Theorem xrnegiso 11405

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 9921	. . . . . 6 |- ( ( T. /\ x e. RR* ) -> -e x e. RR* )
4		simpr 110	. . . . . . 7 |- ( ( T. /\ y e. RR* ) -> y e. RR* )
5	4	xnegcld 9921	. . . . . 6 |- ( ( T. /\ y e. RR* ) -> -e y e. RR* )
6		xnegneg 9899	. . . . . . . . . . 11 |- ( x e. RR* -> -e -e x = x )
7	6	eqeq2d 2205	. . . . . . . . . 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 2195	. . . . . . . . 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 9898	. . . . . . . . . 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 9900	. . . . . . . . 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 6122	. . . . 5 |- ( T. -> ( F : RR* -1-1-onto-> RR* /\ `' F = ( y e. RR* |-> -e y ) ) )
19	18	mptru 1373	. . . 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 9921	. . . . . 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 9921	. . . . . 6 |- ( ( z e. RR* /\ y e. RR* ) -> -e y e. RR* )
25		brcnvg 4843	. . . . . 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 9893	. . . . . . 7 |- ( x = z -> -e x = -e z )
28	1, 27, 21, 22	fvmptd3 5651	. . . . . 6 |- ( ( z e. RR* /\ y e. RR* ) -> ( F ` z ) = -e z )
29		xnegeq 9893	. . . . . . 7 |- ( x = y -> -e x = -e y )
30	1, 29, 23, 24	fvmptd3 5651	. . . . . 6 |- ( ( z e. RR* /\ y e. RR* ) -> ( F ` y ) = -e y )
31	28, 30	breq12d 4042	. . . . 5 |- ( ( z e. RR* /\ y e. RR* ) -> ( ( F ` z ) `' < ( F ` y ) <-> -e z `' < -e y ) )
32		xltneg 9902	. . . . 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 2548	. . 3 |- A. z e. RR* A. y e. RR* ( z < y <-> ( F ` z ) `' < ( F ` y ) )
35		df-isom 5263	. . 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 944	. 2 |- F Isom < , `' < ( RR* , RR* )
37		xnegeq 9893	. . . 4 |- ( y = x -> -e y = -e x )
38	37	cbvmptv 4125	. . 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 2224	. 2 |- `' F = F
41	36, 40	pm3.2i 272	1 |- ( F Isom < , `' < ( RR* , RR* ) /\ `' F = F )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1364   T. wtru 1365   e. wcel 2164   A.wral 2472   class class class wbr 4029   |-> cmpt 4090   `'ccnv 4658   -1-1-onto->wf1o 5253   ` cfv 5254   Isom wiso 5255   RR*cxr 8053   < clt 8054   -ecxne 9835

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-addcom 7972  ax-addass 7974  ax-distr 7976  ax-i2m1 7977  ax-0id 7980  ax-rnegex 7981  ax-cnre 7983  ax-pre-ltadd 7988

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-isom 5263  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-sub 8192  df-neg 8193  df-xneg 9838

This theorem is referenced by:  infxrnegsupex  11406