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Theorem rpnnen1lem5OLD 11775
Description: Lemma for rpnnen1OLD 11776. (Contributed by Mario Carneiro, 12-May-2013.) Obsolete version of rpnnen1lem5 11769 as of 13-Aug-2021. (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
rpnnen1.1OLD 𝑇 = {𝑛 ∈ ℤ ∣ (𝑛 / 𝑘) < 𝑥}
rpnnen1.2OLD 𝐹 = (𝑥 ∈ ℝ ↦ (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘)))
Assertion
Ref Expression
rpnnen1lem5OLD (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 rpnnen1lem5OLD
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 rpnnen1.1OLD . . . 4 𝑇 = {𝑛 ∈ ℤ ∣ (𝑛 / 𝑘) < 𝑥}
2 rpnnen1.2OLD . . . 4 𝐹 = (𝑥 ∈ ℝ ↦ (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘)))
31, 2rpnnen1lem3OLD 11773 . . 3 (𝑥 ∈ ℝ → ∀𝑛 ∈ ran (𝐹𝑥)𝑛𝑥)
41, 2rpnnen1lem1OLD 11772 . . . . . 6 (𝑥 ∈ ℝ → (𝐹𝑥) ∈ (ℚ ↑𝑚 ℕ))
5 qex 11751 . . . . . . 7 ℚ ∈ V
6 nnex 10977 . . . . . . 7 ℕ ∈ V
75, 6elmap 7837 . . . . . 6 ((𝐹𝑥) ∈ (ℚ ↑𝑚 ℕ) ↔ (𝐹𝑥):ℕ⟶ℚ)
84, 7sylib 208 . . . . 5 (𝑥 ∈ ℝ → (𝐹𝑥):ℕ⟶ℚ)
9 frn 6015 . . . . . 6 ((𝐹𝑥):ℕ⟶ℚ → ran (𝐹𝑥) ⊆ ℚ)
10 qssre 11749 . . . . . 6 ℚ ⊆ ℝ
119, 10syl6ss 3599 . . . . 5 ((𝐹𝑥):ℕ⟶ℚ → ran (𝐹𝑥) ⊆ ℝ)
128, 11syl 17 . . . 4 (𝑥 ∈ ℝ → ran (𝐹𝑥) ⊆ ℝ)
13 1nn 10982 . . . . . . . 8 1 ∈ ℕ
1413ne0ii 3904 . . . . . . 7 ℕ ≠ ∅
15 fdm 6013 . . . . . . . 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 5307 . . . . . . 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 4622 . . . . . . 7 (𝑦 = 𝑥 → (𝑛𝑦𝑛𝑥))
2322ralbidv 2981 . . . . . 6 (𝑦 = 𝑥 → (∀𝑛 ∈ ran (𝐹𝑥)𝑛𝑦 ↔ ∀𝑛 ∈ ran (𝐹𝑥)𝑛𝑥))
2423rspcev 3298 . . . . 5 ((𝑥 ∈ ℝ ∧ ∀𝑛 ∈ ran (𝐹𝑥)𝑛𝑥) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ ran (𝐹𝑥)𝑛𝑦)
253, 24mpdan 701 . . . 4 (𝑥 ∈ ℝ → ∃𝑦 ∈ ℝ ∀𝑛 ∈ ran (𝐹𝑥)𝑛𝑦)
26 id 22 . . . 4 (𝑥 ∈ ℝ → 𝑥 ∈ ℝ)
27 suprleub 10940 . . . 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))
293, 28mpbird 247 . 2 (𝑥 ∈ ℝ → sup(ran (𝐹𝑥), ℝ, < ) ≤ 𝑥)
301, 2rpnnen1lem4OLD 11774 . . . . . . . . 9 (𝑥 ∈ ℝ → sup(ran (𝐹𝑥), ℝ, < ) ∈ ℝ)
31 resubcl 10296 . . . . . . . . 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 10472 . . . . . . . . . 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 10543 . . . . . . 7 ((𝑥 ∈ ℝ ∧ sup(ran (𝐹𝑥), ℝ, < ) < 𝑥) → (𝑥 − sup(ran (𝐹𝑥), ℝ, < )) ≠ 0)
3833, 37rereccld 10803 . . . . . 6 ((𝑥 ∈ ℝ ∧ sup(ran (𝐹𝑥), ℝ, < ) < 𝑥) → (1 / (𝑥 − sup(ran (𝐹𝑥), ℝ, < ))) ∈ ℝ)
39 arch 11240 . . . . . 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 11765 . . . . . . . . 9 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → sup(𝑇, ℝ, < ) ∈ ℤ)
4342zred 11433 . . . . . . . 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 10905 . . . . . 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 10978 . . . . . . . . . . . . . 14 (𝑘 ∈ ℕ → 𝑘 ∈ ℝ)
48 nngt0 11000 . . . . . . . . . . . . . 14 (𝑘 ∈ ℕ → 0 < 𝑘)
4947, 48jca 554 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ → (𝑘 ∈ ℝ ∧ 0 < 𝑘))
50 ltrec1 10861 . . . . . . . . . . . . 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 11008 . . . . . . . . . . . . . . 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 10579 . . . . . . . . . . . . 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 6007 . . . . . . . . . . . . . . . . . . 19 ((𝐹𝑥):ℕ⟶ℚ → (𝐹𝑥) Fn ℕ)
