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Theorem rpnnen1lem5 12003
Description: Lemma for rpnnen1 12005. (Contributed by Mario Carneiro, 12-May-2013.) (Revised by NM, 13-Aug-2021.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
rpnnen1lem.1 𝑇 = {𝑛 ∈ ℤ ∣ (𝑛 / 𝑘) < 𝑥}
rpnnen1lem.2 𝐹 = (𝑥 ∈ ℝ ↦ (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘)))
rpnnen1lem.n ℕ ∈ V
rpnnen1lem.q ℚ ∈ V
Assertion
Ref Expression
rpnnen1lem5 (𝑥 ∈ ℝ → sup(ran (𝐹𝑥), ℝ, < ) = 𝑥)
Distinct variable groups:   𝑘,𝐹,𝑛,𝑥   𝑇,𝑛
Allowed substitution hints:   𝑇(𝑥,𝑘)

Proof of Theorem rpnnen1lem5
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 rpnnen1lem.1 . . . 4 𝑇 = {𝑛 ∈ ℤ ∣ (𝑛 / 𝑘) < 𝑥}
2 rpnnen1lem.2 . . . 4 𝐹 = (𝑥 ∈ ℝ ↦ (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘)))
3 rpnnen1lem.n . . . 4 ℕ ∈ V
4 rpnnen1lem.q . . . 4 ℚ ∈ V
51, 2, 3, 4rpnnen1lem3 12001 . . 3 (𝑥 ∈ ℝ → ∀𝑛 ∈ ran (𝐹𝑥)𝑛𝑥)
61, 2, 3, 4rpnnen1lem1 12000 . . . . . 6 (𝑥 ∈ ℝ → (𝐹𝑥) ∈ (ℚ ↑𝑚 ℕ))
74, 3elmap 8044 . . . . . 6 ((𝐹𝑥) ∈ (ℚ ↑𝑚 ℕ) ↔ (𝐹𝑥):ℕ⟶ℚ)
86, 7sylib 208 . . . . 5 (𝑥 ∈ ℝ → (𝐹𝑥):ℕ⟶ℚ)
9 frn 6206 . . . . . 6 ((𝐹𝑥):ℕ⟶ℚ → ran (𝐹𝑥) ⊆ ℚ)
10 qssre 11983 . . . . . 6 ℚ ⊆ ℝ
119, 10syl6ss 3748 . . . . 5 ((𝐹𝑥):ℕ⟶ℚ → ran (𝐹𝑥) ⊆ ℝ)
128, 11syl 17 . . . 4 (𝑥 ∈ ℝ → ran (𝐹𝑥) ⊆ ℝ)
13 1nn 11215 . . . . . . . 8 1 ∈ ℕ
1413ne0ii 4058 . . . . . . 7 ℕ ≠ ∅
15 fdm 6204 . . . . . . . 8 ((𝐹𝑥):ℕ⟶ℚ → dom (𝐹𝑥) = ℕ)
1615neeq1d 2983 . . . . . . 7 ((𝐹𝑥):ℕ⟶ℚ → (dom (𝐹𝑥) ≠ ∅ ↔ ℕ ≠ ∅))
1714, 16mpbiri 248 . . . . . 6 ((𝐹𝑥):ℕ⟶ℚ → dom (𝐹𝑥) ≠ ∅)
18 dm0rn0 5489 . . . . . . 7 (dom (𝐹𝑥) = ∅ ↔ ran (𝐹𝑥) = ∅)
1918necon3bii 2976 . . . . . 6 (dom (𝐹𝑥) ≠ ∅ ↔ ran (𝐹𝑥) ≠ ∅)
2017, 19sylib 208 . . . . 5 ((𝐹𝑥):ℕ⟶ℚ → ran (𝐹𝑥) ≠ ∅)
218, 20syl 17 . . . 4 (𝑥 ∈ ℝ → ran (𝐹𝑥) ≠ ∅)
22 breq2 4800 . . . . . . 7 (𝑦 = 𝑥 → (𝑛𝑦𝑛𝑥))
2322ralbidv 3116 . . . . . 6 (𝑦 = 𝑥 → (∀𝑛 ∈ ran (𝐹𝑥)𝑛𝑦 ↔ ∀𝑛 ∈ ran (𝐹𝑥)𝑛𝑥))
2423rspcev 3441 . . . . 5 ((𝑥 ∈ ℝ ∧ ∀𝑛 ∈ ran (𝐹𝑥)𝑛𝑥) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ ran (𝐹𝑥)𝑛𝑦)
255, 24mpdan 705 . . . 4 (𝑥 ∈ ℝ → ∃𝑦 ∈ ℝ ∀𝑛 ∈ ran (𝐹𝑥)𝑛𝑦)
26 id 22 . . . 4 (𝑥 ∈ ℝ → 𝑥 ∈ ℝ)
27 suprleub 11173 . . . 4 (((ran (𝐹𝑥) ⊆ ℝ ∧ ran (𝐹𝑥) ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑛 ∈ ran (𝐹𝑥)𝑛𝑦) ∧ 𝑥 ∈ ℝ) → (sup(ran (𝐹𝑥), ℝ, < ) ≤ 𝑥 ↔ ∀𝑛 ∈ ran (𝐹𝑥)𝑛𝑥))
2812, 21, 25, 26, 27syl31anc 1476 . . 3 (𝑥 ∈ ℝ → (sup(ran (𝐹𝑥), ℝ, < ) ≤ 𝑥 ↔ ∀𝑛 ∈ ran (𝐹𝑥)𝑛𝑥))
295, 28mpbird 247 . 2 (𝑥 ∈ ℝ → sup(ran (𝐹𝑥), ℝ, < ) ≤ 𝑥)
301, 2, 3, 4rpnnen1lem4 12002 . . . . . . . . 9 (𝑥 ∈ ℝ → sup(ran (𝐹𝑥), ℝ, < ) ∈ ℝ)
31 resubcl 10529 . . . . . . . . 9 ((𝑥 ∈ ℝ ∧ sup(ran (𝐹𝑥), ℝ, < ) ∈ ℝ) → (𝑥 − sup(ran (𝐹𝑥), ℝ, < )) ∈ ℝ)
3230, 31mpdan 705 . . . . . . . 8 (𝑥 ∈ ℝ → (𝑥 − sup(ran (𝐹𝑥), ℝ, < )) ∈ ℝ)
