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Mirrors > Home > MPE Home > Th. List > ruclem7 | Structured version Visualization version GIF version |
Description: Lemma for ruc 15584. 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 485 | . . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0) → 𝑁 ∈ ℕ0) | |
2 | nn0uz 12268 | . . . . 5 ⊢ ℕ0 = (ℤ≥‘0) | |
3 | 1, 2 | eleqtrdi 2920 | . . . 4 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0) → 𝑁 ∈ (ℤ≥‘0)) |
4 | seqp1 13372 | . . . 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 6664 | . . 3 ⊢ (𝐺‘(𝑁 + 1)) = (seq0(𝐷, 𝐶)‘(𝑁 + 1)) |
8 | 6 | fveq1i 6664 | . . . 4 ⊢ (𝐺‘𝑁) = (seq0(𝐷, 𝐶)‘𝑁) |
9 | 8 | oveq1i 7155 | . . 3 ⊢ ((𝐺‘𝑁)𝐷(𝐶‘(𝑁 + 1))) = ((seq0(𝐷, 𝐶)‘𝑁)𝐷(𝐶‘(𝑁 + 1))) |
10 | 5, 7, 9 | 3eqtr4g 2878 | . 2 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0) → (𝐺‘(𝑁 + 1)) = ((𝐺‘𝑁)𝐷(𝐶‘(𝑁 + 1)))) |
11 | nn0p1nn 11924 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ) | |
12 | 11 | adantl 482 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0) → (𝑁 + 1) ∈ ℕ) |
13 | 12 | nnne0d 11675 | . . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0) → (𝑁 + 1) ≠ 0) |
14 | 13 | necomd 3068 | . . . 4 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0) → 0 ≠ (𝑁 + 1)) |
15 | ruc.4 | . . . . . . 7 ⊢ 𝐶 = ({〈0, 〈0, 1〉〉} ∪ 𝐹) | |
16 | 15 | equncomi 4128 | . . . . . 6 ⊢ 𝐶 = (𝐹 ∪ {〈0, 〈0, 1〉〉}) |
17 | 16 | fveq1i 6664 | . . . . 5 ⊢ (𝐶‘(𝑁 + 1)) = ((𝐹 ∪ {〈0, 〈0, 1〉〉})‘(𝑁 + 1)) |
18 | fvunsn 6933 | . . . . 5 ⊢ (0 ≠ (𝑁 + 1) → ((𝐹 ∪ {〈0, 〈0, 1〉〉})‘(𝑁 + 1)) = (𝐹‘(𝑁 + 1))) | |
19 | 17, 18 | syl5eq 2865 | . . . 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 7161 | . 2 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0) → ((𝐺‘𝑁)𝐷(𝐶‘(𝑁 + 1))) = ((𝐺‘𝑁)𝐷(𝐹‘(𝑁 + 1)))) |
22 | 10, 21 | eqtrd 2853 | 1 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0) → (𝐺‘(𝑁 + 1)) = ((𝐺‘𝑁)𝐷(𝐹‘(𝑁 + 1)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 ⦋csb 3880 ∪ cun 3931 ifcif 4463 {csn 4557 〈cop 4563 class class class wbr 5057 × cxp 5546 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 ∈ cmpo 7147 1st c1st 7676 2nd c2nd 7677 ℝcr 10524 0cc0 10525 1c1 10526 + caddc 10528 < clt 10663 / cdiv 11285 ℕcn 11626 2c2 11680 ℕ0cn0 11885 ℤ≥cuz 12231 seqcseq 13357 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-n0 11886 df-z 11970 df-uz 12232 df-seq 13358 |
This theorem is referenced by: ruclem8 15578 ruclem9 15579 ruclem12 15582 |
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