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Mirrors > Home > MPE Home > Th. List > ruclem4 | Structured version Visualization version GIF version |
Description: Lemma for ruc 14972. 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 6192 | . 2 ⊢ (𝐺‘0) = (seq0(𝐷, 𝐶)‘0) |
3 | 0z 11388 | . . 3 ⊢ 0 ∈ ℤ | |
4 | ruc.4 | . . . . . 6 ⊢ 𝐶 = ({〈0, 〈0, 1〉〉} ∪ 𝐹) | |
5 | dfn2 11305 | . . . . . . . . 9 ⊢ ℕ = (ℕ0 ∖ {0}) | |
6 | 5 | reseq2i 5393 | . . . . . . . 8 ⊢ (𝐹 ↾ ℕ) = (𝐹 ↾ (ℕ0 ∖ {0})) |
7 | ruc.1 | . . . . . . . . 9 ⊢ (𝜑 → 𝐹:ℕ⟶ℝ) | |
8 | ffn 6045 | . . . . . . . . 9 ⊢ (𝐹:ℕ⟶ℝ → 𝐹 Fn ℕ) | |
9 | fnresdm 6000 | . . . . . . . . 9 ⊢ (𝐹 Fn ℕ → (𝐹 ↾ ℕ) = 𝐹) | |
10 | 7, 8, 9 | 3syl 18 | . . . . . . . 8 ⊢ (𝜑 → (𝐹 ↾ ℕ) = 𝐹) |
11 | 6, 10 | syl5reqr 2671 | . . . . . . 7 ⊢ (𝜑 → 𝐹 = (𝐹 ↾ (ℕ0 ∖ {0}))) |
12 | 11 | uneq2d 3767 | . . . . . 6 ⊢ (𝜑 → ({〈0, 〈0, 1〉〉} ∪ 𝐹) = ({〈0, 〈0, 1〉〉} ∪ (𝐹 ↾ (ℕ0 ∖ {0})))) |
13 | 4, 12 | syl5eq 2668 | . . . . 5 ⊢ (𝜑 → 𝐶 = ({〈0, 〈0, 1〉〉} ∪ (𝐹 ↾ (ℕ0 ∖ {0})))) |
14 | 13 | fveq1d 6193 | . . . 4 ⊢ (𝜑 → (𝐶‘0) = (({〈0, 〈0, 1〉〉} ∪ (𝐹 ↾ (ℕ0 ∖ {0})))‘0)) |
15 | c0ex 10034 | . . . . 5 ⊢ 0 ∈ V | |
16 | opex 4932 | . . . . 5 ⊢ 〈0, 1〉 ∈ V | |
17 | eqid 2622 | . . . . 5 ⊢ ({〈0, 〈0, 1〉〉} ∪ (𝐹 ↾ (ℕ0 ∖ {0}))) = ({〈0, 〈0, 1〉〉} ∪ (𝐹 ↾ (ℕ0 ∖ {0}))) | |
18 | 15, 16, 17 | fvsnun1 6448 | . . . 4 ⊢ (({〈0, 〈0, 1〉〉} ∪ (𝐹 ↾ (ℕ0 ∖ {0})))‘0) = 〈0, 1〉 |
19 | 14, 18 | syl6eq 2672 | . . 3 ⊢ (𝜑 → (𝐶‘0) = 〈0, 1〉) |
20 | 3, 19 | seq1i 12815 | . 2 ⊢ (𝜑 → (seq0(𝐷, 𝐶)‘0) = 〈0, 1〉) |
21 | 2, 20 | syl5eq 2668 | 1 ⊢ (𝜑 → (𝐺‘0) = 〈0, 1〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ⦋csb 3533 ∖ cdif 3571 ∪ cun 3572 ifcif 4086 {csn 4177 〈cop 4183 class class class wbr 4653 × cxp 5112 ↾ cres 5116 Fn wfn 5883 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ↦ cmpt2 6652 1st c1st 7166 2nd c2nd 7167 ℝcr 9935 0cc0 9936 1c1 9937 + caddc 9939 < clt 10074 / cdiv 10684 ℕcn 11020 2c2 11070 ℕ0cn0 11292 seqcseq 12801 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-seq 12802 |
This theorem is referenced by: ruclem6 14964 ruclem8 14966 ruclem11 14969 |
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