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Theorem dvgt0lem2 25891
Description: Lemma for dvgt0 25892 and dvlt0 25893. (Contributed by Mario Carneiro, 19-Feb-2015.)
Hypotheses
Ref Expression
dvgt0.a (πœ‘ β†’ 𝐴 ∈ ℝ)
dvgt0.b (πœ‘ β†’ 𝐡 ∈ ℝ)
dvgt0.f (πœ‘ β†’ 𝐹 ∈ ((𝐴[,]𝐡)–cn→ℝ))
dvgt0lem.d (πœ‘ β†’ (ℝ D 𝐹):(𝐴(,)𝐡)βŸΆπ‘†)
dvgt0lem.o 𝑂 Or ℝ
dvgt0lem.i (((πœ‘ ∧ (π‘₯ ∈ (𝐴[,]𝐡) ∧ 𝑦 ∈ (𝐴[,]𝐡))) ∧ π‘₯ < 𝑦) β†’ (πΉβ€˜π‘₯)𝑂(πΉβ€˜π‘¦))
Assertion
Ref Expression
dvgt0lem2 (πœ‘ β†’ 𝐹 Isom < , 𝑂 ((𝐴[,]𝐡), ran 𝐹))
Distinct variable groups:   π‘₯,𝑦,𝐴   π‘₯,𝑂,𝑦   πœ‘,π‘₯,𝑦   π‘₯,𝐡,𝑦   π‘₯,𝐹,𝑦
Allowed substitution hints:   𝑆(π‘₯,𝑦)

Proof of Theorem dvgt0lem2
StepHypRef Expression
1 dvgt0lem.i . . . . . 6 (((πœ‘ ∧ (π‘₯ ∈ (𝐴[,]𝐡) ∧ 𝑦 ∈ (𝐴[,]𝐡))) ∧ π‘₯ < 𝑦) β†’ (πΉβ€˜π‘₯)𝑂(πΉβ€˜π‘¦))
21ex 412 . . . . 5 ((πœ‘ ∧ (π‘₯ ∈ (𝐴[,]𝐡) ∧ 𝑦 ∈ (𝐴[,]𝐡))) β†’ (π‘₯ < 𝑦 β†’ (πΉβ€˜π‘₯)𝑂(πΉβ€˜π‘¦)))
32ralrimivva 3194 . . . 4 (πœ‘ β†’ βˆ€π‘₯ ∈ (𝐴[,]𝐡)βˆ€π‘¦ ∈ (𝐴[,]𝐡)(π‘₯ < 𝑦 β†’ (πΉβ€˜π‘₯)𝑂(πΉβ€˜π‘¦)))
4 dvgt0.a . . . . . . 7 (πœ‘ β†’ 𝐴 ∈ ℝ)
5 dvgt0.b . . . . . . 7 (πœ‘ β†’ 𝐡 ∈ ℝ)
6 iccssre 13412 . . . . . . 7 ((𝐴 ∈ ℝ ∧ 𝐡 ∈ ℝ) β†’ (𝐴[,]𝐡) βŠ† ℝ)
74, 5, 6syl2anc 583 . . . . . 6 (πœ‘ β†’ (𝐴[,]𝐡) βŠ† ℝ)
8 ltso 11298 . . . . . 6 < Or ℝ
9 soss 5601 . . . . . 6 ((𝐴[,]𝐡) βŠ† ℝ β†’ ( < Or ℝ β†’ < Or (𝐴[,]𝐡)))
107, 8, 9mpisyl 21 . . . . 5 (πœ‘ β†’ < Or (𝐴[,]𝐡))
11 dvgt0lem.o . . . . . 6 𝑂 Or ℝ
1211a1i 11 . . . . 5 (πœ‘ β†’ 𝑂 Or ℝ)
13 dvgt0.f . . . . . 6 (πœ‘ β†’ 𝐹 ∈ ((𝐴[,]𝐡)–cn→ℝ))
14 cncff 24768 . . . . . 6 (𝐹 ∈ ((𝐴[,]𝐡)–cn→ℝ) β†’ 𝐹:(𝐴[,]𝐡)βŸΆβ„)
1513, 14syl 17 . . . . 5 (πœ‘ β†’ 𝐹:(𝐴[,]𝐡)βŸΆβ„)
16 ssidd 4000 . . . . 5 (πœ‘ β†’ (𝐴[,]𝐡) βŠ† (𝐴[,]𝐡))
17 soisores 7320 . . . . 5 ((( < Or (𝐴[,]𝐡) ∧ 𝑂 Or ℝ) ∧ (𝐹:(𝐴[,]𝐡)βŸΆβ„ ∧ (𝐴[,]𝐡) βŠ† (𝐴[,]𝐡))) β†’ ((𝐹 β†Ύ (𝐴[,]𝐡)) Isom < , 𝑂 ((𝐴[,]𝐡), (𝐹 β€œ (𝐴[,]𝐡))) ↔ βˆ€π‘₯ ∈ (𝐴[,]𝐡)βˆ€π‘¦ ∈ (𝐴[,]𝐡)(π‘₯ < 𝑦 β†’ (πΉβ€˜π‘₯)𝑂(πΉβ€˜π‘¦))))
