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Theorem ruclem11 16179

Description: Lemma for ruc 16182. 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 16174	. . . 4 ⊢ (𝜑 → 𝐺:ℕ₀⟶(ℝ × ℝ))
6		1stcof 7975	. . . 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 6680	. 2 ⊢ (𝜑 → ran (1^st ∘ 𝐺) ⊆ ℝ)
9	7	fdmd 6682	. . . 4 ⊢ (𝜑 → dom (1^st ∘ 𝐺) = ℕ₀)
10		0nn0 12430	. . . . 5 ⊢ 0 ∈ ℕ₀
11		ne0i 4295	. . . . 5 ⊢ (0 ∈ ℕ₀ → ℕ₀ ≠ ∅)
12	10, 11	mp1i 13	. . . 4 ⊢ (𝜑 → ℕ₀ ≠ ∅)
13	9, 12	eqnetrd 3000	. . 3 ⊢ (𝜑 → dom (1^st ∘ 𝐺) ≠ ∅)
14		dm0rn0 5883	. . . 4 ⊢ (dom (1^st ∘ 𝐺) = ∅ ↔ ran (1^st ∘ 𝐺) = ∅)
15	14	necon3bii 2985	. . 3 ⊢ (dom (1^st ∘ 𝐺) ≠ ∅ ↔ ran (1^st ∘ 𝐺) ≠ ∅)
16	13, 15	sylib 218	. 2 ⊢ (𝜑 → ran (1^st ∘ 𝐺) ≠ ∅)
17		fvco3 6943	. . . . . 6 ⊢ ((𝐺:ℕ₀⟶(ℝ × ℝ) ∧ 𝑛 ∈ ℕ₀) → ((1^st ∘ 𝐺)‘𝑛) = (1^st ‘(𝐺‘𝑛)))
18	5, 17	sylan 581	. . . . 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 16178	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → (1^st ‘(𝐺‘𝑛)) < (2^nd ‘(𝐺‘0)))
24	1, 2, 3, 4	ruclem4 16173	. . . . . . . . . 10 ⊢ (𝜑 → (𝐺‘0) = ⟨0, 1⟩)
25	24	fveq2d 6848	. . . . . . . . 9 ⊢ (𝜑 → (2^nd ‘(𝐺‘0)) = (2^nd ‘⟨0, 1⟩))
26		c0ex 11140	. . . . . . . . . 10 ⊢ 0 ∈ V
27		1ex 11142	. . . . . . . . . 10 ⊢ 1 ∈ V
28	26, 27	op2nd 7954	. . . . . . . . 9 ⊢ (2^nd ‘⟨0, 1⟩) = 1
29	25, 28	eqtrdi 2788	. . . . . . . 8 ⊢ (𝜑 → (2^nd ‘(𝐺‘0)) = 1)
30	29	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → (2^nd ‘(𝐺‘0)) = 1)
31	23, 30	breqtrd 5126	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → (1^st ‘(𝐺‘𝑛)) < 1)
32	5	ffvelcdmda 7040	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → (𝐺‘𝑛) ∈ (ℝ × ℝ))
33		xp1st 7977	. . . . . . . 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 11146	. . . . . . 7 ⊢ 1 ∈ ℝ
36		ltle 11235	. . . . . . 7 ⊢ (((1^st ‘(𝐺‘𝑛)) ∈ ℝ ∧ 1 ∈ ℝ) → ((1^st ‘(𝐺‘𝑛)) < 1 → (1^st ‘(𝐺‘𝑛)) ≤ 1))
37	34, 35, 36	sylancl 587	. . . . . 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 5122	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → ((1^st ∘ 𝐺)‘𝑛) ≤ 1)
40	39	ralrimiva 3130	. . 3 ⊢ (𝜑 → ∀𝑛 ∈ ℕ₀ ((1^st ∘ 𝐺)‘𝑛) ≤ 1)
41	7	ffnd 6673	. . . 4 ⊢ (𝜑 → (1^st ∘ 𝐺) Fn ℕ₀)
42		breq1 5103	. . . . 5 ⊢ (𝑧 = ((1^st ∘ 𝐺)‘𝑛) → (𝑧 ≤ 1 ↔ ((1^st ∘ 𝐺)‘𝑛) ≤ 1))
43	42	ralrn 7044	. . . 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 1129	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 1087 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 ∀wral 3052 ⦋csb 3851 ∪ cun 3901 ⊆ wss 3903 ∅c0 4287 ifcif 4481 {csn 4582 ⟨cop 4588 class class class wbr 5100 × cxp 5632 dom cdm 5634 ran crn 5635 ∘ ccom 5638 Fn wfn 6497 ⟶wf 6498 ‘cfv 6502 (class class class)co 7370 ∈ cmpo 7372 1^st c1st 7943 2^nd c2nd 7944 ℝcr 11039 0cc0 11040 1c1 11041 + caddc 11043 < clt 11180 ≤ cle 11181 / cdiv 11808 ℕcn 12159 2c2 12214 ℕ₀cn0 12415 seqcseq 13938

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5245 ax-nul 5255 ax-pow 5314 ax-pr 5381 ax-un 7692 ax-cnex 11096 ax-resscn 11097 ax-1cn 11098 ax-icn 11099 ax-addcl 11100 ax-addrcl 11101 ax-mulcl 11102 ax-mulrcl 11103 ax-mulcom 11104 ax-addass 11105 ax-mulass 11106 ax-distr 11107 ax-i2m1 11108 ax-1ne0 11109 ax-1rid 11110 ax-rnegex 11111 ax-rrecex 11112 ax-cnre 11113 ax-pre-lttri 11114 ax-pre-lttrn 11115 ax-pre-ltadd 11116 ax-pre-mulgt0 11117

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5529 df-eprel 5534 df-po 5542 df-so 5543 df-fr 5587 df-we 5589 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6269 df-ord 6330 df-on 6331 df-lim 6332 df-suc 6333 df-iota 6458 df-fun 6504 df-fn 6505 df-f 6506 df-f1 6507 df-fo 6508 df-f1o 6509 df-fv 6510 df-riota 7327 df-ov 7373 df-oprab 7374 df-mpo 7375 df-om 7821 df-1st 7945 df-2nd 7946 df-frecs 8235 df-wrecs 8266 df-recs 8315 df-rdg 8353 df-er 8647 df-en 8898 df-dom 8899 df-sdom 8900 df-pnf 11182 df-mnf 11183 df-xr 11184 df-ltxr 11185 df-le 11186 df-sub 11380 df-neg 11381 df-div 11809 df-nn 12160 df-2 12222 df-n0 12416 df-z 12503 df-uz 12766 df-fz 13438 df-seq 13939

This theorem is referenced by: ruclem12 16180

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