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Theorem ruclem11 15431

Description: Lemma for ruc 15434. 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 15426	. . . 4 ⊢ (𝜑 → 𝐺:ℕ₀⟶(ℝ × ℝ))
6		1stcof 7580	. . . 4 ⊢ (𝐺:ℕ₀⟶(ℝ × ℝ) → (1^st ∘ 𝐺):ℕ₀⟶ℝ)
7	5, 6	syl 17	. . 3 ⊢ (𝜑 → (1^st ∘ 𝐺):ℕ₀⟶ℝ)
8	7	frnd 6394	. 2 ⊢ (𝜑 → ran (1^st ∘ 𝐺) ⊆ ℝ)
9	7	fdmd 6396	. . . 4 ⊢ (𝜑 → dom (1^st ∘ 𝐺) = ℕ₀)
10		0nn0 11765	. . . . 5 ⊢ 0 ∈ ℕ₀
11		ne0i 4224	. . . . 5 ⊢ (0 ∈ ℕ₀ → ℕ₀ ≠ ∅)
12	10, 11	mp1i 13	. . . 4 ⊢ (𝜑 → ℕ₀ ≠ ∅)
13	9, 12	eqnetrd 3051	. . 3 ⊢ (𝜑 → dom (1^st ∘ 𝐺) ≠ ∅)
14		dm0rn0 5684	. . . 4 ⊢ (dom (1^st ∘ 𝐺) = ∅ ↔ ran (1^st ∘ 𝐺) = ∅)
15	14	necon3bii 3036	. . 3 ⊢ (dom (1^st ∘ 𝐺) ≠ ∅ ↔ ran (1^st ∘ 𝐺) ≠ ∅)
16	13, 15	sylib 219	. 2 ⊢ (𝜑 → ran (1^st ∘ 𝐺) ≠ ∅)
17		fvco3 6632	. . . . . 6 ⊢ ((𝐺:ℕ₀⟶(ℝ × ℝ) ∧ 𝑛 ∈ ℕ₀) → ((1^st ∘ 𝐺)‘𝑛) = (1^st ‘(𝐺‘𝑛)))
18	5, 17	sylan 580	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → ((1^st ∘ 𝐺)‘𝑛) = (1^st ‘(𝐺‘𝑛)))
19	1	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → 𝐹:ℕ⟶ℝ)
20	2	adantr 481	. . . . . . . 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 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → 𝑛 ∈ ℕ₀)
22	10	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → 0 ∈ ℕ₀)
23	19, 20, 3, 4, 21, 22	ruclem10 15430	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → (1^st ‘(𝐺‘𝑛)) < (2^nd ‘(𝐺‘0)))
24	1, 2, 3, 4	ruclem4 15425	. . . . . . . . . 10 ⊢ (𝜑 → (𝐺‘0) = ⟨0, 1⟩)
25	24	fveq2d 6547	. . . . . . . . 9 ⊢ (𝜑 → (2^nd ‘(𝐺‘0)) = (2^nd ‘⟨0, 1⟩))
26		c0ex 10486	. . . . . . . . . 10 ⊢ 0 ∈ V
27		1ex 10488	. . . . . . . . . 10 ⊢ 1 ∈ V
28	26, 27	op2nd 7559	. . . . . . . . 9 ⊢ (2^nd ‘⟨0, 1⟩) = 1
29	25, 28	syl6eq 2847	. . . . . . . 8 ⊢ (𝜑 → (2^nd ‘(𝐺‘0)) = 1)
30	29	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → (2^nd ‘(𝐺‘0)) = 1)
31	23, 30	breqtrd 4992	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → (1^st ‘(𝐺‘𝑛)) < 1)
32	5	ffvelrnda 6721	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → (𝐺‘𝑛) ∈ (ℝ × ℝ))
33		xp1st 7582	. . . . . . . 8 ⊢ ((𝐺‘𝑛) ∈ (ℝ × ℝ) → (1^st ‘(𝐺‘𝑛)) ∈ ℝ)
34	32, 33	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → (1^st ‘(𝐺‘𝑛)) ∈ ℝ)
35		1re 10492	. . . . . . 7 ⊢ 1 ∈ ℝ
36		ltle 10581	. . . . . . 7 ⊢ (((1^st ‘(𝐺‘𝑛)) ∈ ℝ ∧ 1 ∈ ℝ) → ((1^st ‘(𝐺‘𝑛)) < 1 → (1^st ‘(𝐺‘𝑛)) ≤ 1))
37	34, 35, 36	sylancl 586	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → ((1^st ‘(𝐺‘𝑛)) < 1 → (1^st ‘(𝐺‘𝑛)) ≤ 1))
38	31, 37	mpd 15	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → (1^st ‘(𝐺‘𝑛)) ≤ 1)
39	18, 38	eqbrtrd 4988	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → ((1^st ∘ 𝐺)‘𝑛) ≤ 1)
40	39	ralrimiva 3149	. . 3 ⊢ (𝜑 → ∀𝑛 ∈ ℕ₀ ((1^st ∘ 𝐺)‘𝑛) ≤ 1)
41	7	ffnd 6388	. . . 4 ⊢ (𝜑 → (1^st ∘ 𝐺) Fn ℕ₀)
42		breq1 4969	. . . . 5 ⊢ (𝑧 = ((1^st ∘ 𝐺)‘𝑛) → (𝑧 ≤ 1 ↔ ((1^st ∘ 𝐺)‘𝑛) ≤ 1))
43	42	ralrn 6724	. . . 4 ⊢ ((1^st ∘ 𝐺) Fn ℕ₀ → (∀𝑧 ∈ ran (1^st ∘ 𝐺)𝑧 ≤ 1 ↔ ∀𝑛 ∈ ℕ₀ ((1^st ∘ 𝐺)‘𝑛) ≤ 1))
44	41, 43	syl 17	. . 3 ⊢ (𝜑 → (∀𝑧 ∈ ran (1^st ∘ 𝐺)𝑧 ≤ 1 ↔ ∀𝑛 ∈ ℕ₀ ((1^st ∘ 𝐺)‘𝑛) ≤ 1))
45	40, 44	mpbird 258	. 2 ⊢ (𝜑 → ∀𝑧 ∈ ran (1^st ∘ 𝐺)𝑧 ≤ 1)
46	8, 16, 45	3jca 1121	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 207 ∧ wa 396 ∧ w3a 1080 = wceq 1522 ∈ wcel 2081 ≠ wne 2984 ∀wral 3105 ⦋csb 3815 ∪ cun 3861 ⊆ wss 3863 ∅c0 4215 ifcif 4385 {csn 4476 ⟨cop 4482 class class class wbr 4966 × cxp 5446 dom cdm 5448 ran crn 5449 ∘ ccom 5452 Fn wfn 6225 ⟶wf 6226 ‘cfv 6230 (class class class)co 7021 ∈ cmpo 7023 1^st c1st 7548 2^nd c2nd 7549 ℝcr 10387 0cc0 10388 1c1 10389 + caddc 10391 < clt 10526 ≤ cle 10527 / cdiv 11150 ℕcn 11491 2c2 11545 ℕ₀cn0 11750 seqcseq 13224

