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Theorem rpnnen1lem5 11778
Description: Lemma for rpnnen1 11780. (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 11776 . . 3 (𝑥 ∈ ℝ → ∀𝑛 ∈ ran (𝐹𝑥)𝑛𝑥)
61, 2, 3, 4rpnnen1lem1 11775 . . . . . 6 (𝑥 ∈ ℝ → (𝐹𝑥) ∈ (ℚ ↑𝑚 ℕ))
74, 3elmap 7846 . . . . . 6 ((𝐹𝑥) ∈ (ℚ ↑𝑚 ℕ) ↔ (𝐹𝑥):ℕ⟶ℚ)
86, 7sylib 208 . . . . 5 (𝑥 ∈ ℝ → (𝐹𝑥):ℕ⟶ℚ)
9 frn 6020 . . . . . 6 ((𝐹𝑥):ℕ⟶ℚ → ran (𝐹𝑥) ⊆ ℚ)
10 qssre 11758 . . . . . 6 ℚ ⊆ ℝ
119, 10syl6ss 3600 . . . . 5 ((𝐹𝑥):ℕ⟶ℚ → ran (𝐹𝑥) ⊆ ℝ)
128, 11syl 17 . . . 4 (𝑥 ∈ ℝ → ran (𝐹𝑥) ⊆ ℝ)
13 1nn 10991 . . . . . . . 8 1 ∈ ℕ
1413ne0ii 3905 . . . . . . 7 ℕ ≠ ∅
15 fdm 6018 . . . . . . . 8 ((𝐹𝑥):ℕ⟶ℚ → dom (𝐹𝑥) = ℕ)
1615neeq1d 2849 . . . . . . 7 ((𝐹𝑥):ℕ⟶ℚ → (dom (𝐹𝑥) ≠ ∅ ↔ ℕ ≠ ∅))
1714, 16mpbiri 248 . . . . . 6 ((𝐹𝑥):ℕ⟶ℚ → dom (𝐹𝑥) ≠ ∅)
18 dm0rn0 5312 . . . . . . 7 (dom (𝐹𝑥) = ∅ ↔ ran (𝐹𝑥) = ∅)
1918necon3bii 2842 . . . . . 6 (dom (𝐹𝑥) ≠ ∅ ↔ ran (𝐹𝑥) ≠ ∅)
2017, 19sylib 208 . . . . 5 ((𝐹𝑥):ℕ⟶ℚ → ran (𝐹𝑥) ≠ ∅)
218, 20syl 17 . . . 4 (𝑥 ∈ ℝ → ran (𝐹𝑥) ≠ ∅)
22 breq2 4627 . . . . . . 7 (𝑦 = 𝑥 → (𝑛𝑦𝑛𝑥))
2322ralbidv 2982 . . . . . 6 (𝑦 = 𝑥 → (∀𝑛 ∈ ran (𝐹𝑥)𝑛𝑦 ↔ ∀𝑛 ∈ ran (𝐹𝑥)𝑛𝑥))
2423rspcev 3299 . . . . 5 ((𝑥 ∈ ℝ ∧ ∀𝑛 ∈ ran (𝐹𝑥)𝑛𝑥) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ ran (𝐹𝑥)𝑛𝑦)
255, 24mpdan 701 . . . 4 (𝑥 ∈ ℝ → ∃𝑦 ∈ ℝ ∀𝑛 ∈ ran (𝐹𝑥)𝑛𝑦)
26 id 22 . . . 4 (𝑥 ∈ ℝ → 𝑥 ∈ ℝ)
27 suprleub 10949 . . . 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 1326 . . 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 11777 . . . . . . . . 9 (𝑥 ∈ ℝ → sup(ran (𝐹𝑥), ℝ, < ) ∈ ℝ)
31 resubcl 10305 . . . . . . . . 9 ((𝑥 ∈ ℝ ∧ sup(ran (𝐹𝑥), ℝ, < ) ∈ ℝ) → (𝑥 − sup(ran (𝐹𝑥), ℝ, < )) ∈ ℝ)
3230, 31mpdan 701 . . . . . . . 8 (𝑥 ∈ ℝ → (𝑥 − sup(ran (𝐹𝑥), ℝ, < )) ∈ ℝ)
3332adantr 481 . . . . . . 7 ((𝑥 ∈ ℝ ∧ sup(ran (𝐹𝑥), ℝ, < ) < 𝑥) → (𝑥 − sup(ran (𝐹𝑥), ℝ, < )) ∈ ℝ)
34 posdif 10481 . . . . . . . . . 10 ((sup(ran (𝐹𝑥), ℝ, < ) ∈ ℝ ∧ 𝑥 ∈ ℝ) → (sup(ran (𝐹𝑥), ℝ, < ) < 𝑥 ↔ 0 < (𝑥 − sup(ran (𝐹𝑥), ℝ, < ))))
3530, 34mpancom 702 . . . . . . . . 9 (𝑥 ∈ ℝ → (sup(ran (𝐹𝑥), ℝ, < ) < 𝑥 ↔ 0 < (𝑥 − sup(ran (𝐹𝑥), ℝ, < ))))
3635biimpa 501 . . . . . . . 8 ((𝑥 ∈ ℝ ∧ sup(ran (𝐹𝑥), ℝ, < ) < 𝑥) → 0 < (𝑥 − sup(ran (𝐹𝑥), ℝ, < )))
3736gt0ne0d 10552 . . . . . . 7 ((𝑥 ∈ ℝ ∧ sup(ran (𝐹𝑥), ℝ, < ) < 𝑥) → (𝑥 − sup(ran (𝐹𝑥), ℝ, < )) ≠ 0)
3833, 37rereccld 10812 . . . . . 6 ((𝑥 ∈ ℝ ∧ sup(ran (𝐹𝑥), ℝ, < ) < 𝑥) → (1 / (𝑥 − sup(ran (𝐹𝑥), ℝ, < ))) ∈ ℝ)
39 arch 11249 . . . . . 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 450 . . . 4 (𝑥 ∈ ℝ → (sup(ran (𝐹𝑥), ℝ, < ) < 𝑥 → ∃𝑘 ∈ ℕ (1 / (𝑥 − sup(ran (𝐹𝑥), ℝ, < ))) < 𝑘))
