Theorem negiso 9125

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 110	. . . . . . 7 |- ( ( T. /\ x e. RR ) -> x e. RR )
3	2	renegcld 8549	. . . . . 6 |- ( ( T. /\ x e. RR ) -> -u x e. RR )
4		simpr 110	. . . . . . 7 |- ( ( T. /\ y e. RR ) -> y e. RR )
5	4	renegcld 8549	. . . . . 6 |- ( ( T. /\ y e. RR ) -> -u y e. RR )
6		recn 8155	. . . . . . . 8 |- ( x e. RR -> x e. CC )
7		recn 8155	. . . . . . . 8 |- ( y e. RR -> y e. CC )
8		negcon2 8422	. . . . . . . 8 |- ( ( x e. CC /\ y e. CC ) -> ( x = -u y <-> y = -u x ) )
9	6, 7, 8	syl2an 289	. . . . . . 7 |- ( ( x e. RR /\ y e. RR ) -> ( x = -u y <-> y = -u x ) )
10	9	adantl 277	. . . . . 6 |- ( ( T. /\ ( x e. RR /\ y e. RR ) ) -> ( x = -u y <-> y = -u x ) )
11	1, 3, 5, 10	f1ocnv2d 6222	. . . . 5 |- ( T. -> ( F : RR -1-1-onto-> RR /\ `' F = ( y e. RR |-> -u y ) ) )
12	11	mptru 1404	. . . 4 |- ( F : RR -1-1-onto-> RR /\ `' F = ( y e. RR |-> -u y ) )
13	12	simpli 111	. . 3 |- F : RR -1-1-onto-> RR
14		simpl 109	. . . . . . . 8 |- ( ( z e. RR /\ y e. RR ) -> z e. RR )
15	14	recnd 8198	. . . . . . 7 |- ( ( z e. RR /\ y e. RR ) -> z e. CC )
16	15	negcld 8467	. . . . . 6 |- ( ( z e. RR /\ y e. RR ) -> -u z e. CC )
17	7	adantl 277	. . . . . . 7 |- ( ( z e. RR /\ y e. RR ) -> y e. CC )
18	17	negcld 8467	. . . . . 6 |- ( ( z e. RR /\ y e. RR ) -> -u y e. CC )
19		brcnvg 4909	. . . . . 6 |- ( ( -u z e. CC /\ -u y e. CC ) -> ( -u z `' < -u y <-> -u y < -u z ) )
20	16, 18, 19	syl2anc 411	. . . . 5 |- ( ( z e. RR /\ y e. RR ) -> ( -u z `' < -u y <-> -u y < -u z ) )
21	1	a1i 9	. . . . . . 7 |- ( ( z e. RR /\ y e. RR ) -> F = ( x e. RR |-> -u x ) )
22		negeq 8362	. . . . . . . 8 |- ( x = z -> -u x = -u z )
23	22	adantl 277	. . . . . . 7 |- ( ( ( z e. RR /\ y e. RR ) /\ x = z ) -> -u x = -u z )
24	21, 23, 14, 16	fvmptd 5723	. . . . . 6 |- ( ( z e. RR /\ y e. RR ) -> ( F ` z ) = -u z )
25		negeq 8362	. . . . . . . 8 |- ( x = y -> -u x = -u y )
26	25	adantl 277	. . . . . . 7 |- ( ( ( z e. RR /\ y e. RR ) /\ x = y ) -> -u x = -u y )
27		simpr 110	. . . . . . 7 |- ( ( z e. RR /\ y e. RR ) -> y e. RR )
28	21, 26, 27, 18	fvmptd 5723	. . . . . 6 |- ( ( z e. RR /\ y e. RR ) -> ( F ` y ) = -u y )
29	24, 28	breq12d 4099	. . . . 5 |- ( ( z e. RR /\ y e. RR ) -> ( ( F ` z ) `' < ( F ` y ) <-> -u z `' < -u y ) )
30		ltneg 8632	. . . . 5 |- ( ( z e. RR /\ y e. RR ) -> ( z < y <-> -u y < -u z ) )
31	20, 29, 30	3bitr4rd 221	. . . 4 |- ( ( z e. RR /\ y e. RR ) -> ( z < y <-> ( F ` z ) `' < ( F ` y ) ) )
32	31	rgen2a 2584	. . 3 |- A. z e. RR A. y e. RR ( z < y <-> ( F ` z ) `' < ( F ` y ) )
33		df-isom 5333	. . 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 ) ) ) )
34	13, 32, 33	mpbir2an 948	. 2 |- F Isom < , `' < ( RR , RR )
35		negeq 8362	. . . 4 |- ( y = x -> -u y = -u x )
36	35	cbvmptv 4183	. . 3 |- ( y e. RR |-> -u y ) = ( x e. RR |-> -u x )
37	12	simpri 113	. . 3 |- `' F = ( y e. RR |-> -u y )
38	36, 37, 1	3eqtr4i 2260	. 2 |- `' F = F
39	34, 38	pm3.2i 272	1 |- ( F Isom < , `' < ( RR , RR ) /\ `' F = F )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1395   T. wtru 1396   e. wcel 2200   A.wral 2508   class class class wbr 4086   |-> cmpt 4148   `'ccnv 4722   -1-1-onto->wf1o 5323   ` cfv 5324   Isom wiso 5325   CCcc 8020   RRcr 8021   < clt 8204   -ucneg 8341

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-addcom 8122  ax-addass 8124  ax-distr 8126  ax-i2m1 8127  ax-0id 8130  ax-rnegex 8131  ax-cnre 8133  ax-pre-ltadd 8138

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-isom 5333  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-pnf 8206  df-mnf 8207  df-ltxr 8209  df-sub 8342  df-neg 8343

This theorem is referenced by:  infrenegsupex  9818