Theorem leisorel 11267

Description: Version of isorel 6004 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 1028	. . 3 |- ( ( F Isom < , < ( A , B ) /\ ( A C_ RR* /\ B C_ RR* ) /\ ( C e. A /\ D e. A ) ) -> F Isom < , < ( A , B ) )
2		simp3r 1057	. . 3 |- ( ( F Isom < , < ( A , B ) /\ ( A C_ RR* /\ B C_ RR* ) /\ ( C e. A /\ D e. A ) ) -> D e. A )
3		simp3l 1056	. . 3 |- ( ( F Isom < , < ( A , B ) /\ ( A C_ RR* /\ B C_ RR* ) /\ ( C e. A /\ D e. A ) ) -> C e. A )
4		isorel 6004	. . . 4 |- ( ( F Isom < , < ( A , B ) /\ ( D e. A /\ C e. A ) ) -> ( D < C <-> ( F ` D ) < ( F ` C ) ) )
5	4	notbid 677	. . 3 |- ( ( F Isom < , < ( A , B ) /\ ( D e. A /\ C e. A ) ) -> ( -. D < C <-> -. ( F ` D ) < ( F ` C ) ) )
6	1, 2, 3, 5	syl12anc 1276	. 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 1054	. . . 4 |- ( ( F Isom < , < ( A , B ) /\ ( A C_ RR* /\ B C_ RR* ) /\ ( C e. A /\ D e. A ) ) -> A C_ RR* )
8	7, 3	sseldd 3249	. . 3 |- ( ( F Isom < , < ( A , B ) /\ ( A C_ RR* /\ B C_ RR* ) /\ ( C e. A /\ D e. A ) ) -> C e. RR* )
9	7, 2	sseldd 3249	. . 3 |- ( ( F Isom < , < ( A , B ) /\ ( A C_ RR* /\ B C_ RR* ) /\ ( C e. A /\ D e. A ) ) -> D e. RR* )
10		xrlenlt 8380	. . 3 |- ( ( C e. RR* /\ D e. RR* ) -> ( C <_ D <-> -. D < C ) )
11	8, 9, 10	syl2anc 415	. 2 |- ( ( F Isom < , < ( A , B ) /\ ( A C_ RR* /\ B C_ RR* ) /\ ( C e. A /\ D e. A ) ) -> ( C <_ D <-> -. D < C ) )
12		simp2r 1055	. . . 4 |- ( ( F Isom < , < ( A , B ) /\ ( A C_ RR* /\ B C_ RR* ) /\ ( C e. A /\ D e. A ) ) -> B C_ RR* )
13		isof1o 6003	. . . . . 6 |- ( F Isom < , < ( A , B ) -> F : A -1-1-onto-> B )
14		f1of 5634	. . . . . 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 5835	. . . 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 3249	. . 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 5835	. . . 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 3249	. . 3 |- ( ( F Isom < , < ( A , B ) /\ ( A C_ RR* /\ B C_ RR* ) /\ ( C e. A /\ D e. A ) ) -> ( F ` D ) e. RR* )
20		xrlenlt 8380	. . 3 |- ( ( ( F ` C ) e. RR* /\ ( F ` D ) e. RR* ) -> ( ( F ` C ) <_ ( F ` D ) <-> -. ( F ` D ) < ( F ` C ) ) )
21	17, 19, 20	syl2anc 415	. 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 1009   e. wcel 2209   C_ wss 3220   class class class wbr 4125   -->wf 5368   -1-1-onto->wf1o 5371   ` cfv 5372   Isom wiso 5373   RR*cxr 8349   < clt 8350   <_ cle 8351

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-f1o 5379  df-fv 5380  df-isom 5381  df-le 8356

This theorem is referenced by:  seq3coll  11272  summodclem2a  12126  prodmodclem2a  12321