421, 2rpnnen1lem2 11774 . . . . . . . . 9 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → sup(𝑇, ℝ, < ) ∈ ℤ)
4342zred 11442 . . . . . . . 8 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → sup(𝑇, ℝ, < ) ∈ ℝ)
44433adant3 1079 . . . . . . 7 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ ∧ (1 / (𝑥 − sup(ran (𝐹𝑥), ℝ, < ))) < 𝑘) → sup(𝑇, ℝ, < ) ∈ ℝ)
4544ltp1d 10914 . . . . . 6 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ ∧ (1 / (𝑥 − sup(ran (𝐹𝑥), ℝ, < ))) < 𝑘) → sup(𝑇, ℝ, < ) < (sup(𝑇, ℝ, < ) + 1))
4633, 36jca 554 . . . . . . . . . . . . 13 ((𝑥 ∈ ℝ ∧ sup(ran (𝐹𝑥), ℝ, < ) < 𝑥) → ((𝑥 − sup(ran (𝐹𝑥), ℝ, < )) ∈ ℝ ∧ 0 < (𝑥 − sup(ran (𝐹𝑥), ℝ, < ))))
47 nnre 10987 . . . . . . . . . . . . . 14 (𝑘 ∈ ℕ → 𝑘 ∈ ℝ)
48 nngt0 11009 . . . . . . . . . . . . . 14 (𝑘 ∈ ℕ → 0 < 𝑘)
4947, 48jca 554 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ → (𝑘 ∈ ℝ ∧ 0 < 𝑘))
50 ltrec1 10870 . . . . . . . . . . . . 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 494 . . . . . . . . . . . 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 761 . . . . . . . . . . . . . 14 (((𝑥 ∈ ℝ ∧ sup(ran (𝐹𝑥), ℝ, < ) < 𝑥) ∧ 𝑘 ∈ ℕ) → sup(ran (𝐹𝑥), ℝ, < ) ∈ ℝ)
53 nnrecre 11017 . . . . . . . . . . . . . . 15 (𝑘 ∈ ℕ → (1 / 𝑘) ∈ ℝ)
5453adantl 482 . . . . . . . . . . . . . 14 (((𝑥 ∈ ℝ ∧ sup(ran (𝐹𝑥), ℝ, < ) < 𝑥) ∧ 𝑘 ∈ ℕ) → (1 / 𝑘) ∈ ℝ)
55 simpll 789 . . . . . . . . . . . . . 14 (((𝑥 ∈ ℝ ∧ sup(ran (𝐹𝑥), ℝ, < ) < 𝑥) ∧ 𝑘 ∈ ℕ) → 𝑥 ∈ ℝ)
5652, 54, 55ltaddsub2d 10588 . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ran (𝐹𝑥) ⊆ ℝ)
58 ffn 6012 . . . . . . . . . . . . . . . . . . 19 ((𝐹𝑥):ℕ⟶ℚ → (𝐹𝑥) Fn ℕ)
598, 58syl 17 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ ℝ → (𝐹𝑥) Fn ℕ)
60 fnfvelrn 6322 . . . . . . . . . . . . . . . . . 18 (((𝐹𝑥) Fn ℕ ∧ 𝑘 ∈ ℕ) → ((𝐹𝑥)‘𝑘) ∈ ran (𝐹𝑥))
6159, 60sylan 488 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ((𝐹𝑥)‘𝑘) ∈ ran (𝐹𝑥))
6257, 61sseldd 3589 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ((𝐹𝑥)‘𝑘) ∈ ℝ)
6330adantr 481 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → sup(ran (𝐹𝑥), ℝ, < ) ∈ ℝ)
6453adantl 482 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (1 / 𝑘) ∈ ℝ)
6512, 21, 253jca 1240 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ ℝ → (ran (𝐹𝑥) ⊆ ℝ ∧ ran (𝐹𝑥) ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑛 ∈ ran (𝐹𝑥)𝑛𝑦))
6665adantr 481 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (ran (𝐹𝑥) ⊆ ℝ ∧ ran (𝐹𝑥) ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑛 ∈ ran (𝐹𝑥)𝑛𝑦))
