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Theorem dvgt0lem2 25980
Description: Lemma for dvgt0 25981 and dvlt0 25982. (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 411 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝑥 < 𝑦 → (𝐹𝑥)𝑂(𝐹𝑦)))
32ralrimivva 3190 . . . 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 13441 . . . . . . 7 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
74, 5, 6syl2anc 582 . . . . . 6 (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
8 ltso 11326 . . . . . 6 < Or ℝ
9 soss 5610 . . . . . 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 24857 . . . . . 6 (𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ) → 𝐹:(𝐴[,]𝐵)⟶ℝ)
1513, 14syl 17 . . . . 5 (𝜑𝐹:(𝐴[,]𝐵)⟶ℝ)
16 ssidd 4000 . . . . 5 (𝜑 → (𝐴[,]𝐵) ⊆ (𝐴[,]𝐵))
17 soisores 7334 . . . . 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 837 . . . 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 256 . . 3 (𝜑 → (𝐹 ↾ (𝐴[,]𝐵)) Isom < , 𝑂 ((𝐴[,]𝐵), (𝐹 “ (𝐴[,]𝐵))))
20 ffn 6723 . . . . 5 (𝐹:(𝐴[,]𝐵)⟶ℝ → 𝐹 Fn (𝐴[,]𝐵))
2113, 14, 203syl 18 . . . 4 (𝜑𝐹 Fn (𝐴[,]𝐵))
22 fnresdm 6675 . . . 4 (𝐹 Fn (𝐴[,]𝐵) → (𝐹 ↾ (𝐴[,]𝐵)) = 𝐹)
23 isoeq1 7324 . . . 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 231 . 2 (𝜑𝐹 Isom < , 𝑂 ((𝐴[,]𝐵), (𝐹 “ (𝐴[,]𝐵))))
26 fnima 6686 . . 3 (𝐹 Fn (𝐴[,]𝐵) → (𝐹 “ (𝐴[,]𝐵)) = ran 𝐹)
27 isoeq5 7328 . . 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 231 1 (𝜑𝐹 Isom < , 𝑂 ((𝐴[,]𝐵), ran 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1533  wcel 2098  wral 3050  wss 3944   class class class wbr 5149   Or wor 5589  ran crn 5679  cres 5680  cima 5681   Fn wfn 6544  wf 6545  cfv 6549   Isom wiso 6550  (class class class)co 7419  cr 11139   < clt 11280  (,)cioo 13359  [,]cicc 13362  cnccncf 24840   D cdv 25836
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-cnex 11196  ax-resscn 11197  ax-pre-lttri 11214  ax-pre-lttrn 11215
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-po 5590  df-so 5591  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-isom 6558  df-ov 7422  df-oprab 7423  df-mpo 7424  df-er 8725  df-map 8847  df-en 8965  df-dom 8966  df-sdom 8967  df-pnf 11282  df-mnf 11283  df-xr 11284  df-ltxr 11285  df-le 11286  df-icc 13366  df-cncf 24842
This theorem is referenced by:  dvgt0  25981  dvlt0  25982
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