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Theorem dvgt0lem2 24527
Description: Lemma for dvgt0 24528 and dvlt0 24529. (Contributed by Mario Carneiro, 19-Feb-2015.)
Hypotheses
Ref Expression
dvgt0.a (𝜑𝐴 ∈ ℝ)
dvgt0.b (𝜑𝐵 ∈ ℝ)
dvgt0.f (𝜑𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ))
dvgt0lem.d (𝜑 → (ℝ D 𝐹):(𝐴(,)𝐵)⟶𝑆)
dvgt0lem.o 𝑂 Or ℝ
dvgt0lem.i (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (𝐹𝑥)𝑂(𝐹𝑦))
Assertion
Ref Expression
dvgt0lem2 (𝜑𝐹 Isom < , 𝑂 ((𝐴[,]𝐵), ran 𝐹))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝑂,𝑦   𝜑,𝑥,𝑦   𝑥,𝐵,𝑦   𝑥,𝐹,𝑦
Allowed substitution hints:   𝑆(𝑥,𝑦)

Proof of Theorem dvgt0lem2
StepHypRef Expression
1 dvgt0lem.i . . . . . 6 (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (𝐹𝑥)𝑂(𝐹𝑦))
21ex 413 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝑥 < 𝑦 → (𝐹𝑥)𝑂(𝐹𝑦)))
32ralrimivva 3188 . . . 4 (𝜑 → ∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(𝑥 < 𝑦 → (𝐹𝑥)𝑂(𝐹𝑦)))
4 dvgt0.a . . . . . . 7 (𝜑𝐴 ∈ ℝ)
5 dvgt0.b . . . . . . 7 (𝜑𝐵 ∈ ℝ)
6 iccssre 12806 . . . . . . 7 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
74, 5, 6syl2anc 584 . . . . . 6 (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
8 ltso 10709 . . . . . 6 < Or ℝ
9 soss 5486 . . . . . 6 ((𝐴[,]𝐵) ⊆ ℝ → ( < Or ℝ → < Or (𝐴[,]𝐵)))
107, 8, 9mpisyl 21 . . . . 5 (𝜑 → < Or (𝐴[,]𝐵))
11 dvgt0lem.o . . . . . 6 𝑂 Or ℝ
1211a1i 11 . . . . 5 (𝜑𝑂 Or ℝ)
13 dvgt0.f . . . . . 6 (𝜑𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ))
14 cncff 23428 . . . . . 6 (𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ) → 𝐹:(𝐴[,]𝐵)⟶ℝ)
1513, 14syl 17 . . . . 5 (𝜑𝐹:(𝐴[,]𝐵)⟶ℝ)
16 ssidd 3987 . . . . 5 (𝜑 → (𝐴[,]𝐵) ⊆ (𝐴[,]𝐵))
17 soisores 7069 . . . . 5 ((( < Or (𝐴[,]𝐵) ∧ 𝑂 Or ℝ) ∧ (𝐹:(𝐴[,]𝐵)⟶ℝ ∧ (𝐴[,]𝐵) ⊆ (𝐴[,]𝐵))) → ((𝐹 ↾ (𝐴[,]𝐵)) Isom < , 𝑂 ((𝐴[,]𝐵), (𝐹 “ (𝐴[,]𝐵))) ↔ ∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(𝑥 < 𝑦 → (𝐹𝑥)𝑂(𝐹𝑦))))
1810, 12, 15, 16, 17syl22anc 834 . . . 4 (𝜑 → ((𝐹 ↾ (𝐴[,]𝐵)) Isom < , 𝑂 ((𝐴[,]𝐵), (𝐹 “ (𝐴[,]𝐵))) ↔ ∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(𝑥 < 𝑦 → (𝐹𝑥)𝑂(𝐹𝑦))))
193, 18mpbird 258 . . 3 (𝜑 → (𝐹 ↾ (𝐴[,]𝐵)) Isom < , 𝑂 ((𝐴[,]𝐵), (𝐹 “ (𝐴[,]𝐵))))
20 ffn 6507 . . . . 5 (𝐹:(𝐴[,]𝐵)⟶ℝ → 𝐹 Fn (𝐴[,]𝐵))
2113, 14, 203syl 18 . . . 4 (𝜑𝐹 Fn (𝐴[,]𝐵))
22 fnresdm 6459 . . . 4 (𝐹 Fn (𝐴[,]𝐵) → (𝐹 ↾ (𝐴[,]𝐵)) = 𝐹)
23 isoeq1 7059 . . . 4 ((𝐹 ↾ (𝐴[,]𝐵)) = 𝐹 → ((𝐹 ↾ (𝐴[,]𝐵)) Isom < , 𝑂 ((𝐴[,]𝐵), (𝐹 “ (𝐴[,]𝐵))) ↔ 𝐹 Isom < , 𝑂 ((𝐴[,]𝐵), (𝐹 “ (𝐴[,]𝐵)))))
2421, 22, 233syl 18 . . 3 (𝜑 → ((𝐹 ↾ (𝐴[,]𝐵)) Isom < , 𝑂 ((𝐴[,]𝐵), (𝐹 “ (𝐴[,]𝐵))) ↔ 𝐹 Isom < , 𝑂 ((𝐴[,]𝐵), (𝐹 “ (𝐴[,]𝐵)))))
2519, 24mpbid 233 . 2 (𝜑𝐹 Isom < , 𝑂 ((𝐴[,]𝐵), (𝐹 “ (𝐴[,]𝐵))))
26 fnima 6471 . . 3 (𝐹 Fn (𝐴[,]𝐵) → (𝐹 “ (𝐴[,]𝐵)) = ran 𝐹)
27 isoeq5 7063 . . 3 ((𝐹 “ (𝐴[,]𝐵)) = ran 𝐹 → (𝐹 Isom < , 𝑂 ((𝐴[,]𝐵), (𝐹 “ (𝐴[,]𝐵))) ↔ 𝐹 Isom < , 𝑂 ((𝐴[,]𝐵), ran 𝐹)))
2821, 26, 273syl 18 . 2 (𝜑 → (𝐹 Isom < , 𝑂 ((𝐴[,]𝐵), (𝐹 “ (𝐴[,]𝐵))) ↔ 𝐹 Isom < , 𝑂 ((𝐴[,]𝐵), ran 𝐹)))
2925, 28mpbid 233 1 (𝜑𝐹 Isom < , 𝑂 ((𝐴[,]𝐵), ran 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  wral 3135  wss 3933   class class class wbr 5057   Or wor 5466  ran crn 5549  cres 5550  cima 5551   Fn wfn 6343  wf 6344  cfv 6348   Isom wiso 6349  (class class class)co 7145  cr 10524   < clt 10663  (,)cioo 12726  [,]cicc 12729  cnccncf 23411   D cdv 24388
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-pre-lttri 10599  ax-pre-lttrn 10600
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-po 5467  df-so 5468  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-ov 7148  df-oprab 7149  df-mpo 7150  df-er 8278  df-map 8397  df-en 8498  df-dom 8499  df-sdom 8500  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-icc 12733  df-cncf 23413
This theorem is referenced by:  dvgt0  24528  dvlt0  24529
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