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Mirrors > Home > ILE Home > Th. List > iseqp1 | GIF version |
Description: Value of the sequence builder function at a successor. (Contributed by Jim Kingdon, 31-May-2020.) |
Ref | Expression |
---|---|
iseqp1.m | ⊢ (φ → 𝑁 ∈ (ℤ≥‘𝑀)) |
iseqp1.ex | ⊢ (φ → 𝑆 ∈ 𝑉) |
iseqp1.f | ⊢ ((φ ∧ x ∈ (ℤ≥‘𝑀)) → (𝐹‘x) ∈ 𝑆) |
iseqp1.pl | ⊢ ((φ ∧ (x ∈ 𝑆 ∧ y ∈ 𝑆)) → (x + y) ∈ 𝑆) |
Ref | Expression |
---|---|
iseqp1 | ⊢ (φ → (seq𝑀( + , 𝐹, 𝑆)‘(𝑁 + 1)) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑁) + (𝐹‘(𝑁 + 1)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iseqp1.m | . . 3 ⊢ (φ → 𝑁 ∈ (ℤ≥‘𝑀)) | |
2 | eluzel2 8254 | . . . . 5 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ) | |
3 | 1, 2 | syl 14 | . . . 4 ⊢ (φ → 𝑀 ∈ ℤ) |
4 | eqid 2037 | . . . 4 ⊢ frec((x ∈ ℤ ↦ (x + 1)), 𝑀) = frec((x ∈ ℤ ↦ (x + 1)), 𝑀) | |
5 | iseqp1.ex | . . . 4 ⊢ (φ → 𝑆 ∈ 𝑉) | |
6 | uzid 8263 | . . . . . 6 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ≥‘𝑀)) | |
7 | 3, 6 | syl 14 | . . . . 5 ⊢ (φ → 𝑀 ∈ (ℤ≥‘𝑀)) |
8 | iseqp1.f | . . . . . 6 ⊢ ((φ ∧ x ∈ (ℤ≥‘𝑀)) → (𝐹‘x) ∈ 𝑆) | |
9 | 8 | ralrimiva 2386 | . . . . 5 ⊢ (φ → ∀x ∈ (ℤ≥‘𝑀)(𝐹‘x) ∈ 𝑆) |
10 | fveq2 5121 | . . . . . . 7 ⊢ (x = 𝑀 → (𝐹‘x) = (𝐹‘𝑀)) | |
11 | 10 | eleq1d 2103 | . . . . . 6 ⊢ (x = 𝑀 → ((𝐹‘x) ∈ 𝑆 ↔ (𝐹‘𝑀) ∈ 𝑆)) |
12 | 11 | rspcv 2646 | . . . . 5 ⊢ (𝑀 ∈ (ℤ≥‘𝑀) → (∀x ∈ (ℤ≥‘𝑀)(𝐹‘x) ∈ 𝑆 → (𝐹‘𝑀) ∈ 𝑆)) |
13 | 7, 9, 12 | sylc 56 | . . . 4 ⊢ (φ → (𝐹‘𝑀) ∈ 𝑆) |
14 | iseqp1.pl | . . . . 5 ⊢ ((φ ∧ (x ∈ 𝑆 ∧ y ∈ 𝑆)) → (x + y) ∈ 𝑆) | |
15 | 8, 14 | iseqovex 8899 | . . . 4 ⊢ ((φ ∧ (x ∈ (ℤ≥‘𝑀) ∧ y ∈ 𝑆)) → (x(z ∈ (ℤ≥‘𝑀), w ∈ 𝑆 ↦ (w + (𝐹‘(z + 1))))y) ∈ 𝑆) |
16 | eqid 2037 | . . . 4 ⊢ 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)〉) | |
17 | 16, 8, 14 | iseqval 8900 | . . . 4 ⊢ (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)〉)) |
18 | 3, 4, 5, 13, 15, 16, 17 | frecuzrdgsuc 8882 | . . 3 ⊢ ((φ ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → (seq𝑀( + , 𝐹, 𝑆)‘(𝑁 + 1)) = (𝑁(z ∈ (ℤ≥‘𝑀), w ∈ 𝑆 ↦ (w + (𝐹‘(z + 1))))(seq𝑀( + , 𝐹, 𝑆)‘𝑁))) |
