Theorem leisorel 11147

Description: Version of isorel 5959 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 1024	. . 3 ⊢ ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ* ∧ 𝐵 ⊆ ℝ*) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → 𝐹 Isom < , < (𝐴, 𝐵))
2		simp3r 1053	. . 3 ⊢ ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ* ∧ 𝐵 ⊆ ℝ*) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → 𝐷 ∈ 𝐴)
3		simp3l 1052	. . 3 ⊢ ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ* ∧ 𝐵 ⊆ ℝ*) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → 𝐶 ∈ 𝐴)
4		isorel 5959	. . . 4 ⊢ ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐷 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐷 < 𝐶 ↔ (𝐹‘𝐷) < (𝐹‘𝐶)))
5	4	notbid 673	. . 3 ⊢ ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐷 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (¬ 𝐷 < 𝐶 ↔ ¬ (𝐹‘𝐷) < (𝐹‘𝐶)))
6	1, 2, 3, 5	syl12anc 1272	. 2 ⊢ ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ* ∧ 𝐵 ⊆ ℝ*) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (¬ 𝐷 < 𝐶 ↔ ¬ (𝐹‘𝐷) < (𝐹‘𝐶)))
7		simp2l 1050	. . . 4 ⊢ ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ* ∧ 𝐵 ⊆ ℝ*) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → 𝐴 ⊆ ℝ*)
8	7, 3	sseldd 3229	. . 3 ⊢ ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ* ∧ 𝐵 ⊆ ℝ*) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → 𝐶 ∈ ℝ*)
9	7, 2	sseldd 3229	. . 3 ⊢ ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ* ∧ 𝐵 ⊆ ℝ*) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → 𝐷 ∈ ℝ*)
10		xrlenlt 8286	. . 3 ⊢ ((𝐶 ∈ ℝ* ∧ 𝐷 ∈ ℝ*) → (𝐶 ≤ 𝐷 ↔ ¬ 𝐷 < 𝐶))
11	8, 9, 10	syl2anc 411	. 2 ⊢ ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ* ∧ 𝐵 ⊆ ℝ*) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝐶 ≤ 𝐷 ↔ ¬ 𝐷 < 𝐶))
12		simp2r 1051	. . . 4 ⊢ ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ* ∧ 𝐵 ⊆ ℝ*) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → 𝐵 ⊆ ℝ*)
13		isof1o 5958	. . . . . 6 ⊢ (𝐹 Isom < , < (𝐴, 𝐵) → 𝐹:𝐴–1-1-onto→𝐵)
14		f1of 5592	. . . . . 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 5791	. . . 4 ⊢ ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ* ∧ 𝐵 ⊆ ℝ*) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝐹‘𝐶) ∈ 𝐵)
17	12, 16	sseldd 3229	. . 3 ⊢ ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ* ∧ 𝐵 ⊆ ℝ*) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝐹‘𝐶) ∈ ℝ*)
18	15, 2	ffvelcdmd 5791	. . . 4 ⊢ ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ* ∧ 𝐵 ⊆ ℝ*) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝐹‘𝐷) ∈ 𝐵)
19	12, 18	sseldd 3229	. . 3 ⊢ ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ* ∧ 𝐵 ⊆ ℝ*) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝐹‘𝐷) ∈ ℝ*)
20		xrlenlt 8286	. . 3 ⊢ (((𝐹‘𝐶) ∈ ℝ* ∧ (𝐹‘𝐷) ∈ ℝ*) → ((𝐹‘𝐶) ≤ (𝐹‘𝐷) ↔ ¬ (𝐹‘𝐷) < (𝐹‘𝐶)))
21	17, 19, 20	syl2anc 411	. 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)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1005   ∈ wcel 2202   ⊆ wss 3201   class class class wbr 4093  ⟶wf 5329  –1-1-onto→wf1o 5332  ‘cfv 5333   Isom wiso 5334  ℝ^*cxr 8255   < clt 8256   ≤ cle 8257

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-sbc 3033  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-f1o 5340  df-fv 5341  df-isom 5342  df-le 8262

This theorem is referenced by:  seq3coll  11152  summodclem2a  12005  prodmodclem2a  12200