Theorem leisorel 11213

Description: Version of isorel 5983 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 5983	. . . 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 3241	. . 3 ⊢ ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ* ∧ 𝐵 ⊆ ℝ*) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → 𝐶 ∈ ℝ*)
9	7, 2	sseldd 3241	. . 3 ⊢ ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ* ∧ 𝐵 ⊆ ℝ*) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → 𝐷 ∈ ℝ*)
10		xrlenlt 8340	. . 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 5982	. . . . . 6 ⊢ (𝐹 Isom < , < (𝐴, 𝐵) → 𝐹:𝐴–1-1-onto→𝐵)
14		f1of 5616	. . . . . 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 5815	. . . 4 ⊢ ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ* ∧ 𝐵 ⊆ ℝ*) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝐹‘𝐶) ∈ 𝐵)
17	12, 16	sseldd 3241	. . 3 ⊢ ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ* ∧ 𝐵 ⊆ ℝ*) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝐹‘𝐶) ∈ ℝ*)
18	15, 2	ffvelcdmd 5815	. . . 4 ⊢ ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ* ∧ 𝐵 ⊆ ℝ*) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝐹‘𝐷) ∈ 𝐵)
19	12, 18	sseldd 3241	. . 3 ⊢ ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ* ∧ 𝐵 ⊆ ℝ*) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝐹‘𝐷) ∈ ℝ*)
20		xrlenlt 8340	. . 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 2205   ⊆ wss 3213   class class class wbr 4111  ⟶wf 5350  –1-1-onto→wf1o 5353  ‘cfv 5354   Isom wiso 5355  ℝ^*cxr 8309   < clt 8310   ≤ cle 8311

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 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3045  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-br 4112  df-opab 4174  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-f1o 5361  df-fv 5362  df-isom 5363  df-le 8316

This theorem is referenced by:  seq3coll  11218  summodclem2a  12071  prodmodclem2a  12266