19 | 1, 18 | mpdan 398 | . 2 ⊢ (φ → (seq𝑀( + , 𝐹, 𝑆)‘(𝑁 + 1)) = (𝑁(z ∈ (ℤ≥‘𝑀), w ∈ 𝑆 ↦ (w + (𝐹‘(z + 1))))(seq𝑀( + , 𝐹, 𝑆)‘𝑁))) |
20 | 1, 5, 8, 14 | iseqcl 8903 | . . 3 ⊢ (φ → (seq𝑀( + , 𝐹, 𝑆)‘𝑁) ∈ 𝑆) |
21 | peano2uz 8302 | . . . . . 6 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑁 + 1) ∈ (ℤ≥‘𝑀)) | |
22 | 1, 21 | syl 14 | . . . . 5 ⊢ (φ → (𝑁 + 1) ∈ (ℤ≥‘𝑀)) |
23 | fveq2 5121 | . . . . . . 7 ⊢ (x = (𝑁 + 1) → (𝐹‘x) = (𝐹‘(𝑁 + 1))) | |
24 | 23 | eleq1d 2103 | . . . . . 6 ⊢ (x = (𝑁 + 1) → ((𝐹‘x) ∈ 𝑆 ↔ (𝐹‘(𝑁 + 1)) ∈ 𝑆)) |
25 | 24 | rspcv 2646 | . . . . 5 ⊢ ((𝑁 + 1) ∈ (ℤ≥‘𝑀) → (∀x ∈ (ℤ≥‘𝑀)(𝐹‘x) ∈ 𝑆 → (𝐹‘(𝑁 + 1)) ∈ 𝑆)) |
26 | 22, 9, 25 | sylc 56 | . . . 4 ⊢ (φ → (𝐹‘(𝑁 + 1)) ∈ 𝑆) |
27 | 14, 20, 26 | caovcld 5596 | . . 3 ⊢ (φ → ((seq𝑀( + , 𝐹, 𝑆)‘𝑁) + (𝐹‘(𝑁 + 1))) ∈ 𝑆) |
28 | oveq1 5462 | . . . . . 6 ⊢ (z = 𝑁 → (z + 1) = (𝑁 + 1)) | |
29 | 28 | fveq2d 5125 | . . . . 5 ⊢ (z = 𝑁 → (𝐹‘(z + 1)) = (𝐹‘(𝑁 + 1))) |
30 | 29 | oveq2d 5471 | . . . 4 ⊢ (z = 𝑁 → (w + (𝐹‘(z + 1))) = (w + (𝐹‘(𝑁 + 1)))) |
31 | oveq1 5462 | . . . 4 ⊢ (w = (seq𝑀( + , 𝐹, 𝑆)‘𝑁) → (w + (𝐹‘(𝑁 + 1))) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑁) + (𝐹‘(𝑁 + 1)))) | |
32 | eqid 2037 | . . . 4 ⊢ (z ∈ (ℤ≥‘𝑀), w ∈ 𝑆 ↦ (w + (𝐹‘(z + 1)))) = (z ∈ (ℤ≥‘𝑀), w ∈ 𝑆 ↦ (w + (𝐹‘(z + 1)))) | |
33 | 30, 31, 32 | ovmpt2g 5577 | . . 3 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (seq𝑀( + , 𝐹, 𝑆)‘𝑁) ∈ 𝑆 ∧ ((seq𝑀( + , 𝐹, 𝑆)‘𝑁) + (𝐹‘(𝑁 + 1))) ∈ 𝑆) → (𝑁(z ∈ (ℤ≥‘𝑀), w ∈ 𝑆 ↦ (w + (𝐹‘(z + 1))))(seq𝑀( + , 𝐹, 𝑆)‘𝑁)) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑁) + (𝐹‘(𝑁 + 1)))) |
34 | 1, 20, 27, 33 | syl3anc 1134 | . 2 ⊢ (φ → (𝑁(z ∈ (ℤ≥‘𝑀), w ∈ 𝑆 ↦ (w + (𝐹‘(z + 1))))(seq𝑀( + , 𝐹, 𝑆)‘𝑁)) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑁) + (𝐹‘(𝑁 + 1)))) |
35 | 19, 34 | eqtrd 2069 | 1 ⊢ (φ → (seq𝑀( + , 𝐹, 𝑆)‘(𝑁 + 1)) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑁) + (𝐹‘(𝑁 + 1)))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 = wceq 1242 ∈ wcel 1390 ∀wral 2300 〈cop 3370 ↦ cmpt 3809 ‘cfv 4845 (class class class)co 5455 ↦ cmpt2 5457 freccfrec 5917 1c1 6712 + caddc 6714 ℤcz 8021 