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Theorem rpnnen1lem5 11694
Description: Lemma for rpnnen1 11696. (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 11692 . . 3 (𝑥 ∈ ℝ → ∀𝑛 ∈ ran (𝐹𝑥)𝑛𝑥)
61, 2, 3, 4rpnnen1lem1 11691 . . . . . 6 (𝑥 ∈ ℝ → (𝐹𝑥) ∈ (ℚ ↑𝑚 ℕ))
74, 3elmap 7772 . . . . . 6 ((𝐹𝑥) ∈ (ℚ ↑𝑚 ℕ) ↔ (𝐹𝑥):ℕ⟶ℚ)
86, 7sylib 207 . . . . 5 (𝑥 ∈ ℝ → (𝐹𝑥):ℕ⟶ℚ)
9 frn 5966 . . . . . 6 ((𝐹𝑥):ℕ⟶ℚ → ran (𝐹𝑥) ⊆ ℚ)
10 qssre 11674 . . . . . 6 ℚ ⊆ ℝ
119, 10syl6ss 3580 . . . . 5 ((𝐹𝑥):ℕ⟶ℚ → ran (𝐹𝑥) ⊆ ℝ)
128, 11syl 17 . . . 4 (𝑥 ∈ ℝ → ran (𝐹𝑥) ⊆ ℝ)
13 1nn 10908 . . . . . . . 8 1 ∈ ℕ
1413ne0ii 3882 . . . . . . 7 ℕ ≠ ∅
15 fdm 5964 . . . . . . . 8 ((𝐹𝑥):ℕ⟶ℚ → dom (𝐹𝑥) = ℕ)
1615neeq1d 2841 . . . . . . 7 ((𝐹𝑥):ℕ⟶ℚ → (dom (𝐹𝑥) ≠ ∅ ↔ ℕ ≠ ∅))
1714, 16mpbiri 247 . . . . . 6 ((𝐹𝑥):ℕ⟶ℚ → dom (𝐹𝑥) ≠ ∅)
18 dm0rn0 5263 . . . . . . 7 (dom (𝐹𝑥) = ∅ ↔ ran (𝐹𝑥) = ∅)
1918necon3bii 2834 . . . . . 6 (dom (𝐹𝑥) ≠ ∅ ↔ ran (𝐹𝑥) ≠ ∅)
2017, 19sylib 207 . . . . 5 ((𝐹𝑥):ℕ⟶ℚ → ran (𝐹𝑥) ≠ ∅)
218, 20syl 17 . . . 4 (𝑥 ∈ ℝ → ran (𝐹𝑥) ≠ ∅)
22 breq2 4587 . . . . . . 7 (𝑦 = 𝑥 → (𝑛𝑦𝑛𝑥))
2322ralbidv 2969 . . . . . 6 (𝑦 = 𝑥 → (∀𝑛 ∈ ran (𝐹𝑥)𝑛𝑦 ↔ ∀𝑛 ∈ ran (𝐹𝑥)𝑛𝑥))
2423rspcev 3282 . . . . 5 ((𝑥 ∈ ℝ ∧ ∀𝑛 ∈ ran (𝐹𝑥)𝑛𝑥) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ ran (𝐹𝑥)𝑛𝑦)
255, 24mpdan 699 . . . 4 (𝑥 ∈ ℝ → ∃𝑦 ∈ ℝ ∀𝑛 ∈ ran (𝐹𝑥)𝑛𝑦)
26 id 22 . . . 4 (𝑥 ∈ ℝ → 𝑥 ∈ ℝ)
27 suprleub 10866 . . . 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 1321 . . 3 (𝑥 ∈ ℝ → (sup(ran (𝐹𝑥), ℝ, < ) ≤ 𝑥 ↔ ∀𝑛 ∈ ran (𝐹𝑥)𝑛𝑥))
295, 28mpbird 246 . 2 (𝑥 ∈ ℝ → sup(ran (𝐹𝑥), ℝ, < ) ≤ 𝑥)
301, 2, 3, 4rpnnen1lem4 11693 . . . . . . . . 9 (𝑥 ∈ ℝ → sup(ran (𝐹𝑥), ℝ, < ) ∈ ℝ)
31 resubcl 10224 . . . . . . . . 9 ((𝑥 ∈ ℝ ∧ sup(ran (𝐹𝑥), ℝ, < ) ∈ ℝ) → (𝑥 − sup(ran (𝐹𝑥), ℝ, < )) ∈ ℝ)
3230, 31mpdan 699 . . . . . . . 8 (𝑥 ∈ ℝ → (𝑥 − sup(ran (𝐹𝑥), ℝ, < )) ∈ ℝ)
3332adantr 480 . . . . . . 7 ((𝑥 ∈ ℝ ∧ sup(ran (𝐹𝑥), ℝ, < ) < 𝑥) → (𝑥 − sup(ran (𝐹𝑥), ℝ, < )) ∈ ℝ)
34 posdif 10400 . . . . . . . . . 10 ((sup(ran (𝐹𝑥), ℝ, < ) ∈ ℝ ∧ 𝑥 ∈ ℝ) → (sup(ran (𝐹𝑥), ℝ, < ) < 𝑥 ↔ 0 < (𝑥 − sup(ran (𝐹𝑥), ℝ, < ))))
3530, 34mpancom 700 . . . . . . . . 9 (𝑥 ∈ ℝ → (sup(ran (𝐹𝑥), ℝ, < ) < 𝑥 ↔ 0 < (𝑥 − sup(ran (𝐹𝑥), ℝ, < ))))
3635biimpa 500 . . . . . . . 8 ((𝑥 ∈ ℝ ∧ sup(ran (𝐹𝑥), ℝ, < ) < 𝑥) → 0 < (𝑥 − sup(ran (𝐹𝑥), ℝ, < )))
3736gt0ne0d 10471 . . . . . . 7 ((𝑥 ∈ ℝ ∧ sup(ran (𝐹𝑥), ℝ, < ) < 𝑥) → (𝑥 − sup(ran (𝐹𝑥), ℝ, < )) ≠ 0)
