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Mirrors > Home > MPE Home > Th. List > ruclem7 | Structured version Visualization version GIF version |
Description: Lemma for ruc 16275. 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 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0) → 𝑁 ∈ ℕ0) | |
2 | nn0uz 12917 | . . . . 5 ⊢ ℕ0 = (ℤ≥‘0) | |
3 | 1, 2 | eleqtrdi 2848 | . . . 4 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0) → 𝑁 ∈ (ℤ≥‘0)) |
4 | seqp1 14053 | . . . 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 6907 | . . 3 ⊢ (𝐺‘(𝑁 + 1)) = (seq0(𝐷, 𝐶)‘(𝑁 + 1)) |
8 | 6 | fveq1i 6907 | . . . 4 ⊢ (𝐺‘𝑁) = (seq0(𝐷, 𝐶)‘𝑁) |
9 | 8 | oveq1i 7440 | . . 3 ⊢ ((𝐺‘𝑁)𝐷(𝐶‘(𝑁 + 1))) = ((seq0(𝐷, 𝐶)‘𝑁)𝐷(𝐶‘(𝑁 + 1))) |
10 | 5, 7, 9 | 3eqtr4g 2799 | . 2 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0) → (𝐺‘(𝑁 + 1)) = ((𝐺‘𝑁)𝐷(𝐶‘(𝑁 + 1)))) |
11 | nn0p1nn 12562 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ) | |
12 | 11 | adantl 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0) → (𝑁 + 1) ∈ ℕ) |
13 | 12 | nnne0d 12313 | . . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0) → (𝑁 + 1) ≠ 0) |
14 | 13 | necomd 2993 | . . . 4 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0) → 0 ≠ (𝑁 + 1)) |
15 | ruc.4 | . . . . . . 7 ⊢ 𝐶 = ({〈0, 〈0, 1〉〉} ∪ 𝐹) | |
16 | 15 | equncomi 4169 | . . . . . 6 ⊢ 𝐶 = (𝐹 ∪ {〈0, 〈0, 1〉〉}) |
17 | 16 | fveq1i 6907 | . . . . 5 ⊢ (𝐶‘(𝑁 + 1)) = ((𝐹 ∪ {〈0, 〈0, 1〉〉})‘(𝑁 + 1)) |
18 | fvunsn 7198 | . . . . 5 ⊢ (0 ≠ (𝑁 + 1) → ((𝐹 ∪ {〈0, 〈0, 1〉〉})‘(𝑁 + 1)) = (𝐹‘(𝑁 + 1))) | |
19 | 17, 18 | eqtrid 2786 | . . . 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 7446 | . 2 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0) → ((𝐺‘𝑁)𝐷(𝐶‘(𝑁 + 1))) = ((𝐺‘𝑁)𝐷(𝐹‘(𝑁 + 1)))) |
22 | 10, 21 | eqtrd 2774 | 1 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0) → (𝐺‘(𝑁 + 1)) = ((𝐺‘𝑁)𝐷(𝐹‘(𝑁 + 1)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1536 ∈ wcel 2105 ≠ wne 2937 ⦋csb 3907 ∪ cun 3960 ifcif 4530 {csn 4630 〈cop 4636 class class class wbr 5147 × cxp 5686 ⟶wf 6558 ‘cfv 6562 (class class class)co 7430 ∈ cmpo 7432 1st c1st 8010 2nd c2nd 8011 ℝcr 11151 0cc0 11152 1c1 11153 + caddc 11155 < clt 11292 / cdiv 11917 ℕcn 12263 2c2 12318 ℕ0cn0 12523 ℤ≥cuz 12875 seqcseq 14038 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-nn 12264 df-n0 12524 df-z 12611 df-uz 12876 df-seq 14039 |
This theorem is referenced by: ruclem8 16269 ruclem9 16270 ruclem12 16273 |
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