ℤ≥cuz 8249 seqcseq 8892 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-13 1401 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-coll 3863 ax-sep 3866 ax-nul 3874 ax-pow 3918 ax-pr 3935 ax-un 4136 ax-setind 4220 ax-iinf 4254 ax-cnex 6774 ax-resscn 6775 ax-1cn 6776 ax-1re 6777 ax-icn 6778 ax-addcl 6779 ax-addrcl 6780 ax-mulcl 6781 ax-addcom 6783 ax-addass 6785 ax-distr 6787 ax-i2m1 6788 ax-0id 6791 ax-rnegex 6792 ax-cnre 6794 ax-pre-ltirr 6795 ax-pre-ltwlin 6796 ax-pre-lttrn 6797 ax-pre-ltadd 6799 |
This theorem depends on definitions: df-bi 110 df-dc 742 df-3or 885 df-3an 886 df-tru 1245 df-fal 1248 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ne 2203 df-nel 2204 df-ral 2305 df-rex 2306 df-reu 2307 df-rab 2309 df-v 2553 df-sbc 2759 df-csb 2847 df-dif 2914 df-un 2916 df-in 2918 df-ss 2925 df-nul 3219 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-int 3607 df-iun 3650 df-br 3756 df-opab 3810 df-mpt 3811 df-tr 3846 df-eprel 4017 df-id 4021 df-po 4024 df-iso 4025 df-iord 4069 df-on 4071 df-suc 4074 df-iom 4257 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-rn 4299 df-res 4300 df-ima 4301 df-iota 4810 df-fun 4847 df-fn 4848 df-f 4849 df-f1 4850 df-fo 4851 df-f1o 4852 df-fv 4853 df-riota 5411 df-ov 5458 df-oprab 5459 df-mpt2 5460 df-1st 5709 df-2nd 5710 df-recs 5861 df-irdg 5897 df-frec 5918 df-1o 5940 df-2o 5941 df-oadd 5944 df-omul 5945 df-er 6042 df-ec 6044 df-qs 6048 df-ni 6288 df-pli 6289 df-mi 6290 df-lti 6291 df-plpq 6328 df-mpq 6329 df-enq 6331 df-nqqs 6332 df-plqqs 6333 df-mqqs 6334 df-1nqqs 6335 df-rq 6336 df-ltnqqs 6337 df-enq0 6407 df-nq0 6408 df-0nq0 6409 df-plq0 6410 df-mq0 6411 df-inp 6449 df-i1p 6450 df-iplp 6451 df-iltp 6453 df-enr 6654 df-nr 6655 df-ltr 6658 df-0r 6659 df-1r 6660 df-0 6718 df-1 6719 df-r 6721 df-lt 6724 df-pnf 6859 df-mnf 6860 df-xr 6861 df-ltxr 6862 df-le 6863 df-sub 6981 df-neg 6982 df-inn 7696 df-n0 7958 df-z 8022 df-uz 8250 df-iseq 8893 |
This theorem is referenced by: iseqfveq2 8905 expivallem 8910 expp1 8916 |
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