ruclem11 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > ruclem11		Structured version Visualization version GIF version

Theorem ruclem11 16296

Description: Lemma for ruc 16299. Closure lemmas for supremum. (Contributed by Mario Carneiro, 28-May-2014.)

**Hypotheses**
Ref	Expression
ruc.1	⊢ (𝜑 → 𝐹:ℕ⟶ℝ)
ruc.2	⊢ (𝜑 → 𝐷 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ ⦋(((1^st ‘𝑥) + (2^nd ‘𝑥)) / 2) / 𝑚⦌if(𝑚 < 𝑦, ⟨(1^st ‘𝑥), 𝑚⟩, ⟨((𝑚 + (2^nd ‘𝑥)) / 2), (2^nd ‘𝑥)⟩)))
ruc.4	⊢ 𝐶 = ({⟨0, ⟨0, 1⟩⟩} ∪ 𝐹)
ruc.5	⊢ 𝐺 = seq0(𝐷, 𝐶)

**Assertion**
Ref	Expression
ruclem11	⊢ (𝜑 → (ran (1^st ∘ 𝐺) ⊆ ℝ ∧ ran (1^st ∘ 𝐺) ≠ ∅ ∧ ∀𝑧 ∈ ran (1^st ∘ 𝐺)𝑧 ≤ 1))

Distinct variable groups: 𝑥,𝑚,𝑦 𝑧,𝐶 𝑧,𝑚,𝐹,𝑥,𝑦 𝑚,𝐺,𝑥,𝑦,𝑧 𝜑,𝑧 𝑧,𝐷 Allowed substitution hints: 𝜑(𝑥,𝑦,𝑚) 𝐶(𝑥,𝑦,𝑚) 𝐷(𝑥,𝑦,𝑚)

