Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > ruclem4 | Structured version Visualization version GIF version |
Description: Lemma for ruc 15595. Initial value of the interval sequence. (Contributed by Mario Carneiro, 28-May-2014.) |
Ref | Expression |
---|---|
ruc.1 | ⊢ (𝜑 → 𝐹:ℕ⟶ℝ) |
ruc.2 | ⊢ (𝜑 → 𝐷 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ ⦋(((1st ‘𝑥) + (2nd ‘𝑥)) / 2) / 𝑚⦌if(𝑚 < 𝑦, 〈(1st ‘𝑥), 𝑚〉, 〈((𝑚 + (2nd ‘𝑥)) / 2), (2nd ‘𝑥)〉))) |
ruc.4 | ⊢ 𝐶 = ({〈0, 〈0, 1〉〉} ∪ 𝐹) |
ruc.5 | ⊢ 𝐺 = seq0(𝐷, 𝐶) |
Ref | Expression |
---|---|
ruclem4 | ⊢ (𝜑 → (𝐺‘0) = 〈0, 1〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ruc.5 | . . 3 ⊢ 𝐺 = seq0(𝐷, 𝐶) | |
2 | 1 | fveq1i 6670 | . 2 ⊢ (𝐺‘0) = (seq0(𝐷, 𝐶)‘0) |
3 | 0z 11991 | . . 3 ⊢ 0 ∈ ℤ | |
4 | ruc.4 | . . . . . 6 ⊢ 𝐶 = ({〈0, 〈0, 1〉〉} ∪ 𝐹) | |
5 | dfn2 11909 | . . . . . . . . 9 ⊢ ℕ = (ℕ0 ∖ {0}) | |
6 | 5 | reseq2i 5849 | . . . . . . . 8 ⊢ (𝐹 ↾ ℕ) = (𝐹 ↾ (ℕ0 ∖ {0})) |
7 | ruc.1 | . . . . . . . . 9 ⊢ (𝜑 → 𝐹:ℕ⟶ℝ) | |
8 | ffn 6513 | . . . . . . . . 9 ⊢ (𝐹:ℕ⟶ℝ → 𝐹 Fn ℕ) | |
9 | fnresdm 6465 | . . . . . . . . 9 ⊢ (𝐹 Fn ℕ → (𝐹 ↾ ℕ) = 𝐹) | |
10 | 7, 8, 9 | 3syl 18 | . . . . . . . 8 ⊢ (𝜑 → (𝐹 ↾ ℕ) = 𝐹) |
11 | 6, 10 | syl5reqr 2871 | . . . . . . 7 ⊢ (𝜑 → 𝐹 = (𝐹 ↾ (ℕ0 ∖ {0}))) |
12 | 11 | uneq2d 4138 | . . . . . 6 ⊢ (𝜑 → ({〈0, 〈0, 1〉〉} ∪ 𝐹) = ({〈0, 〈0, 1〉〉} ∪ (𝐹 ↾ (ℕ0 ∖ {0})))) |
13 | 4, 12 | syl5eq 2868 | . . . . 5 ⊢ (𝜑 → 𝐶 = ({〈0, 〈0, 1〉〉} ∪ (𝐹 ↾ (ℕ0 ∖ {0})))) |
14 | 13 | fveq1d 6671 | . . . 4 ⊢ (𝜑 → (𝐶‘0) = (({〈0, 〈0, 1〉〉} ∪ (𝐹 ↾ (ℕ0 ∖ {0})))‘0)) |
15 | c0ex 10634 | . . . . . . 7 ⊢ 0 ∈ V | |
16 | 15 | a1i 11 | . . . . . 6 ⊢ (⊤ → 0 ∈ V) |
17 | opex 5355 | . . . . . . 7 ⊢ 〈0, 1〉 ∈ V | |
18 | 17 | a1i 11 | . . . . . 6 ⊢ (⊤ → 〈0, 1〉 ∈ V) |
19 | eqid 2821 | . . . . . 6 ⊢ ({〈0, 〈0, 1〉〉} ∪ (𝐹 ↾ (ℕ0 ∖ {0}))) = ({〈0, 〈0, 1〉〉} ∪ (𝐹 ↾ (ℕ0 ∖ {0}))) | |
20 | 16, 18, 19 | fvsnun1 6943 | . . . . 5 ⊢ (⊤ → (({〈0, 〈0, 1〉〉} ∪ (𝐹 ↾ (ℕ0 ∖ {0})))‘0) = 〈0, 1〉) |
21 | 20 | mptru 1540 | . . . 4 ⊢ (({〈0, 〈0, 1〉〉} ∪ (𝐹 ↾ (ℕ0 ∖ {0})))‘0) = 〈0, 1〉 |
22 | 14, 21 | syl6eq 2872 | . . 3 ⊢ (𝜑 → (𝐶‘0) = 〈0, 1〉) |
23 | 3, 22 | seq1i 13382 | . 2 ⊢ (𝜑 → (seq0(𝐷, 𝐶)‘0) = 〈0, 1〉) |
24 | 2, 23 | syl5eq 2868 | 1 ⊢ (𝜑 → (𝐺‘0) = 〈0, 1〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ⊤wtru 1534 ∈ wcel 2110 Vcvv 3494 ⦋csb 3882 ∖ cdif 3932 ∪ cun 3933 ifcif 4466 {csn 4566 〈cop 4572 class class class wbr 5065 × cxp 5552 ↾ cres 5556 Fn wfn 6349 ⟶wf 6350 ‘cfv 6354 (class class class)co 7155 ∈ cmpo 7157 1st c1st 7686 2nd c2nd 7687 ℝcr 10535 0cc0 10536 1c1 10537 + caddc 10539 < clt 10674 / cdiv 11296 ℕcn 11637 2c2 11691 ℕ0cn0 11896 seqcseq 13368 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-nn 11638 df-n0 11897 df-z 11981 df-uz 12243 df-seq 13369 |
This theorem is referenced by: ruclem6 15587 ruclem8 15589 ruclem11 15592 |
Copyright terms: Public domain | W3C validator |