Theorem xrnegiso 11152

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 9752	. . . . . 6 |- ( ( T. /\ x e. RR* ) -> -e x e. RR* )
4		simpr 109	. . . . . . 7 |- ( ( T. /\ y e. RR* ) -> y e. RR* )
5	4	xnegcld 9752	. . . . . 6 |- ( ( T. /\ y e. RR* ) -> -e y e. RR* )
6		xnegneg 9730	. . . . . . . . . . 11 |- ( x e. RR* -> -e -e x = x )
7	6	eqeq2d 2169	. . . . . . . . . 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 2159	. . . . . . . . 9 |- ( -e y = x <-> x = -e y )
10	8, 9	bitrdi 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 9729	. . . . . . . . . 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 9731	. . . . . . . . 9 |- ( ( y e. RR* /\ -e x e. RR* ) -> ( -e y = -e -e x <-> y = -e x ) )
15	11, 13, 14	syl2anc 409	. . . . . . . 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 6021	. . . . 5 |- ( T. -> ( F : RR* -1-1-onto-> RR* /\ `' F = ( y e. RR* |-> -e y ) ) )
19	18	mptru 1344	. . . 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 9752	. . . . . 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 9752	. . . . . 6 |- ( ( z e. RR* /\ y e. RR* ) -> -e y e. RR* )
25		brcnvg 4766	. . . . . 6 |- ( ( -e z e. RR* /\ -e y e. RR* ) -> ( -e z `' < -e y <-> -e y < -e z ) )
26	22, 24, 25	syl2anc 409	. . . . 5 |- ( ( z e. RR* /\ y e. RR* ) -> ( -e z `' < -e y <-> -e y < -e z ) )
27		xnegeq 9724	. . . . . . 7 |- ( x = z -> -e x = -e z )
28	1, 27, 21, 22	fvmptd3 5560	. . . . . 6 |- ( ( z e. RR* /\ y e. RR* ) -> ( F ` z ) = -e z )
29		xnegeq 9724	. . . . . . 7 |- ( x = y -> -e x = -e y )
30	1, 29, 23, 24	fvmptd3 5560	. . . . . 6 |- ( ( z e. RR* /\ y e. RR* ) -> ( F ` y ) = -e y )
31	28, 30	breq12d 3978	. . . . 5 |- ( ( z e. RR* /\ y e. RR* ) -> ( ( F ` z ) `' < ( F ` y ) <-> -e z `' < -e y ) )
32		xltneg 9733	. . . . 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 2511	. . 3 |- A. z e. RR* A. y e. RR* ( z < y <-> ( F ` z ) `' < ( F ` y ) )
35		df-isom 5178	. . 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 927	. 2 |- F Isom < , `' < ( RR* , RR* )
37		xnegeq 9724	. . . 4 |- ( y = x -> -e y = -e x )
38	37	cbvmptv 4060	. . 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 2188	. 2 |- `' F = F
41	36, 40	pm3.2i 270	1 |- ( F Isom < , `' < ( RR* , RR* ) /\ `' F = F )

Syntax hints:   /\ wa 103   <-> wb 104   = wceq 1335   T. wtru 1336   e. wcel 2128   A.wral 2435   class class class wbr 3965   |-> cmpt 4025   `'ccnv 4584   -1-1-onto->wf1o 5168   ` cfv 5169   Isom wiso 5170   RR*cxr 7905   < clt 7906   -ecxne 9669

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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4495  ax-cnex 7817  ax-resscn 7818  ax-1cn 7819  ax-1re 7820  ax-icn 7821  ax-addcl 7822  ax-addrcl 7823  ax-mulcl 7824  ax-addcom 7826  ax-addass 7828  ax-distr 7830  ax-i2m1 7831  ax-0id 7834  ax-rnegex 7835  ax-cnre 7837  ax-pre-ltadd 7842

This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-if 3506  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-mpt 4027  df-id 4253  df-xp 4591  df-rel 4592  df-cnv 4593  df-co 4594  df-dm 4595  df-rn 4596  df-iota 5134  df-fun 5171  df-fn 5172  df-f 5173  df-f1 5174  df-fo 5175  df-f1o 5176  df-fv 5177  df-isom 5178  df-riota 5777  df-ov 5824  df-oprab 5825  df-mpo 5826  df-pnf 7908  df-mnf 7909  df-xr 7910  df-ltxr 7911  df-sub 8042  df-neg 8043  df-xneg 9672

This theorem is referenced by:  infxrnegsupex  11153