3332adantr 472 . . . . . . 7 ((𝑥 ∈ ℝ ∧ sup(ran (𝐹𝑥), ℝ, < ) < 𝑥) → (𝑥 − sup(ran (𝐹𝑥), ℝ, < )) ∈ ℝ)
34 posdif 10705 . . . . . . . . . 10 ((sup(ran (𝐹𝑥), ℝ, < ) ∈ ℝ ∧ 𝑥 ∈ ℝ) → (sup(ran (𝐹𝑥), ℝ, < ) < 𝑥 ↔ 0 < (𝑥 − sup(ran (𝐹𝑥), ℝ, < ))))
3530, 34mpancom 706 . . . . . . . . 9 (𝑥 ∈ ℝ → (sup(ran (𝐹𝑥), ℝ, < ) < 𝑥 ↔ 0 < (𝑥 − sup(ran (𝐹𝑥), ℝ, < ))))
3635biimpa 502 . . . . . . . 8 ((𝑥 ∈ ℝ ∧ sup(ran (𝐹𝑥), ℝ, < ) < 𝑥) → 0 < (𝑥 − sup(ran (𝐹𝑥), ℝ, < )))
3736gt0ne0d 10776 . . . . . . 7 ((𝑥 ∈ ℝ ∧ sup(ran (𝐹𝑥), ℝ, < ) < 𝑥) → (𝑥 − sup(ran (𝐹𝑥), ℝ, < )) ≠ 0)
3833, 37rereccld 11036 . . . . . 6 ((𝑥 ∈ ℝ ∧ sup(ran (𝐹𝑥), ℝ, < ) < 𝑥) → (1 / (𝑥 − sup(ran (𝐹𝑥), ℝ, < ))) ∈ ℝ)
39 arch 11473 . . . . . 6 ((1 / (𝑥 − sup(ran (𝐹𝑥), ℝ, < ))) ∈ ℝ → ∃𝑘 ∈ ℕ (1 / (𝑥 − sup(ran (𝐹𝑥), ℝ, < ))) < 𝑘)
4038, 39syl 17 . . . . 5 ((𝑥 ∈ ℝ ∧ sup(ran (𝐹𝑥), ℝ, < ) < 𝑥) → ∃𝑘 ∈ ℕ (1 / (𝑥 − sup(ran (𝐹𝑥), ℝ, < ))) < 𝑘)
4140ex 449 . . . 4 (𝑥 ∈ ℝ → (sup(ran (𝐹𝑥), ℝ, < ) < 𝑥 → ∃𝑘 ∈ ℕ (1 / (𝑥 − sup(ran (𝐹𝑥), ℝ, < ))) < 𝑘))
421, 2rpnnen1lem2 11999 . . . . . . . . 9 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → sup(𝑇, ℝ, < ) ∈ ℤ)
4342zred 11666 . . . . . . . 8 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → sup(𝑇, ℝ, < ) ∈ ℝ)
44433adant3 1126 . . . . . . 7 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ ∧ (1 / (𝑥 − sup(ran (𝐹𝑥), ℝ, < ))) < 𝑘) → sup(𝑇, ℝ, < ) ∈ ℝ)
4544ltp1d 11138 . . . . . 6 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ ∧ (1 / (𝑥 − sup(ran (𝐹𝑥), ℝ, < ))) < 𝑘) → sup(𝑇, ℝ, < ) < (sup(𝑇, ℝ, < ) + 1))
4633, 36jca 555 . . . . . . . . . . . . 13 ((𝑥 ∈ ℝ ∧ sup(ran (𝐹𝑥), ℝ, < ) < 𝑥) → ((𝑥 − sup(ran (𝐹𝑥), ℝ, < )) ∈ ℝ ∧ 0 < (𝑥 − sup(ran (𝐹𝑥), ℝ, < ))))
47 nnre 11211 . . . . . . . . . . . . . 14 (𝑘 ∈ ℕ → 𝑘 ∈ ℝ)
48 nngt0 11233 . . . . . . . . . . . . . 14 (𝑘 ∈ ℕ → 0 < 𝑘)
4947, 48jca 555 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ → (𝑘 ∈ ℝ ∧ 0 < 𝑘))
50 ltrec1 11094 . . . . . . . . . . . . 13 ((((𝑥 − sup(ran (𝐹𝑥), ℝ, < )) ∈ ℝ ∧ 0 < (𝑥 − sup(ran (𝐹𝑥), ℝ, < ))) ∧ (𝑘 ∈ ℝ ∧ 0 < 𝑘)) → ((1 / (𝑥 − sup(ran (𝐹𝑥), ℝ, < ))) < 𝑘 ↔ (1 / 𝑘) < (𝑥 − sup(ran (𝐹𝑥), ℝ, < ))))
5146, 49, 50syl2an 495 . . . . . . . . . . . 12 (((𝑥 ∈ ℝ ∧ sup(ran (𝐹𝑥), ℝ, < ) < 𝑥) ∧ 𝑘 ∈ ℕ) → ((1 / (𝑥 − sup(ran (𝐹𝑥), ℝ, < ))) < 𝑘 ↔ (1 / 𝑘) < (𝑥 − sup(ran (𝐹𝑥), ℝ, < ))))
5230ad2antrr 764 . . . . . . . . . . . . . 14 (((𝑥 ∈ ℝ ∧ sup(ran (𝐹𝑥), ℝ, < ) < 𝑥) ∧ 𝑘 ∈ ℕ) → sup(ran (𝐹𝑥), ℝ, < ) ∈ ℝ)
53 nnrecre 11241 . . . . . . . . . . . . . . 15 (𝑘 ∈ ℕ → (1 / 𝑘) ∈ ℝ)
5453adantl 473 . . . . . . . . . . . . . 14 (((𝑥 ∈ ℝ ∧ sup(ran (𝐹𝑥), ℝ, < ) < 𝑥) ∧ 𝑘 ∈ ℕ) → (1 / 𝑘) ∈ ℝ)
55 simpll 807 . . . . . . . . . . . . . 14 (((𝑥 ∈ ℝ ∧ sup(ran (𝐹𝑥), ℝ, < ) < 𝑥) ∧ 𝑘 ∈ ℕ) → 𝑥 ∈ ℝ)
