Theorem leisorel

Description: Version of isorel for strictly increasing functions on the reals. (Contributed by Mario Carneiro, 6-Apr-2015) (Revised by Mario Carneiro, 9-Sep-2015)

	Ref	Expression
Assertion	leisorel	$$⊢\left(F{Isom}_{<,<}\left(A,B\right)\wedge \left(A\subseteq {ℝ}^{*}\wedge B\subseteq {ℝ}^{*}\right)\wedge \left(C\in A\wedge D\in A\right)\right)\to \left(C\le D\leftrightarrow F\left(C\right)\le F\left(D\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		leiso	$$⊢\left(A\subseteq {ℝ}^{*}\wedge B\subseteq {ℝ}^{*}\right)\to \left(F{Isom}_{<,<}\left(A,B\right)\leftrightarrow F{Isom}_{\le ,\le }\left(A,B\right)\right)$$
2	1	biimpcd	$$⊢F{Isom}_{<,<}\left(A,B\right)\to \left(\left(A\subseteq {ℝ}^{*}\wedge B\subseteq {ℝ}^{*}\right)\to F{Isom}_{\le ,\le }\left(A,B\right)\right)$$
3		isorel	$$⊢\left(F{Isom}_{\le ,\le }\left(A,B\right)\wedge \left(C\in A\wedge D\in A\right)\right)\to \left(C\le D\leftrightarrow F\left(C\right)\le F\left(D\right)\right)$$
4	3	ex	$$⊢F{Isom}_{\le ,\le }\left(A,B\right)\to \left(\left(C\in A\wedge D\in A\right)\to \left(C\le D\leftrightarrow F\left(C\right)\le F\left(D\right)\right)\right)$$
5	2 4	syl6	$$⊢F{Isom}_{<,<}\left(A,B\right)\to \left(\left(A\subseteq {ℝ}^{*}\wedge B\subseteq {ℝ}^{*}\right)\to \left(\left(C\in A\wedge D\in A\right)\to \left(C\le D\leftrightarrow F\left(C\right)\le F\left(D\right)\right)\right)\right)$$
6	5	3imp	$$⊢\left(F{Isom}_{<,<}\left(A,B\right)\wedge \left(A\subseteq {ℝ}^{*}\wedge B\subseteq {ℝ}^{*}\right)\wedge \left(C\in A\wedge D\in A\right)\right)\to \left(C\le D\leftrightarrow F\left(C\right)\le F\left(D\right)\right)$$