3833, 37rereccld 10731 . . . . . 6 ((𝑥 ∈ ℝ ∧ sup(ran (𝐹𝑥), ℝ, < ) < 𝑥) → (1 / (𝑥 − sup(ran (𝐹𝑥), ℝ, < ))) ∈ ℝ)
39 arch 11166 . . . . . 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 11690 . . . . . . . . 9 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → sup(𝑇, ℝ, < ) ∈ ℤ)
4342zred 11358 . . . . . . . 8 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → sup(𝑇, ℝ, < ) ∈ ℝ)
44433adant3 1074 . . . . . . 7 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ ∧ (1 / (𝑥 − sup(ran (𝐹𝑥), ℝ, < ))) < 𝑘) → sup(𝑇, ℝ, < ) ∈ ℝ)
4544ltp1d 10833 . . . . . 6 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ ∧ (1 / (𝑥 − sup(ran (𝐹𝑥), ℝ, < ))) < 𝑘) → sup(𝑇, ℝ, < ) < (sup(𝑇, ℝ, < ) + 1))
4633, 36jca 553 . . . . . . . . . . . . 13 ((𝑥 ∈ ℝ ∧ sup(ran (𝐹𝑥), ℝ, < ) < 𝑥) → ((𝑥 − sup(ran (𝐹𝑥), ℝ, < )) ∈ ℝ ∧ 0 < (𝑥 − sup(ran (𝐹𝑥), ℝ, < ))))
47 nnre 10904 . . . . . . . . . . . . . 14 (𝑘 ∈ ℕ → 𝑘 ∈ ℝ)
48 nngt0 10926 . . . . . . . . . . . . . 14 (𝑘 ∈ ℕ → 0 < 𝑘)
4947, 48jca 553 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ → (𝑘 ∈ ℝ ∧ 0 < 𝑘))
50 ltrec1 10789 . . . . . . . . . . . . 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 493 . . . . . . . . . . . 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 758 . . . . . . . . . . . . . 14 (((𝑥 ∈ ℝ ∧ sup(ran (𝐹𝑥), ℝ, < ) < 𝑥) ∧ 𝑘 ∈ ℕ) → sup(ran (𝐹𝑥), ℝ, < ) ∈ ℝ)
53 nnrecre 10934 . . . . . . . . . . . . . . 15 (𝑘 ∈ ℕ → (1 / 𝑘) ∈ ℝ)
5453adantl 481 . . . . . . . . . . . . . 14 (((𝑥 ∈ ℝ ∧ sup(ran (𝐹𝑥), ℝ, < ) < 𝑥) ∧ 𝑘 ∈ ℕ) → (1 / 𝑘) ∈ ℝ)
55 simpll 786 . . . . . . . . . . . . . 14 (((𝑥 ∈ ℝ ∧ sup(ran (𝐹𝑥), ℝ, < ) < 𝑥) ∧ 𝑘 ∈ ℕ) → 𝑥 ∈ ℝ)
5652, 54, 55ltaddsub2d 10507 . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ran (𝐹𝑥) ⊆ ℝ)
58 ffn 5958 . . . . . . . . . . . . . . . . . . 19 ((𝐹𝑥):ℕ⟶ℚ → (𝐹𝑥) Fn ℕ)
598, 58syl 17 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ ℝ → (𝐹𝑥) Fn ℕ)
60 fnfvelrn 6264 . . . . . . . . . . . . . . . . . 18 (((𝐹𝑥) Fn ℕ ∧ 𝑘 ∈ ℕ) → ((𝐹𝑥)‘𝑘) ∈ ran (𝐹𝑥))
6159, 60sylan 487 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ((𝐹𝑥)‘𝑘) ∈ ran (𝐹𝑥))
6257, 61sseldd 3569 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ((𝐹𝑥)‘𝑘) ∈ ℝ)
6330adantr 480 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → sup(ran (𝐹𝑥), ℝ, < ) ∈ ℝ)
6453adantl 481 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (1 / 𝑘) ∈ ℝ)