5652, 54, 55ltaddsub2d 10812 . . . . . . . . . . . . 13 (((𝑥 ∈ ℝ ∧ sup(ran (𝐹𝑥), ℝ, < ) < 𝑥) ∧ 𝑘 ∈ ℕ) → ((sup(ran (𝐹𝑥), ℝ, < ) + (1 / 𝑘)) < 𝑥 ↔ (1 / 𝑘) < (𝑥 − sup(ran (𝐹𝑥), ℝ, < ))))
5712adantr 472 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ran (𝐹𝑥) ⊆ ℝ)
58 ffn 6198 . . . . . . . . . . . . . . . . . . 19 ((𝐹𝑥):ℕ⟶ℚ → (𝐹𝑥) Fn ℕ)
598, 58syl 17 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ ℝ → (𝐹𝑥) Fn ℕ)
60 fnfvelrn 6511 . . . . . . . . . . . . . . . . . 18 (((𝐹𝑥) Fn ℕ ∧ 𝑘 ∈ ℕ) → ((𝐹𝑥)‘𝑘) ∈ ran (𝐹𝑥))
6159, 60sylan 489 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ((𝐹𝑥)‘𝑘) ∈ ran (𝐹𝑥))
6257, 61sseldd 3737 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ((𝐹𝑥)‘𝑘) ∈ ℝ)
6330adantr 472 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → sup(ran (𝐹𝑥), ℝ, < ) ∈ ℝ)
6453adantl 473 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (1 / 𝑘) ∈ ℝ)
6512, 21, 253jca 1122 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ ℝ → (ran (𝐹𝑥) ⊆ ℝ ∧ ran (𝐹𝑥) ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑛 ∈ ran (𝐹𝑥)𝑛𝑦))
6665adantr 472 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (ran (𝐹𝑥) ⊆ ℝ ∧ ran (𝐹𝑥) ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑛 ∈ ran (𝐹𝑥)𝑛𝑦))
67 suprub 11168 . . . . . . . . . . . . . . . . 17 (((ran (𝐹𝑥) ⊆ ℝ ∧ ran (𝐹𝑥) ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑛 ∈ ran (𝐹𝑥)𝑛𝑦) ∧ ((𝐹𝑥)‘𝑘) ∈ ran (𝐹𝑥)) → ((𝐹𝑥)‘𝑘) ≤ sup(ran (𝐹𝑥), ℝ, < ))
6866, 61, 67syl2anc 696 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ((𝐹𝑥)‘𝑘) ≤ sup(ran (𝐹𝑥), ℝ, < ))
6962, 63, 64, 68leadd1dd 10825 . . . . . . . . . . . . . . 15 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (((𝐹𝑥)‘𝑘) + (1 / 𝑘)) ≤ (sup(ran (𝐹𝑥), ℝ, < ) + (1 / 𝑘)))
7062, 64readdcld 10253 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (((𝐹𝑥)‘𝑘) + (1 / 𝑘)) ∈ ℝ)
71 readdcl 10203 . . . . . . . . . . . . . . . . 17 ((sup(ran (𝐹𝑥), ℝ, < ) ∈ ℝ ∧ (1 / 𝑘) ∈ ℝ) → (sup(ran (𝐹𝑥), ℝ, < ) + (1 / 𝑘)) ∈ ℝ)
7230, 53, 71syl2an 495 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (sup(ran (𝐹𝑥), ℝ, < ) + (1 / 𝑘)) ∈ ℝ)
73 simpl 474 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → 𝑥 ∈ ℝ)
74 lelttr 10312 . . . . . . . . . . . . . . . . 17 (((((𝐹𝑥)‘𝑘) + (1 / 𝑘)) ∈ ℝ ∧ (sup(ran (𝐹𝑥), ℝ, < ) + (1 / 𝑘)) ∈ ℝ ∧ 𝑥 ∈ ℝ) → (((((𝐹𝑥)‘𝑘) + (1 / 𝑘)) ≤ (sup(ran (𝐹𝑥), ℝ, < ) + (1 / 𝑘)) ∧ (sup(ran (𝐹𝑥), ℝ, < ) + (1 / 𝑘)) < 𝑥) → (((𝐹𝑥)‘𝑘) + (1 / 𝑘)) < 𝑥))
7574expd 451 . . . . . . . . . . . . . . . 16 (((((𝐹𝑥)‘𝑘) + (1 / 𝑘)) ∈ ℝ ∧ (sup(ran (𝐹𝑥), ℝ, < ) + (1 / 𝑘)) ∈ ℝ ∧ 𝑥 ∈ ℝ) → ((((𝐹𝑥)‘𝑘) + (1 / 𝑘)) ≤ (sup(ran (𝐹𝑥), ℝ, < ) + (1 / 𝑘)) → ((sup(ran (𝐹𝑥), ℝ, < ) + (1 / 𝑘)) < 𝑥 → (((𝐹𝑥)‘𝑘) + (1 / 𝑘)) < 𝑥)))
