Theorem redc0 13599

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 7872	. . . . 5 |- 0 e. RR
2		eqeq1 2164	. . . . . . 7 |- ( x = z -> ( x = y <-> z = y ) )
3	2	dcbid 824	. . . . . 6 |- ( x = z -> (DECID x = y <-> DECID z = y ) )
4		eqeq2 2167	. . . . . . 7 |- ( y = 0 -> ( z = y <-> z = 0 ) )
5	4	dcbid 824	. . . . . 6 |- ( y = 0 -> (DECID z = y <-> DECID z = 0 ) )
6	3, 5	rspc2v 2829	. . . . 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 2530	. 2 |- ( A. x e. RR A. y e. RR DECID x = y -> A. z e. RR DECID z = 0 )
10		eqeq1 2164	. . . . . 6 |- ( z = ( x - y ) -> ( z = 0 <-> ( x - y ) = 0 ) )
11	10	dcbid 824	. . . . 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 8133	. . . . . 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 2821	. . . 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 7900	. . . . . 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 7900	. . . . . 6 |- ( ( A. z e. RR DECID z = 0 /\ ( x e. RR /\ y e. RR ) ) -> y e. CC )
20	17, 19	subeq0ad 8190	. . . . 5 |- ( ( A. z e. RR DECID z = 0 /\ ( x e. RR /\ y e. RR ) ) -> ( ( x - y ) = 0 <-> x = y ) )
21	20	dcbid 824	. . . 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 2539	. 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 820   = wceq 1335   e. wcel 2128   A.wral 2435  (class class class)co 5821   RRcr 7725   0cc0 7726   - cmin 8040

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-setind 4495  ax-resscn 7818  ax-1cn 7819  ax-1re 7820  ax-icn 7821  ax-addcl 7822  ax-addrcl 7823  ax-mulcl 7824  ax-addcom 7826  ax-addass 7828  ax-distr 7830  ax-i2m1 7831  ax-0id 7834  ax-rnegex 7835  ax-cnre 7837

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-id 4253  df-xp 4591  df-rel 4592  df-cnv 4593  df-co 4594  df-dm 4595  df-iota 5134  df-fun 5171  df-fv 5177  df-riota 5777  df-ov 5824  df-oprab 5825  df-mpo 5826  df-sub 8042  df-neg 8043

This theorem is referenced by:  dcapnconstALT  13603