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Theorem leisorel 10819
Description: Version of isorel 5811 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 997 . . 3 ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ*𝐵 ⊆ ℝ*) ∧ (𝐶𝐴𝐷𝐴)) → 𝐹 Isom < , < (𝐴, 𝐵))
2 simp3r 1026 . . 3 ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ*𝐵 ⊆ ℝ*) ∧ (𝐶𝐴𝐷𝐴)) → 𝐷𝐴)
3 simp3l 1025 . . 3 ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ*𝐵 ⊆ ℝ*) ∧ (𝐶𝐴𝐷𝐴)) → 𝐶𝐴)
4 isorel 5811 . . . 4 ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐷𝐴𝐶𝐴)) → (𝐷 < 𝐶 ↔ (𝐹𝐷) < (𝐹𝐶)))
54notbid 667 . . 3 ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐷𝐴𝐶𝐴)) → (¬ 𝐷 < 𝐶 ↔ ¬ (𝐹𝐷) < (𝐹𝐶)))
61, 2, 3, 5syl12anc 1236 . 2 ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ*𝐵 ⊆ ℝ*) ∧ (𝐶𝐴𝐷𝐴)) → (¬ 𝐷 < 𝐶 ↔ ¬ (𝐹𝐷) < (𝐹𝐶)))
7 simp2l 1023 . . . 4 ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ*𝐵 ⊆ ℝ*) ∧ (𝐶𝐴𝐷𝐴)) → 𝐴 ⊆ ℝ*)
87, 3sseldd 3158 . . 3 ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ*𝐵 ⊆ ℝ*) ∧ (𝐶𝐴𝐷𝐴)) → 𝐶 ∈ ℝ*)
97, 2sseldd 3158 . . 3 ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ*𝐵 ⊆ ℝ*) ∧ (𝐶𝐴𝐷𝐴)) → 𝐷 ∈ ℝ*)
10 xrlenlt 8024 . . 3 ((𝐶 ∈ ℝ*𝐷 ∈ ℝ*) → (𝐶𝐷 ↔ ¬ 𝐷 < 𝐶))
118, 9, 10syl2anc 411 . 2 ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ*𝐵 ⊆ ℝ*) ∧ (𝐶𝐴𝐷𝐴)) → (𝐶𝐷 ↔ ¬ 𝐷 < 𝐶))
12 simp2r 1024 . . . 4 ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ*𝐵 ⊆ ℝ*) ∧ (𝐶𝐴𝐷𝐴)) → 𝐵 ⊆ ℝ*)
13 isof1o 5810 . . . . . 6 (𝐹 Isom < , < (𝐴, 𝐵) → 𝐹:𝐴1-1-onto𝐵)
14 f1of 5463 . . . . . 6 (𝐹:𝐴1-1-onto𝐵𝐹:𝐴𝐵)
151, 13, 143syl 17 . . . . 5 ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ*𝐵 ⊆ ℝ*) ∧ (𝐶𝐴𝐷𝐴)) → 𝐹:𝐴𝐵)
1615, 3ffvelcdmd 5654 . . . 4 ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ*𝐵 ⊆ ℝ*) ∧ (𝐶𝐴𝐷𝐴)) → (𝐹𝐶) ∈ 𝐵)
1712, 16sseldd 3158 . . 3 ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ*𝐵 ⊆ ℝ*) ∧ (𝐶𝐴𝐷𝐴)) → (𝐹𝐶) ∈ ℝ*)
1815, 2ffvelcdmd 5654 . . . 4 ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ*𝐵 ⊆ ℝ*) ∧ (𝐶𝐴𝐷𝐴)) → (𝐹𝐷) ∈ 𝐵)
1912, 18sseldd 3158 . . 3 ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ*𝐵 ⊆ ℝ*) ∧ (𝐶𝐴𝐷𝐴)) → (𝐹𝐷) ∈ ℝ*)
20 xrlenlt 8024 . . 3 (((𝐹𝐶) ∈ ℝ* ∧ (𝐹𝐷) ∈ ℝ*) → ((𝐹𝐶) ≤ (𝐹𝐷) ↔ ¬ (𝐹𝐷) < (𝐹𝐶)))
2117, 19, 20syl2anc 411 . 2 ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ*𝐵 ⊆ ℝ*) ∧ (𝐶𝐴𝐷𝐴)) → ((𝐹𝐶) ≤ (𝐹𝐷) ↔ ¬ (𝐹𝐷) < (𝐹𝐶)))
226, 11, 213bitr4d 220 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 104  wb 105  w3a 978  wcel 2148  wss 3131   class class class wbr 4005  wf 5214  1-1-ontowf1o 5217  cfv 5218   Isom wiso 5219  *cxr 7993   < clt 7994  cle 7995
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-f1o 5225  df-fv 5226  df-isom 5227  df-le 8000
This theorem is referenced by:  seq3coll  10824  summodclem2a  11391  prodmodclem2a  11586
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