( + , 𝐹)) | |
6 | 5 | fveq1d 6503 | . . . . 5 ⊢ (𝑁 = 𝑀 → (seq𝑁( + , 𝐹)‘𝑁) = (seq𝑀( + , 𝐹)‘𝑁)) |
7 | 6 | eqeq1d 2780 | . . . 4 ⊢ (𝑁 = 𝑀 → ((seq𝑁( + , 𝐹)‘𝑁) = (𝐹‘𝑁) ↔ (seq𝑀( + , 𝐹)‘𝑁) = (𝐹‘𝑁))) |
8 | 4, 7 | syl5ibcom 237 | . . 3 ⊢ (𝜑 → (𝑁 = 𝑀 → (seq𝑀( + , 𝐹)‘𝑁) = (𝐹‘𝑁))) |
9 | eluzel2 12066 | . . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ) | |
10 | 1, 9 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℤ) |
11 | seqm1 13205 | . . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 1))) → (seq𝑀( + , 𝐹)‘𝑁) = ((seq𝑀( + , 𝐹)‘(𝑁 − 1)) + (𝐹‘𝑁))) | |
12 | 10, 11 | sylan 572 | . . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 1))) → (seq𝑀( + , 𝐹)‘𝑁) = ((seq𝑀( + , 𝐹)‘(𝑁 − 1)) + (𝐹‘𝑁))) |
13 | oveq2 6986 | . . . . . . . . . 10 ⊢ (𝑥 = 𝑍 → (𝑍 + 𝑥) = (𝑍 + 𝑍)) | |
14 | id 22 | . . . . . . . . . 10 ⊢ (𝑥 = 𝑍 → 𝑥 = 𝑍) | |
15 | 13, 14 | eqeq12d 2793 | . . . . . . . . 9 ⊢ (𝑥 = 𝑍 → ((𝑍 + 𝑥) = 𝑥 ↔ (𝑍 + 𝑍) = 𝑍)) |
16 | seqid.1 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝑍 + 𝑥) = 𝑥) | |
17 | 16 | ralrimiva 3132 | . . . . . . . . 9 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 (𝑍 + 𝑥) = 𝑥) |
18 | seqid.2 | . . . . . . . . 9 ⊢ (𝜑 → 𝑍 ∈ 𝑆) | |
19 | 15, 17, 18 | rspcdva 3541 | . . . . . . . 8 ⊢ (𝜑 → (𝑍 + 𝑍) = 𝑍) |
20 | 19 | adantr 473 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 1))) → (𝑍 + 𝑍) = 𝑍) |
21 | eluzp1m1 12085 | . . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 1))) → (𝑁 − 1) ∈ (ℤ≥‘𝑀)) | |
22 | 10, 21 | sylan 572 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 1))) → (𝑁 − 1) ∈ (ℤ≥‘𝑀)) |
23 | seqid.5 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...(𝑁 − 1))) → (𝐹‘𝑥) = 𝑍) | |
24 | 23 | adantlr 702 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 1))) ∧ 𝑥 ∈ (𝑀...(𝑁 − 1))) → (𝐹‘𝑥) = 𝑍) |
25 | 20, 22, 24 | seqid3 13232 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 1))) → (seq𝑀( + , 𝐹)‘(𝑁 − 1)) = 𝑍) |
26 | 25 | oveq1d 6993 | . . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 1))) → ((seq𝑀( + , 𝐹)‘(𝑁 − 1)) + (𝐹‘𝑁)) = (𝑍 + (𝐹‘𝑁))) |
27 | oveq2 6986 | . . . . . . 7 ⊢ (𝑥 = (𝐹‘𝑁) → (𝑍 + 𝑥) = (𝑍 + (𝐹‘𝑁))) | |
28 | id 22 | . . . . . . 7 ⊢ (𝑥 = (𝐹‘𝑁) → 𝑥 = (𝐹‘𝑁)) | |
29 | 27, 28 | eqeq12d 2793 | . . . . . 6 ⊢ (𝑥 = (𝐹‘𝑁) → ((𝑍 + 𝑥) = 𝑥 ↔ (𝑍 + (𝐹‘𝑁)) = (𝐹‘𝑁))) |
30 | 17 | adantr 473 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 1))) → ∀𝑥 ∈ 𝑆 (𝑍 + 𝑥) = 𝑥) |
31 | seqid.4 | . . . . . . 7 ⊢ (𝜑 → (𝐹‘𝑁) ∈ 𝑆) | |
32 | 31 | adantr 473 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 1))) → (𝐹‘𝑁) ∈ 𝑆) |
33 | 29, 30, 32 | rspcdva 3541 | . . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 1))) → (𝑍 + (𝐹‘𝑁)) = (𝐹‘𝑁)) |
