Theorem negiso 11003

Description: Negation is an order anti-isomorphism of the real numbers, which is its own inverse. (Contributed by Mario Carneiro, 24-Dec-2016.)

Hypothesis

Ref	Expression
negiso.1	|- F = ( x e. RR |-> -u x )

Assertion

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

Proof of Theorem negiso

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

Step	Hyp	Ref	Expression
1		negiso.1	. . . . . 6 |- F = ( x e. RR |-> -u x )
2		simpr 477	. . . . . . 7 |- ( ( T. /\ x e. RR ) -> x e. RR )
3	2	renegcld 10457	. . . . . 6 |- ( ( T. /\ x e. RR ) -> -u x e. RR )
4		simpr 477	. . . . . . 7 |- ( ( T. /\ y e. RR ) -> y e. RR )
5	4	renegcld 10457	. . . . . 6 |- ( ( T. /\ y e. RR ) -> -u y e. RR )
6		recn 10026	. . . . . . . 8 |- ( x e. RR -> x e. CC )
7		recn 10026	. . . . . . . 8 |- ( y e. RR -> y e. CC )
8		negcon2 10334	. . . . . . . 8 |- ( ( x e. CC /\ y e. CC ) -> ( x = -u y <-> y = -u x ) )
9	6, 7, 8	syl2an 494	. . . . . . 7 |- ( ( x e. RR /\ y e. RR ) -> ( x = -u y <-> y = -u x ) )
10	9	adantl 482	. . . . . 6 |- ( ( T. /\ ( x e. RR /\ y e. RR ) ) -> ( x = -u y <-> y = -u x ) )
11	1, 3, 5, 10	f1ocnv2d 6886	. . . . 5 |- ( T. -> ( F : RR -1-1-onto-> RR /\ `' F = ( y e. RR |-> -u y ) ) )
12	11	trud 1493	. . . 4 |- ( F : RR -1-1-onto-> RR /\ `' F = ( y e. RR |-> -u y ) )
13	12	simpli 474	. . 3 |- F : RR -1-1-onto-> RR
14		ltneg 10528	. . . . . 6 |- ( ( z e. RR /\ y e. RR ) -> ( z < y <-> -u y < -u z ) )
15		negex 10279	. . . . . . 7 |- -u z e. _V
16		negex 10279	. . . . . . 7 |- -u y e. _V
17	15, 16	brcnv 5305	. . . . . 6 |- ( -u z `' < -u y <-> -u y < -u z )
18	14, 17	syl6bbr 278	. . . . 5 |- ( ( z e. RR /\ y e. RR ) -> ( z < y <-> -u z `' < -u y ) )
19		negeq 10273	. . . . . . 7 |- ( x = z -> -u x = -u z )
20	19, 1, 15	fvmpt 6282	. . . . . 6 |- ( z e. RR -> ( F ` z ) = -u z )
21		negeq 10273	. . . . . . 7 |- ( x = y -> -u x = -u y )
22	21, 1, 16	fvmpt 6282	. . . . . 6 |- ( y e. RR -> ( F ` y ) = -u y )
23	20, 22	breqan12d 4669	. . . . 5 |- ( ( z e. RR /\ y e. RR ) -> ( ( F ` z ) `' < ( F ` y ) <-> -u z `' < -u y ) )
24	18, 23	bitr4d 271	. . . 4 |- ( ( z e. RR /\ y e. RR ) -> ( z < y <-> ( F ` z ) `' < ( F ` y ) ) )
25	24	rgen2a 2977	. . 3 |- A. z e. RR A. y e. RR ( z < y <-> ( F ` z ) `' < ( F ` y ) )
26		df-isom 5897	. . 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 ) ) ) )
27	13, 25, 26	mpbir2an 955	. 2 |- F Isom < , `' < ( RR , RR )
28		negeq 10273	. . . 4 |- ( y = x -> -u y = -u x )
29	28	cbvmptv 4750	. . 3 |- ( y e. RR |-> -u y ) = ( x e. RR |-> -u x )
30	12	simpri 478	. . 3 |- `' F = ( y e. RR |-> -u y )
31	29, 30, 1	3eqtr4i 2654	. 2 |- `' F = F
32	27, 31	pm3.2i 471	1 |- ( F Isom < , `' < ( RR , RR ) /\ `' F = F )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   T. wtru 1484   e. wcel 1990   A.wral 2912   class class class wbr 4653   |-> cmpt 4729   `'ccnv 5113   -1-1-onto->wf1o 5887   ` cfv 5888   Isom wiso 5889   CCcc 9934   RRcr 9935   < clt 10074   -ucneg 10267

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269

This theorem is referenced by:  infrenegsup  11006