Theorem xrnegiso 11606

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 9979	. . . . . 6 |- ( ( T. /\ x e. RR* ) -> -e x e. RR* )
4		simpr 110	. . . . . . 7 |- ( ( T. /\ y e. RR* ) -> y e. RR* )
5	4	xnegcld 9979	. . . . . 6 |- ( ( T. /\ y e. RR* ) -> -e y e. RR* )
6		xnegneg 9957	. . . . . . . . . . 11 |- ( x e. RR* -> -e -e x = x )
7	6	eqeq2d 2217	. . . . . . . . . 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 2207	. . . . . . . . 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 9956	. . . . . . . . . 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 9958	. . . . . . . . 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 6152	. . . . 5 |- ( T. -> ( F : RR* -1-1-onto-> RR* /\ `' F = ( y e. RR* |-> -e y ) ) )
19	18	mptru 1382	. . . 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 9979	. . . . . 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 9979	. . . . . 6 |- ( ( z e. RR* /\ y e. RR* ) -> -e y e. RR* )
25		brcnvg 4860	. . . . . 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 9951	. . . . . . 7 |- ( x = z -> -e x = -e z )
28	1, 27, 21, 22	fvmptd3 5675	. . . . . 6 |- ( ( z e. RR* /\ y e. RR* ) -> ( F ` z ) = -e z )
29		xnegeq 9951	. . . . . . 7 |- ( x = y -> -e x = -e y )
30	1, 29, 23, 24	fvmptd3 5675	. . . . . 6 |- ( ( z e. RR* /\ y e. RR* ) -> ( F ` y ) = -e y )
31	28, 30	breq12d 4058	. . . . 5 |- ( ( z e. RR* /\ y e. RR* ) -> ( ( F ` z ) `' < ( F ` y ) <-> -e z `' < -e y ) )
32		xltneg 9960	. . . . 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 2560	. . 3 |- A. z e. RR* A. y e. RR* ( z < y <-> ( F ` z ) `' < ( F ` y ) )
35		df-isom 5281	. . 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 945	. 2 |- F Isom < , `' < ( RR* , RR* )
37		xnegeq 9951	. . . 4 |- ( y = x -> -e y = -e x )
38	37	cbvmptv 4141	. . 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 2236	. 2 |- `' F = F
41	36, 40	pm3.2i 272	1 |- ( F Isom < , `' < ( RR* , RR* ) /\ `' F = F )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1373   T. wtru 1374   e. wcel 2176   A.wral 2484   class class class wbr 4045   |-> cmpt 4106   `'ccnv 4675   -1-1-onto->wf1o 5271   ` cfv 5272   Isom wiso 5273   RR*cxr 8108   < clt 8109   -ecxne 9893

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-pow 4219  ax-pr 4254  ax-un 4481  ax-setind 4586  ax-cnex 8018  ax-resscn 8019  ax-1cn 8020  ax-1re 8021  ax-icn 8022  ax-addcl 8023  ax-addrcl 8024  ax-mulcl 8025  ax-addcom 8027  ax-addass 8029  ax-distr 8031  ax-i2m1 8032  ax-0id 8035  ax-rnegex 8036  ax-cnre 8038  ax-pre-ltadd 8043

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-if 3572  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4046  df-opab 4107  df-mpt 4108  df-id 4341  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-rn 4687  df-iota 5233  df-fun 5274  df-fn 5275  df-f 5276  df-f1 5277  df-fo 5278  df-f1o 5279  df-fv 5280  df-isom 5281  df-riota 5901  df-ov 5949  df-oprab 5950  df-mpo 5951  df-pnf 8111  df-mnf 8112  df-xr 8113  df-ltxr 8114  df-sub 8247  df-neg 8248  df-xneg 9896

This theorem is referenced by:  infxrnegsupex  11607