Theorem xrnegiso 11822

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 110	. . . . . . 7 |- ( ( T. /\ x e. RR* ) -> x e. RR* )
3	2	xnegcld 10089	. . . . . 6 |- ( ( T. /\ x e. RR* ) -> -e x e. RR* )
4		simpr 110	. . . . . . 7 |- ( ( T. /\ y e. RR* ) -> y e. RR* )
5	4	xnegcld 10089	. . . . . 6 |- ( ( T. /\ y e. RR* ) -> -e y e. RR* )
6		xnegneg 10067	. . . . . . . . . . 11 |- ( x e. RR* -> -e -e x = x )
7	6	eqeq2d 2243	. . . . . . . . . 10 |- ( x e. RR* -> ( -e y = -e -e x <-> -e y = x ) )
8	7	adantr 276	. . . . . . . . 9 |- ( ( x e. RR* /\ y e. RR* ) -> ( -e y = -e -e x <-> -e y = x ) )
9		eqcom 2233	. . . . . . . . 9 |- ( -e y = x <-> x = -e y )
10	8, 9	bitrdi 196	. . . . . . . 8 |- ( ( x e. RR* /\ y e. RR* ) -> ( -e y = -e -e x <-> x = -e y ) )
11		simpr 110	. . . . . . . . 9 |- ( ( x e. RR* /\ y e. RR* ) -> y e. RR* )
12		xnegcl 10066	. . . . . . . . . 10 |- ( x e. RR* -> -e x e. RR* )
13	12	adantr 276	. . . . . . . . 9 |- ( ( x e. RR* /\ y e. RR* ) -> -e x e. RR* )
14		xneg11 10068	. . . . . . . . 9 |- ( ( y e. RR* /\ -e x e. RR* ) -> ( -e y = -e -e x <-> y = -e x ) )
15	11, 13, 14	syl2anc 411	. . . . . . . 8 |- ( ( x e. RR* /\ y e. RR* ) -> ( -e y = -e -e x <-> y = -e x ) )
16	10, 15	bitr3d 190	. . . . . . 7 |- ( ( x e. RR* /\ y e. RR* ) -> ( x = -e y <-> y = -e x ) )
17	16	adantl 277	. . . . . 6 |- ( ( T. /\ ( x e. RR* /\ y e. RR* ) ) -> ( x = -e y <-> y = -e x ) )
18	1, 3, 5, 17	f1ocnv2d 6226	. . . . 5 |- ( T. -> ( F : RR* -1-1-onto-> RR* /\ `' F = ( y e. RR* |-> -e y ) ) )
19	18	mptru 1406	. . . 4 |- ( F : RR* -1-1-onto-> RR* /\ `' F = ( y e. RR* |-> -e y ) )
20	19	simpli 111	. . 3 |- F : RR* -1-1-onto-> RR*
21		simpl 109	. . . . . . 7 |- ( ( z e. RR* /\ y e. RR* ) -> z e. RR* )
22	21	xnegcld 10089	. . . . . 6 |- ( ( z e. RR* /\ y e. RR* ) -> -e z e. RR* )
23		simpr 110	. . . . . . 7 |- ( ( z e. RR* /\ y e. RR* ) -> y e. RR* )
24	23	xnegcld 10089	. . . . . 6 |- ( ( z e. RR* /\ y e. RR* ) -> -e y e. RR* )
25		brcnvg 4911	. . . . . 6 |- ( ( -e z e. RR* /\ -e y e. RR* ) -> ( -e z `' < -e y <-> -e y < -e z ) )
26	22, 24, 25	syl2anc 411	. . . . 5 |- ( ( z e. RR* /\ y e. RR* ) -> ( -e z `' < -e y <-> -e y < -e z ) )
27		xnegeq 10061	. . . . . . 7 |- ( x = z -> -e x = -e z )
28	1, 27, 21, 22	fvmptd3 5740	. . . . . 6 |- ( ( z e. RR* /\ y e. RR* ) -> ( F ` z ) = -e z )
29		xnegeq 10061	. . . . . . 7 |- ( x = y -> -e x = -e y )
30	1, 29, 23, 24	fvmptd3 5740	. . . . . 6 |- ( ( z e. RR* /\ y e. RR* ) -> ( F ` y ) = -e y )
31	28, 30	breq12d 4101	. . . . 5 |- ( ( z e. RR* /\ y e. RR* ) -> ( ( F ` z ) `' < ( F ` y ) <-> -e z `' < -e y ) )
32		xltneg 10070	. . . . 5 |- ( ( z e. RR* /\ y e. RR* ) -> ( z < y <-> -e y < -e z ) )
33	26, 31, 32	3bitr4rd 221	. . . 4 |- ( ( z e. RR* /\ y e. RR* ) -> ( z < y <-> ( F ` z ) `' < ( F ` y ) ) )
34	33	rgen2a 2586	. . 3 |- A. z e. RR* A. y e. RR* ( z < y <-> ( F ` z ) `' < ( F ` y ) )
35		df-isom 5335	. . 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 950	. 2 |- F Isom < , `' < ( RR* , RR* )
37		xnegeq 10061	. . . 4 |- ( y = x -> -e y = -e x )
38	37	cbvmptv 4185	. . 3 |- ( y e. RR* |-> -e y ) = ( x e. RR* |-> -e x )
39	19	simpri 113	. . 3 |- `' F = ( y e. RR* |-> -e y )
40	38, 39, 1	3eqtr4i 2262	. 2 |- `' F = F
41	36, 40	pm3.2i 272	1 |- ( F Isom < , `' < ( RR* , RR* ) /\ `' F = F )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1397   T. wtru 1398   e. wcel 2202   A.wral 2510   class class class wbr 4088   |-> cmpt 4150   `'ccnv 4724   -1-1-onto->wf1o 5325   ` cfv 5326   Isom wiso 5327   RR*cxr 8212   < clt 8213   -ecxne 10003

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-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-addass 8133  ax-distr 8135  ax-i2m1 8136  ax-0id 8139  ax-rnegex 8140  ax-cnre 8142  ax-pre-ltadd 8147

This theorem depends on definitions:  df-bi 117  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  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-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  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-fo 5332  df-f1o 5333  df-fv 5334  df-isom 5335  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-sub 8351  df-neg 8352  df-xneg 10006

This theorem is referenced by:  infxrnegsupex  11823