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Mirrors > Home > MPE Home > Th. List > ruclem4 | Structured version Visualization version GIF version |
Description: Lemma for ruc 15767. 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 6696 | . 2 ⊢ (𝐺‘0) = (seq0(𝐷, 𝐶)‘0) |
3 | 0z 12152 | . . 3 ⊢ 0 ∈ ℤ | |
4 | ruc.4 | . . . . . 6 ⊢ 𝐶 = ({〈0, 〈0, 1〉〉} ∪ 𝐹) | |
5 | ruc.1 | . . . . . . . . 9 ⊢ (𝜑 → 𝐹:ℕ⟶ℝ) | |
6 | ffn 6523 | . . . . . . . . 9 ⊢ (𝐹:ℕ⟶ℝ → 𝐹 Fn ℕ) | |
7 | fnresdm 6474 | . . . . . . . . 9 ⊢ (𝐹 Fn ℕ → (𝐹 ↾ ℕ) = 𝐹) | |
8 | 5, 6, 7 | 3syl 18 | . . . . . . . 8 ⊢ (𝜑 → (𝐹 ↾ ℕ) = 𝐹) |
9 | dfn2 12068 | . . . . . . . . 9 ⊢ ℕ = (ℕ0 ∖ {0}) | |
10 | 9 | reseq2i 5833 | . . . . . . . 8 ⊢ (𝐹 ↾ ℕ) = (𝐹 ↾ (ℕ0 ∖ {0})) |
11 | 8, 10 | eqtr3di 2786 | . . . . . . 7 ⊢ (𝜑 → 𝐹 = (𝐹 ↾ (ℕ0 ∖ {0}))) |
12 | 11 | uneq2d 4063 | . . . . . 6 ⊢ (𝜑 → ({〈0, 〈0, 1〉〉} ∪ 𝐹) = ({〈0, 〈0, 1〉〉} ∪ (𝐹 ↾ (ℕ0 ∖ {0})))) |
13 | 4, 12 | syl5eq 2783 | . . . . 5 ⊢ (𝜑 → 𝐶 = ({〈0, 〈0, 1〉〉} ∪ (𝐹 ↾ (ℕ0 ∖ {0})))) |
14 | 13 | fveq1d 6697 | . . . 4 ⊢ (𝜑 → (𝐶‘0) = (({〈0, 〈0, 1〉〉} ∪ (𝐹 ↾ (ℕ0 ∖ {0})))‘0)) |
15 | c0ex 10792 | . . . . . . 7 ⊢ 0 ∈ V | |
16 | 15 | a1i 11 | . . . . . 6 ⊢ (⊤ → 0 ∈ V) |
17 | opex 5333 | . . . . . . 7 ⊢ 〈0, 1〉 ∈ V | |
18 | 17 | a1i 11 | . . . . . 6 ⊢ (⊤ → 〈0, 1〉 ∈ V) |
19 | eqid 2736 | . . . . . 6 ⊢ ({〈0, 〈0, 1〉〉} ∪ (𝐹 ↾ (ℕ0 ∖ {0}))) = ({〈0, 〈0, 1〉〉} ∪ (𝐹 ↾ (ℕ0 ∖ {0}))) | |
20 | 16, 18, 19 | fvsnun1 6975 | . . . . 5 ⊢ (⊤ → (({〈0, 〈0, 1〉〉} ∪ (𝐹 ↾ (ℕ0 ∖ {0})))‘0) = 〈0, 1〉) |
21 | 20 | mptru 1550 | . . . 4 ⊢ (({〈0, 〈0, 1〉〉} ∪ (𝐹 ↾ (ℕ0 ∖ {0})))‘0) = 〈0, 1〉 |
22 | 14, 21 | eqtrdi 2787 | . . 3 ⊢ (𝜑 → (𝐶‘0) = 〈0, 1〉) |
23 | 3, 22 | seq1i 13553 | . 2 ⊢ (𝜑 → (seq0(𝐷, 𝐶)‘0) = 〈0, 1〉) |
24 | 2, 23 | syl5eq 2783 | 1 ⊢ (𝜑 → (𝐺‘0) = 〈0, 1〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ⊤wtru 1544 ∈ wcel 2112 Vcvv 3398 ⦋csb 3798 ∖ cdif 3850 ∪ cun 3851 ifcif 4425 {csn 4527 〈cop 4533 class class class wbr 5039 × cxp 5534 ↾ cres 5538 Fn wfn 6353 ⟶wf 6354 ‘cfv 6358 (class class class)co 7191 ∈ cmpo 7193 1st c1st 7737 2nd c2nd 7738 ℝcr 10693 0cc0 10694 1c1 10695 + caddc 10697 < clt 10832 / cdiv 11454 ℕcn 11795 2c2 11850 ℕ0cn0 12055 seqcseq 13539 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-2nd 7740 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-nn 11796 df-n0 12056 df-z 12142 df-uz 12404 df-seq 13540 |
This theorem is referenced by: ruclem6 15759 ruclem8 15761 ruclem11 15764 |
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