7670, 72, 73, 75syl3anc 1473 . . . . . . . . . . . . . . 15 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ((((𝐹𝑥)‘𝑘) + (1 / 𝑘)) ≤ (sup(ran (𝐹𝑥), ℝ, < ) + (1 / 𝑘)) → ((sup(ran (𝐹𝑥), ℝ, < ) + (1 / 𝑘)) < 𝑥 → (((𝐹𝑥)‘𝑘) + (1 / 𝑘)) < 𝑥)))
7769, 76mpd 15 . . . . . . . . . . . . . 14 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ((sup(ran (𝐹𝑥), ℝ, < ) + (1 / 𝑘)) < 𝑥 → (((𝐹𝑥)‘𝑘) + (1 / 𝑘)) < 𝑥))
7877adantlr 753 . . . . . . . . . . . . 13 (((𝑥 ∈ ℝ ∧ sup(ran (𝐹𝑥), ℝ, < ) < 𝑥) ∧ 𝑘 ∈ ℕ) → ((sup(ran (𝐹𝑥), ℝ, < ) + (1 / 𝑘)) < 𝑥 → (((𝐹𝑥)‘𝑘) + (1 / 𝑘)) < 𝑥))
7956, 78sylbird 250 . . . . . . . . . . . 12 (((𝑥 ∈ ℝ ∧ sup(ran (𝐹𝑥), ℝ, < ) < 𝑥) ∧ 𝑘 ∈ ℕ) → ((1 / 𝑘) < (𝑥 − sup(ran (𝐹𝑥), ℝ, < )) → (((𝐹𝑥)‘𝑘) + (1 / 𝑘)) < 𝑥))
8051, 79sylbid 230 . . . . . . . . . . 11 (((𝑥 ∈ ℝ ∧ sup(ran (𝐹𝑥), ℝ, < ) < 𝑥) ∧ 𝑘 ∈ ℕ) → ((1 / (𝑥 − sup(ran (𝐹𝑥), ℝ, < ))) < 𝑘 → (((𝐹𝑥)‘𝑘) + (1 / 𝑘)) < 𝑥))
8142peano2zd 11669 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (sup(𝑇, ℝ, < ) + 1) ∈ ℤ)
82 oveq1 6812 . . . . . . . . . . . . . . . . . . 19 (𝑛 = (sup(𝑇, ℝ, < ) + 1) → (𝑛 / 𝑘) = ((sup(𝑇, ℝ, < ) + 1) / 𝑘))
8382breq1d 4806 . . . . . . . . . . . . . . . . . 18 (𝑛 = (sup(𝑇, ℝ, < ) + 1) → ((𝑛 / 𝑘) < 𝑥 ↔ ((sup(𝑇, ℝ, < ) + 1) / 𝑘) < 𝑥))
8483, 1elrab2 3499 . . . . . . . . . . . . . . . . 17 ((sup(𝑇, ℝ, < ) + 1) ∈ 𝑇 ↔ ((sup(𝑇, ℝ, < ) + 1) ∈ ℤ ∧ ((sup(𝑇, ℝ, < ) + 1) / 𝑘) < 𝑥))
8584biimpri 218 . . . . . . . . . . . . . . . 16 (((sup(𝑇, ℝ, < ) + 1) ∈ ℤ ∧ ((sup(𝑇, ℝ, < ) + 1) / 𝑘) < 𝑥) → (sup(𝑇, ℝ, < ) + 1) ∈ 𝑇)
8681, 85sylan 489 . . . . . . . . . . . . . . 15 (((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) ∧ ((sup(𝑇, ℝ, < ) + 1) / 𝑘) < 𝑥) → (sup(𝑇, ℝ, < ) + 1) ∈ 𝑇)
87 ssrab2 3820 . . . . . . . . . . . . . . . . . . . 20 {𝑛 ∈ ℤ ∣ (𝑛 / 𝑘) < 𝑥} ⊆ ℤ
881, 87eqsstri 3768 . . . . . . . . . . . . . . . . . . 19 𝑇 ⊆ ℤ
89 zssre 11568 . . . . . . . . . . . . . . . . . . 19 ℤ ⊆ ℝ
9088, 89sstri 3745 . . . . . . . . . . . . . . . . . 18 𝑇 ⊆ ℝ
9190a1i 11 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → 𝑇 ⊆ ℝ)
92 remulcl 10205 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑘 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (𝑘 · 𝑥) ∈ ℝ)
9392ancoms 468 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℝ) → (𝑘 · 𝑥) ∈ ℝ)
9447, 93sylan2 492 . . . . . . . . . . . . . . . . . . . . 21 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (𝑘 · 𝑥) ∈ ℝ)
95 btwnz 11663 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑘 · 𝑥) ∈ ℝ → (∃𝑛 ∈ ℤ 𝑛 < (𝑘 · 𝑥) ∧ ∃𝑛 ∈ ℤ (𝑘 · 𝑥) < 𝑛))
9695simpld 477 . . . . . . . . . . . . . . . . . . . . 21 ((𝑘 · 𝑥) ∈ ℝ → ∃𝑛 ∈ ℤ 𝑛 < (𝑘 · 𝑥))