598, 58syl 17 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ ℝ → (𝐹𝑥) Fn ℕ)
60 fnfvelrn 6317 . . . . . . . . . . . . . . . . . 18 (((𝐹𝑥) Fn ℕ ∧ 𝑘 ∈ ℕ) → ((𝐹𝑥)‘𝑘) ∈ ran (𝐹𝑥))
6159, 60sylan 488 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ((𝐹𝑥)‘𝑘) ∈ ran (𝐹𝑥))
6257, 61sseldd 3588 . . . . . . . . . . . . . . . 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 10935 . . . . . . . . . . . . . . . . 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 10592 . . . . . . . . . . . . . . 15 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (((𝐹𝑥)‘𝑘) + (1 / 𝑘)) ≤ (sup(ran (𝐹𝑥), ℝ, < ) + (1 / 𝑘)))
7062, 64readdcld 10020 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (((𝐹𝑥)‘𝑘) + (1 / 𝑘)) ∈ ℝ)
71 readdcl 9970 . . . . . . . . . . . . . . . . 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 10079 . . . . . . . . . . . . . . . . 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 11436 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (sup(𝑇, ℝ, < ) + 1) ∈ ℤ)
82 oveq1 6617 . . . . . . . . . . . . . . . . . . 19 (𝑛 = (sup(𝑇, ℝ, < ) + 1) → (𝑛 / 𝑘) = ((sup(𝑇, ℝ, < ) + 1) / 𝑘))
8382breq1d 4628 . . . . . . . . . . . . . . . . . 18 (𝑛 = (sup(𝑇, ℝ, < ) + 1) → ((𝑛 / 𝑘) < 𝑥 ↔ ((sup(𝑇, ℝ, < ) + 1) / 𝑘) < 𝑥))
8483, 1elrab2 3352 . . . . . . . . . . . . . . . . 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 3671 . . . . . . . . . . . . . . . . . . . 20 {𝑛 ∈ ℤ ∣ (𝑛 / 𝑘) < 𝑥} ⊆ ℤ
881, 87eqsstri 3619 . . . . . . . . . . . . . . . . . . 19 𝑇 ⊆ ℤ
89 zssre 11335 . . . . . . . . . . . . . . . . . . 19 ℤ ⊆ ℝ
9088, 89sstri 3596 . . . . . . . . . . . . . . . . . 18 𝑇 ⊆ ℝ
9190a1i 11 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → 𝑇 ⊆ ℝ)
92 remulcl 9972 . . . . . . . . . . . . . . . . . . . . . . 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 11430 . . . . . . . . . . . . . . . . . . . . . 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 11332 . . . . . . . . . . . . . . . . . . . . . . 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 10849 . . . . . . . . . . . . . . . . . . . . . 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 3043 . . . . . . . . . . . . . . . . . . . 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 3937 . . . . . . . . . . . . . . . . . . 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 3186 . . . . . . . . . . . . . . . . . . . 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 10077 . . . . . . . . . . . . . . . . . . . . . . 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 2961 . . . . . . . . . . . . . . . . . 18 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ∀𝑛𝑇 𝑛 ≤ (𝑘 · 𝑥))
119 breq2 4622 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = (𝑘 · 𝑥) → (𝑛𝑦𝑛 ≤ (𝑘 · 𝑥)))
120119ralbidv 2981 . . . . . . . . . . . . . . . . . . 19 (𝑦 = (𝑘 · 𝑥) → (∀𝑛𝑇 𝑛𝑦 ↔ ∀𝑛𝑇 𝑛 ≤ (𝑘 · 𝑥)))
