Theorem leisorel 11100

Description: Version of isorel 5948 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 C_ RR* /\ B C_ RR* ) /\ ( C e. A /\ D e. A ) ) -> ( C <_ D <-> ( F ` C ) <_ ( F ` D ) ) )

Proof of Theorem leisorel

Step	Hyp	Ref	Expression
1		simp1 1023	. . 3 |- ( ( F Isom < , < ( A , B ) /\ ( A C_ RR* /\ B C_ RR* ) /\ ( C e. A /\ D e. A ) ) -> F Isom < , < ( A , B ) )
2		simp3r 1052	. . 3 |- ( ( F Isom < , < ( A , B ) /\ ( A C_ RR* /\ B C_ RR* ) /\ ( C e. A /\ D e. A ) ) -> D e. A )
3		simp3l 1051	. . 3 |- ( ( F Isom < , < ( A , B ) /\ ( A C_ RR* /\ B C_ RR* ) /\ ( C e. A /\ D e. A ) ) -> C e. A )
4		isorel 5948	. . . 4 |- ( ( F Isom < , < ( A , B ) /\ ( D e. A /\ C e. A ) ) -> ( D < C <-> ( F ` D ) < ( F ` C ) ) )
5	4	notbid 673	. . 3 |- ( ( F Isom < , < ( A , B ) /\ ( D e. A /\ C e. A ) ) -> ( -. D < C <-> -. ( F ` D ) < ( F ` C ) ) )
6	1, 2, 3, 5	syl12anc 1271	. 2 |- ( ( F Isom < , < ( A , B ) /\ ( A C_ RR* /\ B C_ RR* ) /\ ( C e. A /\ D e. A ) ) -> ( -. D < C <-> -. ( F ` D ) < ( F ` C ) ) )
7		simp2l 1049	. . . 4 |- ( ( F Isom < , < ( A , B ) /\ ( A C_ RR* /\ B C_ RR* ) /\ ( C e. A /\ D e. A ) ) -> A C_ RR* )
8	7, 3	sseldd 3228	. . 3 |- ( ( F Isom < , < ( A , B ) /\ ( A C_ RR* /\ B C_ RR* ) /\ ( C e. A /\ D e. A ) ) -> C e. RR* )
9	7, 2	sseldd 3228	. . 3 |- ( ( F Isom < , < ( A , B ) /\ ( A C_ RR* /\ B C_ RR* ) /\ ( C e. A /\ D e. A ) ) -> D e. RR* )
10		xrlenlt 8243	. . 3 |- ( ( C e. RR* /\ D e. RR* ) -> ( C <_ D <-> -. D < C ) )
11	8, 9, 10	syl2anc 411	. 2 |- ( ( F Isom < , < ( A , B ) /\ ( A C_ RR* /\ B C_ RR* ) /\ ( C e. A /\ D e. A ) ) -> ( C <_ D <-> -. D < C ) )
12		simp2r 1050	. . . 4 |- ( ( F Isom < , < ( A , B ) /\ ( A C_ RR* /\ B C_ RR* ) /\ ( C e. A /\ D e. A ) ) -> B C_ RR* )
13		isof1o 5947	. . . . . 6 |- ( F Isom < , < ( A , B ) -> F : A -1-1-onto-> B )
14		f1of 5583	. . . . . 6 |- ( F : A -1-1-onto-> B -> F : A --> B )
15	1, 13, 14	3syl 17	. . . . 5 |- ( ( F Isom < , < ( A , B ) /\ ( A C_ RR* /\ B C_ RR* ) /\ ( C e. A /\ D e. A ) ) -> F : A --> B )
16	15, 3	ffvelcdmd 5783	. . . 4 |- ( ( F Isom < , < ( A , B ) /\ ( A C_ RR* /\ B C_ RR* ) /\ ( C e. A /\ D e. A ) ) -> ( F ` C ) e. B )
17	12, 16	sseldd 3228	. . 3 |- ( ( F Isom < , < ( A , B ) /\ ( A C_ RR* /\ B C_ RR* ) /\ ( C e. A /\ D e. A ) ) -> ( F ` C ) e. RR* )
18	15, 2	ffvelcdmd 5783	. . . 4 |- ( ( F Isom < , < ( A , B ) /\ ( A C_ RR* /\ B C_ RR* ) /\ ( C e. A /\ D e. A ) ) -> ( F ` D ) e. B )
19	12, 18	sseldd 3228	. . 3 |- ( ( F Isom < , < ( A , B ) /\ ( A C_ RR* /\ B C_ RR* ) /\ ( C e. A /\ D e. A ) ) -> ( F ` D ) e. RR* )
20		xrlenlt 8243	. . 3 |- ( ( ( F ` C ) e. RR* /\ ( F ` D ) e. RR* ) -> ( ( F ` C ) <_ ( F ` D ) <-> -. ( F ` D ) < ( F ` C ) ) )
21	17, 19, 20	syl2anc 411	. 2 |- ( ( F Isom < , < ( A , B ) /\ ( A C_ RR* /\ B C_ RR* ) /\ ( C e. A /\ D e. A ) ) -> ( ( F ` C ) <_ ( F ` D ) <-> -. ( F ` D ) < ( F ` C ) ) )
22	6, 11, 21	3bitr4d 220	1 |- ( ( F Isom < , < ( A , B ) /\ ( A C_ RR* /\ B C_ RR* ) /\ ( C e. A /\ D e. A ) ) -> ( C <_ D <-> ( F ` C ) <_ ( F ` D ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1004   e. wcel 2202   C_ wss 3200   class class class wbr 4088   -->wf 5322   -1-1-onto->wf1o 5325   ` cfv 5326   Isom wiso 5327   RR*cxr 8212   < clt 8213   <_ cle 8214

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-f1o 5333  df-fv 5334  df-isom 5335  df-le 8219

This theorem is referenced by:  seq3coll  11105  summodclem2a  11941  prodmodclem2a  12136