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Theorem leisorel 14385

Description: Version of isorel 7267 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 14384	. . . 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 7267	. . . 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 2109 ⊆ wss 3905 class class class wbr 5095 ‘cfv 6486 Isom wiso 6487 ℝ^*cxr 11167 < clt 11168 ≤ cle 11169

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5238 ax-nul 5248 ax-pr 5374

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-ne 2926 df-ral 3045 df-rex 3054 df-rab 3397 df-v 3440 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-nul 4287 df-if 4479 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4862 df-br 5096 df-opab 5158 df-id 5518 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-isom 6495 df-le 11174

This theorem is referenced by: seqcoll 14389 isercolllem2 15591 isercoll 15593 summolem2a 15640 prodmolem2a 15859 xrhmeo 24860 fourierdlem52 46143

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