![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > ruclem11 | Structured version Visualization version GIF version |
Description: Lemma for ruc 16193. Closure lemmas for supremum. (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 |
---|---|
ruclem11 | ⊢ (𝜑 → (ran (1st ∘ 𝐺) ⊆ ℝ ∧ ran (1st ∘ 𝐺) ≠ ∅ ∧ ∀𝑧 ∈ ran (1st ∘ 𝐺)𝑧 ≤ 1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ruc.1 | . . . . 5 ⊢ (𝜑 → 𝐹:ℕ⟶ℝ) | |
2 | ruc.2 | . . . . 5 ⊢ (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)〉))) | |
3 | ruc.4 | . . . . 5 ⊢ 𝐶 = ({〈0, 〈0, 1〉〉} ∪ 𝐹) | |
4 | ruc.5 | . . . . 5 ⊢ 𝐺 = seq0(𝐷, 𝐶) | |
5 | 1, 2, 3, 4 | ruclem6 16185 | . . . 4 ⊢ (𝜑 → 𝐺:ℕ0⟶(ℝ × ℝ)) |
6 | 1stcof 8009 | . . . 4 ⊢ (𝐺:ℕ0⟶(ℝ × ℝ) → (1st ∘ 𝐺):ℕ0⟶ℝ) | |
7 | 5, 6 | syl 17 | . . 3 ⊢ (𝜑 → (1st ∘ 𝐺):ℕ0⟶ℝ) |
8 | 7 | frnd 6725 | . 2 ⊢ (𝜑 → ran (1st ∘ 𝐺) ⊆ ℝ) |
9 | 7 | fdmd 6728 | . . . 4 ⊢ (𝜑 → dom (1st ∘ 𝐺) = ℕ0) |
10 | 0nn0 12494 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
11 | ne0i 4334 | . . . . 5 ⊢ (0 ∈ ℕ0 → ℕ0 ≠ ∅) | |
12 | 10, 11 | mp1i 13 | . . . 4 ⊢ (𝜑 → ℕ0 ≠ ∅) |
13 | 9, 12 | eqnetrd 3007 | . . 3 ⊢ (𝜑 → dom (1st ∘ 𝐺) ≠ ∅) |
14 | dm0rn0 5924 | . . . 4 ⊢ (dom (1st ∘ 𝐺) = ∅ ↔ ran (1st ∘ 𝐺) = ∅) | |
15 | 14 | necon3bii 2992 | . . 3 ⊢ (dom (1st ∘ 𝐺) ≠ ∅ ↔ ran (1st ∘ 𝐺) ≠ ∅) |
16 | 13, 15 | sylib 217 | . 2 ⊢ (𝜑 → ran (1st ∘ 𝐺) ≠ ∅) |
17 | fvco3 6990 | . . . . . 6 ⊢ ((𝐺:ℕ0⟶(ℝ × ℝ) ∧ 𝑛 ∈ ℕ0) → ((1st ∘ 𝐺)‘𝑛) = (1st ‘(𝐺‘𝑛))) | |
18 | 5, 17 | sylan 579 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → ((1st ∘ 𝐺)‘𝑛) = (1st ‘(𝐺‘𝑛))) |
19 | 1 | adantr 480 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → 𝐹:ℕ⟶ℝ) |
20 | 2 | adantr 480 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → 𝐷 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ ⦋(((1st ‘𝑥) + (2nd ‘𝑥)) / 2) / 𝑚⦌if(𝑚 < 𝑦, 〈(1st ‘𝑥), 𝑚〉, 〈((𝑚 + (2nd ‘𝑥)) / 2), (2nd ‘𝑥)〉))) |
21 | simpr 484 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈ ℕ0) | |
22 | 10 | a1i 11 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → 0 ∈ ℕ0) |
23 | 19, 20, 3, 4, 21, 22 | ruclem10 16189 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (1st ‘(𝐺‘𝑛)) < (2nd ‘(𝐺‘0))) |
24 | 1, 2, 3, 4 | ruclem4 16184 | . . . . . . . . . 10 ⊢ (𝜑 → (𝐺‘0) = 〈0, 1〉) |
25 | 24 | fveq2d 6895 | . . . . . . . . 9 ⊢ (𝜑 → (2nd ‘(𝐺‘0)) = (2nd ‘〈0, 1〉)) |
26 | c0ex 11215 | . . . . . . . . . 10 ⊢ 0 ∈ V | |
27 | 1ex 11217 | . . . . . . . . . 10 ⊢ 1 ∈ V | |
