Theorem negiso 8858

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 8286	. . . . . 6 |- ( ( T. /\ x e. RR ) -> -u x e. RR )
4		simpr 109	. . . . . . 7 |- ( ( T. /\ y e. RR ) -> y e. RR )
5	4	renegcld 8286	. . . . . 6 |- ( ( T. /\ y e. RR ) -> -u y e. RR )
6		recn 7894	. . . . . . . 8 |- ( x e. RR -> x e. CC )
7		recn 7894	. . . . . . . 8 |- ( y e. RR -> y e. CC )
8		negcon2 8159	. . . . . . . 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 6050	. . . . 5 |- ( T. -> ( F : RR -1-1-onto-> RR /\ `' F = ( y e. RR |-> -u y ) ) )
12	11	mptru 1357	. . . 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 7935	. . . . . . 7 |- ( ( z e. RR /\ y e. RR ) -> z e. CC )
16	15	negcld 8204	. . . . . 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 8204	. . . . . 6 |- ( ( z e. RR /\ y e. RR ) -> -u y e. CC )
19		brcnvg 4790	. . . . . 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 8099	. . . . . . . 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 5575	. . . . . 6 |- ( ( z e. RR /\ y e. RR ) -> ( F ` z ) = -u z )
25		negeq 8099	. . . . . . . 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 5575	. . . . . 6 |- ( ( z e. RR /\ y e. RR ) -> ( F ` y ) = -u y )
29	24, 28	breq12d 4000	. . . . 5 |- ( ( z e. RR /\ y e. RR ) -> ( ( F ` z ) `' < ( F ` y ) <-> -u z `' < -u y ) )
30		ltneg 8368	. . . . 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 2524	. . 3 |- A. z e. RR A. y e. RR ( z < y <-> ( F ` z ) `' < ( F ` y ) )
33		df-isom 5205	. . 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 937	. 2 |- F Isom < , `' < ( RR , RR )
35		negeq 8099	. . . 4 |- ( y = x -> -u y = -u x )
36	35	cbvmptv 4083	. . 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 2201	. 2 |- `' F = F
39	34, 38	pm3.2i 270	1 |- ( F Isom < , `' < ( RR , RR ) /\ `' F = F )

Syntax hints:   /\ wa 103   <-> wb 104   = wceq 1348   T. wtru 1349   e. wcel 2141   A.wral 2448   class class class wbr 3987   |-> cmpt 4048   `'ccnv 4608   -1-1-onto->wf1o 5195   ` cfv 5196   Isom wiso 5197   CCcc 7759   RRcr 7760   < clt 7941   -ucneg 8078

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-cnex 7852  ax-resscn 7853  ax-1cn 7854  ax-1re 7855  ax-icn 7856  ax-addcl 7857  ax-addrcl 7858  ax-mulcl 7859  ax-addcom 7861  ax-addass 7863  ax-distr 7865  ax-i2m1 7866  ax-0id 7869  ax-rnegex 7870  ax-cnre 7872  ax-pre-ltadd 7877

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-br 3988  df-opab 4049  df-mpt 4050  df-id 4276  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-isom 5205  df-riota 5806  df-ov 5853  df-oprab 5854  df-mpo 5855  df-pnf 7943  df-mnf 7944  df-ltxr 7946  df-sub 8079  df-neg 8080

This theorem is referenced by:  infrenegsupex  9540