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Theorem dvgt0lem2 23886
Description: Lemma for dvgt0 23887 and dvlt0 23888. (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 449 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝑥 < 𝑦 → (𝐹𝑥)𝑂(𝐹𝑦)))
32ralrimivva 3073 . . . 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 12369 . . . . . . 7 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
74, 5, 6syl2anc 696 . . . . . 6 (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
8 ltso 10231 . . . . . 6 < Or ℝ
9 soss 5157 . . . . . 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 22818 . . . . . 6 (𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ) → 𝐹:(𝐴[,]𝐵)⟶ℝ)
1513, 14syl 17 . . . . 5 (𝜑𝐹:(𝐴[,]𝐵)⟶ℝ)
16 ssid 3730 . . . . . 6 (𝐴[,]𝐵) ⊆ (𝐴[,]𝐵)
1716a1i 11 . . . . 5 (𝜑 → (𝐴[,]𝐵) ⊆ (𝐴[,]𝐵))
18 soisores 6692 . . . . 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))))
1910, 12, 15, 17, 18syl22anc 1440 . . . 4 (𝜑 → ((𝐹 ↾ (𝐴[,]𝐵)) Isom < , 𝑂 ((𝐴[,]𝐵), (𝐹 “ (𝐴[,]𝐵))) ↔ ∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(𝑥 < 𝑦 → (𝐹𝑥)𝑂(𝐹𝑦))))
203, 19mpbird 247 . . 3 (𝜑 → (𝐹 ↾ (𝐴[,]𝐵)) Isom < , 𝑂 ((𝐴[,]𝐵), (𝐹 “ (𝐴[,]𝐵))))
21 ffn 6158 . . . . 5 (𝐹:(𝐴[,]𝐵)⟶ℝ → 𝐹 Fn (𝐴[,]𝐵))
2213, 14, 213syl 18 . . . 4 (𝜑𝐹 Fn (𝐴[,]𝐵))
23 fnresdm 6113 . . . 4 (𝐹 Fn (𝐴[,]𝐵) → (𝐹 ↾ (𝐴[,]𝐵)) = 𝐹)
24 isoeq1 6682 . . . 4 ((𝐹 ↾ (𝐴[,]𝐵)) = 𝐹 → ((𝐹 ↾ (𝐴[,]𝐵)) Isom < , 𝑂 ((𝐴[,]𝐵), (𝐹 “ (𝐴[,]𝐵))) ↔ 𝐹 Isom < , 𝑂 ((𝐴[,]𝐵), (𝐹 “ (𝐴[,]𝐵)))))
2522, 23, 243syl 18 . . 3 (𝜑 → ((𝐹 ↾ (𝐴[,]𝐵)) Isom < , 𝑂 ((𝐴[,]𝐵), (𝐹 “ (𝐴[,]𝐵))) ↔ 𝐹 Isom < , 𝑂 ((𝐴[,]𝐵), (𝐹 “ (𝐴[,]𝐵)))))
2620, 25mpbid 222 . 2 (𝜑𝐹 Isom < , 𝑂 ((𝐴[,]𝐵), (𝐹 “ (𝐴[,]𝐵))))
27 fnima 6123 . . 3 (𝐹 Fn (𝐴[,]𝐵) → (𝐹 “ (𝐴[,]𝐵)) = ran 𝐹)
28 isoeq5 6686 . . 3 ((𝐹 “ (𝐴[,]𝐵)) = ran 𝐹 → (𝐹 Isom < , 𝑂 ((𝐴[,]𝐵), (𝐹 “ (𝐴[,]𝐵))) ↔ 𝐹 Isom < , 𝑂 ((𝐴[,]𝐵), ran 𝐹)))
2922, 27, 283syl 18 . 2 (𝜑 → (𝐹 Isom < , 𝑂 ((𝐴[,]𝐵), (𝐹 “ (𝐴[,]𝐵))) ↔ 𝐹 Isom < , 𝑂 ((𝐴[,]𝐵), ran 𝐹)))
3026, 29mpbid 222 1 (𝜑𝐹 Isom < , 𝑂 ((𝐴[,]𝐵), ran 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1596  wcel 2103  wral 3014  wss 3680   class class class wbr 4760   Or wor 5138  ran crn 5219  cres 5220  cima 5221   Fn wfn 5996  wf 5997  cfv 6001   Isom wiso 6002  (class class class)co 6765  cr 10048   < clt 10187  (,)cioo 12289  [,]cicc 12292  cnccncf 22801   D cdv 23747
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066  ax-cnex 10105  ax-resscn 10106  ax-pre-lttri 10123  ax-pre-lttrn 10124
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-nel 3000  df-ral 3019  df-rex 3020  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-br 4761  df-opab 4821  df-mpt 4838  df-id 5128  df-po 5139  df-so 5140  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-isom 6010  df-ov 6768  df-oprab 6769  df-mpt2 6770  df-er 7862  df-map 7976  df-en 8073  df-dom 8074  df-sdom 8075  df-pnf 10189  df-mnf 10190  df-xr 10191  df-ltxr 10192  df-le 10193  df-icc 12296  df-cncf 22803
This theorem is referenced by:  dvgt0  23887  dvlt0  23888
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