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Mirrors > Home > ILE Home > Th. List > seqclg | GIF version |
Description: Closure properties of the recursive sequence builder. (Contributed by Mario Carneiro, 2-Jul-2013.) (Revised by Mario Carneiro, 27-May-2014.) |
Ref | Expression |
---|---|
seqcl.1 | ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) |
seqcl.2 | ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹‘𝑥) ∈ 𝑆) |
seqcl.3 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) |
seqclg.f | ⊢ (𝜑 → 𝐹 ∈ 𝑉) |
seqclg.p | ⊢ (𝜑 → + ∈ 𝑊) |
Ref | Expression |
---|---|
seqclg | ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | seqcl.1 | . 2 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) | |
2 | seqclg.f | . . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝑉) | |
3 | 2 | adantr 276 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → 𝐹 ∈ 𝑉) |
4 | vex 2763 | . . 3 ⊢ 𝑥 ∈ V | |
5 | fvexg 5565 | . . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑥 ∈ V) → (𝐹‘𝑥) ∈ V) | |
6 | 3, 4, 5 | sylancl 413 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ V) |
7 | seqcl.2 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹‘𝑥) ∈ 𝑆) | |
8 | seqcl.3 | . 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) | |
9 | ssv 3201 | . . 3 ⊢ 𝑆 ⊆ V | |
10 | 9 | a1i 9 | . 2 ⊢ (𝜑 → 𝑆 ⊆ V) |
11 | seqclg.p | . . 3 ⊢ (𝜑 → + ∈ 𝑊) | |
12 | simprr 531 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)) → 𝑦 ∈ V) | |
13 | ovexg 5944 | . . 3 ⊢ ((𝑥 ∈ V ∧ + ∈ 𝑊 ∧ 𝑦 ∈ V) → (𝑥 + 𝑦) ∈ V) | |
14 | 4, 11, 12, 13 | mp3an2ani 1355 | . 2 ⊢ ((𝜑 ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)) → (𝑥 + 𝑦) ∈ V) |
15 | 1, 6, 7, 8, 10, 14 | seq3clss 10532 | 1 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) ∈ 𝑆) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∈ wcel 2164 Vcvv 2760 ⊆ wss 3153 ‘cfv 5246 (class class class)co 5910 ℤ≥cuz 9582 ...cfz 10064 seqcseq 10508 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-coll 4144 ax-sep 4147 ax-nul 4155 ax-pow 4203 ax-pr 4238 ax-un 4462 ax-setind 4565 ax-iinf 4616 ax-cnex 7953 ax-resscn 7954 ax-1cn 7955 ax-1re 7956 ax-icn 7957 ax-addcl 7958 ax-addrcl 7959 ax-mulcl 7960 ax-addcom 7962 ax-addass 7964 ax-distr 7966 ax-i2m1 7967 ax-0lt1 7968 ax-0id 7970 ax-rnegex 7971 ax-cnre 7973 ax-pre-ltirr 7974 ax-pre-ltwlin 7975 ax-pre-lttrn 7976 ax-pre-ltadd 7978 |
This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rab 2481 df-v 2762 df-sbc 2986 df-csb 3081 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-nul 3447 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-int 3871 df-iun 3914 df-br 4030 df-opab 4091 df-mpt 4092 df-tr 4128 df-id 4322 df-iord 4395 df-on 4397 df-ilim 4398 df-suc 4400 df-iom 4619 df-xp 4661 df-rel 4662 df-cnv 4663 df-co 4664 df-dm 4665 df-rn 4666 df-res 4667 df-ima 4668 df-iota 5207 df-fun 5248 df-fn 5249 df-f 5250 df-f1 5251 df-fo 5252 df-f1o 5253 df-fv 5254 df-riota 5865 df-ov 5913 df-oprab 5914 df-mpo 5915 df-1st 6184 df-2nd 6185 df-recs 6349 df-frec 6435 df-pnf 8046 df-mnf 8047 df-xr 8048 df-ltxr 8049 df-le 8050 df-sub 8182 df-neg 8183 df-inn 8973 df-n0 9231 df-z 9308 df-uz 9583 df-fz 10065 df-fzo 10199 df-seqfrec 10509 |
This theorem is referenced by: seqsplitg 10550 seqcaopr2g 10555 seqf1oglem2a 10579 seqf1oglem2 10581 seqhomog 10591 gsumfzsubmcl 13397 |
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