28 | 26, 27 | op2nd 7988 | . . . . . . . . 9 ⊢ (2nd ‘〈0, 1〉) = 1 |
29 | 25, 28 | eqtrdi 2787 | . . . . . . . 8 ⊢ (𝜑 → (2nd ‘(𝐺‘0)) = 1) |
30 | 29 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (2nd ‘(𝐺‘0)) = 1) |
31 | 23, 30 | breqtrd 5174 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (1st ‘(𝐺‘𝑛)) < 1) |
32 | 5 | ffvelcdmda 7086 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (𝐺‘𝑛) ∈ (ℝ × ℝ)) |
33 | xp1st 8011 | . . . . . . . 8 ⊢ ((𝐺‘𝑛) ∈ (ℝ × ℝ) → (1st ‘(𝐺‘𝑛)) ∈ ℝ) | |
34 | 32, 33 | syl 17 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (1st ‘(𝐺‘𝑛)) ∈ ℝ) |
35 | 1re 11221 | . . . . . . 7 ⊢ 1 ∈ ℝ | |
36 | ltle 11309 | . . . . . . 7 ⊢ (((1st ‘(𝐺‘𝑛)) ∈ ℝ ∧ 1 ∈ ℝ) → ((1st ‘(𝐺‘𝑛)) < 1 → (1st ‘(𝐺‘𝑛)) ≤ 1)) | |
37 | 34, 35, 36 | sylancl 585 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → ((1st ‘(𝐺‘𝑛)) < 1 → (1st ‘(𝐺‘𝑛)) ≤ 1)) |
38 | 31, 37 | mpd 15 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (1st ‘(𝐺‘𝑛)) ≤ 1) |
39 | 18, 38 | eqbrtrd 5170 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → ((1st ∘ 𝐺)‘𝑛) ≤ 1) |
40 | 39 | ralrimiva 3145 | . . 3 ⊢ (𝜑 → ∀𝑛 ∈ ℕ0 ((1st ∘ 𝐺)‘𝑛) ≤ 1) |
41 | 7 | ffnd 6718 | . . . 4 ⊢ (𝜑 → (1st ∘ 𝐺) Fn ℕ0) |
42 | breq1 5151 | . . . . 5 ⊢ (𝑧 = ((1st ∘ 𝐺)‘𝑛) → (𝑧 ≤ 1 ↔ ((1st ∘ 𝐺)‘𝑛) ≤ 1)) | |
43 | 42 | ralrn 7089 | . . . 4 ⊢ ((1st ∘ 𝐺) Fn ℕ0 → (∀𝑧 ∈ ran (1st ∘ 𝐺)𝑧 ≤ 1 ↔ ∀𝑛 ∈ ℕ0 ((1st ∘ 𝐺)‘𝑛) ≤ 1)) |
44 | 41, 43 | syl 17 | . . 3 ⊢ (𝜑 → (∀𝑧 ∈ ran (1st ∘ 𝐺)𝑧 ≤ 1 ↔ ∀𝑛 ∈ ℕ0 ((1st ∘ 𝐺)‘𝑛) ≤ 1)) |
45 | 40, 44 | mpbird 257 | . 2 ⊢ (𝜑 → ∀𝑧 ∈ ran (1st ∘ 𝐺)𝑧 ≤ 1) |
46 | 8, 16, 45 | 3jca 1127 | 1 ⊢ (𝜑 → (ran (1st ∘ 𝐺) ⊆ ℝ ∧ ran (1st ∘ 𝐺) ≠ ∅ ∧ ∀𝑧 ∈ ran (1st ∘ 𝐺)𝑧 ≤ 1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 ≠ wne 2939 ∀wral 3060 ⦋csb 3893 ∪ cun 3946 ⊆ wss 3948 ∅c0 4322 ifcif 4528 {csn 4628 〈cop 4634 class class class wbr 5148 × cxp 5674 dom cdm 5676 ran crn 5677 ∘ ccom 5680 Fn wfn 6538 ⟶wf 6539 ‘cfv 6543 (class class class)co 7412 ∈ cmpo 7414 1st c1st 7977 2nd c2nd 7978 ℝcr 11115 0cc0 11116 1c1 11117 + caddc 11119 < clt 11255 ≤ cle 11256 / cdiv 11878 ℕcn 12219 2c2 12274 ℕ0cn0 12479 seqcseq 13973 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-er 8709 df-en 8946 df-dom 8947 df-sdom 8948 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-div 11879 df-nn 12220 df-2 12282 df-n0 12480 df-z 12566 df-uz 12830 df-fz 13492 df-seq 13974 |
This theorem is referenced by: ruclem12 16191 |
Copyright terms: Public domain | W3C validator |