![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > seq3-1 | GIF version |
Description: Value of the sequence builder function at its initial value. (Contributed by Jim Kingdon, 3-Oct-2022.) |
Ref | Expression |
---|---|
seq3-1.m | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
seq3-1.f | ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆) |
seq3-1.pl | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) |
Ref | Expression |
---|---|
seq3-1 | ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑀) = (𝐹‘𝑀)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | seq3-1.m | . 2 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
2 | fveq2 5516 | . . . 4 ⊢ (𝑥 = 𝑀 → (𝐹‘𝑥) = (𝐹‘𝑀)) | |
3 | 2 | eleq1d 2246 | . . 3 ⊢ (𝑥 = 𝑀 → ((𝐹‘𝑥) ∈ 𝑆 ↔ (𝐹‘𝑀) ∈ 𝑆)) |
4 | seq3-1.f | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆) | |
5 | 4 | ralrimiva 2550 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ (ℤ≥‘𝑀)(𝐹‘𝑥) ∈ 𝑆) |
6 | uzid 9542 | . . . 4 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ≥‘𝑀)) | |
7 | 1, 6 | syl 14 | . . 3 ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘𝑀)) |
8 | 3, 5, 7 | rspcdva 2847 | . 2 ⊢ (𝜑 → (𝐹‘𝑀) ∈ 𝑆) |
9 | ssv 3178 | . . 3 ⊢ 𝑆 ⊆ V | |
10 | 9 | a1i 9 | . 2 ⊢ (𝜑 → 𝑆 ⊆ V) |
11 | seq3-1.pl | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) | |
12 | 4, 11 | iseqovex 10456 | . 2 ⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝑀) ∧ 𝑦 ∈ 𝑆)) → (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦) ∈ 𝑆) |
13 | iseqvalcbv 10457 | . 2 ⊢ 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)⟩) | |
14 | 1, 13, 4, 11 | seq3val 10458 | . 2 ⊢ (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)⟩)) |
15 | 1, 8, 10, 12, 13, 14 | frecuzrdg0t 10422 | 1 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑀) = (𝐹‘𝑀)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1353 ∈ wcel 2148 Vcvv 2738 ⊆ wss 3130 ⟨cop 3596 ‘cfv 5217 (class class class)co 5875 ∈ cmpo 5877 freccfrec 6391 1c1 7812 + caddc 7814 ℤcz 9253 ℤ≥cuz 9528 seqcseq 10445 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4119 ax-sep 4122 ax-nul 4130 ax-pow 4175 ax-pr 4210 ax-un 4434 ax-setind 4537 ax-iinf 4588 ax-cnex 7902 ax-resscn 7903 ax-1cn 7904 ax-1re 7905 ax-icn 7906 ax-addcl 7907 ax-addrcl 7908 ax-mulcl 7909 ax-addcom 7911 ax-addass 7913 ax-distr 7915 ax-i2m1 7916 ax-0lt1 7917 ax-0id 7919 ax-rnegex 7920 ax-cnre 7922 ax-pre-ltirr 7923 ax-pre-ltwlin 7924 ax-pre-lttrn 7925 ax-pre-ltadd 7927 |
This theorem depends on definitions: df-bi 117 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2740 df-sbc 2964 df-csb 3059 df-dif 3132 df-un 3134 df-in 3136 df-ss 3143 df-nul 3424 df-pw 3578 df-sn 3599 df-pr 3600 df-op 3602 df-uni 3811 df-int 3846 df-iun 3889 df-br 4005 df-opab 4066 df-mpt 4067 df-tr 4103 df-id 4294 df-iord 4367 df-on 4369 df-ilim 4370 df-suc 4372 df-iom 4591 df-xp 4633 df-rel 4634 df-cnv 4635 df-co 4636 df-dm 4637 df-rn 4638 df-res 4639 df-ima 4640 df-iota 5179 df-fun 5219 df-fn 5220 df-f 5221 df-f1 5222 df-fo 5223 df-f1o 5224 df-fv 5225 df-riota 5831 df-ov 5878 df-oprab 5879 df-mpo 5880 df-1st 6141 df-2nd 6142 df-recs 6306 df-frec 6392 df-pnf 7994 df-mnf 7995 df-xr 7996 df-ltxr 7997 df-le 7998 df-sub 8130 df-neg 8131 df-inn 8920 df-n0 9177 df-z 9254 df-uz 9529 df-seqfrec 10446 |
This theorem is referenced by: seq3clss 10467 seq3fveq2 10469 seq3fveq 10471 seq3shft2 10473 seq3split 10479 seq3-1p 10480 seq3caopr3 10481 seq3id3 10507 seq3id 10508 seq3homo 10510 seq3z 10511 ser3ge0 10517 exp3vallem 10521 exp1 10526 fac1 10709 bcn2 10744 seq3coll 10822 resqrexlemf1 11017 sumsnf 11417 isumrpcl 11502 clim2prod 11547 prodfap0 11553 prodfrecap 11554 prodsnf 11600 ef0lem 11668 ege2le3 11679 efgt1p2 11703 efgt1p 11704 ialgr0 12044 pcmpt 12341 mulg1 12990 |
Copyright terms: Public domain | W3C validator |