Theorem negf1o 8489

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 3195	. . . . . 6 |- ( A C_ RR -> ( x e. A -> x e. RR ) )
3		renegcl 8368	. . . . . 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 124	. . . 4 |- ( ( A C_ RR /\ x e. A ) -> -u x e. RR )
6	2	imp 124	. . . . 5 |- ( ( A C_ RR /\ x e. A ) -> x e. RR )
7		recn 8093	. . . . . . . . 9 |- ( x e. RR -> x e. CC )
8		negneg 8357	. . . . . . . . . 10 |- ( x e. CC -> -u -u x = x )
9	8	eqcomd 2213	. . . . . . . . 9 |- ( x e. CC -> x = -u -u x )
10	7, 9	syl 14	. . . . . . . 8 |- ( x e. RR -> x = -u -u x )
11	10	eleq1d 2276	. . . . . . 7 |- ( x e. RR -> ( x e. A <-> -u -u x e. A ) )
12	11	biimpcd 159	. . . . . 6 |- ( x e. A -> ( x e. RR -> -u -u x e. A ) )
13	12	adantl 277	. . . . 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 8300	. . . . . 6 |- ( n = -u x -> -u n = -u -u x )
16	15	eleq1d 2276	. . . . 5 |- ( n = -u x -> ( -u n e. A <-> -u -u x e. A ) )
17	16	elrab 2936	. . . 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 417	. . 3 |- ( ( A C_ RR /\ x e. A ) -> -u x e. { n e. RR | -u n e. A } )
19		negeq 8300	. . . . . . 7 |- ( n = y -> -u n = -u y )
20	19	eleq1d 2276	. . . . . 6 |- ( n = y -> ( -u n e. A <-> -u y e. A ) )
21	20	elrab 2936	. . . . 5 |- ( y e. { n e. RR | -u n e. A } <-> ( y e. RR /\ -u y e. A ) )
22		simpr 110	. . . . . 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	biimtrid 152	. . . 4 |- ( A C_ RR -> ( y e. { n e. RR | -u n e. A } -> -u y e. A ) )
25	24	imp 124	. . 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 277	. . . . . . . . 9 |- ( ( ( y e. RR /\ -u y e. A ) /\ x e. A ) -> ( A C_ RR -> x e. CC ) )
28	27	imp 124	. . . . . . . 8 |- ( ( ( ( y e. RR /\ -u y e. A ) /\ x e. A ) /\ A C_ RR ) -> x e. CC )
29		recn 8093	. . . . . . . . 9 |- ( y e. RR -> y e. CC )
30	29	ad3antrrr 492	. . . . . . . 8 |- ( ( ( ( y e. RR /\ -u y e. A ) /\ x e. A ) /\ A C_ RR ) -> y e. CC )
31		negcon2 8360	. . . . . . . 8 |- ( ( x e. CC /\ y e. CC ) -> ( x = -u y <-> y = -u x ) )
32	28, 30, 31	syl2anc 411	. . . . . . 7 |- ( ( ( ( y e. RR /\ -u y e. A ) /\ x e. A ) /\ A C_ RR ) -> ( x = -u y <-> y = -u x ) )
33	32	exp31 364	. . . . . 6 |- ( ( y e. RR /\ -u y e. A ) -> ( x e. A -> ( A C_ RR -> ( x = -u y <-> y = -u x ) ) ) )
34	21, 33	sylbi 121	. . . . 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 125	. . . 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 125	. . 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 6173	. 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 112	1 |- ( A C_ RR -> F : A -1-1-onto-> { n e. RR | -u n e. A } )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   e. wcel 2178   {crab 2490   C_ wss 3174   |-> cmpt 4121   `'ccnv 4692   -1-1-onto->wf1o 5289   CCcc 7958   RRcr 7959   -ucneg 8279

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-setind 4603  ax-resscn 8052  ax-1cn 8053  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-addcom 8060  ax-addass 8062  ax-distr 8064  ax-i2m1 8065  ax-0id 8068  ax-rnegex 8069  ax-cnre 8071

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-sub 8280  df-neg 8281

This theorem is referenced by:  negfi  11654