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Mirrors > Home > MPE Home > Th. List > ruclem7 | Structured version Visualization version GIF version |
Description: Lemma for ruc 16291. 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 12945 | . . . . 5 ⊢ ℕ0 = (ℤ≥‘0) | |
3 | 1, 2 | eleqtrdi 2854 | . . . 4 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0) → 𝑁 ∈ (ℤ≥‘0)) |
4 | seqp1 14067 | . . . 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 6921 | . . 3 ⊢ (𝐺‘(𝑁 + 1)) = (seq0(𝐷, 𝐶)‘(𝑁 + 1)) |
8 | 6 | fveq1i 6921 | . . . 4 ⊢ (𝐺‘𝑁) = (seq0(𝐷, 𝐶)‘𝑁) |
9 | 8 | oveq1i 7458 | . . 3 ⊢ ((𝐺‘𝑁)𝐷(𝐶‘(𝑁 + 1))) = ((seq0(𝐷, 𝐶)‘𝑁)𝐷(𝐶‘(𝑁 + 1))) |
10 | 5, 7, 9 | 3eqtr4g 2805 | . 2 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0) → (𝐺‘(𝑁 + 1)) = ((𝐺‘𝑁)𝐷(𝐶‘(𝑁 + 1)))) |
11 | nn0p1nn 12592 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ) | |
12 | 11 | adantl 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0) → (𝑁 + 1) ∈ ℕ) |
13 | 12 | nnne0d 12343 | . . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0) → (𝑁 + 1) ≠ 0) |
14 | 13 | necomd 3002 | . . . 4 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0) → 0 ≠ (𝑁 + 1)) |
15 | ruc.4 | . . . . . . 7 ⊢ 𝐶 = ({〈0, 〈0, 1〉〉} ∪ 𝐹) | |
16 | 15 | equncomi 4183 | . . . . . 6 ⊢ 𝐶 = (𝐹 ∪ {〈0, 〈0, 1〉〉}) |
17 | 16 | fveq1i 6921 | . . . . 5 ⊢ (𝐶‘(𝑁 + 1)) = ((𝐹 ∪ {〈0, 〈0, 1〉〉})‘(𝑁 + 1)) |
18 | fvunsn 7213 | . . . . 5 ⊢ (0 ≠ (𝑁 + 1) → ((𝐹 ∪ {〈0, 〈0, 1〉〉})‘(𝑁 + 1)) = (𝐹‘(𝑁 + 1))) | |
19 | 17, 18 | eqtrid 2792 | . . . 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 7464 | . 2 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0) → ((𝐺‘𝑁)𝐷(𝐶‘(𝑁 + 1))) = ((𝐺‘𝑁)𝐷(𝐹‘(𝑁 + 1)))) |
22 | 10, 21 | eqtrd 2780 | 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 1537 ∈ wcel 2108 ≠ wne 2946 ⦋csb 3921 ∪ cun 3974 ifcif 4548 {csn 4648 〈cop 4654 class class class wbr 5166 × cxp 5698 ⟶wf 6569 ‘cfv 6573 (class class class)co 7448 ∈ cmpo 7450 1st c1st 8028 2nd c2nd 8029 ℝcr 11183 0cc0 11184 1c1 11185 + caddc 11187 < clt 11324 / cdiv 11947 ℕcn 12293 2c2 12348 ℕ0cn0 12553 ℤ≥cuz 12903 seqcseq 14052 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-er 8763 df-en 9004 df-dom 9005 df-sdom 9006 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-nn 12294 df-n0 12554 df-z 12640 df-uz 12904 df-seq 14053 |
This theorem is referenced by: ruclem8 16285 ruclem9 16286 ruclem12 16289 |
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