67 suprub 10944 . . . . . . . . . . . . . . . . 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 692 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ((𝐹𝑥)‘𝑘) ≤ sup(ran (𝐹𝑥), ℝ, < ))
6962, 63, 64, 68leadd1dd 10601 . . . . . . . . . . . . . . 15 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (((𝐹𝑥)‘𝑘) + (1 / 𝑘)) ≤ (sup(ran (𝐹𝑥), ℝ, < ) + (1 / 𝑘)))
7062, 64readdcld 10029 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (((𝐹𝑥)‘𝑘) + (1 / 𝑘)) ∈ ℝ)
71 readdcl 9979 . . . . . . . . . . . . . . . . 17 ((sup(ran (𝐹𝑥), ℝ, < ) ∈ ℝ ∧ (1 / 𝑘) ∈ ℝ) → (sup(ran (𝐹𝑥), ℝ, < ) + (1 / 𝑘)) ∈ ℝ)
7230, 53, 71syl2an 494 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (sup(ran (𝐹𝑥), ℝ, < ) + (1 / 𝑘)) ∈ ℝ)
73 simpl 473 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → 𝑥 ∈ ℝ)
74 lelttr 10088 . . . . . . . . . . . . . . . . 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 452 . . . . . . . . . . . . . . . 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 1323 . . . . . . . . . . . . . . 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 750 . . . . . . . . . . . . 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 11445 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (sup(𝑇, ℝ, < ) + 1) ∈ ℤ)
82 oveq1 6622 . . . . . . . . . . . . . . . . . . 19 (𝑛 = (sup(𝑇, ℝ, < ) + 1) → (𝑛 / 𝑘) = ((sup(𝑇, ℝ, < ) + 1) / 𝑘))
8382breq1d 4633 . . . . . . . . . . . . . . . . . 18 (𝑛 = (sup(𝑇, ℝ, < ) + 1) → ((𝑛 / 𝑘) < 𝑥 ↔ ((sup(𝑇, ℝ, < ) + 1) / 𝑘) < 𝑥))
8483, 1elrab2 3353 . . . . . . . . . . . . . . . . 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 488 . . . . . . . . . . . . . . 15 (((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) ∧ ((sup(𝑇, ℝ, < ) + 1) / 𝑘) < 𝑥) → (sup(𝑇, ℝ, < ) + 1) ∈ 𝑇)
87 ssrab2 3672 . . . . . . . . . . . . . . . . . . . 20 {𝑛 ∈ ℤ ∣ (𝑛 / 𝑘) < 𝑥} ⊆ ℤ
881, 87eqsstri 3620 . . . . . . . . . . . . . . . . . . 19 𝑇 ⊆ ℤ
89 zssre 11344 . . . . . . . . . . . . . . . . . . 19 ℤ ⊆ ℝ
9088, 89sstri 3597 . . . . . . . . . . . . . . . . . 18 𝑇 ⊆ ℝ
9190a1i 11 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → 𝑇 ⊆ ℝ)
92 remulcl 9981 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑘 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (𝑘 · 𝑥) ∈ ℝ)
9392ancoms 469 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℝ) → (𝑘 · 𝑥) ∈ ℝ)
9447, 93sylan2 491 . . . . . . . . . . . . . . . . . . . . 21 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (𝑘 · 𝑥) ∈ ℝ)
95 btwnz 11439 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑘 · 𝑥) ∈ ℝ → (∃𝑛 ∈ ℤ 𝑛 < (𝑘 · 𝑥) ∧ ∃𝑛 ∈ ℤ (𝑘 · 𝑥) < 𝑛))
9695simpld 475 . . . . . . . . . . . . . . . . . . . . 21 ((𝑘 · 𝑥) ∈ ℝ → ∃𝑛 ∈ ℤ 𝑛 < (𝑘 · 𝑥))
