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Theorem dvgt0lem2 26057
Description: Lemma for dvgt0 26058 and dvlt0 26059. (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 412 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝑥 < 𝑦 → (𝐹𝑥)𝑂(𝐹𝑦)))
32ralrimivva 3200 . . . 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 13466 . . . . . . 7 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
74, 5, 6syl2anc 584 . . . . . 6 (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
8 ltso 11339 . . . . . 6 < Or ℝ
9 soss 5617 . . . . . 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 24933 . . . . . 6 (𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ) → 𝐹:(𝐴[,]𝐵)⟶ℝ)
1513, 14syl 17 . . . . 5 (𝜑𝐹:(𝐴[,]𝐵)⟶ℝ)
16 ssidd 4019 . . . . 5 (𝜑 → (𝐴[,]𝐵) ⊆ (𝐴[,]𝐵))
17 soisores 7347 . . . . 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 839 . . . 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 257 . . 3 (𝜑 → (𝐹 ↾ (𝐴[,]𝐵)) Isom < , 𝑂 ((𝐴[,]𝐵), (𝐹 “ (𝐴[,]𝐵))))
20 ffn 6737 . . . . 5 (𝐹:(𝐴[,]𝐵)⟶ℝ → 𝐹 Fn (𝐴[,]𝐵))
2113, 14, 203syl 18 . . . 4 (𝜑𝐹 Fn (𝐴[,]𝐵))
22 fnresdm 6688 . . . 4 (𝐹 Fn (𝐴[,]𝐵) → (𝐹 ↾ (𝐴[,]𝐵)) = 𝐹)
23 isoeq1 7337 . . . 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 232 . 2 (𝜑𝐹 Isom < , 𝑂 ((𝐴[,]𝐵), (𝐹 “ (𝐴[,]𝐵))))
26 fnima 6699 . . 3 (𝐹 Fn (𝐴[,]𝐵) → (𝐹 “ (𝐴[,]𝐵)) = ran 𝐹)
27 isoeq5 7341 . . 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 232 1 (𝜑𝐹 Isom < , 𝑂 ((𝐴[,]𝐵), ran 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wral 3059  wss 3963   class class class wbr 5148   Or wor 5596  ran crn 5690  cres 5691  cima 5692   Fn wfn 6558  wf 6559  cfv 6563   Isom wiso 6564  (class class class)co 7431  cr 11152   < clt 11293  (,)cioo 13384  [,]cicc 13387  cnccncf 24916   D cdv 25913
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-pre-lttri 11227  ax-pre-lttrn 11228
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-po 5597  df-so 5598  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-ov 7434  df-oprab 7435  df-mpo 7436  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-icc 13391  df-cncf 24918
This theorem is referenced by:  dvgt0  26058  dvlt0  26059
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