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Theorem dvgt0lem2 24585
 Description: Lemma for dvgt0 24586 and dvlt0 24587. (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 415 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝑥 < 𝑦 → (𝐹𝑥)𝑂(𝐹𝑦)))
32ralrimivva 3178 . . . 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 12798 . . . . . . 7 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
74, 5, 6syl2anc 586 . . . . . 6 (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
8 ltso 10699 . . . . . 6 < Or ℝ
9 soss 5469 . . . . . 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 23477 . . . . . 6 (𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ) → 𝐹:(𝐴[,]𝐵)⟶ℝ)
1513, 14syl 17 . . . . 5 (𝜑𝐹:(𝐴[,]𝐵)⟶ℝ)
16 ssidd 3969 . . . . 5 (𝜑 → (𝐴[,]𝐵) ⊆ (𝐴[,]𝐵))
17 soisores 7057 . . . . 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 836 . . . 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 259 . . 3 (𝜑 → (𝐹 ↾ (𝐴[,]𝐵)) Isom < , 𝑂 ((𝐴[,]𝐵), (𝐹 “ (𝐴[,]𝐵))))
20 ffn 6490 . . . . 5 (𝐹:(𝐴[,]𝐵)⟶ℝ → 𝐹 Fn (𝐴[,]𝐵))
2113, 14, 203syl 18 . . . 4 (𝜑𝐹 Fn (𝐴[,]𝐵))
22 fnresdm 6442 . . . 4 (𝐹 Fn (𝐴[,]𝐵) → (𝐹 ↾ (𝐴[,]𝐵)) = 𝐹)
23 isoeq1 7047 . . . 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 234 . 2 (𝜑𝐹 Isom < , 𝑂 ((𝐴[,]𝐵), (𝐹 “ (𝐴[,]𝐵))))
26 fnima 6454 . . 3 (𝐹 Fn (𝐴[,]𝐵) → (𝐹 “ (𝐴[,]𝐵)) = ran 𝐹)
27 isoeq5 7051 . . 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 234 1 (𝜑𝐹 Isom < , 𝑂 ((𝐴[,]𝐵), ran 𝐹))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∀wral 3125   ⊆ wss 3913   class class class wbr 5042   Or wor 5449  ran crn 5532   ↾ cres 5533   “ cima 5534   Fn wfn 6326  ⟶wf 6327  ‘cfv 6331   Isom wiso 6332  (class class class)co 7133  ℝcr 10514   < clt 10653  (,)cioo 12717  [,]cicc 12720  –cn→ccncf 23460   D cdv 24445 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5179  ax-nul 5186  ax-pow 5242  ax-pr 5306  ax-un 7439  ax-cnex 10571  ax-resscn 10572  ax-pre-lttri 10589  ax-pre-lttrn 10590 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-nel 3111  df-ral 3130  df-rex 3131  df-rab 3134  df-v 3475  df-sbc 3753  df-csb 3861  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-pw 4517  df-sn 4544  df-pr 4546  df-op 4550  df-uni 4815  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5436  df-po 5450  df-so 5451  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-rn 5542  df-res 5543  df-ima 5544  df-iota 6290  df-fun 6333  df-fn 6334  df-f 6335  df-f1 6336  df-fo 6337  df-f1o 6338  df-fv 6339  df-isom 6340  df-ov 7136  df-oprab 7137  df-mpo 7138  df-er 8267  df-map 8386  df-en 8488  df-dom 8489  df-sdom 8490  df-pnf 10655  df-mnf 10656  df-xr 10657  df-ltxr 10658  df-le 10659  df-icc 12724  df-cncf 23462 This theorem is referenced by:  dvgt0  24586  dvlt0  24587
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