9794, 96syl 17 . . . . . . . . . . . . . . . . . . . 20 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ∃𝑛 ∈ ℤ 𝑛 < (𝑘 · 𝑥))
98 zre 11341 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛 ∈ ℤ → 𝑛 ∈ ℝ)
9998adantl 482 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ ℤ) → 𝑛 ∈ ℝ)
100 simpll 789 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ ℤ) → 𝑥 ∈ ℝ)
10149ad2antlr 762 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ ℤ) → (𝑘 ∈ ℝ ∧ 0 < 𝑘))
102 ltdivmul 10858 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑛 ∈ ℝ ∧ 𝑥 ∈ ℝ ∧ (𝑘 ∈ ℝ ∧ 0 < 𝑘)) → ((𝑛 / 𝑘) < 𝑥𝑛 < (𝑘 · 𝑥)))
10399, 100, 101, 102syl3anc 1323 . . . . . . . . . . . . . . . . . . . . 21 (((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ ℤ) → ((𝑛 / 𝑘) < 𝑥𝑛 < (𝑘 · 𝑥)))
104103rexbidva 3044 . . . . . . . . . . . . . . . . . . . 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 3938 . . . . . . . . . . . . . . . . . . 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 2854 . . . . . . . . . . . . . . . . . 18 (𝑇 ≠ ∅ ↔ {𝑛 ∈ ℤ ∣ (𝑛 / 𝑘) < 𝑥} ≠ ∅)
109107, 108sylibr 224 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → 𝑇 ≠ ∅)
1101rabeq2i 3187 . . . . . . . . . . . . . . . . . . . 20 (𝑛𝑇 ↔ (𝑛 ∈ ℤ ∧ (𝑛 / 𝑘) < 𝑥))
11147ad2antlr 762 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ ℤ) → 𝑘 ∈ ℝ)
112111, 100, 92syl2anc 692 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ ℤ) → (𝑘 · 𝑥) ∈ ℝ)
113 ltle 10086 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑛 ∈ ℝ ∧ (𝑘 · 𝑥) ∈ ℝ) → (𝑛 < (𝑘 · 𝑥) → 𝑛 ≤ (𝑘 · 𝑥)))
11499, 112, 113syl2anc 692 . . . . . . . . . . . . . . . . . . . . . 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 648 . . . . . . . . . . . . . . . . . . . 20 (((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) ∧ (𝑛 ∈ ℤ ∧ (𝑛 / 𝑘) < 𝑥)) → 𝑛 ≤ (𝑘 · 𝑥))
117110, 116sylan2b 492 . . . . . . . . . . . . . . . . . . 19 (((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) ∧ 𝑛𝑇) → 𝑛 ≤ (𝑘 · 𝑥))
118117ralrimiva 2962 . . . . . . . . . . . . . . . . . 18 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ∀𝑛𝑇 𝑛 ≤ (𝑘 · 𝑥))
119 breq2 4627 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = (𝑘 · 𝑥) → (𝑛𝑦𝑛 ≤ (𝑘 · 𝑥)))
120119ralbidv 2982 . . . . . . . . . . . . . . . . . . 19 (𝑦 = (𝑘 · 𝑥) → (∀𝑛𝑇 𝑛𝑦 ↔ ∀𝑛𝑇 𝑛 ≤ (𝑘 · 𝑥)))
121120rspcev 3299 . . . . . . . . . . . . . . . . . 18 (((𝑘 · 𝑥) ∈ ℝ ∧ ∀𝑛𝑇 𝑛 ≤ (𝑘 · 𝑥)) → ∃𝑦 ∈ ℝ ∀𝑛𝑇 𝑛𝑦)
12294, 118, 121syl2anc 692 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ∃𝑦 ∈ ℝ ∀𝑛𝑇 𝑛𝑦)
12391, 109, 1223jca 1240 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (𝑇 ⊆ ℝ ∧ 𝑇 ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑛𝑇 𝑛𝑦))
