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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 13969 | . . . 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 6899 | . . 3 ⊢ (𝑀 = if(𝑀 ∈ ℤ, 𝑀, 0) → (seq𝑀( + , 𝐹)‘𝑀) = (seqif(𝑀 ∈ ℤ, 𝑀, 0)( + , 𝐹)‘if(𝑀 ∈ ℤ, 𝑀, 0))) |
4 | fveq2 6892 | . . 3 ⊢ (𝑀 = if(𝑀 ∈ ℤ, 𝑀, 0) → (𝐹‘𝑀) = (𝐹‘if(𝑀 ∈ ℤ, 𝑀, 0))) | |
5 | 3, 4 | eqeq12d 2749 | . 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 12569 | . . . 4 ⊢ 0 ∈ ℤ | |
7 | 6 | elimel 4598 | . . 3 ⊢ if(𝑀 ∈ ℤ, 𝑀, 0) ∈ ℤ |
8 | eqid 2733 | . . 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 6905 | . . 3 ⊢ (𝐹‘if(𝑀 ∈ ℤ, 𝑀, 0)) ∈ V | |
10 | eqid 2733 | . . 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 13977 | . . 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 13924 | . 2 ⊢ (seqif(𝑀 ∈ ℤ, 𝑀, 0)( + , 𝐹)‘if(𝑀 ∈ ℤ, 𝑀, 0)) = (𝐹‘if(𝑀 ∈ ℤ, 𝑀, 0)) |
13 | 5, 12 | dedth 4587 | 1 ⊢ (𝑀 ∈ ℤ → (seq𝑀( + , 𝐹)‘𝑀) = (𝐹‘𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 Vcvv 3475 ifcif 4529 ⟨cop 4635 ↦ cmpt 5232 ↾ cres 5679 ‘cfv 6544 (class class class)co 7409 ∈ cmpo 7411 ωcom 7855 reccrdg 8409 0cc0 11110 1c1 11111 + caddc 11113 ℤcz 12558 seqcseq 13966 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-nn 12213 df-n0 12473 df-z 12559 df-uz 12823 df-seq 13967 |
This theorem is referenced by: seq1i 13980 seqexw 13982 seqcl2 13986 seqfveq2 13990 seqfveq 13992 seqshft2 13994 seqsplit 14001 seq1p 14002 seqcaopr3 14003 seqf1olem2a 14006 seqf1olem2 14008 seqf1o 14009 seqid 14013 seqhomo 14015 seqz 14016 exp1 14033 fac1 14237 bcn2 14279 seqcoll 14425 isumrpcl 15789 clim2prod 15834 prodfn0 15840 prodfrec 15841 ruclem6 16178 sadc0 16395 smup0 16420 seq1st 16508 algr0 16509 eulerthlem2 16715 pcmpt 16825 gsumsplit1r 18606 gsumprval 18607 mulgfval 18952 voliunlem1 25067 volsup 25073 abelthlem6 25948 abelthlem9 25952 leibpi 26447 bposlem5 26791 opsqrlem2 31394 esumfzf 33067 sseqp1 33394 rrvsum 33453 cvmliftlem4 34279 iprodefisumlem 34710 faclimlem1 34713 heiborlem4 36682 fmul01 44296 fmuldfeq 44299 fmul01lt1lem1 44300 stoweidlem3 44719 wallispilem4 44784 wallispi2lem1 44787 wallispi2lem2 44788 stirlinglem7 44796 stirlinglem11 44800 sge0isum 45143 ackval0 47366 |
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