Theorem eqreznegel 10014

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 3242	. . . . . . . 8 |- ( A C_ ZZ -> ( -u w e. A -> -u w e. ZZ ) )
2		recn 8312	. . . . . . . . 9 |- ( w e. RR -> w e. CC )
3		negid 8573	. . . . . . . . . . . 12 |- ( w e. CC -> ( w + -u w ) = 0 )
4		0z 9655	. . . . . . . . . . . 12 |- 0 e. ZZ
5	3, 4	eqeltrdi 2329	. . . . . . . . . . 11 |- ( w e. CC -> ( w + -u w ) e. ZZ )
6	5	pm4.71i 395	. . . . . . . . . 10 |- ( w e. CC <-> ( w e. CC /\ ( w + -u w ) e. ZZ ) )
7		zrevaddcl 9695	. . . . . . . . . 10 |- ( -u w e. ZZ -> ( ( w e. CC /\ ( w + -u w ) e. ZZ ) <-> w e. ZZ ) )
8	6, 7	bitrid 192	. . . . . . . . 9 |- ( -u w e. ZZ -> ( w e. CC <-> w e. ZZ ) )
9	2, 8	imbitrid 154	. . . . . . . 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 254	. . . . 5 |- ( A C_ ZZ -> ( ( w e. RR /\ -u w e. A ) -> w e. ZZ ) )
13		simpr 110	. . . . . 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 307	. . . 4 |- ( A C_ ZZ -> ( ( w e. RR /\ -u w e. A ) -> ( w e. ZZ /\ -u w e. A ) ) )
16		zre 9648	. . . . 5 |- ( w e. ZZ -> w e. RR )
17	16	anim1i 340	. . . 4 |- ( ( w e. ZZ /\ -u w e. A ) -> ( w e. RR /\ -u w e. A ) )
18	15, 17	impbid1 142	. . 3 |- ( A C_ ZZ -> ( ( w e. RR /\ -u w e. A ) <-> ( w e. ZZ /\ -u w e. A ) ) )
19		negeq 8519	. . . . 5 |- ( z = w -> -u z = -u w )
20	19	eleq1d 2307	. . . 4 |- ( z = w -> ( -u z e. A <-> -u w e. A ) )
21	20	elrab 2982	. . 3 |- ( w e. { z e. RR | -u z e. A } <-> ( w e. RR /\ -u w e. A ) )
22	20	elrab 2982	. . 3 |- ( w e. { z e. ZZ | -u z e. A } <-> ( w e. ZZ /\ -u w e. A ) )
23	18, 21, 22	3bitr4g 223	. 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 2236	1 |- ( A C_ ZZ -> { z e. RR | -u z e. A } = { z e. ZZ | -u z e. A } )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   {crab 2532   C_ wss 3220  (class class class)co 6085   CCcc 8177   RRcr 8178   0cc0 8179   + caddc 8182   -ucneg 8498   ZZcz 9644

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-0id 8287  ax-rnegex 8288  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-ltadd 8295

This proof depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4131  df-opab 4193  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-iota 5337  df-fun 5379  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-inn 9305  df-n0 9564  df-z 9645

This theorem is used by: (None)