Theorem negf1o 8454

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 3187	. . . . . 6 |- ( A C_ RR -> ( x e. A -> x e. RR ) )
3		renegcl 8333	. . . . . 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 8058	. . . . . . . . 9 |- ( x e. RR -> x e. CC )
8		negneg 8322	. . . . . . . . . 10 |- ( x e. CC -> -u -u x = x )
9	8	eqcomd 2211	. . . . . . . . 9 |- ( x e. CC -> x = -u -u x )
10	7, 9	syl 14	. . . . . . . 8 |- ( x e. RR -> x = -u -u x )
11	10	eleq1d 2274	. . . . . . 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 8265	. . . . . 6 |- ( n = -u x -> -u n = -u -u x )
16	15	eleq1d 2274	. . . . 5 |- ( n = -u x -> ( -u n e. A <-> -u -u x e. A ) )
17	16	elrab 2929	. . . 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 8265	. . . . . . 7 |- ( n = y -> -u n = -u y )
20	19	eleq1d 2274	. . . . . 6 |- ( n = y -> ( -u n e. A <-> -u y e. A ) )
21	20	elrab 2929	. . . . 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 8058	. . . . . . . . 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 8325	. . . . . . . 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 6150	. 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 2176   {crab 2488   C_ wss 3166   |-> cmpt 4105   `'ccnv 4674   -1-1-onto->wf1o 5270   CCcc 7923   RRcr 7924   -ucneg 8244

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-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-setind 4585  ax-resscn 8017  ax-1cn 8018  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-addcom 8025  ax-addass 8027  ax-distr 8029  ax-i2m1 8030  ax-0id 8033  ax-rnegex 8034  ax-cnre 8036

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-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  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 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-sub 8245  df-neg 8246

This theorem is referenced by:  negfi  11539