9794, 96syl 17 . . . . . . . . . . . . . . . . . . . 20 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ∃𝑛 ∈ ℤ 𝑛 < (𝑘 · 𝑥))
98 zre 11565 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛 ∈ ℤ → 𝑛 ∈ ℝ)
9998adantl 473 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ ℤ) → 𝑛 ∈ ℝ)
100 simpll 807 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ ℤ) → 𝑥 ∈ ℝ)
10149ad2antlr 765 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ ℤ) → (𝑘 ∈ ℝ ∧ 0 < 𝑘))
102 ltdivmul 11082 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑛 ∈ ℝ ∧ 𝑥 ∈ ℝ ∧ (𝑘 ∈ ℝ ∧ 0 < 𝑘)) → ((𝑛 / 𝑘) < 𝑥𝑛 < (𝑘 · 𝑥)))
10399, 100, 101, 102syl3anc 1473 . . . . . . . . . . . . . . . . . . . . 21 (((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ ℤ) → ((𝑛 / 𝑘) < 𝑥𝑛 < (𝑘 · 𝑥)))
104103rexbidva 3179 . . . . . . . . . . . . . . . . . . . 20 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (∃𝑛 ∈ ℤ (𝑛 / 𝑘) < 𝑥 ↔ ∃𝑛 ∈ ℤ 𝑛 < (𝑘 · 𝑥)))
10597, 104mpbird 247 . . . . . . . . . . . . . . . . . . 19 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ∃𝑛 ∈ ℤ (𝑛 / 𝑘) < 𝑥)
106 rabn0 4093 . . . . . . . . . . . . . . . . . . 19 ({𝑛 ∈ ℤ ∣ (𝑛 / 𝑘) < 𝑥} ≠ ∅ ↔ ∃𝑛 ∈ ℤ (𝑛 / 𝑘) < 𝑥)
107105, 106sylibr 224 . . . . . . . . . . . . . . . . . 18 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → {𝑛 ∈ ℤ ∣ (𝑛 / 𝑘) < 𝑥} ≠ ∅)
1081neeq1i 2988 . . . . . . . . . . . . . . . . . 18 (𝑇 ≠ ∅ ↔ {𝑛 ∈ ℤ ∣ (𝑛 / 𝑘) < 𝑥} ≠ ∅)
109107, 108sylibr 224 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → 𝑇 ≠ ∅)
1101rabeq2i 3329 . . . . . . . . . . . . . . . . . . . 20 (𝑛𝑇 ↔ (𝑛 ∈ ℤ ∧ (𝑛 / 𝑘) < 𝑥))
11147ad2antlr 765 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ ℤ) → 𝑘 ∈ ℝ)
112111, 100, 92syl2anc 696 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ ℤ) → (𝑘 · 𝑥) ∈ ℝ)
113 ltle 10310 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑛 ∈ ℝ ∧ (𝑘 · 𝑥) ∈ ℝ) → (𝑛 < (𝑘 · 𝑥) → 𝑛 ≤ (𝑘 · 𝑥)))
11499, 112, 113syl2anc 696 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ ℤ) → (𝑛 < (𝑘 · 𝑥) → 𝑛 ≤ (𝑘 · 𝑥)))
115103, 114sylbid 230 . . . . . . . . . . . . . . . . . . . . 21 (((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ ℤ) → ((𝑛 / 𝑘) < 𝑥𝑛 ≤ (𝑘 · 𝑥)))
116115impr 650 . . . . . . . . . . . . . . . . . . . 20 (((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) ∧ (𝑛 ∈ ℤ ∧ (𝑛 / 𝑘) < 𝑥)) → 𝑛 ≤ (𝑘 · 𝑥))
117110, 116sylan2b 493 . . . . . . . . . . . . . . . . . . 19 (((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) ∧ 𝑛𝑇) → 𝑛 ≤ (𝑘 · 𝑥))
118117ralrimiva 3096 . . . . . . . . . . . . . . . . . 18 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ∀𝑛𝑇 𝑛 ≤ (𝑘 · 𝑥))
119 breq2 4800 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = (𝑘 · 𝑥) → (𝑛𝑦𝑛 ≤ (𝑘 · 𝑥)))