121120rspcev 3298 . . . . . . . . . . . . . . . . . 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 10935 . . . . . . . . . . . . . . . 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 11434 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → sup(𝑇, ℝ, < ) ∈ ℂ)
129 1cnd 10007 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → 1 ∈ ℂ)
130 nncn 10979 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ ℕ → 𝑘 ∈ ℂ)
131 nnne0 11004 . . . . . . . . . . . . . . . . . 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 10661 . . . . . . . . . . . . . . . 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)))
1366mptex 6446 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘)) ∈ V
1372fvmpt2 6253 . . . . . . . . . . . . . . . . . . 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 6155 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ ℝ → ((𝐹𝑥)‘𝑘) = ((𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘))‘𝑘))
140 ovex 6638 . . . . . . . . . . . . . . . . . 18 (sup(𝑇, ℝ, < ) / 𝑘) ∈ V
141 eqid 2621 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘)) = (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘))
142141fvmpt2 6253 . . . . . . . . . . . . . . . . . 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 6625 . . . . . . . . . . . . . . 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 4628 . . . . . . . . . . . . 13 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (((sup(𝑇, ℝ, < ) + 1) / 𝑘) < 𝑥 ↔ (((𝐹𝑥)‘𝑘) + (1 / 𝑘)) < 𝑥))
14881zred 11433 . . . . . . . . . . . . . 14 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (sup(𝑇, ℝ, < ) + 1) ∈ ℝ)
149148, 43lenltd 10134 . . . . . . . . . . . . 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 3027 . . . 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 10076 . . 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 2907  wrex 2908  {crab 2911  Vcvv 3189  wss 3559  c0 3896   class class class wbr 4618  cmpt 4678  dom cdm 5079  ran crn 5080   Fn wfn 5847  wf 5848  cfv 5852  (class class class)co 6610  𝑚 cmap 7809  supcsup 8297  cc 9885  cr 9886  0cc0 9887  1c1 9888   + caddc 9890   · cmul 9892   < clt 10025  cle 10026  cmin 10217   / cdiv 10635  cn 10971  cz 11328  cq 11739
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 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-cnex 9943  ax-resscn 9944  ax-1cn 9945  ax-icn 9946  ax-addcl 9947  ax-addrcl 9948  ax-mulcl 9949  ax-mulrcl 9950  ax-mulcom 9951  ax-addass 9952  ax-mulass 9953  ax-distr 9954  ax-i2m1 9955  ax-1ne0 9956  ax-1rid 9957  ax-rnegex 9958  ax-rrecex 9959  ax-cnre 9960  ax-pre-lttri 9961  ax-pre-lttrn 9962  ax-pre-ltadd 9963  ax-pre-mulgt0 9964  ax-pre-sup 9965
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 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-er 7694  df-map 7811  df-en 7907  df-dom 7908  df-sdom 7909  df-sup 8299  df-pnf 10027  df-mnf 10028  df-xr 10029  df-ltxr 10030  df-le 10031  df-sub 10219  df-neg 10220  df-div 10636  df-nn 10972  df-n0 11244  df-z 11329  df-q 11740
This theorem is referenced by:  rpnnen1OLD  11776
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