6512, 21, 253jca 1235 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ ℝ → (ran (𝐹𝑥) ⊆ ℝ ∧ ran (𝐹𝑥) ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑛 ∈ ran (𝐹𝑥)𝑛𝑦))
6665adantr 480 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (ran (𝐹𝑥) ⊆ ℝ ∧ ran (𝐹𝑥) ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑛 ∈ ran (𝐹𝑥)𝑛𝑦))
67 suprub 10863 . . . . . . . . . . . . . . . . 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 691 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ((𝐹𝑥)‘𝑘) ≤ sup(ran (𝐹𝑥), ℝ, < ))
6962, 63, 64, 68leadd1dd 10520 . . . . . . . . . . . . . . 15 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (((𝐹𝑥)‘𝑘) + (1 / 𝑘)) ≤ (sup(ran (𝐹𝑥), ℝ, < ) + (1 / 𝑘)))
7062, 64readdcld 9948 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (((𝐹𝑥)‘𝑘) + (1 / 𝑘)) ∈ ℝ)
71 readdcl 9898 . . . . . . . . . . . . . . . . 17 ((sup(ran (𝐹𝑥), ℝ, < ) ∈ ℝ ∧ (1 / 𝑘) ∈ ℝ) → (sup(ran (𝐹𝑥), ℝ, < ) + (1 / 𝑘)) ∈ ℝ)
7230, 53, 71syl2an 493 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (sup(ran (𝐹𝑥), ℝ, < ) + (1 / 𝑘)) ∈ ℝ)
73 simpl 472 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → 𝑥 ∈ ℝ)
74 lelttr 10007 . . . . . . . . . . . . . . . . 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 1318 . . . . . . . . . . . . . . 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 747 . . . . . . . . . . . . 13 (((𝑥 ∈ ℝ ∧ sup(ran (𝐹𝑥), ℝ, < ) < 𝑥) ∧ 𝑘 ∈ ℕ) → ((sup(ran (𝐹𝑥), ℝ, < ) + (1 / 𝑘)) < 𝑥 → (((𝐹𝑥)‘𝑘) + (1 / 𝑘)) < 𝑥))
7956, 78sylbird 249 . . . . . . . . . . . 12 (((𝑥 ∈ ℝ ∧ sup(ran (𝐹𝑥), ℝ, < ) < 𝑥) ∧ 𝑘 ∈ ℕ) → ((1 / 𝑘) < (𝑥 − sup(ran (𝐹𝑥), ℝ, < )) → (((𝐹𝑥)‘𝑘) + (1 / 𝑘)) < 𝑥))
8051, 79sylbid 229 . . . . . . . . . . 11 (((𝑥 ∈ ℝ ∧ sup(ran (𝐹𝑥), ℝ, < ) < 𝑥) ∧ 𝑘 ∈ ℕ) → ((1 / (𝑥 − sup(ran (𝐹𝑥), ℝ, < ))) < 𝑘 → (((𝐹𝑥)‘𝑘) + (1 / 𝑘)) < 𝑥))
8142peano2zd 11361 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (sup(𝑇, ℝ, < ) + 1) ∈ ℤ)
82 oveq1 6556 . . . . . . . . . . . . . . . . . . 19 (𝑛 = (sup(𝑇, ℝ, < ) + 1) → (𝑛 / 𝑘) = ((sup(𝑇, ℝ, < ) + 1) / 𝑘))
8382breq1d 4593 . . . . . . . . . . . . . . . . . 18 (𝑛 = (sup(𝑇, ℝ, < ) + 1) → ((𝑛 / 𝑘) < 𝑥 ↔ ((sup(𝑇, ℝ, < ) + 1) / 𝑘) < 𝑥))
8483, 1elrab2 3333 . . . . . . . . . . . . . . . . 17 ((sup(𝑇, ℝ, < ) + 1) ∈ 𝑇 ↔ ((sup(𝑇, ℝ, < ) + 1) ∈ ℤ ∧ ((sup(𝑇, ℝ, < ) + 1) / 𝑘) < 𝑥))
8584biimpri 217 . . . . . . . . . . . . . . . 16 (((sup(𝑇, ℝ, < ) + 1) ∈ ℤ ∧ ((sup(𝑇, ℝ, < ) + 1) / 𝑘) < 𝑥) → (sup(𝑇, ℝ, < ) + 1) ∈ 𝑇)