120119ralbidv 3116 . . . . . . . . . . . . . . . . . . 19 (𝑦 = (𝑘 · 𝑥) → (∀𝑛𝑇 𝑛𝑦 ↔ ∀𝑛𝑇 𝑛 ≤ (𝑘 · 𝑥)))
121120rspcev 3441 . . . . . . . . . . . . . . . . . 18 (((𝑘 · 𝑥) ∈ ℝ ∧ ∀𝑛𝑇 𝑛 ≤ (𝑘 · 𝑥)) → ∃𝑦 ∈ ℝ ∀𝑛𝑇 𝑛𝑦)
12294, 118, 121syl2anc 696 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ∃𝑦 ∈ ℝ ∀𝑛𝑇 𝑛𝑦)
12391, 109, 1223jca 1122 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (𝑇 ⊆ ℝ ∧ 𝑇 ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑛𝑇 𝑛𝑦))
124 suprub 11168 . . . . . . . . . . . . . . . 16 (((𝑇 ⊆ ℝ ∧ 𝑇 ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑛𝑇 𝑛𝑦) ∧ (sup(𝑇, ℝ, < ) + 1) ∈ 𝑇) → (sup(𝑇, ℝ, < ) + 1) ≤ sup(𝑇, ℝ, < ))
125123, 124sylan 489 . . . . . . . . . . . . . . 15 (((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) ∧ (sup(𝑇, ℝ, < ) + 1) ∈ 𝑇) → (sup(𝑇, ℝ, < ) + 1) ≤ sup(𝑇, ℝ, < ))
12686, 125syldan 488 . . . . . . . . . . . . . 14 (((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) ∧ ((sup(𝑇, ℝ, < ) + 1) / 𝑘) < 𝑥) → (sup(𝑇, ℝ, < ) + 1) ≤ sup(𝑇, ℝ, < ))
127126ex 449 . . . . . . . . . . . . 13 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (((sup(𝑇, ℝ, < ) + 1) / 𝑘) < 𝑥 → (sup(𝑇, ℝ, < ) + 1) ≤ sup(𝑇, ℝ, < )))
12842zcnd 11667 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → sup(𝑇, ℝ, < ) ∈ ℂ)
129 1cnd 10240 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → 1 ∈ ℂ)
130 nncn 11212 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ ℕ → 𝑘 ∈ ℂ)
131 nnne0 11237 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ ℕ → 𝑘 ≠ 0)
132130, 131jca 555 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ ℕ → (𝑘 ∈ ℂ ∧ 𝑘 ≠ 0))
133132adantl 473 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (𝑘 ∈ ℂ ∧ 𝑘 ≠ 0))
134 divdir 10894 . . . . . . . . . . . . . . . 16 ((sup(𝑇, ℝ, < ) ∈ ℂ ∧ 1 ∈ ℂ ∧ (𝑘 ∈ ℂ ∧ 𝑘 ≠ 0)) → ((sup(𝑇, ℝ, < ) + 1) / 𝑘) = ((sup(𝑇, ℝ, < ) / 𝑘) + (1 / 𝑘)))
135128, 129, 133, 134syl3anc 1473 . . . . . . . . . . . . . . 15 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ((sup(𝑇, ℝ, < ) + 1) / 𝑘) = ((sup(𝑇, ℝ, < ) / 𝑘) + (1 / 𝑘)))
1363mptex 6642 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘)) ∈ V
1372fvmpt2 6445 . . . . . . . . . . . . . . . . . . 19 ((𝑥 ∈ ℝ ∧ (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘)) ∈ V) → (𝐹𝑥) = (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘)))
138136, 137mpan2 709 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ ℝ → (𝐹𝑥) = (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘)))
139138fveq1d 6346 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ ℝ → ((𝐹𝑥)‘𝑘) = ((𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘))‘𝑘))
140 ovex 6833 . . . . . . . . . . . . . . . . . 18 (sup(𝑇, ℝ, < ) / 𝑘) ∈ V
