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Mirrors > Home > MPE Home > Th. List > ruclem7 | Structured version Visualization version GIF version |
Description: Lemma for ruc 15016. Successor value for 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 |
---|---|
ruclem7 | ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0) → (𝐺‘(𝑁 + 1)) = ((𝐺‘𝑁)𝐷(𝐹‘(𝑁 + 1)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 476 | . . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0) → 𝑁 ∈ ℕ0) | |
2 | nn0uz 11760 | . . . . 5 ⊢ ℕ0 = (ℤ≥‘0) | |
3 | 1, 2 | syl6eleq 2740 | . . . 4 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0) → 𝑁 ∈ (ℤ≥‘0)) |
4 | seqp1 12856 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘0) → (seq0(𝐷, 𝐶)‘(𝑁 + 1)) = ((seq0(𝐷, 𝐶)‘𝑁)𝐷(𝐶‘(𝑁 + 1)))) | |
5 | 3, 4 | syl 17 | . . 3 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0) → (seq0(𝐷, 𝐶)‘(𝑁 + 1)) = ((seq0(𝐷, 𝐶)‘𝑁)𝐷(𝐶‘(𝑁 + 1)))) |
6 | ruc.5 | . . . 4 ⊢ 𝐺 = seq0(𝐷, 𝐶) | |
7 | 6 | fveq1i 6230 | . . 3 ⊢ (𝐺‘(𝑁 + 1)) = (seq0(𝐷, 𝐶)‘(𝑁 + 1)) |
8 | 6 | fveq1i 6230 | . . . 4 ⊢ (𝐺‘𝑁) = (seq0(𝐷, 𝐶)‘𝑁) |
9 | 8 | oveq1i 6700 | . . 3 ⊢ ((𝐺‘𝑁)𝐷(𝐶‘(𝑁 + 1))) = ((seq0(𝐷, 𝐶)‘𝑁)𝐷(𝐶‘(𝑁 + 1))) |
10 | 5, 7, 9 | 3eqtr4g 2710 | . 2 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0) → (𝐺‘(𝑁 + 1)) = ((𝐺‘𝑁)𝐷(𝐶‘(𝑁 + 1)))) |
11 | nn0p1nn 11370 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ) | |
12 | 11 | adantl 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0) → (𝑁 + 1) ∈ ℕ) |
13 | 12 | nnne0d 11103 | . . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0) → (𝑁 + 1) ≠ 0) |
14 | 13 | necomd 2878 | . . . 4 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0) → 0 ≠ (𝑁 + 1)) |
15 | ruc.4 | . . . . . . 7 ⊢ 𝐶 = ({〈0, 〈0, 1〉〉} ∪ 𝐹) | |
16 | 15 | equncomi 3792 | . . . . . 6 ⊢ 𝐶 = (𝐹 ∪ {〈0, 〈0, 1〉〉}) |
17 | 16 | fveq1i 6230 | . . . . 5 ⊢ (𝐶‘(𝑁 + 1)) = ((𝐹 ∪ {〈0, 〈0, 1〉〉})‘(𝑁 + 1)) |
18 | fvunsn 6486 | . . . . 5 ⊢ (0 ≠ (𝑁 + 1) → ((𝐹 ∪ {〈0, 〈0, 1〉〉})‘(𝑁 + 1)) = (𝐹‘(𝑁 + 1))) | |
19 | 17, 18 | syl5eq 2697 | . . . 4 ⊢ (0 ≠ (𝑁 + 1) → (𝐶‘(𝑁 + 1)) = (𝐹‘(𝑁 + 1))) |
20 | 14, 19 | syl 17 | . . 3 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0) → (𝐶‘(𝑁 + 1)) = (𝐹‘(𝑁 + 1))) |
21 | 20 | oveq2d 6706 | . 2 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0) → ((𝐺‘𝑁)𝐷(𝐶‘(𝑁 + 1))) = ((𝐺‘𝑁)𝐷(𝐹‘(𝑁 + 1)))) |
22 | 10, 21 | eqtrd 2685 | 1 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0) → (𝐺‘(𝑁 + 1)) = ((𝐺‘𝑁)𝐷(𝐹‘(𝑁 + 1)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1523 ∈ wcel 2030 ≠ wne 2823 ⦋csb 3566 ∪ cun 3605 ifcif 4119 {csn 4210 〈cop 4216 class class class wbr 4685 × cxp 5141 ⟶wf 5922 ‘cfv 5926 (class class class)co 6690 ↦ cmpt2 6692 1st c1st 7208 2nd c2nd 7209 ℝcr 9973 0cc0 9974 1c1 9975 + caddc 9977 < clt 10112 / cdiv 10722 ℕcn 11058 2c2 11108 ℕ0cn0 11330 ℤ≥cuz 11725 seqcseq 12841 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-2nd 7211 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-nn 11059 df-n0 11331 df-z 11416 df-uz 11726 df-seq 12842 |
This theorem is referenced by: ruclem8 15010 ruclem9 15011 ruclem12 15014 |
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