Proof of Theorem ruclem11 Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ruc.1	. . . . 5 ⊢ (𝜑 → 𝐹:ℕ⟶ℝ)
2		ruc.2	. . . . 5 ⊢ (𝜑 → 𝐷 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ ⦋(((1^st ‘𝑥) + (2^nd ‘𝑥)) / 2) / 𝑚⦌if(𝑚 < 𝑦, ⟨(1^st ‘𝑥), 𝑚⟩, ⟨((𝑚 + (2^nd ‘𝑥)) / 2), (2^nd ‘𝑥)⟩)))
3		ruc.4	. . . . 5 ⊢ 𝐶 = ({⟨0, ⟨0, 1⟩⟩} ∪ 𝐹)
4		ruc.5	. . . . 5 ⊢ 𝐺 = seq0(𝐷, 𝐶)
5	1, 2, 3, 4	ruclem6 16291	. . . 4 ⊢ (𝜑 → 𝐺:ℕ₀⟶(ℝ × ℝ))
6		1stcof 8016	. . . 4 ⊢ (𝐺:ℕ₀⟶(ℝ × ℝ) → (1^st ∘ 𝐺):ℕ₀⟶ℝ)
7	5, 6	syl 18	. . 3 ⊢ (𝜑 → (1^st ∘ 𝐺):ℕ₀⟶ℝ)
8	7	frnd 6715	. 2 ⊢ (𝜑 → ran (1^st ∘ 𝐺) ⊆ ℝ)
9	7	fdmd 6717	. . . 4 ⊢ (𝜑 → dom (1^st ∘ 𝐺) = ℕ₀)
10		0nn0 12519	. . . . 5 ⊢ 0 ∈ ℕ₀
11		ne0i 4300	. . . . 5 ⊢ (0 ∈ ℕ₀ → ℕ₀ ≠ ∅)
12	10, 11	mp1i 14	. . . 4 ⊢ (𝜑 → ℕ₀ ≠ ∅)
13	9, 12	eqnetrd 3031	. . 3 ⊢ (𝜑 → dom (1^st ∘ 𝐺) ≠ ∅)
14		dm0rn0 5915	. . . 4 ⊢ (dom (1^st ∘ 𝐺) = ∅ ↔ ran (1^st ∘ 𝐺) = ∅)
15	14	necon3bii 3016	. . 3 ⊢ (dom (1^st ∘ 𝐺) ≠ ∅ ↔ ran (1^st ∘ 𝐺) ≠ ∅)
16	13, 15	sylib 221	. 2 ⊢ (𝜑 → ran (1^st ∘ 𝐺) ≠ ∅)
17		fvco3 6982	. . . . . 6 ⊢ ((𝐺:ℕ₀⟶(ℝ × ℝ) ∧ 𝑛 ∈ ℕ₀) → ((1^st ∘ 𝐺)‘𝑛) = (1^st ‘(𝐺‘𝑛)))
18	5, 17	sylan 591	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → ((1^st ∘ 𝐺)‘𝑛) = (1^st ‘(𝐺‘𝑛)))
19	1	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → 𝐹:ℕ⟶ℝ)
20	2	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → 𝐷 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ ⦋(((1^st ‘𝑥) + (2^nd ‘𝑥)) / 2) / 𝑚⦌if(𝑚 < 𝑦, ⟨(1^st ‘𝑥), 𝑚⟩, ⟨((𝑚 + (2^nd ‘𝑥)) / 2), (2^nd ‘𝑥)⟩)))
21		simpr 489	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → 𝑛 ∈ ℕ₀)
22	10	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → 0 ∈ ℕ₀)
23	19, 20, 3, 4, 21, 22	ruclem10 16295	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → (1^st ‘(𝐺‘𝑛)) < (2^nd ‘(𝐺‘0)))
24	1, 2, 3, 4	ruclem4 16290	. . . . . . . . . 10 ⊢ (𝜑 → (𝐺‘0) = ⟨0, 1⟩)
25	24	fveq2d 6886	. . . . . . . . 9 ⊢ (𝜑 → (2^nd ‘(𝐺‘0)) = (2^nd ‘⟨0, 1⟩))
26		c0ex 11200	. . . . . . . . . 10 ⊢ 0 ∈ V
27		1ex 11203	. . . . . . . . . 10 ⊢ 1 ∈ V
28	26, 27	op2nd 7995	. . . . . . . . 9 ⊢ (2^nd ‘⟨0, 1⟩) = 1
29	25, 28	eqtrdi 2820	. . . . . . . 8 ⊢ (𝜑 → (2^nd ‘(𝐺‘0)) = 1)
30	29	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → (2^nd ‘(𝐺‘0)) = 1)
31	23, 30	breqtrd 5139	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → (1^st ‘(𝐺‘𝑛)) < 1)
32	5	ffvelcdmda 7080	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → (𝐺‘𝑛) ∈ (ℝ × ℝ))
33		xp1st 8018	. . . . . . . 8 ⊢ ((𝐺‘𝑛) ∈ (ℝ × ℝ) → (1^st ‘(𝐺‘𝑛)) ∈ ℝ)
34	32, 33	syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → (1^st ‘(𝐺‘𝑛)) ∈ ℝ)
35		1re 11208	. . . . . . 7 ⊢ 1 ∈ ℝ
36		ltle 11298	. . . . . . 7 ⊢ (((1^st ‘(𝐺‘𝑛)) ∈ ℝ ∧ 1 ∈ ℝ) → ((1^st ‘(𝐺‘𝑛)) < 1 → (1^st ‘(𝐺‘𝑛)) ≤ 1))
37	34, 35, 36	sylancl 597	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → ((1^st ‘(𝐺‘𝑛)) < 1 → (1^st ‘(𝐺‘𝑛)) ≤ 1))
38	31, 37	mpd 16	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → (1^st ‘(𝐺‘𝑛)) ≤ 1)
39	18, 38	eqbrtrd 5135	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → ((1^st ∘ 𝐺)‘𝑛) ≤ 1)
40	39	ralrimiva 3163	. . 3 ⊢ (𝜑 → ∀𝑛 ∈ ℕ₀ ((1^st ∘ 𝐺)‘𝑛) ≤ 1)
41	7	ffnd 6707	. . . 4 ⊢ (𝜑 → (1^st ∘ 𝐺) Fn ℕ₀)
42		breq1 5114	. . . . 5 ⊢ (𝑧 = ((1^st ∘ 𝐺)‘𝑛) → (𝑧 ≤ 1 ↔ ((1^st ∘ 𝐺)‘𝑛) ≤ 1))
43	42	ralrn 7084	. . . 4 ⊢ ((1^st ∘ 𝐺) Fn ℕ₀ → (∀𝑧 ∈ ran (1^st ∘ 𝐺)𝑧 ≤ 1 ↔ ∀𝑛 ∈ ℕ₀ ((1^st ∘ 𝐺)‘𝑛) ≤ 1))
44	41, 43	syl 18	. . 3 ⊢ (𝜑 → (∀𝑧 ∈ ran (1^st ∘ 𝐺)𝑧 ≤ 1 ↔ ∀𝑛 ∈ ℕ₀ ((1^st ∘ 𝐺)‘𝑛) ≤ 1))
45	40, 44	mpbird 260	. 2 ⊢ (𝜑 → ∀𝑧 ∈ ran (1^st ∘ 𝐺)𝑧 ≤ 1)
46	8, 16, 45	3jca 1144	1 ⊢ (𝜑 → (ran (1^st ∘ 𝐺) ⊆ ℝ ∧ ran (1^st ∘ 𝐺) ≠ ∅ ∧ ∀𝑧 ∈ ran (1^st ∘ 𝐺)𝑧 ≤ 1))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 ∧ w3a 1101 = wceq 1567 ∈ wcel 2149 ≠ wne 2964 ∀wral 3085 ⦋csb 3859 ∪ cun 3909 ⊆ wss 3911 ∅c0 4292 ifcif 4490 {csn 4592 ⟨cop 4598 class class class wbr 5111 × cxp 5660 dom cdm 5662 ran crn 5663 ∘ ccom 5666 Fn wfn 6532 ⟶wf 6533 ‘cfv 6537 (class class class)co 7411 ∈ cmpo 7413 1^st c1st 7984 2^nd c2nd 7985 ℝcr 11099 0cc0 11100 1c1 11101 + caddc 11103 < clt 11243 ≤ cle 11244 / cdiv 11871 ℕcn 12233 2c2 12295 ℕ₀cn0 12504 seqcseq 14037