8681, 85sylan 487 . . . . . . . . . . . . . . 15 (((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) ∧ ((sup(𝑇, ℝ, < ) + 1) / 𝑘) < 𝑥) → (sup(𝑇, ℝ, < ) + 1) ∈ 𝑇)
87 ssrab2 3650 . . . . . . . . . . . . . . . . . . . 20 {𝑛 ∈ ℤ ∣ (𝑛 / 𝑘) < 𝑥} ⊆ ℤ
881, 87eqsstri 3598 . . . . . . . . . . . . . . . . . . 19 𝑇 ⊆ ℤ
89 zssre 11261 . . . . . . . . . . . . . . . . . . 19 ℤ ⊆ ℝ
9088, 89sstri 3577 . . . . . . . . . . . . . . . . . 18 𝑇 ⊆ ℝ
9190a1i 11 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → 𝑇 ⊆ ℝ)
92 remulcl 9900 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑘 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (𝑘 · 𝑥) ∈ ℝ)
9392ancoms 468 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℝ) → (𝑘 · 𝑥) ∈ ℝ)
9447, 93sylan2 490 . . . . . . . . . . . . . . . . . . . . 21 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (𝑘 · 𝑥) ∈ ℝ)
95 btwnz 11355 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑘 · 𝑥) ∈ ℝ → (∃𝑛 ∈ ℤ 𝑛 < (𝑘 · 𝑥) ∧ ∃𝑛 ∈ ℤ (𝑘 · 𝑥) < 𝑛))
9695simpld 474 . . . . . . . . . . . . . . . . . . . . 21 ((𝑘 · 𝑥) ∈ ℝ → ∃𝑛 ∈ ℤ 𝑛 < (𝑘 · 𝑥))
9794, 96syl 17 . . . . . . . . . . . . . . . . . . . 20 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ∃𝑛 ∈ ℤ 𝑛 < (𝑘 · 𝑥))
98 zre 11258 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛 ∈ ℤ → 𝑛 ∈ ℝ)
9998adantl 481 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ ℤ) → 𝑛 ∈ ℝ)
100 simpll 786 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ ℤ) → 𝑥 ∈ ℝ)
10149ad2antlr 759 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ ℤ) → (𝑘 ∈ ℝ ∧ 0 < 𝑘))
102 ltdivmul 10777 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑛 ∈ ℝ ∧ 𝑥 ∈ ℝ ∧ (𝑘 ∈ ℝ ∧ 0 < 𝑘)) → ((𝑛 / 𝑘) < 𝑥𝑛 < (𝑘 · 𝑥)))
10399, 100, 101, 102syl3anc 1318 . . . . . . . . . . . . . . . . . . . . 21 (((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ ℤ) → ((𝑛 / 𝑘) < 𝑥𝑛 < (𝑘 · 𝑥)))
104103rexbidva 3031 . . . . . . . . . . . . . . . . . . . 20 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (∃𝑛 ∈ ℤ (𝑛 / 𝑘) < 𝑥 ↔ ∃𝑛 ∈ ℤ 𝑛 < (𝑘 · 𝑥)))
10597, 104mpbird 246 . . . . . . . . . . . . . . . . . . 19 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ∃𝑛 ∈ ℤ (𝑛 / 𝑘) < 𝑥)
106 rabn0 3912 . . . . . . . . . . . . . . . . . . 19 ({𝑛 ∈ ℤ ∣ (𝑛 / 𝑘) < 𝑥} ≠ ∅ ↔ ∃𝑛 ∈ ℤ (𝑛 / 𝑘) < 𝑥)
107105, 106sylibr 223 . . . . . . . . . . . . . . . . . 18 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → {𝑛 ∈ ℤ ∣ (𝑛 / 𝑘) < 𝑥} ≠ ∅)