141 eqid 2752 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘)) = (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘))
142141fvmpt2 6445 . . . . . . . . . . . . . . . . . 18 ((𝑘 ∈ ℕ ∧ (sup(𝑇, ℝ, < ) / 𝑘) ∈ V) → ((𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘))‘𝑘) = (sup(𝑇, ℝ, < ) / 𝑘))
143140, 142mpan2 709 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ ℕ → ((𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘))‘𝑘) = (sup(𝑇, ℝ, < ) / 𝑘))
144139, 143sylan9eq 2806 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ((𝐹𝑥)‘𝑘) = (sup(𝑇, ℝ, < ) / 𝑘))
145144oveq1d 6820 . . . . . . . . . . . . . . 15 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (((𝐹𝑥)‘𝑘) + (1 / 𝑘)) = ((sup(𝑇, ℝ, < ) / 𝑘) + (1 / 𝑘)))
146135, 145eqtr4d 2789 . . . . . . . . . . . . . 14 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ((sup(𝑇, ℝ, < ) + 1) / 𝑘) = (((𝐹𝑥)‘𝑘) + (1 / 𝑘)))
147146breq1d 4806 . . . . . . . . . . . . 13 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (((sup(𝑇, ℝ, < ) + 1) / 𝑘) < 𝑥 ↔ (((𝐹𝑥)‘𝑘) + (1 / 𝑘)) < 𝑥))
14881zred 11666 . . . . . . . . . . . . . 14 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (sup(𝑇, ℝ, < ) + 1) ∈ ℝ)
149148, 43lenltd 10367 . . . . . . . . . . . . 13 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ((sup(𝑇, ℝ, < ) + 1) ≤ sup(𝑇, ℝ, < ) ↔ ¬ sup(𝑇, ℝ, < ) < (sup(𝑇, ℝ, < ) + 1)))
150127, 147, 1493imtr3d 282 . . . . . . . . . . . 12 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ((((𝐹𝑥)‘𝑘) + (1 / 𝑘)) < 𝑥 → ¬ sup(𝑇, ℝ, < ) < (sup(𝑇, ℝ, < ) + 1)))
151150adantlr 753 . . . . . . . . . . 11 (((𝑥 ∈ ℝ ∧ sup(ran (𝐹𝑥), ℝ, < ) < 𝑥) ∧ 𝑘 ∈ ℕ) → ((((𝐹𝑥)‘𝑘) + (1 / 𝑘)) < 𝑥 → ¬ sup(𝑇, ℝ, < ) < (sup(𝑇, ℝ, < ) + 1)))
15280, 151syld 47 . . . . . . . . . 10 (((𝑥 ∈ ℝ ∧ sup(ran (𝐹𝑥), ℝ, < ) < 𝑥) ∧ 𝑘 ∈ ℕ) → ((1 / (𝑥 − sup(ran (𝐹𝑥), ℝ, < ))) < 𝑘 → ¬ sup(𝑇, ℝ, < ) < (sup(𝑇, ℝ, < ) + 1)))
153152exp31 631 . . . . . . . . 9 (𝑥 ∈ ℝ → (sup(ran (𝐹𝑥), ℝ, < ) < 𝑥 → (𝑘 ∈ ℕ → ((1 / (𝑥 − sup(ran (𝐹𝑥), ℝ, < ))) < 𝑘 → ¬ sup(𝑇, ℝ, < ) < (sup(𝑇, ℝ, < ) + 1)))))
154153com4l 92 . . . . . . . 8 (sup(ran (𝐹𝑥), ℝ, < ) < 𝑥 → (𝑘 ∈ ℕ → ((1 / (𝑥 − sup(ran (𝐹𝑥), ℝ, < ))) < 𝑘 → (𝑥 ∈ ℝ → ¬ sup(𝑇, ℝ, < ) < (sup(𝑇, ℝ, < ) + 1)))))
155154com14 96 . . . . . . 7 (𝑥 ∈ ℝ → (𝑘 ∈ ℕ → ((1 / (𝑥 − sup(ran (𝐹𝑥), ℝ, < ))) < 𝑘 → (sup(ran (𝐹𝑥), ℝ, < ) < 𝑥 → ¬ sup(𝑇, ℝ, < ) < (sup(𝑇, ℝ, < ) + 1)))))
1561553imp 1101 . . . . . 6 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ ∧ (1 / (𝑥 − sup(ran (𝐹𝑥), ℝ, < ))) < 𝑘) → (sup(ran (𝐹𝑥), ℝ, < ) < 𝑥 → ¬ sup(𝑇, ℝ, < ) < (sup(𝑇, ℝ, < ) + 1)))
15745, 156mt2d 131 . . . . 5 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ ∧ (1 / (𝑥 − sup(ran (𝐹𝑥), ℝ, < ))) < 𝑘) → ¬ sup(ran (𝐹𝑥), ℝ, < ) < 𝑥)
