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Mirrors > Home > ILE Home > Th. List > seq3p1 | GIF version |
Description: Value of the sequence builder function at a successor. (Contributed by Jim Kingdon, 30-Apr-2022.) |
Ref | Expression |
---|---|
seq3p1.m | ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) |
seq3p1.f | ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆) |
seq3p1.pl | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) |
Ref | Expression |
---|---|
seq3p1 | ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘(𝑁 + 1)) = ((seq𝑀( + , 𝐹)‘𝑁) + (𝐹‘(𝑁 + 1)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | seq3p1.m | . . 3 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) | |
2 | eluzel2 9492 | . . . . 5 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ) | |
3 | 1, 2 | syl 14 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ ℤ) |
4 | fveq2 5496 | . . . . . 6 ⊢ (𝑥 = 𝑀 → (𝐹‘𝑥) = (𝐹‘𝑀)) | |
5 | 4 | eleq1d 2239 | . . . . 5 ⊢ (𝑥 = 𝑀 → ((𝐹‘𝑥) ∈ 𝑆 ↔ (𝐹‘𝑀) ∈ 𝑆)) |
6 | seq3p1.f | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆) | |
7 | 6 | ralrimiva 2543 | . . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ (ℤ≥‘𝑀)(𝐹‘𝑥) ∈ 𝑆) |
8 | uzid 9501 | . . . . . 6 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ≥‘𝑀)) | |
9 | 3, 8 | syl 14 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘𝑀)) |
10 | 5, 7, 9 | rspcdva 2839 | . . . 4 ⊢ (𝜑 → (𝐹‘𝑀) ∈ 𝑆) |
11 | ssv 3169 | . . . . 5 ⊢ 𝑆 ⊆ V | |
12 | 11 | a1i 9 | . . . 4 ⊢ (𝜑 → 𝑆 ⊆ V) |
13 | seq3p1.pl | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) | |
14 | 6, 13 | iseqovex 10412 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝑀) ∧ 𝑦 ∈ 𝑆)) → (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦) ∈ 𝑆) |
15 | iseqvalcbv 10413 | . . . 4 ⊢ frec((𝑎 ∈ (ℤ≥‘𝑀), 𝑏 ∈ V ↦ 〈(𝑎 + 1), (𝑎(𝑐 ∈ (ℤ≥‘𝑀), 𝑑 ∈ 𝑆 ↦ (𝑑 + (𝐹‘(𝑐 + 1))))𝑏)〉), 〈𝑀, (𝐹‘𝑀)〉) = frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)〉), 〈𝑀, (𝐹‘𝑀)〉) | |
16 | 3, 15, 6, 13 | seq3val 10414 | . . . 4 ⊢ (𝜑 → seq𝑀( + , 𝐹) = ran frec((𝑎 ∈ (ℤ≥‘𝑀), 𝑏 ∈ V ↦ 〈(𝑎 + 1), (𝑎(𝑐 ∈ (ℤ≥‘𝑀), 𝑑 ∈ 𝑆 ↦ (𝑑 + (𝐹‘(𝑐 + 1))))𝑏)〉), 〈𝑀, (𝐹‘𝑀)〉)) |
17 | 3, 10, 12, 14, 15, 16 | frecuzrdgsuct 10380 | . . 3 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → (seq𝑀( + , 𝐹)‘(𝑁 + 1)) = (𝑁(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(seq𝑀( + , 𝐹)‘𝑁))) |
18 | 1, 17 | mpdan 419 | . 2 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘(𝑁 + 1)) = (𝑁(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(seq𝑀( + , 𝐹)‘𝑁))) |
19 | eqid 2170 | . . . . 5 ⊢ (ℤ≥‘𝑀) = (ℤ≥‘𝑀) | |
20 | 19, 3, 6, 13 | seqf 10417 | . . . 4 ⊢ (𝜑 → seq𝑀( + , 𝐹):(ℤ≥‘𝑀)⟶𝑆) |
21 | 20, 1 | ffvelrnd 5632 | . . 3 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) ∈ 𝑆) |
