Theorem negf1o 8301

Description: Negation is an isomorphism of a subset of the real numbers to the negated elements of the subset. (Contributed by AV, 9-Aug-2020.)

Hypothesis

Ref	Expression
negf1o.1	|- F = ( x e. A |-> -u x )

Assertion

Ref	Expression
negf1o	|- ( A C_ RR -> F : A -1-1-onto-> { n e. RR | -u n e. A } )

Distinct variable group:   A, n, x

Allowed substitution hints:   F( x, n)

Proof of Theorem negf1o

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		negf1o.1	. . 3 |- F = ( x e. A |-> -u x )
2		ssel 3141	. . . . . 6 |- ( A C_ RR -> ( x e. A -> x e. RR ) )
3		renegcl 8180	. . . . . 6 |- ( x e. RR -> -u x e. RR )
4	2, 3	syl6 33	. . . . 5 |- ( A C_ RR -> ( x e. A -> -u x e. RR ) )
5	4	imp 123	. . . 4 |- ( ( A C_ RR /\ x e. A ) -> -u x e. RR )
6	2	imp 123	. . . . 5 |- ( ( A C_ RR /\ x e. A ) -> x e. RR )
7		recn 7907	. . . . . . . . 9 |- ( x e. RR -> x e. CC )
8		negneg 8169	. . . . . . . . . 10 |- ( x e. CC -> -u -u x = x )
9	8	eqcomd 2176	. . . . . . . . 9 |- ( x e. CC -> x = -u -u x )
10	7, 9	syl 14	. . . . . . . 8 |- ( x e. RR -> x = -u -u x )
11	10	eleq1d 2239	. . . . . . 7 |- ( x e. RR -> ( x e. A <-> -u -u x e. A ) )
12	11	biimpcd 158	. . . . . 6 |- ( x e. A -> ( x e. RR -> -u -u x e. A ) )
13	12	adantl 275	. . . . 5 |- ( ( A C_ RR /\ x e. A ) -> ( x e. RR -> -u -u x e. A ) )
14	6, 13	mpd 13	. . . 4 |- ( ( A C_ RR /\ x e. A ) -> -u -u x e. A )
15		negeq 8112	. . . . . 6 |- ( n = -u x -> -u n = -u -u x )
16	15	eleq1d 2239	. . . . 5 |- ( n = -u x -> ( -u n e. A <-> -u -u x e. A ) )
17	16	elrab 2886	. . . 4 |- ( -u x e. { n e. RR | -u n e. A } <-> ( -u x e. RR /\ -u -u x e. A ) )
18	5, 14, 17	sylanbrc 415	. . 3 |- ( ( A C_ RR /\ x e. A ) -> -u x e. { n e. RR | -u n e. A } )
19		negeq 8112	. . . . . . 7 |- ( n = y -> -u n = -u y )
20	19	eleq1d 2239	. . . . . 6 |- ( n = y -> ( -u n e. A <-> -u y e. A ) )
21	20	elrab 2886	. . . . 5 |- ( y e. { n e. RR | -u n e. A } <-> ( y e. RR /\ -u y e. A ) )
22		simpr 109	. . . . . 6 |- ( ( y e. RR /\ -u y e. A ) -> -u y e. A )
23	22	a1i 9	. . . . 5 |- ( A C_ RR -> ( ( y e. RR /\ -u y e. A ) -> -u y e. A ) )
24	21, 23	syl5bi 151	. . . 4 |- ( A C_ RR -> ( y e. { n e. RR | -u n e. A } -> -u y e. A ) )
25	24	imp 123	. . 3 |- ( ( A C_ RR /\ y e. { n e. RR | -u n e. A } ) -> -u y e. A )
26	2, 7	syl6com 35	. . . . . . . . . 10 |- ( x e. A -> ( A C_ RR -> x e. CC ) )
27	26	adantl 275	. . . . . . . . 9 |- ( ( ( y e. RR /\ -u y e. A ) /\ x e. A ) -> ( A C_ RR -> x e. CC ) )
28	27	imp 123	. . . . . . . 8 |- ( ( ( ( y e. RR /\ -u y e. A ) /\ x e. A ) /\ A C_ RR ) -> x e. CC )
29		recn 7907	. . . . . . . . 9 |- ( y e. RR -> y e. CC )
30	29	ad3antrrr 489	. . . . . . . 8 |- ( ( ( ( y e. RR /\ -u y e. A ) /\ x e. A ) /\ A C_ RR ) -> y e. CC )
31		negcon2 8172	. . . . . . . 8 |- ( ( x e. CC /\ y e. CC ) -> ( x = -u y <-> y = -u x ) )
32	28, 30, 31	syl2anc 409	. . . . . . 7 |- ( ( ( ( y e. RR /\ -u y e. A ) /\ x e. A ) /\ A C_ RR ) -> ( x = -u y <-> y = -u x ) )
33	32	exp31 362	. . . . . 6 |- ( ( y e. RR /\ -u y e. A ) -> ( x e. A -> ( A C_ RR -> ( x = -u y <-> y = -u x ) ) ) )
34	21, 33	sylbi 120	. . . . 5 |- ( y e. { n e. RR | -u n e. A } -> ( x e. A -> ( A C_ RR -> ( x = -u y <-> y = -u x ) ) ) )
35	34	impcom 124	. . . 4 |- ( ( x e. A /\ y e. { n e. RR | -u n e. A } ) -> ( A C_ RR -> ( x = -u y <-> y = -u x ) ) )
36	35	impcom 124	. . 3 |- ( ( A C_ RR /\ ( x e. A /\ y e. { n e. RR | -u n e. A } ) ) -> ( x = -u y <-> y = -u x ) )
37	1, 18, 25, 36	f1ocnv2d 6053	. 2 |- ( A C_ RR -> ( F : A -1-1-onto-> { n e. RR | -u n e. A } /\ `' F = ( y e. { n e. RR | -u n e. A } |-> -u y ) ) )
38	37	simpld 111	1 |- ( A C_ RR -> F : A -1-1-onto-> { n e. RR | -u n e. A } )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1348   e. wcel 2141   {crab 2452   C_ wss 3121   |-> cmpt 4050   `'ccnv 4610   -1-1-onto->wf1o 5197   CCcc 7772   RRcr 7773   -ucneg 8091

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-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-setind 4521  ax-resscn 7866  ax-1cn 7867  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-addcom 7874  ax-addass 7876  ax-distr 7878  ax-i2m1 7879  ax-0id 7882  ax-rnegex 7883  ax-cnre 7885

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-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-sub 8092  df-neg 8093

This theorem is referenced by:  negfi  11191