![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > ruclem7 | Structured version Visualization version GIF version |
Description: Lemma for ruc 15456. 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 477 | . . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0) → 𝑁 ∈ ℕ0) | |
2 | nn0uz 12094 | . . . . 5 ⊢ ℕ0 = (ℤ≥‘0) | |
3 | 1, 2 | syl6eleq 2877 | . . . 4 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0) → 𝑁 ∈ (ℤ≥‘0)) |
4 | seqp1 13199 | . . . 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 6500 | . . 3 ⊢ (𝐺‘(𝑁 + 1)) = (seq0(𝐷, 𝐶)‘(𝑁 + 1)) |
8 | 6 | fveq1i 6500 | . . . 4 ⊢ (𝐺‘𝑁) = (seq0(𝐷, 𝐶)‘𝑁) |
9 | 8 | oveq1i 6986 | . . 3 ⊢ ((𝐺‘𝑁)𝐷(𝐶‘(𝑁 + 1))) = ((seq0(𝐷, 𝐶)‘𝑁)𝐷(𝐶‘(𝑁 + 1))) |
10 | 5, 7, 9 | 3eqtr4g 2840 | . 2 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0) → (𝐺‘(𝑁 + 1)) = ((𝐺‘𝑁)𝐷(𝐶‘(𝑁 + 1)))) |
11 | nn0p1nn 11748 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ) | |
12 | 11 | adantl 474 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0) → (𝑁 + 1) ∈ ℕ) |
13 | 12 | nnne0d 11490 | . . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0) → (𝑁 + 1) ≠ 0) |
14 | 13 | necomd 3023 | . . . 4 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0) → 0 ≠ (𝑁 + 1)) |
15 | ruc.4 | . . . . . . 7 ⊢ 𝐶 = ({〈0, 〈0, 1〉〉} ∪ 𝐹) | |
16 | 15 | equncomi 4021 | . . . . . 6 ⊢ 𝐶 = (𝐹 ∪ {〈0, 〈0, 1〉〉}) |
17 | 16 | fveq1i 6500 | . . . . 5 ⊢ (𝐶‘(𝑁 + 1)) = ((𝐹 ∪ {〈0, 〈0, 1〉〉})‘(𝑁 + 1)) |
18 | fvunsn 6764 | . . . . 5 ⊢ (0 ≠ (𝑁 + 1) → ((𝐹 ∪ {〈0, 〈0, 1〉〉})‘(𝑁 + 1)) = (𝐹‘(𝑁 + 1))) | |
19 | 17, 18 | syl5eq 2827 | . . . 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 6992 | . 2 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0) → ((𝐺‘𝑁)𝐷(𝐶‘(𝑁 + 1))) = ((𝐺‘𝑁)𝐷(𝐹‘(𝑁 + 1)))) |
22 | 10, 21 | eqtrd 2815 | 1 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0) → (𝐺‘(𝑁 + 1)) = ((𝐺‘𝑁)𝐷(𝐹‘(𝑁 + 1)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 = wceq 1507 ∈ wcel 2050 ≠ wne 2968 ⦋csb 3787 ∪ cun 3828 ifcif 4350 {csn 4441 〈cop 4447 class class class wbr 4929 × cxp 5405 ⟶wf 6184 ‘cfv 6188 (class class class)co 6976 ∈ cmpo 6978 1st c1st 7499 2nd c2nd 7500 ℝcr 10334 0cc0 10335 1c1 10336 + caddc 10338 < clt 10474 / cdiv 11098 ℕcn 11439 2c2 11495 ℕ0cn0 11707 ℤ≥cuz 12058 seqcseq 13184 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2751 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 ax-cnex 10391 ax-resscn 10392 ax-1cn 10393 ax-icn 10394 ax-addcl 10395 ax-addrcl 10396 ax-mulcl 10397 ax-mulrcl 10398 ax-mulcom 10399 ax-addass 10400 ax-mulass 10401 ax-distr 10402 ax-i2m1 10403 ax-1ne0 10404 ax-1rid 10405 ax-rnegex 10406 ax-rrecex 10407 ax-cnre 10408 ax-pre-lttri 10409 ax-pre-lttrn 10410 ax-pre-ltadd 10411 ax-pre-mulgt0 10412 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2760 df-cleq 2772 df-clel 2847 df-nfc 2919 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3418 df-sbc 3683 df-csb 3788 df-dif 3833 df-un 3835 df-in 3837 df-ss 3844 df-pss 3846 df-nul 4180 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-tp 4446 df-op 4448 df-uni 4713 df-iun 4794 df-br 4930 df-opab 4992 df-mpt 5009 df-tr 5031 df-id 5312 df-eprel 5317 df-po 5326 df-so 5327 df-fr 5366 df-we 5368 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-pred 5986 df-ord 6032 df-on 6033 df-lim 6034 df-suc 6035 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-riota 6937 df-ov 6979 df-oprab 6980 df-mpo 6981 df-om 7397 df-2nd 7502 df-wrecs 7750 df-recs 7812 df-rdg 7850 df-er 8089 df-en 8307 df-dom 8308 df-sdom 8309 df-pnf 10476 df-mnf 10477 df-xr 10478 df-ltxr 10479 df-le 10480 df-sub 10672 df-neg 10673 df-nn 11440 df-n0 11708 df-z 11794 df-uz 12059 df-seq 13185 |
This theorem is referenced by: ruclem8 15450 ruclem9 15451 ruclem12 15454 |
Copyright terms: Public domain | W3C validator |