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Mirrors > Home > MPE Home > Th. List > seq1 | Structured version Visualization version GIF version |
Description: Value of the sequence builder function at its initial value. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Mario Carneiro, 15-Sep-2013.) |
Ref | Expression |
---|---|
seq1 | ⊢ (𝑀 ∈ ℤ → (seq𝑀( + , 𝐹)‘𝑀) = (𝐹‘𝑀)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | seqeq1 14041 | . . . 4 ⊢ (𝑀 = if(𝑀 ∈ ℤ, 𝑀, 0) → seq𝑀( + , 𝐹) = seqif(𝑀 ∈ ℤ, 𝑀, 0)( + , 𝐹)) | |
2 | id 22 | . . . 4 ⊢ (𝑀 = if(𝑀 ∈ ℤ, 𝑀, 0) → 𝑀 = if(𝑀 ∈ ℤ, 𝑀, 0)) | |
3 | 1, 2 | fveq12d 6913 | . . 3 ⊢ (𝑀 = if(𝑀 ∈ ℤ, 𝑀, 0) → (seq𝑀( + , 𝐹)‘𝑀) = (seqif(𝑀 ∈ ℤ, 𝑀, 0)( + , 𝐹)‘if(𝑀 ∈ ℤ, 𝑀, 0))) |
4 | fveq2 6906 | . . 3 ⊢ (𝑀 = if(𝑀 ∈ ℤ, 𝑀, 0) → (𝐹‘𝑀) = (𝐹‘if(𝑀 ∈ ℤ, 𝑀, 0))) | |
5 | 3, 4 | eqeq12d 2750 | . 2 ⊢ (𝑀 = if(𝑀 ∈ ℤ, 𝑀, 0) → ((seq𝑀( + , 𝐹)‘𝑀) = (𝐹‘𝑀) ↔ (seqif(𝑀 ∈ ℤ, 𝑀, 0)( + , 𝐹)‘if(𝑀 ∈ ℤ, 𝑀, 0)) = (𝐹‘if(𝑀 ∈ ℤ, 𝑀, 0)))) |
6 | 0z 12621 | . . . 4 ⊢ 0 ∈ ℤ | |
7 | 6 | elimel 4599 | . . 3 ⊢ if(𝑀 ∈ ℤ, 𝑀, 0) ∈ ℤ |
8 | eqid 2734 | . . 3 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 + 1)), if(𝑀 ∈ ℤ, 𝑀, 0)) ↾ ω) = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), if(𝑀 ∈ ℤ, 𝑀, 0)) ↾ ω) | |
9 | fvex 6919 | . . 3 ⊢ (𝐹‘if(𝑀 ∈ ℤ, 𝑀, 0)) ∈ V | |
10 | eqid 2734 | . . 3 ⊢ (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) | |
11 | 10 | seqval 14049 | . . 3 ⊢ 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) |
12 | 7, 8, 9, 10, 11 | uzrdg0i 13996 | . 2 ⊢ (seqif(𝑀 ∈ ℤ, 𝑀, 0)( + , 𝐹)‘if(𝑀 ∈ ℤ, 𝑀, 0)) = (𝐹‘if(𝑀 ∈ ℤ, 𝑀, 0)) |
13 | 5, 12 | dedth 4588 | 1 ⊢ (𝑀 ∈ ℤ → (seq𝑀( + , 𝐹)‘𝑀) = (𝐹‘𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2105 Vcvv 3477 ifcif 4530 〈cop 4636 ↦ cmpt 5230 ↾ cres 5690 ‘cfv 6562 (class class class)co 7430 ∈ cmpo 7432 ωcom 7886 reccrdg 8447 0cc0 11152 1c1 11153 + caddc 11155 ℤcz 12610 seqcseq 14038 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-nn 12264 df-n0 12524 df-z 12611 df-uz 12876 df-seq 14039 |
This theorem is referenced by: seq1i 14052 seqexw 14054 seqcl2 14057 seqfveq2 14061 seqfveq 14063 seqshft2 14065 seqsplit 14072 seq1p 14073 seqcaopr3 14074 seqf1olem2a 14077 seqf1olem2 14079 seqf1o 14080 seqid 14084 seqhomo 14086 seqz 14087 exp1 14104 fac1 14312 bcn2 14354 seqcoll 14499 isumrpcl 15875 clim2prod 15920 prodfn0 15926 prodfrec 15927 ruclem6 16267 sadc0 16487 smup0 16512 seq1st 16604 algr0 16605 eulerthlem2 16815 pcmpt 16925 gsumsplit1r 18712 gsumprval 18713 mulgfval 19099 voliunlem1 25598 volsup 25604 abelthlem6 26494 abelthlem9 26498 leibpi 26999 bposlem5 27346 opsqrlem2 32169 esumfzf 34049 sseqp1 34376 rrvsum 34435 cvmliftlem4 35272 iprodefisumlem 35719 faclimlem1 35722 heiborlem4 37800 fmul01 45535 fmuldfeq 45538 fmul01lt1lem1 45539 stoweidlem3 45958 wallispilem4 46023 wallispi2lem1 46026 wallispi2lem2 46027 stirlinglem7 46035 stirlinglem11 46039 sge0isum 46382 ackval0 48529 |
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