124 suprub 10944 . . . . . . . . . . . . . . . 16 (((𝑇 ⊆ ℝ ∧ 𝑇 ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑛𝑇 𝑛𝑦) ∧ (sup(𝑇, ℝ, < ) + 1) ∈ 𝑇) → (sup(𝑇, ℝ, < ) + 1) ≤ sup(𝑇, ℝ, < ))
125123, 124sylan 488 . . . . . . . . . . . . . . 15 (((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) ∧ (sup(𝑇, ℝ, < ) + 1) ∈ 𝑇) → (sup(𝑇, ℝ, < ) + 1) ≤ sup(𝑇, ℝ, < ))
12686, 125syldan 487 . . . . . . . . . . . . . 14 (((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) ∧ ((sup(𝑇, ℝ, < ) + 1) / 𝑘) < 𝑥) → (sup(𝑇, ℝ, < ) + 1) ≤ sup(𝑇, ℝ, < ))
127126ex 450 . . . . . . . . . . . . 13 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (((sup(𝑇, ℝ, < ) + 1) / 𝑘) < 𝑥 → (sup(𝑇, ℝ, < ) + 1) ≤ sup(𝑇, ℝ, < )))
12842zcnd 11443 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → sup(𝑇, ℝ, < ) ∈ ℂ)
129 1cnd 10016 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → 1 ∈ ℂ)
130 nncn 10988 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ ℕ → 𝑘 ∈ ℂ)
131 nnne0 11013 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ ℕ → 𝑘 ≠ 0)
132130, 131jca 554 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ ℕ → (𝑘 ∈ ℂ ∧ 𝑘 ≠ 0))
133132adantl 482 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (𝑘 ∈ ℂ ∧ 𝑘 ≠ 0))
134 divdir 10670 . . . . . . . . . . . . . . . 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 1323 . . . . . . . . . . . . . . 15 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ((sup(𝑇, ℝ, < ) + 1) / 𝑘) = ((sup(𝑇, ℝ, < ) / 𝑘) + (1 / 𝑘)))
1363mptex 6451 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘)) ∈ V
1372fvmpt2 6258 . . . . . . . . . . . . . . . . . . 19 ((𝑥 ∈ ℝ ∧ (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘)) ∈ V) → (𝐹𝑥) = (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘)))
138136, 137mpan2 706 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ ℝ → (𝐹𝑥) = (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘)))
139138fveq1d 6160 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ ℝ → ((𝐹𝑥)‘𝑘) = ((𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘))‘𝑘))
140 ovex 6643 . . . . . . . . . . . . . . . . . 18 (sup(𝑇, ℝ, < ) / 𝑘) ∈ V
141 eqid 2621 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘)) = (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘))
142141fvmpt2 6258 . . . . . . . . . . . . . . . . . 18 ((𝑘 ∈ ℕ ∧ (sup(𝑇, ℝ, < ) / 𝑘) ∈ V) → ((𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘))‘𝑘) = (sup(𝑇, ℝ, < ) / 𝑘))
143140, 142mpan2 706 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ ℕ → ((𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘))‘𝑘) = (sup(𝑇, ℝ, < ) / 𝑘))
