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Theorem ruclem11 16202

Description: Lemma for ruc 16205. 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 16197	. . . 4 ⊢ (𝜑 → 𝐺:ℕ₀⟶(ℝ × ℝ))
6		1stcof 7965	. . . 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 6667	. 2 ⊢ (𝜑 → ran (1^st ∘ 𝐺) ⊆ ℝ)
9	7	fdmd 6669	. . . 4 ⊢ (𝜑 → dom (1^st ∘ 𝐺) = ℕ₀)
10		0nn0 12447	. . . . 5 ⊢ 0 ∈ ℕ₀
11		ne0i 4272	. . . . 5 ⊢ (0 ∈ ℕ₀ → ℕ₀ ≠ ∅)
12	10, 11	mp1i 13	. . . 4 ⊢ (𝜑 → ℕ₀ ≠ ∅)
13	9, 12	eqnetrd 3003	. . 3 ⊢ (𝜑 → dom (1^st ∘ 𝐺) ≠ ∅)
14		dm0rn0 5873	. . . 4 ⊢ (dom (1^st ∘ 𝐺) = ∅ ↔ ran (1^st ∘ 𝐺) = ∅)
15	14	necon3bii 2988	. . 3 ⊢ (dom (1^st ∘ 𝐺) ≠ ∅ ↔ ran (1^st ∘ 𝐺) ≠ ∅)
16	13, 15	sylib 220	. 2 ⊢ (𝜑 → ran (1^st ∘ 𝐺) ≠ ∅)
17		fvco3 6931	. . . . . 6 ⊢ ((𝐺:ℕ₀⟶(ℝ × ℝ) ∧ 𝑛 ∈ ℕ₀) → ((1^st ∘ 𝐺)‘𝑛) = (1^st ‘(𝐺‘𝑛)))
18	5, 17	sylan 587	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → ((1^st ∘ 𝐺)‘𝑛) = (1^st ‘(𝐺‘𝑛)))
19	1	adantr 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → 𝐹:ℕ⟶ℝ)
20	2	adantr 482	. . . . . . . 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 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → 𝑛 ∈ ℕ₀)
22	10	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → 0 ∈ ℕ₀)
23	19, 20, 3, 4, 21, 22	ruclem10 16201	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → (1^st ‘(𝐺‘𝑛)) < (2^nd ‘(𝐺‘0)))
24	1, 2, 3, 4	ruclem4 16196	. . . . . . . . . 10 ⊢ (𝜑 → (𝐺‘0) = ⟨0, 1⟩)
25	24	fveq2d 6835	. . . . . . . . 9 ⊢ (𝜑 → (2^nd ‘(𝐺‘0)) = (2^nd ‘⟨0, 1⟩))
26		c0ex 11133	. . . . . . . . . 10 ⊢ 0 ∈ V
27		1ex 11135	. . . . . . . . . 10 ⊢ 1 ∈ V
28	26, 27	op2nd 7944	. . . . . . . . 9 ⊢ (2^nd ‘⟨0, 1⟩) = 1
29	25, 28	eqtrdi 2792	. . . . . . . 8 ⊢ (𝜑 → (2^nd ‘(𝐺‘0)) = 1)
30	29	adantr 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → (2^nd ‘(𝐺‘0)) = 1)
31	23, 30	breqtrd 5101	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → (1^st ‘(𝐺‘𝑛)) < 1)
32	5	ffvelcdmda 7029	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → (𝐺‘𝑛) ∈ (ℝ × ℝ))
33		xp1st 7967	. . . . . . . 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 11139	. . . . . . 7 ⊢ 1 ∈ ℝ
36		ltle 11229	. . . . . . 7 ⊢ (((1^st ‘(𝐺‘𝑛)) ∈ ℝ ∧ 1 ∈ ℝ) → ((1^st ‘(𝐺‘𝑛)) < 1 → (1^st ‘(𝐺‘𝑛)) ≤ 1))
37	34, 35, 36	sylancl 593	. . . . . 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 5097	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → ((1^st ∘ 𝐺)‘𝑛) ≤ 1)
40	39	ralrimiva 3133	. . 3 ⊢ (𝜑 → ∀𝑛 ∈ ℕ₀ ((1^st ∘ 𝐺)‘𝑛) ≤ 1)
41	7	ffnd 6660	. . . 4 ⊢ (𝜑 → (1^st ∘ 𝐺) Fn ℕ₀)
42		breq1 5078	. . . . 5 ⊢ (𝑧 = ((1^st ∘ 𝐺)‘𝑛) → (𝑧 ≤ 1 ↔ ((1^st ∘ 𝐺)‘𝑛) ≤ 1))
43	42	ralrn 7033	. . . 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 259	. 2 ⊢ (𝜑 → ∀𝑧 ∈ ran (1^st ∘ 𝐺)𝑧 ≤ 1)
46	8, 16, 45	3jca 1135	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 208 ∧ wa 397 ∧ w3a 1093 = wceq 1548 ∈ wcel 2121 ≠ wne 2936 ∀wral 3055 ⦋csb 3833 ∪ cun 3883 ⊆ wss 3885 ∅c0 4264 ifcif 4457 {csn 4558 ⟨cop 4564 class class class wbr 5075 × cxp 5619 dom cdm 5621 ran crn 5622 ∘ ccom 5625 Fn wfn 6484 ⟶wf 6485 ‘cfv 6489 (class class class)co 7360 ∈ cmpo 7362 1^st c1st 7933 2^nd c2nd 7934 ℝcr 11032 0cc0 11033 1c1 11034 + caddc 11036 < clt 11174 ≤ cle 11175 / cdiv 11802 ℕcn 12169 2c2 12231 ℕ₀cn0 12432 seqcseq 13958

