Theorem redc0 14808

Description: Two ways to express decidability of real number equality. (Contributed by Jim Kingdon, 23-Jul-2024.)

Assertion

Ref	Expression
redc0	|- ( A. x e. RR A. y e. RR DECID x = y <-> A. z e. RR DECID z = 0 )

Distinct variable group:   x, y, z

Proof of Theorem redc0

Step	Hyp	Ref	Expression
1		0re 7957	. . . . 5 |- 0 e. RR
2		eqeq1 2184	. . . . . . 7 |- ( x = z -> ( x = y <-> z = y ) )
3	2	dcbid 838	. . . . . 6 |- ( x = z -> (DECID x = y <-> DECID z = y ) )
4		eqeq2 2187	. . . . . . 7 |- ( y = 0 -> ( z = y <-> z = 0 ) )
5	4	dcbid 838	. . . . . 6 |- ( y = 0 -> (DECID z = y <-> DECID z = 0 ) )
6	3, 5	rspc2v 2855	. . . . 5 |- ( ( z e. RR /\ 0 e. RR ) -> ( A. x e. RR A. y e. RR DECID x = y -> DECID z = 0 ) )
7	1, 6	mpan2 425	. . . 4 |- ( z e. RR -> ( A. x e. RR A. y e. RR DECID x = y -> DECID z = 0 ) )
8	7	impcom 125	. . 3 |- ( ( A. x e. RR A. y e. RR DECID x = y /\ z e. RR ) -> DECID z = 0 )
9	8	ralrimiva 2550	. 2 |- ( A. x e. RR A. y e. RR DECID x = y -> A. z e. RR DECID z = 0 )
10		eqeq1 2184	. . . . . 6 |- ( z = ( x - y ) -> ( z = 0 <-> ( x - y ) = 0 ) )
11	10	dcbid 838	. . . . 5 |- ( z = ( x - y ) -> (DECID z = 0 <-> DECID ( x - y ) = 0 ) )
12		simpl 109	. . . . 5 |- ( ( A. z e. RR DECID z = 0 /\ ( x e. RR /\ y e. RR ) ) -> A. z e. RR DECID z = 0 )
13		resubcl 8221	. . . . . 6 |- ( ( x e. RR /\ y e. RR ) -> ( x - y ) e. RR )
14	13	adantl 277	. . . . 5 |- ( ( A. z e. RR DECID z = 0 /\ ( x e. RR /\ y e. RR ) ) -> ( x - y ) e. RR )
15	11, 12, 14	rspcdva 2847	. . . 4 |- ( ( A. z e. RR DECID z = 0 /\ ( x e. RR /\ y e. RR ) ) -> DECID ( x - y ) = 0 )
16		simprl 529	. . . . . . 7 |- ( ( A. z e. RR DECID z = 0 /\ ( x e. RR /\ y e. RR ) ) -> x e. RR )
17	16	recnd 7986	. . . . . 6 |- ( ( A. z e. RR DECID z = 0 /\ ( x e. RR /\ y e. RR ) ) -> x e. CC )
18		simprr 531	. . . . . . 7 |- ( ( A. z e. RR DECID z = 0 /\ ( x e. RR /\ y e. RR ) ) -> y e. RR )
19	18	recnd 7986	. . . . . 6 |- ( ( A. z e. RR DECID z = 0 /\ ( x e. RR /\ y e. RR ) ) -> y e. CC )
20	17, 19	subeq0ad 8278	. . . . 5 |- ( ( A. z e. RR DECID z = 0 /\ ( x e. RR /\ y e. RR ) ) -> ( ( x - y ) = 0 <-> x = y ) )
21	20	dcbid 838	. . . 4 |- ( ( A. z e. RR DECID z = 0 /\ ( x e. RR /\ y e. RR ) ) -> (DECID ( x - y ) = 0 <-> DECID x = y ) )
22	15, 21	mpbid 147	. . 3 |- ( ( A. z e. RR DECID z = 0 /\ ( x e. RR /\ y e. RR ) ) -> DECID x = y )
23	22	ralrimivva 2559	. 2 |- ( A. z e. RR DECID z = 0 -> A. x e. RR A. y e. RR DECID x = y )
24	9, 23	impbii 126	1 |- ( A. x e. RR A. y e. RR DECID x = y <-> A. z e. RR DECID z = 0 )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105  DECID wdc 834   = wceq 1353   e. wcel 2148   A.wral 2455  (class class class)co 5875   RRcr 7810   0cc0 7811   - cmin 8128

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-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-setind 4537  ax-resscn 7903  ax-1cn 7904  ax-1re 7905  ax-icn 7906  ax-addcl 7907  ax-addrcl 7908  ax-mulcl 7909  ax-addcom 7911  ax-addass 7913  ax-distr 7915  ax-i2m1 7916  ax-0id 7919  ax-rnegex 7920  ax-cnre 7922

This theorem depends on definitions:  df-bi 117  df-dc 835  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-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-opab 4066  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-iota 5179  df-fun 5219  df-fv 5225  df-riota 5831  df-ov 5878  df-oprab 5879  df-mpo 5880  df-sub 8130  df-neg 8131

This theorem is referenced by:  dcapnconstALT  14812