Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > seqp1 | Structured version Visualization version GIF version |
Description: Value of the sequence builder function at a successor. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Mario Carneiro, 15-Sep-2013.) |
Ref | Expression |
---|---|
seqp1 | ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (seq𝑀( + , 𝐹)‘(𝑁 + 1)) = ((seq𝑀( + , 𝐹)‘𝑁) + (𝐹‘(𝑁 + 1)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluzel2 12249 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ) | |
2 | fveq2 6670 | . . . . . 6 ⊢ (𝑀 = if(𝑀 ∈ ℤ, 𝑀, 0) → (ℤ≥‘𝑀) = (ℤ≥‘if(𝑀 ∈ ℤ, 𝑀, 0))) | |
3 | 2 | eleq2d 2898 | . . . . 5 ⊢ (𝑀 = if(𝑀 ∈ ℤ, 𝑀, 0) → (𝑁 ∈ (ℤ≥‘𝑀) ↔ 𝑁 ∈ (ℤ≥‘if(𝑀 ∈ ℤ, 𝑀, 0)))) |
4 | seqeq1 13373 | . . . . . . 7 ⊢ (𝑀 = if(𝑀 ∈ ℤ, 𝑀, 0) → seq𝑀( + , 𝐹) = seqif(𝑀 ∈ ℤ, 𝑀, 0)( + , 𝐹)) | |
5 | 4 | fveq1d 6672 | . . . . . 6 ⊢ (𝑀 = if(𝑀 ∈ ℤ, 𝑀, 0) → (seq𝑀( + , 𝐹)‘(𝑁 + 1)) = (seqif(𝑀 ∈ ℤ, 𝑀, 0)( + , 𝐹)‘(𝑁 + 1))) |
6 | 4 | fveq1d 6672 | . . . . . . 7 ⊢ (𝑀 = if(𝑀 ∈ ℤ, 𝑀, 0) → (seq𝑀( + , 𝐹)‘𝑁) = (seqif(𝑀 ∈ ℤ, 𝑀, 0)( + , 𝐹)‘𝑁)) |
7 | 6 | oveq2d 7172 | . . . . . 6 ⊢ (𝑀 = if(𝑀 ∈ ℤ, 𝑀, 0) → (𝑁(𝑧 ∈ V, 𝑤 ∈ V ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(seq𝑀( + , 𝐹)‘𝑁)) = (𝑁(𝑧 ∈ V, 𝑤 ∈ V ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(seqif(𝑀 ∈ ℤ, 𝑀, 0)( + , 𝐹)‘𝑁))) |
8 | 5, 7 | eqeq12d 2837 | . . . . 5 ⊢ (𝑀 = if(𝑀 ∈ ℤ, 𝑀, 0) → ((seq𝑀( + , 𝐹)‘(𝑁 + 1)) = (𝑁(𝑧 ∈ V, 𝑤 ∈ V ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(seq𝑀( + , 𝐹)‘𝑁)) ↔ (seqif(𝑀 ∈ ℤ, 𝑀, 0)( + , 𝐹)‘(𝑁 + 1)) = (𝑁(𝑧 ∈ V, 𝑤 ∈ V ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(seqif(𝑀 ∈ ℤ, 𝑀, 0)( + , 𝐹)‘𝑁)))) |
9 | 3, 8 | imbi12d 347 | . . . 4 ⊢ (𝑀 = if(𝑀 ∈ ℤ, 𝑀, 0) → ((𝑁 ∈ (ℤ≥‘𝑀) → (seq𝑀( + , 𝐹)‘(𝑁 + 1)) = (𝑁(𝑧 ∈ V, 𝑤 ∈ V ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(seq𝑀( + , 𝐹)‘𝑁))) ↔ (𝑁 ∈ (ℤ≥‘if(𝑀 ∈ ℤ, 𝑀, 0)) → (seqif(𝑀 ∈ ℤ, 𝑀, 0)( + , 𝐹)‘(𝑁 + 1)) = (𝑁(𝑧 ∈ V, 𝑤 ∈ V ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(seqif(𝑀 ∈ ℤ, 𝑀, 0)( + , 𝐹)‘𝑁))))) |
10 | 0z 11993 | . . . . . 6 ⊢ 0 ∈ ℤ | |
11 | 10 | elimel 4534 | . . . . 5 ⊢ if(𝑀 ∈ ℤ, 𝑀, 0) ∈ ℤ |
12 | eqid 2821 | . . . . 5 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 + 1)), if(𝑀 ∈ ℤ, 𝑀, 0)) ↾ ω) = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), if(𝑀 ∈ ℤ, 𝑀, 0)) ↾ ω) | |
13 | fvex 6683 | . . . . 5 ⊢ (𝐹‘if(𝑀 ∈ ℤ, 𝑀, 0)) ∈ V | |
14 | eqid 2821 | . . . . 5 ⊢ (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑥(𝑧 ∈ V, 𝑤 ∈ V ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)〉), 〈if(𝑀 ∈ ℤ, 𝑀, 0), (𝐹‘if(𝑀 ∈ ℤ, 𝑀, 0))〉) ↾ ω) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑥(𝑧 ∈ V, 𝑤 ∈ V ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)〉), 〈if(𝑀 ∈ ℤ, 𝑀, 0), (𝐹‘if(𝑀 ∈ ℤ, 𝑀, 0))〉) ↾ ω) | |
15 | 14 | seqval 13381 | . . . . 5 ⊢ seqif(𝑀 ∈ ℤ, 𝑀, 0)( + , 𝐹) = ran (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑥(𝑧 ∈ V, 𝑤 ∈ V ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)〉), 〈if(𝑀 ∈ ℤ, 𝑀, 0), (𝐹‘if(𝑀 ∈ ℤ, 𝑀, 0))〉) ↾ ω) |