144139, 143sylan9eq 2675 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ((𝐹𝑥)‘𝑘) = (sup(𝑇, ℝ, < ) / 𝑘))
145144oveq1d 6630 . . . . . . . . . . . . . . 15 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (((𝐹𝑥)‘𝑘) + (1 / 𝑘)) = ((sup(𝑇, ℝ, < ) / 𝑘) + (1 / 𝑘)))
146135, 145eqtr4d 2658 . . . . . . . . . . . . . 14 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ((sup(𝑇, ℝ, < ) + 1) / 𝑘) = (((𝐹𝑥)‘𝑘) + (1 / 𝑘)))
147146breq1d 4633 . . . . . . . . . . . . 13 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (((sup(𝑇, ℝ, < ) + 1) / 𝑘) < 𝑥 ↔ (((𝐹𝑥)‘𝑘) + (1 / 𝑘)) < 𝑥))
14881zred 11442 . . . . . . . . . . . . . 14 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (sup(𝑇, ℝ, < ) + 1) ∈ ℝ)
149148, 43lenltd 10143 . . . . . . . . . . . . 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 750 . . . . . . . . . . 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 629 . . . . . . . . 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 1254 . . . . . 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 3028 . . . 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 10085 . . 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 702 . 2 (𝑥 ∈ ℝ → (sup(ran (𝐹𝑥), ℝ, < ) = 𝑥 ↔ (sup(ran (𝐹𝑥), ℝ, < ) ≤ 𝑥 ∧ ¬ sup(ran (𝐹𝑥), ℝ, < ) < 𝑥)))
16329, 160, 162mpbir2and 956 1 (𝑥 ∈ ℝ → sup(ran (𝐹𝑥), ℝ, < ) = 𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wral 2908  wrex 2909  {crab 2912  Vcvv 3190  wss 3560  c0 3897   class class class wbr 4623  cmpt 4683  dom cdm 5084  ran crn 5085   Fn wfn 5852  wf 5853  cfv 5857  (class class class)co 6615  𝑚 cmap 7817  supcsup 8306  cc 9894  cr 9895  0cc0 9896  1c1 9897   + caddc 9899   · cmul 9901   < clt 10034  cle 10035  cmin 10226   / cdiv 10644  cn 10980  cz 11337  cq 11748
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-resscn 9953  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-addrcl 9957  ax-mulcl 9958  ax-mulrcl 9959  ax-mulcom 9960  ax-addass 9961  ax-mulass 9962  ax-distr 9963  ax-i2m1 9964  ax-1ne0 9965  ax-1rid 9966  ax-rnegex 9967  ax-rrecex 9968  ax-cnre 9969  ax-pre-lttri 9970  ax-pre-lttrn 9971  ax-pre-ltadd 9972  ax-pre-mulgt0 9973  ax-pre-sup 9974
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-er 7702  df-map 7819  df-en 7916  df-dom 7917  df-sdom 7918  df-sup 8308  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040  df-sub 10228  df-neg 10229  df-div 10645  df-nn 10981  df-n0 11253  df-z 11338  df-q 11749
This theorem is referenced by:  rpnnen1lem6  11779
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