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Theorem leisorel 13244

Description: Version of isorel 6576 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 13243	. . . 4 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 ⊆ ℝ^*) → (𝐹 Isom < , < (𝐴, 𝐵) ↔ 𝐹 Isom ≤ , ≤ (𝐴, 𝐵)))
2	1	biimpcd 239	. . 3 ⊢ (𝐹 Isom < , < (𝐴, 𝐵) → ((𝐴 ⊆ ℝ^* ∧ 𝐵 ⊆ ℝ^*) → 𝐹 Isom ≤ , ≤ (𝐴, 𝐵)))
3		isorel 6576	. . . 4 ⊢ ((𝐹 Isom ≤ , ≤ (𝐴, 𝐵) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝐶 ≤ 𝐷 ↔ (𝐹‘𝐶) ≤ (𝐹‘𝐷)))
4	3	ex 450	. . 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 1256	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 196 ∧ wa 384 ∧ w3a 1037 ∈ wcel 1990 ⊆ wss 3574 class class class wbr 4653 ‘cfv 5888 Isom wiso 5889 ℝ^*cxr 10073 < clt 10074 ≤ cle 10075

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-le 10080

This theorem is referenced by: seqcoll 13248 isercolllem2 14396 isercoll 14398 summolem2a 14446 prodmolem2a 14664 xrhmeo 22745 fourierdlem52 40375

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