16 | 11, 12, 13, 14, 15 | uzrdgsuci 13329 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘if(𝑀 ∈ ℤ, 𝑀, 0)) → (seqif(𝑀 ∈ ℤ, 𝑀, 0)( + , 𝐹)‘(𝑁 + 1)) = (𝑁(𝑧 ∈ V, 𝑤 ∈ V ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(seqif(𝑀 ∈ ℤ, 𝑀, 0)( + , 𝐹)‘𝑁))) |
17 | 9, 16 | dedth 4523 | . . 3 ⊢ (𝑀 ∈ ℤ → (𝑁 ∈ (ℤ≥‘𝑀) → (seq𝑀( + , 𝐹)‘(𝑁 + 1)) = (𝑁(𝑧 ∈ V, 𝑤 ∈ V ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(seq𝑀( + , 𝐹)‘𝑁)))) |
18 | 1, 17 | mpcom 38 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (seq𝑀( + , 𝐹)‘(𝑁 + 1)) = (𝑁(𝑧 ∈ V, 𝑤 ∈ V ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(seq𝑀( + , 𝐹)‘𝑁))) |
19 | elex 3512 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ V) | |
20 | fvex 6683 | . . 3 ⊢ (seq𝑀( + , 𝐹)‘𝑁) ∈ V | |
21 | fvoveq1 7179 | . . . . 5 ⊢ (𝑧 = 𝑁 → (𝐹‘(𝑧 + 1)) = (𝐹‘(𝑁 + 1))) | |
22 | 21 | oveq2d 7172 | . . . 4 ⊢ (𝑧 = 𝑁 → (𝑤 + (𝐹‘(𝑧 + 1))) = (𝑤 + (𝐹‘(𝑁 + 1)))) |
23 | oveq1 7163 | . . . 4 ⊢ (𝑤 = (seq𝑀( + , 𝐹)‘𝑁) → (𝑤 + (𝐹‘(𝑁 + 1))) = ((seq𝑀( + , 𝐹)‘𝑁) + (𝐹‘(𝑁 + 1)))) | |
24 | eqid 2821 | . . . 4 ⊢ (𝑧 ∈ V, 𝑤 ∈ V ↦ (𝑤 + (𝐹‘(𝑧 + 1)))) = (𝑧 ∈ V, 𝑤 ∈ V ↦ (𝑤 + (𝐹‘(𝑧 + 1)))) | |
25 | ovex 7189 | . . . 4 ⊢ ((seq𝑀( + , 𝐹)‘𝑁) + (𝐹‘(𝑁 + 1))) ∈ V | |
26 | 22, 23, 24, 25 | ovmpo 7310 | . . 3 ⊢ ((𝑁 ∈ V ∧ (seq𝑀( + , 𝐹)‘𝑁) ∈ V) → (𝑁(𝑧 ∈ V, 𝑤 ∈ V ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(seq𝑀( + , 𝐹)‘𝑁)) = ((seq𝑀( + , 𝐹)‘𝑁) + (𝐹‘(𝑁 + 1)))) |
27 | 19, 20, 26 | sylancl 588 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑁(𝑧 ∈ V, 𝑤 ∈ V ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(seq𝑀( + , 𝐹)‘𝑁)) = ((seq𝑀( + , 𝐹)‘𝑁) + (𝐹‘(𝑁 + 1)))) |
28 | 18, 27 | eqtrd 2856 | 1 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (seq𝑀( + , 𝐹)‘(𝑁 + 1)) = ((seq𝑀( + , 𝐹)‘𝑁) + (𝐹‘(𝑁 + 1)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 Vcvv 3494 ifcif 4467 〈cop 4573 ↦ cmpt 5146 ↾ cres 5557 ‘cfv 6355 (class class class)co 7156 ∈ cmpo 7158 ωcom 7580 reccrdg 8045 0cc0 10537 1c1 10538 + caddc 10540 ℤcz 11982 ℤ≥cuz 12244 seqcseq 13370 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 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 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-n0 11899 df-z 11983 df-uz 12245 df-seq 13371 |
This theorem is referenced by: seqexw 13386 seqp1i 13387 seqm1 13388 seqcl2 13389 seqfveq2 13393 seqshft2 13397 sermono 13403 seqsplit 13404 seqcaopr3 13406 seqf1olem2a 13409 seqf1olem2 13411 seqid2 13417 seqhomo 13418 ser1const 13427 expp1 13437 facp1 13639 seqcoll 13823 relexpsucnnr 14384 climserle 15019 iseraltlem2 15039 iseraltlem3 15040 climcndslem1 15204 climcndslem2 15205 clim2prod 15244 prodfn0 15250 prodfrec 15251 ntrivcvgfvn0 15255 ruclem7 15589 sadcp1 15804 smupp1 15829 seq1st 15915 algrp1 15918 eulerthlem2 16119 pcmpt 16228 gsumsplit1r 17897 gsumprval 17898 mulgfval 18226 mulgnnp1 18236 ovolunlem1a 24097 voliunlem1 24151 volsup 24157 dvnp1 24522 bposlem5 25864 opsqrlem5 29921 esumfzf 31328 esumpcvgval 31337 sseqp1 31653 rrvsum 31712 gsumnunsn 31811 iprodefisumlem 32972 faclimlem1 32975 heiborlem4 35107 heiborlem6 35109 fmul01 41881 fmuldfeqlem1 41883 stoweidlem3 42308 wallispilem4 42373 wallispi2lem1 42376 wallispi2lem2 42377 |
Copyright terms: Public domain | W3C validator |