Theorem xrnegiso 11031

Description: Negation is an order anti-isomorphism of the extended reals, which is its own inverse. (Contributed by Jim Kingdon, 2-May-2023.)

Hypothesis

Ref	Expression
xrnegiso.1	|- F = ( x e. RR* |-> -e x )

Assertion

Ref	Expression
xrnegiso	|- ( F Isom < , `' < ( RR* , RR* ) /\ `' F = F )

Proof of Theorem xrnegiso

Dummy variables y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		xrnegiso.1	. . . . . 6 |- F = ( x e. RR* |-> -e x )
2		simpr 109	. . . . . . 7 |- ( ( T. /\ x e. RR* ) -> x e. RR* )
3	2	xnegcld 9638	. . . . . 6 |- ( ( T. /\ x e. RR* ) -> -e x e. RR* )
4		simpr 109	. . . . . . 7 |- ( ( T. /\ y e. RR* ) -> y e. RR* )
5	4	xnegcld 9638	. . . . . 6 |- ( ( T. /\ y e. RR* ) -> -e y e. RR* )
6		xnegneg 9616	. . . . . . . . . . 11 |- ( x e. RR* -> -e -e x = x )
7	6	eqeq2d 2151	. . . . . . . . . 10 |- ( x e. RR* -> ( -e y = -e -e x <-> -e y = x ) )
8	7	adantr 274	. . . . . . . . 9 |- ( ( x e. RR* /\ y e. RR* ) -> ( -e y = -e -e x <-> -e y = x ) )
9		eqcom 2141	. . . . . . . . 9 |- ( -e y = x <-> x = -e y )
10	8, 9	syl6bb 195	. . . . . . . 8 |- ( ( x e. RR* /\ y e. RR* ) -> ( -e y = -e -e x <-> x = -e y ) )
11		simpr 109	. . . . . . . . 9 |- ( ( x e. RR* /\ y e. RR* ) -> y e. RR* )
12		xnegcl 9615	. . . . . . . . . 10 |- ( x e. RR* -> -e x e. RR* )
13	12	adantr 274	. . . . . . . . 9 |- ( ( x e. RR* /\ y e. RR* ) -> -e x e. RR* )
14		xneg11 9617	. . . . . . . . 9 |- ( ( y e. RR* /\ -e x e. RR* ) -> ( -e y = -e -e x <-> y = -e x ) )
15	11, 13, 14	syl2anc 408	. . . . . . . 8 |- ( ( x e. RR* /\ y e. RR* ) -> ( -e y = -e -e x <-> y = -e x ) )
16	10, 15	bitr3d 189	. . . . . . 7 |- ( ( x e. RR* /\ y e. RR* ) -> ( x = -e y <-> y = -e x ) )
17	16	adantl 275	. . . . . 6 |- ( ( T. /\ ( x e. RR* /\ y e. RR* ) ) -> ( x = -e y <-> y = -e x ) )
18	1, 3, 5, 17	f1ocnv2d 5974	. . . . 5 |- ( T. -> ( F : RR* -1-1-onto-> RR* /\ `' F = ( y e. RR* |-> -e y ) ) )
19	18	mptru 1340	. . . 4 |- ( F : RR* -1-1-onto-> RR* /\ `' F = ( y e. RR* |-> -e y ) )
20	19	simpli 110	. . 3 |- F : RR* -1-1-onto-> RR*
21		simpl 108	. . . . . . 7 |- ( ( z e. RR* /\ y e. RR* ) -> z e. RR* )
22	21	xnegcld 9638	. . . . . 6 |- ( ( z e. RR* /\ y e. RR* ) -> -e z e. RR* )
23		simpr 109	. . . . . . 7 |- ( ( z e. RR* /\ y e. RR* ) -> y e. RR* )
24	23	xnegcld 9638	. . . . . 6 |- ( ( z e. RR* /\ y e. RR* ) -> -e y e. RR* )
25		brcnvg 4720	. . . . . 6 |- ( ( -e z e. RR* /\ -e y e. RR* ) -> ( -e z `' < -e y <-> -e y < -e z ) )
26	22, 24, 25	syl2anc 408	. . . . 5 |- ( ( z e. RR* /\ y e. RR* ) -> ( -e z `' < -e y <-> -e y < -e z ) )
27		xnegeq 9610	. . . . . . 7 |- ( x = z -> -e x = -e z )
28	1, 27, 21, 22	fvmptd3 5514	. . . . . 6 |- ( ( z e. RR* /\ y e. RR* ) -> ( F ` z ) = -e z )
29		xnegeq 9610	. . . . . . 7 |- ( x = y -> -e x = -e y )
30	1, 29, 23, 24	fvmptd3 5514	. . . . . 6 |- ( ( z e. RR* /\ y e. RR* ) -> ( F ` y ) = -e y )
31	28, 30	breq12d 3942	. . . . 5 |- ( ( z e. RR* /\ y e. RR* ) -> ( ( F ` z ) `' < ( F ` y ) <-> -e z `' < -e y ) )
32		xltneg 9619	. . . . 5 |- ( ( z e. RR* /\ y e. RR* ) -> ( z < y <-> -e y < -e z ) )
33	26, 31, 32	3bitr4rd 220	. . . 4 |- ( ( z e. RR* /\ y e. RR* ) -> ( z < y <-> ( F ` z ) `' < ( F ` y ) ) )
34	33	rgen2a 2486	. . 3 |- A. z e. RR* A. y e. RR* ( z < y <-> ( F ` z ) `' < ( F ` y ) )
35		df-isom 5132	. . 3 |- ( F Isom < , `' < ( RR* , RR* ) <-> ( F : RR* -1-1-onto-> RR* /\ A. z e. RR* A. y e. RR* ( z < y <-> ( F ` z ) `' < ( F ` y ) ) ) )
36	20, 34, 35	mpbir2an 926	. 2 |- F Isom < , `' < ( RR* , RR* )
37		xnegeq 9610	. . . 4 |- ( y = x -> -e y = -e x )
38	37	cbvmptv 4024	. . 3 |- ( y e. RR* |-> -e y ) = ( x e. RR* |-> -e x )
39	19	simpri 112	. . 3 |- `' F = ( y e. RR* |-> -e y )
40	38, 39, 1	3eqtr4i 2170	. 2 |- `' F = F
41	36, 40	pm3.2i 270	1 |- ( F Isom < , `' < ( RR* , RR* ) /\ `' F = F )

Syntax hints:   /\ wa 103   <-> wb 104   = wceq 1331   T. wtru 1332   e. wcel 1480   A.wral 2416   class class class wbr 3929   |-> cmpt 3989   `'ccnv 4538   -1-1-onto->wf1o 5122   ` cfv 5123   Isom wiso 5124   RR*cxr 7799   < clt 7800   -ecxne 9556

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-addcom 7720  ax-addass 7722  ax-distr 7724  ax-i2m1 7725  ax-0id 7728  ax-rnegex 7729  ax-cnre 7731  ax-pre-ltadd 7736

This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-isom 5132  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-sub 7935  df-neg 7936  df-xneg 9559

This theorem is referenced by:  infxrnegsupex  11032