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1975 ax-7 2016 ax-8 2123 ax-9 2131 ax-10 2154 ax-11 2170 ax-12 2191 ax-ext 2713 ax-sep 5221 ax-nul 5231 ax-pow 5297 ax-pr 5365 ax-un 7682 ax-cnex 11089 ax-resscn 11090 ax-1cn 11091 ax-icn 11092 ax-addcl 11093 ax-addrcl 11094 ax-mulcl 11095 ax-mulrcl 11096 ax-mulcom 11097 ax-addass 11098 ax-mulass 11099 ax-distr 11100 ax-i2m1 11101 ax-1ne0 11102 ax-1rid 11103 ax-rnegex 11104 ax-rrecex 11105 ax-cnre 11106 ax-pre-lttri 11107 ax-pre-lttrn 11108 ax-pre-ltadd 11109 ax-pre-mulgt0 11110

This theorem depends on definitions: df-bi 209 df-an 398 df-or 855 df-3or 1094 df-3an 1095 df-tru 1551 df-fal 1561 df-ex 1788 df-nf 1792 df-sb 2075 df-mo 2545 df-eu 2575 df-clab 2720 df-cleq 2733 df-clel 2816 df-nfc 2890 df-ne 2937 df-nel 3041 df-ral 3056 df-rex 3066 df-rmo 3346 df-reu 3347 df-rab 3394 df-v 3435 df-sbc 3726 df-csb 3834 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-pss 3905 df-nul 4265 df-if 4458 df-pw 4534 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4842 df-iun 4926 df-br 5076 df-opab 5138 df-mpt 5157 df-tr 5183 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6256 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-1st 7935 df-2nd 7936 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-er 8637 df-en 8888 df-dom 8889 df-sdom 8890 df-pnf 11176 df-mnf 11177 df-xr 11178 df-ltxr 11179 df-le 11180 df-sub 11374 df-neg 11375 df-div 11803 df-nn 12170 df-2 12239 df-n0 12433 df-z 12520 df-uz 12784 df-fz 13457 df-seq 13959

This theorem is referenced by: ruclem12 16203

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