158157rexlimdv3a 3163 . . . 4 (𝑥 ∈ ℝ → (∃𝑘 ∈ ℕ (1 / (𝑥 − sup(ran (𝐹𝑥), ℝ, < ))) < 𝑘 → ¬ sup(ran (𝐹𝑥), ℝ, < ) < 𝑥))
15941, 158syld 47 . . 3 (𝑥 ∈ ℝ → (sup(ran (𝐹𝑥), ℝ, < ) < 𝑥 → ¬ sup(ran (𝐹𝑥), ℝ, < ) < 𝑥))
160159pm2.01d 181 . 2 (𝑥 ∈ ℝ → ¬ sup(ran (𝐹𝑥), ℝ, < ) < 𝑥)
161 eqlelt 10309 . . 3 ((sup(ran (𝐹𝑥), ℝ, < ) ∈ ℝ ∧ 𝑥 ∈ ℝ) → (sup(ran (𝐹𝑥), ℝ, < ) = 𝑥 ↔ (sup(ran (𝐹𝑥), ℝ, < ) ≤ 𝑥 ∧ ¬ sup(ran (𝐹𝑥), ℝ, < ) < 𝑥)))
16230, 161mpancom 706 . 2 (𝑥 ∈ ℝ → (sup(ran (𝐹𝑥), ℝ, < ) = 𝑥 ↔ (sup(ran (𝐹𝑥), ℝ, < ) ≤ 𝑥 ∧ ¬ sup(ran (𝐹𝑥), ℝ, < ) < 𝑥)))
16329, 160, 162mpbir2and 995 1 (𝑥 ∈ ℝ → sup(ran (𝐹𝑥), ℝ, < ) = 𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  w3a 1072   = wceq 1624  wcel 2131  wne 2924  wral 3042  wrex 3043  {crab 3046  Vcvv 3332  wss 3707  c0 4050   class class class wbr 4796  cmpt 4873  dom cdm 5258  ran crn 5259   Fn wfn 6036  wf 6037  cfv 6041  (class class class)co 6805  𝑚 cmap 8015  supcsup 8503  cc 10118  cr 10119  0cc0 10120  1c1 10121   + caddc 10123   · cmul 10125   < clt 10258  cle 10259  cmin 10450   / cdiv 10868  cn 11204  cz 11561  cq 11973
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-rep 4915  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106  ax-resscn 10177  ax-1cn 10178  ax-icn 10179  ax-addcl 10180  ax-addrcl 10181  ax-mulcl 10182  ax-mulrcl 10183  ax-mulcom 10184  ax-addass 10185  ax-mulass 10186  ax-distr 10187  ax-i2m1 10188  ax-1ne0 10189  ax-1rid 10190  ax-rnegex 10191  ax-rrecex 10192  ax-cnre 10193  ax-pre-lttri 10194  ax-pre-lttrn 10195  ax-pre-ltadd 10196  ax-pre-mulgt0 10197  ax-pre-sup 10198
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-nel 3028  df-ral 3047  df-rex 3048  df-reu 3049  df-rmo 3050  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-pss 3723  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-tp 4318  df-op 4320  df-uni 4581  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-tr 4897  df-id 5166  df-eprel 5171  df-po 5179  df-so 5180  df-fr 5217  df-we 5219  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-pred 5833  df-ord 5879  df-on 5880  df-lim 5881  df-suc 5882  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-riota 6766  df-ov 6808  df-oprab 6809  df-mpt2 6810  df-om 7223  df-1st 7325  df-2nd 7326  df-wrecs 7568  df-recs 7629  df-rdg 7667  df-er 7903  df-map 8017  df-en 8114  df-dom 8115  df-sdom 8116  df-sup 8505  df-pnf 10260  df-mnf 10261  df-xr 10262  df-ltxr 10263  df-le 10264  df-sub 10452  df-neg 10453  df-div 10869  df-nn 11205  df-n0 11477  df-z 11562  df-q 11974
This theorem is referenced by:  rpnnen1lem6  12004
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