1810, 12, 15, 16, 17syl22anc 836 . . . 4 (πœ‘ β†’ ((𝐹 β†Ύ (𝐴[,]𝐡)) Isom < , 𝑂 ((𝐴[,]𝐡), (𝐹 β€œ (𝐴[,]𝐡))) ↔ βˆ€π‘₯ ∈ (𝐴[,]𝐡)βˆ€π‘¦ ∈ (𝐴[,]𝐡)(π‘₯ < 𝑦 β†’ (πΉβ€˜π‘₯)𝑂(πΉβ€˜π‘¦))))
193, 18mpbird 257 . . 3 (πœ‘ β†’ (𝐹 β†Ύ (𝐴[,]𝐡)) Isom < , 𝑂 ((𝐴[,]𝐡), (𝐹 β€œ (𝐴[,]𝐡))))
20 ffn 6711 . . . . 5 (𝐹:(𝐴[,]𝐡)βŸΆβ„ β†’ 𝐹 Fn (𝐴[,]𝐡))
2113, 14, 203syl 18 . . . 4 (πœ‘ β†’ 𝐹 Fn (𝐴[,]𝐡))
22 fnresdm 6663 . . . 4 (𝐹 Fn (𝐴[,]𝐡) β†’ (𝐹 β†Ύ (𝐴[,]𝐡)) = 𝐹)
23 isoeq1 7310 . . . 4 ((𝐹 β†Ύ (𝐴[,]𝐡)) = 𝐹 β†’ ((𝐹 β†Ύ (𝐴[,]𝐡)) Isom < , 𝑂 ((𝐴[,]𝐡), (𝐹 β€œ (𝐴[,]𝐡))) ↔ 𝐹 Isom < , 𝑂 ((𝐴[,]𝐡), (𝐹 β€œ (𝐴[,]𝐡)))))
2421, 22, 233syl 18 . . 3 (πœ‘ β†’ ((𝐹 β†Ύ (𝐴[,]𝐡)) Isom < , 𝑂 ((𝐴[,]𝐡), (𝐹 β€œ (𝐴[,]𝐡))) ↔ 𝐹 Isom < , 𝑂 ((𝐴[,]𝐡), (𝐹 β€œ (𝐴[,]𝐡)))))
2519, 24mpbid 231 . 2 (πœ‘ β†’ 𝐹 Isom < , 𝑂 ((𝐴[,]𝐡), (𝐹 β€œ (𝐴[,]𝐡))))
26 fnima 6674 . . 3 (𝐹 Fn (𝐴[,]𝐡) β†’ (𝐹 β€œ (𝐴[,]𝐡)) = ran 𝐹)
27 isoeq5 7314 . . 3 ((𝐹 β€œ (𝐴[,]𝐡)) = ran 𝐹 β†’ (𝐹 Isom < , 𝑂 ((𝐴[,]𝐡), (𝐹 β€œ (𝐴[,]𝐡))) ↔ 𝐹 Isom < , 𝑂 ((𝐴[,]𝐡), ran 𝐹)))
2821, 26, 273syl 18 . 2 (πœ‘ β†’ (𝐹 Isom < , 𝑂 ((𝐴[,]𝐡), (𝐹 β€œ (𝐴[,]𝐡))) ↔ 𝐹 Isom < , 𝑂 ((𝐴[,]𝐡), ran 𝐹)))
2925, 28mpbid 231 1 (πœ‘ β†’ 𝐹 Isom < , 𝑂 ((𝐴[,]𝐡), ran 𝐹))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   = wceq 1533   ∈ wcel 2098  βˆ€wral 3055   βŠ† wss 3943   class class class wbr 5141   Or wor 5580  ran crn 5670   β†Ύ cres 5671   β€œ cima 5672   Fn wfn 6532  βŸΆwf 6533  β€˜cfv 6537   Isom wiso 6538  (class class class)co 7405  β„cr 11111   < clt 11252  (,)cioo 13330  [,]cicc 13333  β€“cnβ†’ccncf 24751   D cdv 25747
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  ax-cnex 11168  ax-resscn 11169  ax-pre-lttri 11186  ax-pre-lttrn 11187
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-po 5581  df-so 5582  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-ov 7408  df-oprab 7409  df-mpo 7410  df-er 8705  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-icc 13337  df-cncf 24753
This theorem is referenced by:  dvgt0  25892  dvlt0  25893
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