Theorem negiso 8850

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 109	. . . . . . 7 |- ( ( T. /\ x e. RR ) -> x e. RR )
3	2	renegcld 8278	. . . . . 6 |- ( ( T. /\ x e. RR ) -> -u x e. RR )
4		simpr 109	. . . . . . 7 |- ( ( T. /\ y e. RR ) -> y e. RR )
5	4	renegcld 8278	. . . . . 6 |- ( ( T. /\ y e. RR ) -> -u y e. RR )
6		recn 7886	. . . . . . . 8 |- ( x e. RR -> x e. CC )
7		recn 7886	. . . . . . . 8 |- ( y e. RR -> y e. CC )
8		negcon2 8151	. . . . . . . 8 |- ( ( x e. CC /\ y e. CC ) -> ( x = -u y <-> y = -u x ) )
9	6, 7, 8	syl2an 287	. . . . . . 7 |- ( ( x e. RR /\ y e. RR ) -> ( x = -u y <-> y = -u x ) )
10	9	adantl 275	. . . . . 6 |- ( ( T. /\ ( x e. RR /\ y e. RR ) ) -> ( x = -u y <-> y = -u x ) )
11	1, 3, 5, 10	f1ocnv2d 6042	. . . . 5 |- ( T. -> ( F : RR -1-1-onto-> RR /\ `' F = ( y e. RR |-> -u y ) ) )
12	11	mptru 1352	. . . 4 |- ( F : RR -1-1-onto-> RR /\ `' F = ( y e. RR |-> -u y ) )
13	12	simpli 110	. . 3 |- F : RR -1-1-onto-> RR
14		simpl 108	. . . . . . . 8 |- ( ( z e. RR /\ y e. RR ) -> z e. RR )
15	14	recnd 7927	. . . . . . 7 |- ( ( z e. RR /\ y e. RR ) -> z e. CC )
16	15	negcld 8196	. . . . . 6 |- ( ( z e. RR /\ y e. RR ) -> -u z e. CC )
17	7	adantl 275	. . . . . . 7 |- ( ( z e. RR /\ y e. RR ) -> y e. CC )
18	17	negcld 8196	. . . . . 6 |- ( ( z e. RR /\ y e. RR ) -> -u y e. CC )
19		brcnvg 4785	. . . . . 6 |- ( ( -u z e. CC /\ -u y e. CC ) -> ( -u z `' < -u y <-> -u y < -u z ) )
20	16, 18, 19	syl2anc 409	. . . . 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 8091	. . . . . . . 8 |- ( x = z -> -u x = -u z )
23	22	adantl 275	. . . . . . 7 |- ( ( ( z e. RR /\ y e. RR ) /\ x = z ) -> -u x = -u z )
24	21, 23, 14, 16	fvmptd 5567	. . . . . 6 |- ( ( z e. RR /\ y e. RR ) -> ( F ` z ) = -u z )
25		negeq 8091	. . . . . . . 8 |- ( x = y -> -u x = -u y )
26	25	adantl 275	. . . . . . 7 |- ( ( ( z e. RR /\ y e. RR ) /\ x = y ) -> -u x = -u y )
27		simpr 109	. . . . . . 7 |- ( ( z e. RR /\ y e. RR ) -> y e. RR )
28	21, 26, 27, 18	fvmptd 5567	. . . . . 6 |- ( ( z e. RR /\ y e. RR ) -> ( F ` y ) = -u y )
29	24, 28	breq12d 3995	. . . . 5 |- ( ( z e. RR /\ y e. RR ) -> ( ( F ` z ) `' < ( F ` y ) <-> -u z `' < -u y ) )
30		ltneg 8360	. . . . 5 |- ( ( z e. RR /\ y e. RR ) -> ( z < y <-> -u y < -u z ) )
31	20, 29, 30	3bitr4rd 220	. . . 4 |- ( ( z e. RR /\ y e. RR ) -> ( z < y <-> ( F ` z ) `' < ( F ` y ) ) )
32	31	rgen2a 2520	. . 3 |- A. z e. RR A. y e. RR ( z < y <-> ( F ` z ) `' < ( F ` y ) )
33		df-isom 5197	. . 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 932	. 2 |- F Isom < , `' < ( RR , RR )
35		negeq 8091	. . . 4 |- ( y = x -> -u y = -u x )
36	35	cbvmptv 4078	. . 3 |- ( y e. RR |-> -u y ) = ( x e. RR |-> -u x )
37	12	simpri 112	. . 3 |- `' F = ( y e. RR |-> -u y )
38	36, 37, 1	3eqtr4i 2196	. 2 |- `' F = F
39	34, 38	pm3.2i 270	1 |- ( F Isom < , `' < ( RR , RR ) /\ `' F = F )

Syntax hints:   /\ wa 103   <-> wb 104   = wceq 1343   T. wtru 1344   e. wcel 2136   A.wral 2444   class class class wbr 3982   |-> cmpt 4043   `'ccnv 4603   -1-1-onto->wf1o 5187   ` cfv 5188   Isom wiso 5189   CCcc 7751   RRcr 7752   < clt 7933   -ucneg 8070

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-addcom 7853  ax-addass 7855  ax-distr 7857  ax-i2m1 7858  ax-0id 7861  ax-rnegex 7862  ax-cnre 7864  ax-pre-ltadd 7869

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-isom 5197  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-pnf 7935  df-mnf 7936  df-ltxr 7938  df-sub 8071  df-neg 8072

This theorem is referenced by:  infrenegsupex  9532