Theorem negiso 9030

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 8454	. . . . . 6 |- ( ( T. /\ x e. RR ) -> -u x e. RR )
4		simpr 110	. . . . . . 7 |- ( ( T. /\ y e. RR ) -> y e. RR )
5	4	renegcld 8454	. . . . . 6 |- ( ( T. /\ y e. RR ) -> -u y e. RR )
6		recn 8060	. . . . . . . 8 |- ( x e. RR -> x e. CC )
7		recn 8060	. . . . . . . 8 |- ( y e. RR -> y e. CC )
8		negcon2 8327	. . . . . . . 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 6152	. . . . 5 |- ( T. -> ( F : RR -1-1-onto-> RR /\ `' F = ( y e. RR |-> -u y ) ) )
12	11	mptru 1382	. . . 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 8103	. . . . . . 7 |- ( ( z e. RR /\ y e. RR ) -> z e. CC )
16	15	negcld 8372	. . . . . 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 8372	. . . . . 6 |- ( ( z e. RR /\ y e. RR ) -> -u y e. CC )
19		brcnvg 4860	. . . . . 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 8267	. . . . . . . 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 5662	. . . . . 6 |- ( ( z e. RR /\ y e. RR ) -> ( F ` z ) = -u z )
25		negeq 8267	. . . . . . . 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 5662	. . . . . 6 |- ( ( z e. RR /\ y e. RR ) -> ( F ` y ) = -u y )
29	24, 28	breq12d 4058	. . . . 5 |- ( ( z e. RR /\ y e. RR ) -> ( ( F ` z ) `' < ( F ` y ) <-> -u z `' < -u y ) )
30		ltneg 8537	. . . . 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 2560	. . 3 |- A. z e. RR A. y e. RR ( z < y <-> ( F ` z ) `' < ( F ` y ) )
33		df-isom 5281	. . 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 945	. 2 |- F Isom < , `' < ( RR , RR )
35		negeq 8267	. . . 4 |- ( y = x -> -u y = -u x )
36	35	cbvmptv 4141	. . 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 2236	. 2 |- `' F = F
39	34, 38	pm3.2i 272	1 |- ( F Isom < , `' < ( RR , RR ) /\ `' F = F )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1373   T. wtru 1374   e. wcel 2176   A.wral 2484   class class class wbr 4045   |-> cmpt 4106   `'ccnv 4675   -1-1-onto->wf1o 5271   ` cfv 5272   Isom wiso 5273   CCcc 7925   RRcr 7926   < clt 8109   -ucneg 8246

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-pow 4219  ax-pr 4254  ax-un 4481  ax-setind 4586  ax-cnex 8018  ax-resscn 8019  ax-1cn 8020  ax-1re 8021  ax-icn 8022  ax-addcl 8023  ax-addrcl 8024  ax-mulcl 8025  ax-addcom 8027  ax-addass 8029  ax-distr 8031  ax-i2m1 8032  ax-0id 8035  ax-rnegex 8036  ax-cnre 8038  ax-pre-ltadd 8043

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4046  df-opab 4107  df-mpt 4108  df-id 4341  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-rn 4687  df-iota 5233  df-fun 5274  df-fn 5275  df-f 5276  df-f1 5277  df-fo 5278  df-f1o 5279  df-fv 5280  df-isom 5281  df-riota 5901  df-ov 5949  df-oprab 5950  df-mpo 5951  df-pnf 8111  df-mnf 8112  df-ltxr 8114  df-sub 8247  df-neg 8248

This theorem is referenced by:  infrenegsupex  9717