1081neeq1i 2846 . . . . . . . . . . . . . . . . . 18 (𝑇 ≠ ∅ ↔ {𝑛 ∈ ℤ ∣ (𝑛 / 𝑘) < 𝑥} ≠ ∅)
109107, 108sylibr 223 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → 𝑇 ≠ ∅)
1101rabeq2i 3170 . . . . . . . . . . . . . . . . . . . 20 (𝑛𝑇 ↔ (𝑛 ∈ ℤ ∧ (𝑛 / 𝑘) < 𝑥))
11147ad2antlr 759 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ ℤ) → 𝑘 ∈ ℝ)
112111, 100, 92syl2anc 691 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ ℤ) → (𝑘 · 𝑥) ∈ ℝ)
113 ltle 10005 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑛 ∈ ℝ ∧ (𝑘 · 𝑥) ∈ ℝ) → (𝑛 < (𝑘 · 𝑥) → 𝑛 ≤ (𝑘 · 𝑥)))
11499, 112, 113syl2anc 691 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ ℤ) → (𝑛 < (𝑘 · 𝑥) → 𝑛 ≤ (𝑘 · 𝑥)))
115103, 114sylbid 229 . . . . . . . . . . . . . . . . . . . . 21 (((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ ℤ) → ((𝑛 / 𝑘) < 𝑥𝑛 ≤ (𝑘 · 𝑥)))
116115impr 647 . . . . . . . . . . . . . . . . . . . 20 (((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) ∧ (𝑛 ∈ ℤ ∧ (𝑛 / 𝑘) < 𝑥)) → 𝑛 ≤ (𝑘 · 𝑥))
117110, 116sylan2b 491 . . . . . . . . . . . . . . . . . . 19 (((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) ∧ 𝑛𝑇) → 𝑛 ≤ (𝑘 · 𝑥))
118117ralrimiva 2949 . . . . . . . . . . . . . . . . . 18 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ∀𝑛𝑇 𝑛 ≤ (𝑘 · 𝑥))
119 breq2 4587 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = (𝑘 · 𝑥) → (𝑛𝑦𝑛 ≤ (𝑘 · 𝑥)))
120119ralbidv 2969 . . . . . . . . . . . . . . . . . . 19 (𝑦 = (𝑘 · 𝑥) → (∀𝑛𝑇 𝑛𝑦 ↔ ∀𝑛𝑇 𝑛 ≤ (𝑘 · 𝑥)))
121120rspcev 3282 . . . . . . . . . . . . . . . . . 18 (((𝑘 · 𝑥) ∈ ℝ ∧ ∀𝑛𝑇 𝑛 ≤ (𝑘 · 𝑥)) → ∃𝑦 ∈ ℝ ∀𝑛𝑇 𝑛𝑦)
12294, 118, 121syl2anc 691 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ∃𝑦 ∈ ℝ ∀𝑛𝑇 𝑛𝑦)
12391, 109, 1223jca 1235 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (𝑇 ⊆ ℝ ∧ 𝑇 ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑛𝑇 𝑛𝑦))
124 suprub 10863 . . . . . . . . . . . . . . . 16 (((𝑇 ⊆ ℝ ∧ 𝑇 ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑛𝑇 𝑛𝑦) ∧ (sup(𝑇, ℝ, < ) + 1) ∈ 𝑇) → (sup(𝑇, ℝ, < ) + 1) ≤ sup(𝑇, ℝ, < ))
125123, 124sylan 487 . . . . . . . . . . . . . . 15 (((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) ∧ (sup(𝑇, ℝ, < ) + 1) ∈ 𝑇) → (sup(𝑇, ℝ, < ) + 1) ≤ sup(𝑇, ℝ, < ))
12686, 125syldan 486 . . . . . . . . . . . . . 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 11359 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → sup(𝑇, ℝ, < ) ∈ ℂ)
