Theorem leisorel 11305

Description: Version of isorel 6014 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*) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝐶 ≤ 𝐷 ↔ (𝐹‘𝐶) ≤ (𝐹‘𝐷)))

Proof of Theorem leisorel

Step	Hyp	Ref	Expression
1		simp1 1028	. . 3 ⊢ ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ* ∧ 𝐵 ⊆ ℝ*) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → 𝐹 Isom < , < (𝐴, 𝐵))
2		simp3r 1057	. . 3 ⊢ ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ* ∧ 𝐵 ⊆ ℝ*) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → 𝐷 ∈ 𝐴)
3		simp3l 1056	. . 3 ⊢ ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ* ∧ 𝐵 ⊆ ℝ*) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → 𝐶 ∈ 𝐴)
4		isorel 6014	. . . 4 ⊢ ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐷 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐷 < 𝐶 ↔ (𝐹‘𝐷) < (𝐹‘𝐶)))
5	4	notbid 677	. . 3 ⊢ ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐷 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (¬ 𝐷 < 𝐶 ↔ ¬ (𝐹‘𝐷) < (𝐹‘𝐶)))
6	1, 2, 3, 5	syl12anc 1276	. 2 ⊢ ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ* ∧ 𝐵 ⊆ ℝ*) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (¬ 𝐷 < 𝐶 ↔ ¬ (𝐹‘𝐷) < (𝐹‘𝐶)))
7		simp2l 1054	. . . 4 ⊢ ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ* ∧ 𝐵 ⊆ ℝ*) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → 𝐴 ⊆ ℝ*)
8	7, 3	sseldd 3249	. . 3 ⊢ ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ* ∧ 𝐵 ⊆ ℝ*) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → 𝐶 ∈ ℝ*)
9	7, 2	sseldd 3249	. . 3 ⊢ ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ* ∧ 𝐵 ⊆ ℝ*) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → 𝐷 ∈ ℝ*)
10		xrlenlt 8391	. . 3 ⊢ ((𝐶 ∈ ℝ* ∧ 𝐷 ∈ ℝ*) → (𝐶 ≤ 𝐷 ↔ ¬ 𝐷 < 𝐶))
11	8, 9, 10	syl2anc 415	. 2 ⊢ ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ* ∧ 𝐵 ⊆ ℝ*) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝐶 ≤ 𝐷 ↔ ¬ 𝐷 < 𝐶))
12		simp2r 1055	. . . 4 ⊢ ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ* ∧ 𝐵 ⊆ ℝ*) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → 𝐵 ⊆ ℝ*)
13		isof1o 6013	. . . . . 6 ⊢ (𝐹 Isom < , < (𝐴, 𝐵) → 𝐹:𝐴–1-1-onto→𝐵)
14		f1of 5639	. . . . . 6 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹:𝐴⟶𝐵)
15	1, 13, 14	3syl 17	. . . . 5 ⊢ ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ* ∧ 𝐵 ⊆ ℝ*) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → 𝐹:𝐴⟶𝐵)
16	15, 3	ffvelcdmd 5844	. . . 4 ⊢ ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ* ∧ 𝐵 ⊆ ℝ*) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝐹‘𝐶) ∈ 𝐵)
17	12, 16	sseldd 3249	. . 3 ⊢ ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ* ∧ 𝐵 ⊆ ℝ*) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝐹‘𝐶) ∈ ℝ*)
18	15, 2	ffvelcdmd 5844	. . . 4 ⊢ ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ* ∧ 𝐵 ⊆ ℝ*) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝐹‘𝐷) ∈ 𝐵)
19	12, 18	sseldd 3249	. . 3 ⊢ ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ* ∧ 𝐵 ⊆ ℝ*) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝐹‘𝐷) ∈ ℝ*)
20		xrlenlt 8391	. . 3 ⊢ (((𝐹‘𝐶) ∈ ℝ* ∧ (𝐹‘𝐷) ∈ ℝ*) → ((𝐹‘𝐶) ≤ (𝐹‘𝐷) ↔ ¬ (𝐹‘𝐷) < (𝐹‘𝐶)))
21	17, 19, 20	syl2anc 415	. 2 ⊢ ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ* ∧ 𝐵 ⊆ ℝ*) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ((𝐹‘𝐶) ≤ (𝐹‘𝐷) ↔ ¬ (𝐹‘𝐷) < (𝐹‘𝐶)))
22	6, 11, 21	3bitr4d 220	1 ⊢ ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ* ∧ 𝐵 ⊆ ℝ*) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝐶 ≤ 𝐷 ↔ (𝐹‘𝐶) ≤ (𝐹‘𝐷)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1009   ∈ wcel 2209   ⊆ wss 3220   class class class wbr 4130  ⟶wf 5373  –1-1-onto→wf1o 5376  ‘cfv 5377   Isom wiso 5378  ℝ^*cxr 8360   < clt 8361   ≤ cle 8362

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-f1o 5384  df-fv 5385  df-isom 5386  df-le 8367

This theorem is used by:  seq3coll  11310  summodclem2a  12167  prodmodclem2a  12362