Theorem eqreznegel 9993

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 8302	. . . . . . . . 9 |- ( w e. RR -> w e. CC )
3		negid 8563	. . . . . . . . . . . 12 |- ( w e. CC -> ( w + -u w ) = 0 )
4		0z 9634	. . . . . . . . . . . 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 9674	. . . . . . . . . 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 9627	. . . . 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 8509	. . . . 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 } )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   {crab 2532   C_ wss 3220  (class class class)co 6075   CCcc 8167   RRcr 8168   0cc0 8169   + caddc 8172   -ucneg 8488   ZZcz 9623

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 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 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-ltadd 8285

This theorem 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 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-iota 5332  df-fun 5374  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-inn 9284  df-n0 9543  df-z 9624

This theorem is referenced by: (None)