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| Description: Lemma for ruc 16280. 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 16272 | . . . 4 ⊢ (𝜑 → 𝐺:ℕ0⟶(ℝ × ℝ)) | 
| 6 | 1stcof 8045 | . . . 4 ⊢ (𝐺:ℕ0⟶(ℝ × ℝ) → (1st ∘ 𝐺):ℕ0⟶ℝ) | |
| 7 | 5, 6 | syl 17 | . . 3 ⊢ (𝜑 → (1st ∘ 𝐺):ℕ0⟶ℝ) | 
| 8 | 7 | frnd 6743 | . 2 ⊢ (𝜑 → ran (1st ∘ 𝐺) ⊆ ℝ) | 
| 9 | 7 | fdmd 6745 | . . . 4 ⊢ (𝜑 → dom (1st ∘ 𝐺) = ℕ0) | 
| 10 | 0nn0 12543 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
| 11 | ne0i 4340 | . . . . 5 ⊢ (0 ∈ ℕ0 → ℕ0 ≠ ∅) | |
| 12 | 10, 11 | mp1i 13 | . . . 4 ⊢ (𝜑 → ℕ0 ≠ ∅) | 
| 13 | 9, 12 | eqnetrd 3007 | . . 3 ⊢ (𝜑 → dom (1st ∘ 𝐺) ≠ ∅) | 
| 14 | dm0rn0 5934 | . . . 4 ⊢ (dom (1st ∘ 𝐺) = ∅ ↔ ran (1st ∘ 𝐺) = ∅) | |
| 15 | 14 | necon3bii 2992 | . . 3 ⊢ (dom (1st ∘ 𝐺) ≠ ∅ ↔ ran (1st ∘ 𝐺) ≠ ∅) | 
| 16 | 13, 15 | sylib 218 | . 2 ⊢ (𝜑 → ran (1st ∘ 𝐺) ≠ ∅) | 
| 17 | fvco3 7007 | . . . . . 6 ⊢ ((𝐺:ℕ0⟶(ℝ × ℝ) ∧ 𝑛 ∈ ℕ0) → ((1st ∘ 𝐺)‘𝑛) = (1st ‘(𝐺‘𝑛))) | |
| 18 | 5, 17 | sylan 580 | . . . . 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 16276 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (1st ‘(𝐺‘𝑛)) < (2nd ‘(𝐺‘0))) | 
| 24 | 1, 2, 3, 4 | ruclem4 16271 | . . . . . . . . . 10 ⊢ (𝜑 → (𝐺‘0) = 〈0, 1〉) | 
| 25 | 24 | fveq2d 6909 | . . . . . . . . 9 ⊢ (𝜑 → (2nd ‘(𝐺‘0)) = (2nd ‘〈0, 1〉)) | 
| 26 | c0ex 11256 | . . . . . . . . . 10 ⊢ 0 ∈ V | |
| 27 | 1ex 11258 | . . . . . . . . . 10 ⊢ 1 ∈ V | |
| 28 | 26, 27 | op2nd 8024 | . . . . . . . . 9 ⊢ (2nd ‘〈0, 1〉) = 1 | 
| 29 | 25, 28 | eqtrdi 2792 | . . . . . . . 8 ⊢ (𝜑 → (2nd ‘(𝐺‘0)) = 1) | 
| 30 | 29 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (2nd ‘(𝐺‘0)) = 1) | 
| 31 | 23, 30 | breqtrd 5168 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (1st ‘(𝐺‘𝑛)) < 1) | 
| 32 | 5 | ffvelcdmda 7103 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (𝐺‘𝑛) ∈ (ℝ × ℝ)) | 
| 33 | xp1st 8047 | . . . . . . . 8 ⊢ ((𝐺‘𝑛) ∈ (ℝ × ℝ) → (1st ‘(𝐺‘𝑛)) ∈ ℝ) | |
| 34 | 32, 33 | syl 17 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (1st ‘(𝐺‘𝑛)) ∈ ℝ) | 
| 35 | 1re 11262 | . . . . . . 7 ⊢ 1 ∈ ℝ | |
| 36 | ltle 11350 | . . . . . . 7 ⊢ (((1st ‘(𝐺‘𝑛)) ∈ ℝ ∧ 1 ∈ ℝ) → ((1st ‘(𝐺‘𝑛)) < 1 → (1st ‘(𝐺‘𝑛)) ≤ 1)) | |
| 37 | 34, 35, 36 | sylancl 586 | . . . . . 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 5164 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → ((1st ∘ 𝐺)‘𝑛) ≤ 1) | 
| 40 | 39 | ralrimiva 3145 | . . 3 ⊢ (𝜑 → ∀𝑛 ∈ ℕ0 ((1st ∘ 𝐺)‘𝑛) ≤ 1) | 
| 41 | 7 | ffnd 6736 | . . . 4 ⊢ (𝜑 → (1st ∘ 𝐺) Fn ℕ0) | 
| 42 | breq1 5145 | . . . . 5 ⊢ (𝑧 = ((1st ∘ 𝐺)‘𝑛) → (𝑧 ≤ 1 ↔ ((1st ∘ 𝐺)‘𝑛) ≤ 1)) | |
| 43 | 42 | ralrn 7107 | . . . 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 1128 | 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 206 ∧ wa 395 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 ≠ wne 2939 ∀wral 3060 ⦋csb 3898 ∪ cun 3948 ⊆ wss 3950 ∅c0 4332 ifcif 4524 {csn 4625 〈cop 4631 class class class wbr 5142 × cxp 5682 dom cdm 5684 ran crn 5685 ∘ ccom 5688 Fn wfn 6555 ⟶wf 6556 ‘cfv 6560 (class class class)co 7432 ∈ cmpo 7434 1st c1st 8013 2nd c2nd 8014 ℝcr 11155 0cc0 11156 1c1 11157 + caddc 11159 < clt 11296 ≤ cle 11297 / cdiv 11921 ℕcn 12267 2c2 12322 ℕ0cn0 12528 seqcseq 14043 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-div 11922 df-nn 12268 df-2 12330 df-n0 12529 df-z 12616 df-uz 12880 df-fz 13549 df-seq 14044 | 
| This theorem is referenced by: ruclem12 16278 | 
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