34 | 12, 26, 33 | 3eqtrd 2818 | . . . 4 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 1))) → (seq𝑀( + , 𝐹)‘𝑁) = (𝐹‘𝑁)) |
35 | 34 | ex 405 | . . 3 ⊢ (𝜑 → (𝑁 ∈ (ℤ≥‘(𝑀 + 1)) → (seq𝑀( + , 𝐹)‘𝑁) = (𝐹‘𝑁))) |
36 | uzp1 12096 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑁 = 𝑀 ∨ 𝑁 ∈ (ℤ≥‘(𝑀 + 1)))) | |
37 | 1, 36 | syl 17 | . . 3 ⊢ (𝜑 → (𝑁 = 𝑀 ∨ 𝑁 ∈ (ℤ≥‘(𝑀 + 1)))) |
38 | 8, 35, 37 | mpjaod 846 | . 2 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = (𝐹‘𝑁)) |
39 | eqidd 2779 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘(𝑁 + 1))) → (𝐹‘𝑥) = (𝐹‘𝑥)) | |
40 | 1, 38, 39 | seqfeq2 13211 | 1 ⊢ (𝜑 → (seq𝑀( + , 𝐹) ↾ (ℤ≥‘𝑁)) = seq𝑁( + , 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 ∨ wo 833 = wceq 1507 ∈ wcel 2050 ∀wral 3088 ↾ cres 5410 ‘cfv 6190 (class class class)co 6978 1c1 10338 + caddc 10340 − cmin 10672 ℤcz 11796 ℤ≥cuz 12061 ...cfz 12711 seqcseq 13187 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-sep 5061 ax-nul 5068 ax-pow 5120 ax-pr 5187 ax-un 7281 ax-cnex 10393 ax-resscn 10394 ax-1cn 10395 ax-icn 10396 ax-addcl 10397 ax-addrcl 10398 ax-mulcl 10399 ax-mulrcl 10400 ax-mulcom 10401 ax-addass 10402 ax-mulass 10403 ax-distr 10404 ax-i2m1 10405 ax-1ne0 10406 ax-1rid 10407 ax-rnegex 10408 ax-rrecex 10409 ax-cnre 10410 ax-pre-lttri 10411 ax-pre-lttrn 10412 ax-pre-ltadd 10413 ax-pre-mulgt0 10414 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2583 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rab 3097 df-v 3417 df-sbc 3684 df-csb 3789 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-pss 3847 df-nul 4181 df-if 4352 df-pw 4425 df-sn 4443 df-pr 4445 df-tp 4447 df-op 4449 df-uni 4714 df-iun 4795 df-br 4931 df-opab 4993 df-mpt 5010 df-tr 5032 df-id 5313 df-eprel 5318 df-po 5327 df-so 5328 df-fr 5367 df-we 5369 df-xp 5414 df-rel 5415 df-cnv 5416 df-co 5417 df-dm 5418 df-rn 5419 df-res 5420 df-ima 5421 df-pred 5988 df-ord 6034 df-on 6035 df-lim 6036 df-suc 6037 df-iota 6154 df-fun 6192 df-fn 6193 df-f 6194 df-f1 6195 df-fo 6196 df-f1o 6197 df-fv 6198 df-riota 6939 df-ov 6981 df-oprab 6982 df-mpo 6983 df-om 7399 df-1st 7503 df-2nd 7504 df-wrecs 7752 df-recs 7814 df-rdg 7852 df-er 8091 df-en 8309 df-dom 8310 df-sdom 8311 df-pnf 10478 df-mnf 10479 df-xr 10480 df-ltxr 10481 df-le 10482 df-sub 10674 df-neg 10675 df-nn 11442 df-n0 11711 df-z 11797 df-uz 12062 df-fz 12712 df-seq 13188 |
This theorem is referenced by: seqcoll 13638 sumrblem 14931 prodrblem 15146 logtayl 24947 leibpilem2 25224 |
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