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Mirrors > Home > MPE Home > Th. List > ruclem4 | Structured version Visualization version GIF version |
Description: Lemma for ruc 14811. 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 6104 | . 2 ⊢ (𝐺‘0) = (seq0(𝐷, 𝐶)‘0) |
3 | 0z 11265 | . . 3 ⊢ 0 ∈ ℤ | |
4 | ruc.4 | . . . . . 6 ⊢ 𝐶 = ({〈0, 〈0, 1〉〉} ∪ 𝐹) | |
5 | dfn2 11182 | . . . . . . . . 9 ⊢ ℕ = (ℕ0 ∖ {0}) | |
6 | 5 | reseq2i 5314 | . . . . . . . 8 ⊢ (𝐹 ↾ ℕ) = (𝐹 ↾ (ℕ0 ∖ {0})) |
7 | ruc.1 | . . . . . . . . 9 ⊢ (𝜑 → 𝐹:ℕ⟶ℝ) | |
8 | ffn 5958 | . . . . . . . . 9 ⊢ (𝐹:ℕ⟶ℝ → 𝐹 Fn ℕ) | |
9 | fnresdm 5914 | . . . . . . . . 9 ⊢ (𝐹 Fn ℕ → (𝐹 ↾ ℕ) = 𝐹) | |
10 | 7, 8, 9 | 3syl 18 | . . . . . . . 8 ⊢ (𝜑 → (𝐹 ↾ ℕ) = 𝐹) |
11 | 6, 10 | syl5reqr 2659 | . . . . . . 7 ⊢ (𝜑 → 𝐹 = (𝐹 ↾ (ℕ0 ∖ {0}))) |
12 | 11 | uneq2d 3729 | . . . . . 6 ⊢ (𝜑 → ({〈0, 〈0, 1〉〉} ∪ 𝐹) = ({〈0, 〈0, 1〉〉} ∪ (𝐹 ↾ (ℕ0 ∖ {0})))) |
13 | 4, 12 | syl5eq 2656 | . . . . 5 ⊢ (𝜑 → 𝐶 = ({〈0, 〈0, 1〉〉} ∪ (𝐹 ↾ (ℕ0 ∖ {0})))) |
14 | 13 | fveq1d 6105 | . . . 4 ⊢ (𝜑 → (𝐶‘0) = (({〈0, 〈0, 1〉〉} ∪ (𝐹 ↾ (ℕ0 ∖ {0})))‘0)) |
15 | c0ex 9913 | . . . . 5 ⊢ 0 ∈ V | |
16 | opex 4859 | . . . . 5 ⊢ 〈0, 1〉 ∈ V | |
17 | eqid 2610 | . . . . 5 ⊢ ({〈0, 〈0, 1〉〉} ∪ (𝐹 ↾ (ℕ0 ∖ {0}))) = ({〈0, 〈0, 1〉〉} ∪ (𝐹 ↾ (ℕ0 ∖ {0}))) | |
18 | 15, 16, 17 | fvsnun1 6353 | . . . 4 ⊢ (({〈0, 〈0, 1〉〉} ∪ (𝐹 ↾ (ℕ0 ∖ {0})))‘0) = 〈0, 1〉 |
19 | 14, 18 | syl6eq 2660 | . . 3 ⊢ (𝜑 → (𝐶‘0) = 〈0, 1〉) |
20 | 3, 19 | seq1i 12677 | . 2 ⊢ (𝜑 → (seq0(𝐷, 𝐶)‘0) = 〈0, 1〉) |
21 | 2, 20 | syl5eq 2656 | 1 ⊢ (𝜑 → (𝐺‘0) = 〈0, 1〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ⦋csb 3499 ∖ cdif 3537 ∪ cun 3538 ifcif 4036 {csn 4125 〈cop 4131 class class class wbr 4583 × cxp 5036 ↾ cres 5040 Fn wfn 5799 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 1st c1st 7057 2nd c2nd 7058 ℝcr 9814 0cc0 9815 1c1 9816 + caddc 9818 < clt 9953 / cdiv 10563 ℕcn 10897 2c2 10947 ℕ0cn0 11169 seqcseq 12663 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-seq 12664 |
This theorem is referenced by: ruclem6 14803 ruclem8 14805 ruclem11 14808 |
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