Theorem negiso 8033

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 108	. . . . . . 7 |- ( ( T. /\ x e. RR ) -> x e. RR )
3	2	renegcld 7484	. . . . . 6 |- ( ( T. /\ x e. RR ) -> -u x e. RR )
4		simpr 108	. . . . . . 7 |- ( ( T. /\ y e. RR ) -> y e. RR )
5	4	renegcld 7484	. . . . . 6 |- ( ( T. /\ y e. RR ) -> -u y e. RR )
6		recn 7106	. . . . . . . 8 |- ( x e. RR -> x e. CC )
7		recn 7106	. . . . . . . 8 |- ( y e. RR -> y e. CC )
8		negcon2 7361	. . . . . . . 8 |- ( ( x e. CC /\ y e. CC ) -> ( x = -u y <-> y = -u x ) )
9	6, 7, 8	syl2an 283	. . . . . . 7 |- ( ( x e. RR /\ y e. RR ) -> ( x = -u y <-> y = -u x ) )
10	9	adantl 271	. . . . . 6 |- ( ( T. /\ ( x e. RR /\ y e. RR ) ) -> ( x = -u y <-> y = -u x ) )
11	1, 3, 5, 10	f1ocnv2d 5724	. . . . 5 |- ( T. -> ( F : RR -1-1-onto-> RR /\ `' F = ( y e. RR |-> -u y ) ) )
12	11	trud 1293	. . . 4 |- ( F : RR -1-1-onto-> RR /\ `' F = ( y e. RR |-> -u y ) )
13	12	simpli 109	. . 3 |- F : RR -1-1-onto-> RR
14		simpl 107	. . . . . . . 8 |- ( ( z e. RR /\ y e. RR ) -> z e. RR )
15	14	recnd 7147	. . . . . . 7 |- ( ( z e. RR /\ y e. RR ) -> z e. CC )
16	15	negcld 7406	. . . . . 6 |- ( ( z e. RR /\ y e. RR ) -> -u z e. CC )
17	7	adantl 271	. . . . . . 7 |- ( ( z e. RR /\ y e. RR ) -> y e. CC )
18	17	negcld 7406	. . . . . 6 |- ( ( z e. RR /\ y e. RR ) -> -u y e. CC )
19		brcnvg 4534	. . . . . 6 |- ( ( -u z e. CC /\ -u y e. CC ) -> ( -u z `' < -u y <-> -u y < -u z ) )
20	16, 18, 19	syl2anc 403	. . . . 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 7301	. . . . . . . 8 |- ( x = z -> -u x = -u z )
23	22	adantl 271	. . . . . . 7 |- ( ( ( z e. RR /\ y e. RR ) /\ x = z ) -> -u x = -u z )
24	21, 23, 14, 16	fvmptd 5274	. . . . . 6 |- ( ( z e. RR /\ y e. RR ) -> ( F ` z ) = -u z )
25		negeq 7301	. . . . . . . 8 |- ( x = y -> -u x = -u y )
26	25	adantl 271	. . . . . . 7 |- ( ( ( z e. RR /\ y e. RR ) /\ x = y ) -> -u x = -u y )
27		simpr 108	. . . . . . 7 |- ( ( z e. RR /\ y e. RR ) -> y e. RR )
28	21, 26, 27, 18	fvmptd 5274	. . . . . 6 |- ( ( z e. RR /\ y e. RR ) -> ( F ` y ) = -u y )
29	24, 28	breq12d 3798	. . . . 5 |- ( ( z e. RR /\ y e. RR ) -> ( ( F ` z ) `' < ( F ` y ) <-> -u z `' < -u y ) )
30		ltneg 7566	. . . . 5 |- ( ( z e. RR /\ y e. RR ) -> ( z < y <-> -u y < -u z ) )
31	20, 29, 30	3bitr4rd 219	. . . 4 |- ( ( z e. RR /\ y e. RR ) -> ( z < y <-> ( F ` z ) `' < ( F ` y ) ) )
32	31	rgen2a 2417	. . 3 |- A. z e. RR A. y e. RR ( z < y <-> ( F ` z ) `' < ( F ` y ) )
33		df-isom 4931	. . 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 883	. 2 |- F Isom < , `' < ( RR , RR )
35		negeq 7301	. . . 4 |- ( y = x -> -u y = -u x )
36	35	cbvmptv 3873	. . 3 |- ( y e. RR |-> -u y ) = ( x e. RR |-> -u x )
37	12	simpri 111	. . 3 |- `' F = ( y e. RR |-> -u y )
38	36, 37, 1	3eqtr4i 2111	. 2 |- `' F = F
39	34, 38	pm3.2i 266	1 |- ( F Isom < , `' < ( RR , RR ) /\ `' F = F )

Syntax hints:   /\ wa 102   <-> wb 103   = wceq 1284   T. wtru 1285   e. wcel 1433   A.wral 2348   class class class wbr 3785   |-> cmpt 3839   `'ccnv 4362   -1-1-onto->wf1o 4921   ` cfv 4922   Isom wiso 4923   CCcc 6979   RRcr 6980   < clt 7153   -ucneg 7280

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-cnex 7067  ax-resscn 7068  ax-1cn 7069  ax-1re 7070  ax-icn 7071  ax-addcl 7072  ax-addrcl 7073  ax-mulcl 7074  ax-addcom 7076  ax-addass 7078  ax-distr 7080  ax-i2m1 7081  ax-0id 7084  ax-rnegex 7085  ax-cnre 7087  ax-pre-ltadd 7092

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-nel 2340  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-mpt 3841  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-isom 4931  df-riota 5488  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-pnf 7155  df-mnf 7156  df-ltxr 7158  df-sub 7281  df-neg 7282

This theorem is referenced by:  infrenegsupex  8682