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Theorem ruclem11 16208

Description: Lemma for ruc 16211. 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 16203	. . . 4 ⊢ (𝜑 → 𝐺:ℕ₀⟶(ℝ × ℝ))
6		1stcof 7998	. . . 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 6696	. 2 ⊢ (𝜑 → ran (1^st ∘ 𝐺) ⊆ ℝ)
9	7	fdmd 6698	. . . 4 ⊢ (𝜑 → dom (1^st ∘ 𝐺) = ℕ₀)
10		0nn0 12457	. . . . 5 ⊢ 0 ∈ ℕ₀
11		ne0i 4304	. . . . 5 ⊢ (0 ∈ ℕ₀ → ℕ₀ ≠ ∅)
12	10, 11	mp1i 13	. . . 4 ⊢ (𝜑 → ℕ₀ ≠ ∅)
13	9, 12	eqnetrd 2992	. . 3 ⊢ (𝜑 → dom (1^st ∘ 𝐺) ≠ ∅)
14		dm0rn0 5888	. . . 4 ⊢ (dom (1^st ∘ 𝐺) = ∅ ↔ ran (1^st ∘ 𝐺) = ∅)
15	14	necon3bii 2977	. . 3 ⊢ (dom (1^st ∘ 𝐺) ≠ ∅ ↔ ran (1^st ∘ 𝐺) ≠ ∅)
16	13, 15	sylib 218	. 2 ⊢ (𝜑 → ran (1^st ∘ 𝐺) ≠ ∅)
17		fvco3 6960	. . . . . 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 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → 𝐹:ℕ⟶ℝ)
20	2	adantr 480	. . . . . . . 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 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 16207	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → (1^st ‘(𝐺‘𝑛)) < (2^nd ‘(𝐺‘0)))
24	1, 2, 3, 4	ruclem4 16202	. . . . . . . . . 10 ⊢ (𝜑 → (𝐺‘0) = ⟨0, 1⟩)
25	24	fveq2d 6862	. . . . . . . . 9 ⊢ (𝜑 → (2^nd ‘(𝐺‘0)) = (2^nd ‘⟨0, 1⟩))
26		c0ex 11168	. . . . . . . . . 10 ⊢ 0 ∈ V
27		1ex 11170	. . . . . . . . . 10 ⊢ 1 ∈ V
28	26, 27	op2nd 7977	. . . . . . . . 9 ⊢ (2^nd ‘⟨0, 1⟩) = 1
29	25, 28	eqtrdi 2780	. . . . . . . 8 ⊢ (𝜑 → (2^nd ‘(𝐺‘0)) = 1)
30	29	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → (2^nd ‘(𝐺‘0)) = 1)
31	23, 30	breqtrd 5133	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → (1^st ‘(𝐺‘𝑛)) < 1)
32	5	ffvelcdmda 7056	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → (𝐺‘𝑛) ∈ (ℝ × ℝ))
33		xp1st 8000	. . . . . . . 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 11174	. . . . . . 7 ⊢ 1 ∈ ℝ
36		ltle 11262	. . . . . . 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 5129	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → ((1^st ∘ 𝐺)‘𝑛) ≤ 1)
40	39	ralrimiva 3125	. . 3 ⊢ (𝜑 → ∀𝑛 ∈ ℕ₀ ((1^st ∘ 𝐺)‘𝑛) ≤ 1)
41	7	ffnd 6689	. . . 4 ⊢ (𝜑 → (1^st ∘ 𝐺) Fn ℕ₀)
42		breq1 5110	. . . . 5 ⊢ (𝑧 = ((1^st ∘ 𝐺)‘𝑛) → (𝑧 ≤ 1 ↔ ((1^st ∘ 𝐺)‘𝑛) ≤ 1))
43	42	ralrn 7060	. . . 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 257	. 2 ⊢ (𝜑 → ∀𝑧 ∈ ran (1^st ∘ 𝐺)𝑧 ≤ 1)
46	8, 16, 45	3jca 1128	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 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 ∀wral 3044 ⦋csb 3862 ∪ cun 3912 ⊆ wss 3914 ∅c0 4296 ifcif 4488 {csn 4589 ⟨cop 4595 class class class wbr 5107 × cxp 5636 dom cdm 5638 ran crn 5639 ∘ ccom 5642 Fn wfn 6506 ⟶wf 6507 ‘cfv 6511 (class class class)co 7387 ∈ cmpo 7389 1^st c1st 7966 2^nd c2nd 7967 ℝcr 11067 0cc0 11068 1c1 11069 + caddc 11071 < clt 11208 ≤ cle 11209 / cdiv 11835 ℕcn 12186 2c2 12241 ℕ₀cn0 12442 seqcseq 13966

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-om 7843 df-1st 7968 df-2nd 7969 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-er 8671 df-en 8919 df-dom 8920 df-sdom 8921 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-div 11836 df-nn 12187 df-2 12249 df-n0 12443 df-z 12530 df-uz 12794 df-fz 13469 df-seq 13967

This theorem is referenced by: ruclem12 16209

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