Theorem redc0 15958

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 8071	. . . . 5 |- 0 e. RR
2		eqeq1 2211	. . . . . . 7 |- ( x = z -> ( x = y <-> z = y ) )
3	2	dcbid 839	. . . . . 6 |- ( x = z -> (DECID x = y <-> DECID z = y ) )
4		eqeq2 2214	. . . . . . 7 |- ( y = 0 -> ( z = y <-> z = 0 ) )
5	4	dcbid 839	. . . . . 6 |- ( y = 0 -> (DECID z = y <-> DECID z = 0 ) )
6	3, 5	rspc2v 2889	. . . . 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 2578	. 2 |- ( A. x e. RR A. y e. RR DECID x = y -> A. z e. RR DECID z = 0 )
10		eqeq1 2211	. . . . . 6 |- ( z = ( x - y ) -> ( z = 0 <-> ( x - y ) = 0 ) )
11	10	dcbid 839	. . . . 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 8335	. . . . . 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 2881	. . . 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 8100	. . . . . 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 8100	. . . . . 6 |- ( ( A. z e. RR DECID z = 0 /\ ( x e. RR /\ y e. RR ) ) -> y e. CC )
20	17, 19	subeq0ad 8392	. . . . 5 |- ( ( A. z e. RR DECID z = 0 /\ ( x e. RR /\ y e. RR ) ) -> ( ( x - y ) = 0 <-> x = y ) )
21	20	dcbid 839	. . . 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 2587	. 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 835   = wceq 1372   e. wcel 2175   A.wral 2483  (class class class)co 5943   RRcr 7923   0cc0 7924   - cmin 8242

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 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-setind 4584  ax-resscn 8016  ax-1cn 8017  ax-1re 8018  ax-icn 8019  ax-addcl 8020  ax-addrcl 8021  ax-mulcl 8022  ax-addcom 8024  ax-addass 8026  ax-distr 8028  ax-i2m1 8029  ax-0id 8032  ax-rnegex 8033  ax-cnre 8035

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-ral 2488  df-rex 2489  df-reu 2490  df-rab 2492  df-v 2773  df-sbc 2998  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-br 4044  df-opab 4105  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-iota 5231  df-fun 5272  df-fv 5278  df-riota 5898  df-ov 5946  df-oprab 5947  df-mpo 5948  df-sub 8244  df-neg 8245

This theorem is referenced by:  dcapnconstALT  15963