129 1cnd 9935 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → 1 ∈ ℂ)
130 nncn 10905 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ ℕ → 𝑘 ∈ ℂ)
131 nnne0 10930 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ ℕ → 𝑘 ≠ 0)
132130, 131jca 553 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ ℕ → (𝑘 ∈ ℂ ∧ 𝑘 ≠ 0))
133132adantl 481 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (𝑘 ∈ ℂ ∧ 𝑘 ≠ 0))
134 divdir 10589 . . . . . . . . . . . . . . . 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 1318 . . . . . . . . . . . . . . 15 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ((sup(𝑇, ℝ, < ) + 1) / 𝑘) = ((sup(𝑇, ℝ, < ) / 𝑘) + (1 / 𝑘)))
1363mptex 6390 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘)) ∈ V
1372fvmpt2 6200 . . . . . . . . . . . . . . . . . . 19 ((𝑥 ∈ ℝ ∧ (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘)) ∈ V) → (𝐹𝑥) = (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘)))
138136, 137mpan2 703 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ ℝ → (𝐹𝑥) = (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘)))
139138fveq1d 6105 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ ℝ → ((𝐹𝑥)‘𝑘) = ((𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘))‘𝑘))
140 ovex 6577 . . . . . . . . . . . . . . . . . 18 (sup(𝑇, ℝ, < ) / 𝑘) ∈ V
141 eqid 2610 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘)) = (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘))
142141fvmpt2 6200 . . . . . . . . . . . . . . . . . 18 ((𝑘 ∈ ℕ ∧ (sup(𝑇, ℝ, < ) / 𝑘) ∈ V) → ((𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘))‘𝑘) = (sup(𝑇, ℝ, < ) / 𝑘))
143140, 142mpan2 703 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ ℕ → ((𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘))‘𝑘) = (sup(𝑇, ℝ, < ) / 𝑘))
144139, 143sylan9eq 2664 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ((𝐹𝑥)‘𝑘) = (sup(𝑇, ℝ, < ) / 𝑘))
145144oveq1d 6564 . . . . . . . . . . . . . . 15 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (((𝐹𝑥)‘𝑘) + (1 / 𝑘)) = ((sup(𝑇, ℝ, < ) / 𝑘) + (1 / 𝑘)))
146135, 145eqtr4d 2647 . . . . . . . . . . . . . 14 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ((sup(𝑇, ℝ, < ) + 1) / 𝑘) = (((𝐹𝑥)‘𝑘) + (1 / 𝑘)))
147146breq1d 4593 . . . . . . . . . . . . 13 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (((sup(𝑇, ℝ, < ) + 1) / 𝑘) < 𝑥 ↔ (((𝐹𝑥)‘𝑘) + (1 / 𝑘)) < 𝑥))
