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Mirrors > Home > ILE Home > Th. List > iseq1 | GIF version |
Description: Value of the sequence builder function at its initial value. (Contributed by Jim Kingdon, 31-May-2020.) |
Ref | Expression |
---|---|
iseqfn.m | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
iseqfn.ex | ⊢ (𝜑 → 𝑆 ∈ 𝑉) |
iseqfn.f | ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆) |
iseqfn.pl | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) |
Ref | Expression |
---|---|
iseq1 | ⊢ (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑀) = (𝐹‘𝑀)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iseqfn.m | . 2 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
2 | eqid 2081 | . 2 ⊢ frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝑀) = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝑀) | |
3 | iseqfn.ex | . 2 ⊢ (𝜑 → 𝑆 ∈ 𝑉) | |
4 | uzid 8633 | . . . 4 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ≥‘𝑀)) | |
5 | 1, 4 | syl 14 | . . 3 ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘𝑀)) |
6 | iseqfn.f | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆) | |
7 | 6 | ralrimiva 2434 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ (ℤ≥‘𝑀)(𝐹‘𝑥) ∈ 𝑆) |
8 | fveq2 5198 | . . . . 5 ⊢ (𝑥 = 𝑀 → (𝐹‘𝑥) = (𝐹‘𝑀)) | |
9 | 8 | eleq1d 2147 | . . . 4 ⊢ (𝑥 = 𝑀 → ((𝐹‘𝑥) ∈ 𝑆 ↔ (𝐹‘𝑀) ∈ 𝑆)) |
10 | 9 | rspcv 2697 | . . 3 ⊢ (𝑀 ∈ (ℤ≥‘𝑀) → (∀𝑥 ∈ (ℤ≥‘𝑀)(𝐹‘𝑥) ∈ 𝑆 → (𝐹‘𝑀) ∈ 𝑆)) |
11 | 5, 7, 10 | sylc 61 | . 2 ⊢ (𝜑 → (𝐹‘𝑀) ∈ 𝑆) |
12 | iseqfn.pl | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) | |
13 | 6, 12 | iseqovex 9439 | . 2 ⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝑀) ∧ 𝑦 ∈ 𝑆)) → (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦) ∈ 𝑆) |
14 | eqid 2081 | . 2 ⊢ frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ 𝑆 ↦ 〈(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)〉), 〈𝑀, (𝐹‘𝑀)〉) = frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ 𝑆 ↦ 〈(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)〉), 〈𝑀, (𝐹‘𝑀)〉) | |
15 | 14, 6, 12 | iseqval 9440 | . 2 ⊢ (𝜑 → seq𝑀( + , 𝐹, 𝑆) = ran frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ 𝑆 ↦ 〈(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)〉), 〈𝑀, (𝐹‘𝑀)〉)) |
16 | 1, 2, 3, 11, 13, 14, 15 | frecuzrdg0 9416 | 1 ⊢ (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑀) = (𝐹‘𝑀)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 = wceq 1284 ∈ wcel 1433 ∀wral 2348 〈cop 3401 ↦ cmpt 3839 ‘cfv 4922 (class class class)co 5532 ↦ cmpt2 5534 freccfrec 6000 1c1 6982 + caddc 6984 ℤcz 8351 ℤ≥cuz 8619 seqcseq 9431 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-coll 3893 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-iinf 4329 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-addcom 7076 ax-addass 7078 ax-distr 7080 ax-i2m1 7081 ax-0lt1 7082 ax-0id 7084 ax-rnegex 7085 ax-cnre 7087 ax-pre-ltirr 7088 ax-pre-ltwlin 7089 ax-pre-lttrn 7090 ax-pre-ltadd 7092 |
This theorem depends on definitions: df-bi 115 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-tr 3876 df-id 4048 df-iord 4121 df-on 4123 df-suc 4126 df-iom 4332 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1st 5787 df-2nd 5788 df-recs 5943 df-frec 6001 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158 df-le 7159 df-sub 7281 df-neg 7282 df-inn 8040 df-n0 8289 df-z 8352 df-uz 8620 df-iseq 9432 |
This theorem is referenced by: iseqss 9446 iseqfveq2 9448 iseqfveq 9450 iseqshft2 9452 iseqsplit 9458 iseq1p 9459 iseqcaopr3 9460 iseqid3s 9466 iseqid 9467 iseqhomo 9468 iseqz 9469 expivallem 9477 exp1 9482 fac1 9656 bcn2 9691 resqrexlemf1 9894 ialgr0 10426 |
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