Theorem redc0 13936

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 7899	. . . . 5 |- 0 e. RR
2		eqeq1 2172	. . . . . . 7 |- ( x = z -> ( x = y <-> z = y ) )
3	2	dcbid 828	. . . . . 6 |- ( x = z -> (DECID x = y <-> DECID z = y ) )
4		eqeq2 2175	. . . . . . 7 |- ( y = 0 -> ( z = y <-> z = 0 ) )
5	4	dcbid 828	. . . . . 6 |- ( y = 0 -> (DECID z = y <-> DECID z = 0 ) )
6	3, 5	rspc2v 2843	. . . . 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 422	. . . 4 |- ( z e. RR -> ( A. x e. RR A. y e. RR DECID x = y -> DECID z = 0 ) )
8	7	impcom 124	. . 3 |- ( ( A. x e. RR A. y e. RR DECID x = y /\ z e. RR ) -> DECID z = 0 )
9	8	ralrimiva 2539	. 2 |- ( A. x e. RR A. y e. RR DECID x = y -> A. z e. RR DECID z = 0 )
10		eqeq1 2172	. . . . . 6 |- ( z = ( x - y ) -> ( z = 0 <-> ( x - y ) = 0 ) )
11	10	dcbid 828	. . . . 5 |- ( z = ( x - y ) -> (DECID z = 0 <-> DECID ( x - y ) = 0 ) )
12		simpl 108	. . . . 5 |- ( ( A. z e. RR DECID z = 0 /\ ( x e. RR /\ y e. RR ) ) -> A. z e. RR DECID z = 0 )
13		resubcl 8162	. . . . . 6 |- ( ( x e. RR /\ y e. RR ) -> ( x - y ) e. RR )
14	13	adantl 275	. . . . 5 |- ( ( A. z e. RR DECID z = 0 /\ ( x e. RR /\ y e. RR ) ) -> ( x - y ) e. RR )
15	11, 12, 14	rspcdva 2835	. . . 4 |- ( ( A. z e. RR DECID z = 0 /\ ( x e. RR /\ y e. RR ) ) -> DECID ( x - y ) = 0 )
16		simprl 521	. . . . . . 7 |- ( ( A. z e. RR DECID z = 0 /\ ( x e. RR /\ y e. RR ) ) -> x e. RR )
17	16	recnd 7927	. . . . . 6 |- ( ( A. z e. RR DECID z = 0 /\ ( x e. RR /\ y e. RR ) ) -> x e. CC )
18		simprr 522	. . . . . . 7 |- ( ( A. z e. RR DECID z = 0 /\ ( x e. RR /\ y e. RR ) ) -> y e. RR )
19	18	recnd 7927	. . . . . 6 |- ( ( A. z e. RR DECID z = 0 /\ ( x e. RR /\ y e. RR ) ) -> y e. CC )
20	17, 19	subeq0ad 8219	. . . . 5 |- ( ( A. z e. RR DECID z = 0 /\ ( x e. RR /\ y e. RR ) ) -> ( ( x - y ) = 0 <-> x = y ) )
21	20	dcbid 828	. . . 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 146	. . 3 |- ( ( A. z e. RR DECID z = 0 /\ ( x e. RR /\ y e. RR ) ) -> DECID x = y )
23	22	ralrimivva 2548	. 2 |- ( A. z e. RR DECID z = 0 -> A. x e. RR A. y e. RR DECID x = y )
24	9, 23	impbii 125	1 |- ( A. x e. RR A. y e. RR DECID x = y <-> A. z e. RR DECID z = 0 )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104  DECID wdc 824   = wceq 1343   e. wcel 2136   A.wral 2444  (class class class)co 5842   RRcr 7752   0cc0 7753   - cmin 8069

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-setind 4514  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-addcom 7853  ax-addass 7855  ax-distr 7857  ax-i2m1 7858  ax-0id 7861  ax-rnegex 7862  ax-cnre 7864

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-iota 5153  df-fun 5190  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-sub 8071  df-neg 8072

This theorem is referenced by:  dcapnconstALT  13940