22 | fveq2 5496 | . . . . . 6 ⊢ (𝑥 = (𝑁 + 1) → (𝐹‘𝑥) = (𝐹‘(𝑁 + 1))) | |
23 | 22 | eleq1d 2239 | . . . . 5 ⊢ (𝑥 = (𝑁 + 1) → ((𝐹‘𝑥) ∈ 𝑆 ↔ (𝐹‘(𝑁 + 1)) ∈ 𝑆)) |
24 | peano2uz 9542 | . . . . . 6 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑁 + 1) ∈ (ℤ≥‘𝑀)) | |
25 | 1, 24 | syl 14 | . . . . 5 ⊢ (𝜑 → (𝑁 + 1) ∈ (ℤ≥‘𝑀)) |
26 | 23, 7, 25 | rspcdva 2839 | . . . 4 ⊢ (𝜑 → (𝐹‘(𝑁 + 1)) ∈ 𝑆) |
27 | 13, 21, 26 | caovcld 6006 | . . 3 ⊢ (𝜑 → ((seq𝑀( + , 𝐹)‘𝑁) + (𝐹‘(𝑁 + 1))) ∈ 𝑆) |
28 | fvoveq1 5876 | . . . . 5 ⊢ (𝑧 = 𝑁 → (𝐹‘(𝑧 + 1)) = (𝐹‘(𝑁 + 1))) | |
29 | 28 | oveq2d 5869 | . . . 4 ⊢ (𝑧 = 𝑁 → (𝑤 + (𝐹‘(𝑧 + 1))) = (𝑤 + (𝐹‘(𝑁 + 1)))) |
30 | oveq1 5860 | . . . 4 ⊢ (𝑤 = (seq𝑀( + , 𝐹)‘𝑁) → (𝑤 + (𝐹‘(𝑁 + 1))) = ((seq𝑀( + , 𝐹)‘𝑁) + (𝐹‘(𝑁 + 1)))) | |
31 | eqid 2170 | . . . 4 ⊢ (𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1)))) = (𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1)))) | |
32 | 29, 30, 31 | ovmpog 5987 | . . 3 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (seq𝑀( + , 𝐹)‘𝑁) ∈ 𝑆 ∧ ((seq𝑀( + , 𝐹)‘𝑁) + (𝐹‘(𝑁 + 1))) ∈ 𝑆) → (𝑁(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(seq𝑀( + , 𝐹)‘𝑁)) = ((seq𝑀( + , 𝐹)‘𝑁) + (𝐹‘(𝑁 + 1)))) |
33 | 1, 21, 27, 32 | syl3anc 1233 | . 2 ⊢ (𝜑 → (𝑁(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(seq𝑀( + , 𝐹)‘𝑁)) = ((seq𝑀( + , 𝐹)‘𝑁) + (𝐹‘(𝑁 + 1)))) |
34 | 18, 33 | eqtrd 2203 | 1 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘(𝑁 + 1)) = ((seq𝑀( + , 𝐹)‘𝑁) + (𝐹‘(𝑁 + 1)))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1348 ∈ wcel 2141 Vcvv 2730 ⊆ wss 3121 〈cop 3586 ‘cfv 5198 (class class class)co 5853 ∈ cmpo 5855 freccfrec 6369 1c1 7775 + caddc 7777 ℤcz 9212 ℤ≥cuz 9487 seqcseq 10401 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-addcom 7874 ax-addass 7876 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-0id 7882 ax-rnegex 7883 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-ltadd 7890 |
This theorem depends on definitions: df-bi 116 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-id 4278 df-iord 4351 df-on 4353 df-ilim 4354 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-recs 6284 df-frec 6370 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-inn 8879 df-n0 9136 df-z 9213 df-uz 9488 df-seqfrec 10402 |
This theorem is referenced by: seq3clss 10423 seq3m1 10424 seq3fveq2 10425 seq3shft2 10429 ser3mono 10434 seq3split 10435 seq3caopr3 10437 seq3id3 10463 seq3id2 10465 seq3homo 10466 seq3z 10467 ser3ge0 10473 exp3vallem 10477 expp1 10483 facp1 10664 seq3coll 10777 resqrexlemfp1 10973 climserle 11308 clim2prod 11502 prodfap0 11508 prodfrecap 11509 ege2le3 11634 efgt1p2 11658 efgt1p 11659 algrp1 12000 pcmpt 12295 nninfdclemp1 12405 |
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