14881zred 11358 . . . . . . . . . . . . . 14 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (sup(𝑇, ℝ, < ) + 1) ∈ ℝ)
149148, 43lenltd 10062 . . . . . . . . . . . . 13 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ((sup(𝑇, ℝ, < ) + 1) ≤ sup(𝑇, ℝ, < ) ↔ ¬ sup(𝑇, ℝ, < ) < (sup(𝑇, ℝ, < ) + 1)))
150127, 147, 1493imtr3d 281 . . . . . . . . . . . 12 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ((((𝐹𝑥)‘𝑘) + (1 / 𝑘)) < 𝑥 → ¬ sup(𝑇, ℝ, < ) < (sup(𝑇, ℝ, < ) + 1)))
151150adantlr 747 . . . . . . . . . . 11 (((𝑥 ∈ ℝ ∧ sup(ran (𝐹𝑥), ℝ, < ) < 𝑥) ∧ 𝑘 ∈ ℕ) → ((((𝐹𝑥)‘𝑘) + (1 / 𝑘)) < 𝑥 → ¬ sup(𝑇, ℝ, < ) < (sup(𝑇, ℝ, < ) + 1)))
15280, 151syld 46 . . . . . . . . . 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 628 . . . . . . . . 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 90 . . . . . . . 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 94 . . . . . . 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 1249 . . . . . 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 130 . . . . 5 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ ∧ (1 / (𝑥 − sup(ran (𝐹𝑥), ℝ, < ))) < 𝑘) → ¬ sup(ran (𝐹𝑥), ℝ, < ) < 𝑥)
158157rexlimdv3a 3015 . . . 4 (𝑥 ∈ ℝ → (∃𝑘 ∈ ℕ (1 / (𝑥 − sup(ran (𝐹𝑥), ℝ, < ))) < 𝑘 → ¬ sup(ran (𝐹𝑥), ℝ, < ) < 𝑥))
15941, 158syld 46 . . 3 (𝑥 ∈ ℝ → (sup(ran (𝐹𝑥), ℝ, < ) < 𝑥 → ¬ sup(ran (𝐹𝑥), ℝ, < ) < 𝑥))
160159pm2.01d 180 . 2 (𝑥 ∈ ℝ → ¬ sup(ran (𝐹𝑥), ℝ, < ) < 𝑥)
161 eqlelt 10004 . . 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 700 . 2 (𝑥 ∈ ℝ → (sup(ran (𝐹𝑥), ℝ, < ) = 𝑥 ↔ (sup(ran (𝐹𝑥), ℝ, < ) ≤ 𝑥 ∧ ¬ sup(ran (𝐹𝑥), ℝ, < ) < 𝑥)))
16329, 160, 162mpbir2and 959 1 (𝑥 ∈ ℝ → sup(ran (𝐹𝑥), ℝ, < ) = 𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 195  wa 383  w3a 1031   = wceq 1475  wcel 1977  wne 2780  wral 2896  wrex 2897  {crab 2900  Vcvv 3173  wss 3540  c0 3874   class class class wbr 4583  cmpt 4643  dom cdm 5038  ran crn 5039   Fn wfn 5799  wf 5800  cfv 5804  (class class class)co 6549  𝑚 cmap 7744  supcsup 8229  cc 9813  cr 9814  0cc0 9815  1c1 9816   + caddc 9818   · cmul 9820   < clt 9953  cle 9954  cmin 10145   / cdiv 10563  cn 10897  cz 11254  cq 11664
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-sup 8231  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-n0 11170  df-z 11255  df-q 11665
This theorem is referenced by:  rpnnen1lem6  11695
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