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Theorem leisorel 10207
Description: Version of isorel 5569 for strictly increasing functions on the reals. (Contributed by Mario Carneiro, 6-Apr-2015.) (Revised by Mario Carneiro, 9-Sep-2015.)
Assertion
Ref Expression
leisorel ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ*𝐵 ⊆ ℝ*) ∧ (𝐶𝐴𝐷𝐴)) → (𝐶𝐷 ↔ (𝐹𝐶) ≤ (𝐹𝐷)))

Proof of Theorem leisorel
StepHypRef Expression
1 simp1 943 . . 3 ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ*𝐵 ⊆ ℝ*) ∧ (𝐶𝐴𝐷𝐴)) → 𝐹 Isom < , < (𝐴, 𝐵))
2 simp3r 972 . . 3 ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ*𝐵 ⊆ ℝ*) ∧ (𝐶𝐴𝐷𝐴)) → 𝐷𝐴)
3 simp3l 971 . . 3 ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ*𝐵 ⊆ ℝ*) ∧ (𝐶𝐴𝐷𝐴)) → 𝐶𝐴)
4 isorel 5569 . . . 4 ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐷𝐴𝐶𝐴)) → (𝐷 < 𝐶 ↔ (𝐹𝐷) < (𝐹𝐶)))
54notbid 627 . . 3 ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐷𝐴𝐶𝐴)) → (¬ 𝐷 < 𝐶 ↔ ¬ (𝐹𝐷) < (𝐹𝐶)))
61, 2, 3, 5syl12anc 1172 . 2 ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ*𝐵 ⊆ ℝ*) ∧ (𝐶𝐴𝐷𝐴)) → (¬ 𝐷 < 𝐶 ↔ ¬ (𝐹𝐷) < (𝐹𝐶)))
7 simp2l 969 . . . 4 ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ*𝐵 ⊆ ℝ*) ∧ (𝐶𝐴𝐷𝐴)) → 𝐴 ⊆ ℝ*)
87, 3sseldd 3024 . . 3 ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ*𝐵 ⊆ ℝ*) ∧ (𝐶𝐴𝐷𝐴)) → 𝐶 ∈ ℝ*)
97, 2sseldd 3024 . . 3 ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ*𝐵 ⊆ ℝ*) ∧ (𝐶𝐴𝐷𝐴)) → 𝐷 ∈ ℝ*)
10 xrlenlt 7530 . . 3 ((𝐶 ∈ ℝ*𝐷 ∈ ℝ*) → (𝐶𝐷 ↔ ¬ 𝐷 < 𝐶))
118, 9, 10syl2anc 403 . 2 ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ*𝐵 ⊆ ℝ*) ∧ (𝐶𝐴𝐷𝐴)) → (𝐶𝐷 ↔ ¬ 𝐷 < 𝐶))
12 simp2r 970 . . . 4 ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ*𝐵 ⊆ ℝ*) ∧ (𝐶𝐴𝐷𝐴)) → 𝐵 ⊆ ℝ*)
13 isof1o 5568 . . . . . 6 (𝐹 Isom < , < (𝐴, 𝐵) → 𝐹:𝐴1-1-onto𝐵)
14 f1of 5237 . . . . . 6 (𝐹:𝐴1-1-onto𝐵𝐹:𝐴𝐵)
151, 13, 143syl 17 . . . . 5 ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ*𝐵 ⊆ ℝ*) ∧ (𝐶𝐴𝐷𝐴)) → 𝐹:𝐴𝐵)
1615, 3ffvelrnd 5419 . . . 4 ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ*𝐵 ⊆ ℝ*) ∧ (𝐶𝐴𝐷𝐴)) → (𝐹𝐶) ∈ 𝐵)
1712, 16sseldd 3024 . . 3 ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ*𝐵 ⊆ ℝ*) ∧ (𝐶𝐴𝐷𝐴)) → (𝐹𝐶) ∈ ℝ*)
1815, 2ffvelrnd 5419 . . . 4 ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ*𝐵 ⊆ ℝ*) ∧ (𝐶𝐴𝐷𝐴)) → (𝐹𝐷) ∈ 𝐵)
1912, 18sseldd 3024 . . 3 ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ*𝐵 ⊆ ℝ*) ∧ (𝐶𝐴𝐷𝐴)) → (𝐹𝐷) ∈ ℝ*)
20 xrlenlt 7530 . . 3 (((𝐹𝐶) ∈ ℝ* ∧ (𝐹𝐷) ∈ ℝ*) → ((𝐹𝐶) ≤ (𝐹𝐷) ↔ ¬ (𝐹𝐷) < (𝐹𝐶)))
2117, 19, 20syl2anc 403 . 2 ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ*𝐵 ⊆ ℝ*) ∧ (𝐶𝐴𝐷𝐴)) → ((𝐹𝐶) ≤ (𝐹𝐷) ↔ ¬ (𝐹𝐷) < (𝐹𝐶)))
226, 11, 213bitr4d 218 1 ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ*𝐵 ⊆ ℝ*) ∧ (𝐶𝐴𝐷𝐴)) → (𝐶𝐷 ↔ (𝐹𝐶) ≤ (𝐹𝐷)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wb 103  w3a 924  wcel 1438  wss 2997   class class class wbr 3837  wf 4998  1-1-ontowf1o 5001  cfv 5002   Isom wiso 5003  *cxr 7500   < clt 7501  cle 7502
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2839  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-f1o 5009  df-fv 5010  df-isom 5011  df-le 7507
This theorem is referenced by:  iseqcoll  10212  isummolem2a  10735
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