Theorem eqreznegel 9616

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 3151	. . . . . . . 8 |- ( A C_ ZZ -> ( -u w e. A -> -u w e. ZZ ) )
2		recn 7946	. . . . . . . . 9 |- ( w e. RR -> w e. CC )
3		negid 8206	. . . . . . . . . . . 12 |- ( w e. CC -> ( w + -u w ) = 0 )
4		0z 9266	. . . . . . . . . . . 12 |- 0 e. ZZ
5	3, 4	eqeltrdi 2268	. . . . . . . . . . 11 |- ( w e. CC -> ( w + -u w ) e. ZZ )
6	5	pm4.71i 391	. . . . . . . . . 10 |- ( w e. CC <-> ( w e. CC /\ ( w + -u w ) e. ZZ ) )
7		zrevaddcl 9305	. . . . . . . . . 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 9259	. . . . 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 8152	. . . . 5 |- ( z = w -> -u z = -u w )
20	19	eleq1d 2246	. . . 4 |- ( z = w -> ( -u z e. A <-> -u w e. A ) )
21	20	elrab 2895	. . 3 |- ( w e. { z e. RR | -u z e. A } <-> ( w e. RR /\ -u w e. A ) )
22	20	elrab 2895	. . 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 2175	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 1353   e. wcel 2148   {crab 2459   C_ wss 3131  (class class class)co 5877   CCcc 7811   RRcr 7812   0cc0 7813   + caddc 7816   -ucneg 8131   ZZcz 9255

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-addcom 7913  ax-addass 7915  ax-distr 7917  ax-i2m1 7918  ax-0lt1 7919  ax-0id 7921  ax-rnegex 7922  ax-cnre 7924  ax-pre-ltirr 7925  ax-pre-ltwlin 7926  ax-pre-lttrn 7927  ax-pre-ltadd 7929

This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-br 4006  df-opab 4067  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-iota 5180  df-fun 5220  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000  df-sub 8132  df-neg 8133  df-inn 8922  df-n0 9179  df-z 9256

This theorem is referenced by: (None)