Theorem eqreznegel 9308

Description: Two ways to express the image under negation of a set of integers. (Contributed by Paul Chapman, 21-Mar-2011.)

Assertion

Ref	Expression
eqreznegel	|- ( A C_ ZZ -> { z e. RR | -u z e. A } = { z e. ZZ | -u z e. A } )

Distinct variable group:   z, A

Proof of Theorem eqreznegel

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssel 3057	. . . . . . . 8 |- ( A C_ ZZ -> ( -u w e. A -> -u w e. ZZ ) )
2		recn 7677	. . . . . . . . 9 |- ( w e. RR -> w e. CC )
3		negid 7932	. . . . . . . . . . . 12 |- ( w e. CC -> ( w + -u w ) = 0 )
4		0z 8969	. . . . . . . . . . . 12 |- 0 e. ZZ
5	3, 4	syl6eqel 2205	. . . . . . . . . . 11 |- ( w e. CC -> ( w + -u w ) e. ZZ )
6	5	pm4.71i 386	. . . . . . . . . 10 |- ( w e. CC <-> ( w e. CC /\ ( w + -u w ) e. ZZ ) )
7		zrevaddcl 9008	. . . . . . . . . 10 |- ( -u w e. ZZ -> ( ( w e. CC /\ ( w + -u w ) e. ZZ ) <-> w e. ZZ ) )
8	6, 7	syl5bb 191	. . . . . . . . 9 |- ( -u w e. ZZ -> ( w e. CC <-> w e. ZZ ) )
9	2, 8	syl5ib 153	. . . . . . . 8 |- ( -u w e. ZZ -> ( w e. RR -> w e. ZZ ) )
10	1, 9	syl6 33	. . . . . . 7 |- ( A C_ ZZ -> ( -u w e. A -> ( w e. RR -> w e. ZZ ) ) )
11	10	com23 78	. . . . . 6 |- ( A C_ ZZ -> ( w e. RR -> ( -u w e. A -> w e. ZZ ) ) )
12	11	impd 252	. . . . 5 |- ( A C_ ZZ -> ( ( w e. RR /\ -u w e. A ) -> w e. ZZ ) )
13		simpr 109	. . . . . 6 |- ( ( w e. RR /\ -u w e. A ) -> -u w e. A )
14	13	a1i 9	. . . . 5 |- ( A C_ ZZ -> ( ( w e. RR /\ -u w e. A ) -> -u w e. A ) )
15	12, 14	jcad 303	. . . 4 |- ( A C_ ZZ -> ( ( w e. RR /\ -u w e. A ) -> ( w e. ZZ /\ -u w e. A ) ) )
16		zre 8962	. . . . 5 |- ( w e. ZZ -> w e. RR )
17	16	anim1i 336	. . . 4 |- ( ( w e. ZZ /\ -u w e. A ) -> ( w e. RR /\ -u w e. A ) )
18	15, 17	impbid1 141	. . 3 |- ( A C_ ZZ -> ( ( w e. RR /\ -u w e. A ) <-> ( w e. ZZ /\ -u w e. A ) ) )
19		negeq 7878	. . . . 5 |- ( z = w -> -u z = -u w )
20	19	eleq1d 2183	. . . 4 |- ( z = w -> ( -u z e. A <-> -u w e. A ) )
21	20	elrab 2809	. . 3 |- ( w e. { z e. RR | -u z e. A } <-> ( w e. RR /\ -u w e. A ) )
22	20	elrab 2809	. . 3 |- ( w e. { z e. ZZ | -u z e. A } <-> ( w e. ZZ /\ -u w e. A ) )
23	18, 21, 22	3bitr4g 222	. 2 |- ( A C_ ZZ -> ( w e. { z e. RR | -u z e. A } <-> w e. { z e. ZZ | -u z e. A } ) )
24	23	eqrdv 2113	1 |- ( A C_ ZZ -> { z e. RR | -u z e. A } = { z e. ZZ | -u z e. A } )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1314   e. wcel 1463   {crab 2394   C_ wss 3037  (class class class)co 5728   CCcc 7545   RRcr 7546   0cc0 7547   + caddc 7550   -ucneg 7857   ZZcz 8958

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-pow 4058  ax-pr 4091  ax-un 4315  ax-setind 4412  ax-cnex 7636  ax-resscn 7637  ax-1cn 7638  ax-1re 7639  ax-icn 7640  ax-addcl 7641  ax-addrcl 7642  ax-mulcl 7643  ax-addcom 7645  ax-addass 7647  ax-distr 7649  ax-i2m1 7650  ax-0lt1 7651  ax-0id 7653  ax-rnegex 7654  ax-cnre 7656  ax-pre-ltirr 7657  ax-pre-ltwlin 7658  ax-pre-lttrn 7659  ax-pre-ltadd 7661

This theorem depends on definitions:  df-bi 116  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ne 2283  df-nel 2378  df-ral 2395  df-rex 2396  df-reu 2397  df-rab 2399  df-v 2659  df-sbc 2879  df-dif 3039  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-int 3738  df-br 3896  df-opab 3950  df-id 4175  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-iota 5046  df-fun 5083  df-fv 5089  df-riota 5684  df-ov 5731  df-oprab 5732  df-mpo 5733  df-pnf 7726  df-mnf 7727  df-xr 7728  df-ltxr 7729  df-le 7730  df-sub 7858  df-neg 7859  df-inn 8631  df-n0 8882  df-z 8959

This theorem is referenced by: (None)