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Theorem leisorel 14362

Description: Version of isorel 7255 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		leiso 14361	. . . 4 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 ⊆ ℝ^*) → (𝐹 Isom < , < (𝐴, 𝐵) ↔ 𝐹 Isom ≤ , ≤ (𝐴, 𝐵)))
2	1	biimpcd 249	. . 3 ⊢ (𝐹 Isom < , < (𝐴, 𝐵) → ((𝐴 ⊆ ℝ^* ∧ 𝐵 ⊆ ℝ^*) → 𝐹 Isom ≤ , ≤ (𝐴, 𝐵)))
3		isorel 7255	. . . 4 ⊢ ((𝐹 Isom ≤ , ≤ (𝐴, 𝐵) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝐶 ≤ 𝐷 ↔ (𝐹‘𝐶) ≤ (𝐹‘𝐷)))
4	3	ex 412	. . 3 ⊢ (𝐹 Isom ≤ , ≤ (𝐴, 𝐵) → ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴) → (𝐶 ≤ 𝐷 ↔ (𝐹‘𝐶) ≤ (𝐹‘𝐷))))
5	2, 4	syl6 35	. 2 ⊢ (𝐹 Isom < , < (𝐴, 𝐵) → ((𝐴 ⊆ ℝ^* ∧ 𝐵 ⊆ ℝ^*) → ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴) → (𝐶 ≤ 𝐷 ↔ (𝐹‘𝐶) ≤ (𝐹‘𝐷)))))
6	5	3imp 1110	1 ⊢ ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ^* ∧ 𝐵 ⊆ ℝ^*) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝐶 ≤ 𝐷 ↔ (𝐹‘𝐶) ≤ (𝐹‘𝐷)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 ∈ wcel 2111 ⊆ wss 3897 class class class wbr 5086 ‘cfv 6476 Isom wiso 6477 ℝ^*cxr 11140 < clt 11141 ≤ cle 11142

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-sep 5229 ax-nul 5239 ax-pr 5365

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-ne 2929 df-ral 3048 df-rex 3057 df-rab 3396 df-v 3438 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-nul 4279 df-if 4471 df-sn 4572 df-pr 4574 df-op 4578 df-uni 4855 df-br 5087 df-opab 5149 df-id 5506 df-xp 5617 df-rel 5618 df-cnv 5619 df-co 5620 df-dm 5621 df-rn 5622 df-res 5623 df-ima 5624 df-iota 6432 df-fun 6478 df-fn 6479 df-f 6480 df-f1 6481 df-fo 6482 df-f1o 6483 df-fv 6484 df-isom 6485 df-le 11147

This theorem is referenced by: seqcoll 14366 isercolllem2 15568 isercoll 15570 summolem2a 15617 prodmolem2a 15836 xrhmeo 24866 fourierdlem52 46196

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