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-sep 5259 ax-nul 5271 ax-pow 5337 ax-pr 5405 ax-un 7733 ax-cnex 11156 ax-resscn 11157 ax-1cn 11158 ax-icn 11159 ax-addcl 11160 ax-addrcl 11161 ax-mulcl 11162 ax-mulrcl 11163 ax-mulcom 11164 ax-addass 11165 ax-mulass 11166 ax-distr 11167 ax-i2m1 11168 ax-1ne0 11169 ax-1rid 11170 ax-rnegex 11171 ax-rrecex 11172 ax-cnre 11173 ax-pre-lttri 11174 ax-pre-lttrn 11175 ax-pre-ltadd 11176 ax-pre-mulgt0 11177

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3375 df-reu 3376 df-rab 3423 df-v 3463 df-sbc 3752 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4491 df-pw 4567 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-iun 4960 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5557 df-eprel 5562 df-po 5570 df-so 5571 df-fr 5615 df-we 5617 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-res 5674 df-ima 5675 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7368 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7863 df-1st 7986 df-2nd 7987 df-frecs 8278 df-wrecs 8309 df-recs 8358 df-rdg 8397 df-er 8694 df-en 8944 df-dom 8945 df-sdom 8946 df-pnf 11245 df-mnf 11246 df-xr 11247 df-ltxr 11248 df-le 11249 df-sub 11443 df-neg 11444 df-div 11872 df-nn 12234 df-2 12303 df-n0 12505 df-z 12592 df-uz 12863 df-fz 13536 df-seq 14038

This theorem is referenced by: ruclem12 16297

W3C validator