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| Mirrors > Home > MPE Home > Th. List > ruclem4 | Structured version Visualization version GIF version | ||
| Description: Lemma for ruc 16279. Initial value of 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 |
|---|---|
| ruclem4 | ⊢ (𝜑 → (𝐺‘0) = 〈0, 1〉) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ruc.5 | . . 3 ⊢ 𝐺 = seq0(𝐷, 𝐶) | |
| 2 | 1 | fveq1i 6907 | . 2 ⊢ (𝐺‘0) = (seq0(𝐷, 𝐶)‘0) |
| 3 | 0z 12624 | . . 3 ⊢ 0 ∈ ℤ | |
| 4 | ruc.4 | . . . . . 6 ⊢ 𝐶 = ({〈0, 〈0, 1〉〉} ∪ 𝐹) | |
| 5 | ruc.1 | . . . . . . . . 9 ⊢ (𝜑 → 𝐹:ℕ⟶ℝ) | |
| 6 | ffn 6736 | . . . . . . . . 9 ⊢ (𝐹:ℕ⟶ℝ → 𝐹 Fn ℕ) | |
| 7 | fnresdm 6687 | . . . . . . . . 9 ⊢ (𝐹 Fn ℕ → (𝐹 ↾ ℕ) = 𝐹) | |
| 8 | 5, 6, 7 | 3syl 18 | . . . . . . . 8 ⊢ (𝜑 → (𝐹 ↾ ℕ) = 𝐹) |
| 9 | dfn2 12539 | . . . . . . . . 9 ⊢ ℕ = (ℕ0 ∖ {0}) | |
| 10 | 9 | reseq2i 5994 | . . . . . . . 8 ⊢ (𝐹 ↾ ℕ) = (𝐹 ↾ (ℕ0 ∖ {0})) |
| 11 | 8, 10 | eqtr3di 2792 | . . . . . . 7 ⊢ (𝜑 → 𝐹 = (𝐹 ↾ (ℕ0 ∖ {0}))) |
| 12 | 11 | uneq2d 4168 | . . . . . 6 ⊢ (𝜑 → ({〈0, 〈0, 1〉〉} ∪ 𝐹) = ({〈0, 〈0, 1〉〉} ∪ (𝐹 ↾ (ℕ0 ∖ {0})))) |
| 13 | 4, 12 | eqtrid 2789 | . . . . 5 ⊢ (𝜑 → 𝐶 = ({〈0, 〈0, 1〉〉} ∪ (𝐹 ↾ (ℕ0 ∖ {0})))) |
| 14 | 13 | fveq1d 6908 | . . . 4 ⊢ (𝜑 → (𝐶‘0) = (({〈0, 〈0, 1〉〉} ∪ (𝐹 ↾ (ℕ0 ∖ {0})))‘0)) |
| 15 | c0ex 11255 | . . . . . . 7 ⊢ 0 ∈ V | |
| 16 | 15 | a1i 11 | . . . . . 6 ⊢ (⊤ → 0 ∈ V) |
| 17 | opex 5469 | . . . . . . 7 ⊢ 〈0, 1〉 ∈ V | |
| 18 | 17 | a1i 11 | . . . . . 6 ⊢ (⊤ → 〈0, 1〉 ∈ V) |
| 19 | eqid 2737 | . . . . . 6 ⊢ ({〈0, 〈0, 1〉〉} ∪ (𝐹 ↾ (ℕ0 ∖ {0}))) = ({〈0, 〈0, 1〉〉} ∪ (𝐹 ↾ (ℕ0 ∖ {0}))) | |
| 20 | 16, 18, 19 | fvsnun1 7202 | . . . . 5 ⊢ (⊤ → (({〈0, 〈0, 1〉〉} ∪ (𝐹 ↾ (ℕ0 ∖ {0})))‘0) = 〈0, 1〉) |
| 21 | 20 | mptru 1547 | . . . 4 ⊢ (({〈0, 〈0, 1〉〉} ∪ (𝐹 ↾ (ℕ0 ∖ {0})))‘0) = 〈0, 1〉 |
| 22 | 14, 21 | eqtrdi 2793 | . . 3 ⊢ (𝜑 → (𝐶‘0) = 〈0, 1〉) |
| 23 | 3, 22 | seq1i 14056 | . 2 ⊢ (𝜑 → (seq0(𝐷, 𝐶)‘0) = 〈0, 1〉) |
| 24 | 2, 23 | eqtrid 2789 | 1 ⊢ (𝜑 → (𝐺‘0) = 〈0, 1〉) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ⊤wtru 1541 ∈ wcel 2108 Vcvv 3480 ⦋csb 3899 ∖ cdif 3948 ∪ cun 3949 ifcif 4525 {csn 4626 〈cop 4632 class class class wbr 5143 × cxp 5683 ↾ cres 5687 Fn wfn 6556 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ∈ cmpo 7433 1st c1st 8012 2nd c2nd 8013 ℝcr 11154 0cc0 11155 1c1 11156 + caddc 11158 < clt 11295 / cdiv 11920 ℕcn 12266 2c2 12321 ℕ0cn0 12526 seqcseq 14042 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-n0 12527 df-z 12614 df-uz 12879 df-seq 14043 |
| This theorem is referenced by: ruclem6 16271 ruclem8 16273 ruclem11 16276 |
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