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-sep 5099 ax-nul 5106 ax-pow 5162 ax-pr 5226 ax-un 7324 ax-cnex 10444 ax-resscn 10445 ax-1cn 10446 ax-icn 10447 ax-addcl 10448 ax-addrcl 10449 ax-mulcl 10450 ax-mulrcl 10451 ax-mulcom 10452 ax-addass 10453 ax-mulass 10454 ax-distr 10455 ax-i2m1 10456 ax-1ne0 10457 ax-1rid 10458 ax-rnegex 10459 ax-rrecex 10460 ax-cnre 10461 ax-pre-lttri 10462 ax-pre-lttrn 10463 ax-pre-ltadd 10464 ax-pre-mulgt0 10465

This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-fal 1535 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3710 df-csb 3816 df-dif 3866 df-un 3868 df-in 3870 df-ss 3878 df-pss 3880 df-nul 4216 df-if 4386 df-pw 4459 df-sn 4477 df-pr 4479 df-tp 4481 df-op 4483 df-uni 4750 df-iun 4831 df-br 4967 df-opab 5029 df-mpt 5046 df-tr 5069 df-id 5353 df-eprel 5358 df-po 5367 df-so 5368 df-fr 5407 df-we 5409 df-xp 5454 df-rel 5455 df-cnv 5456 df-co 5457 df-dm 5458 df-rn 5459 df-res 5460 df-ima 5461 df-pred 6028 df-ord 6074 df-on 6075 df-lim 6076 df-suc 6077 df-iota 6194 df-fun 6232 df-fn 6233 df-f 6234 df-f1 6235 df-fo 6236 df-f1o 6237 df-fv 6238 df-riota 6982 df-ov 7024 df-oprab 7025 df-mpo 7026 df-om 7442 df-1st 7550 df-2nd 7551 df-wrecs 7803 df-recs 7865 df-rdg 7903 df-er 8144 df-en 8363 df-dom 8364 df-sdom 8365 df-pnf 10528 df-mnf 10529 df-xr 10530 df-ltxr 10531 df-le 10532 df-sub 10724 df-neg 10725 df-div 11151 df-nn 11492 df-2 11553 df-n0 11751 df-z 11835 df-uz 12099 df-fz 12748 df-seq 13225

This theorem is referenced by: ruclem12 15432

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