Theorem redc0 14089

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 7920	. . . . 5 |- 0 e. RR
2		eqeq1 2177	. . . . . . 7 |- ( x = z -> ( x = y <-> z = y ) )
3	2	dcbid 833	. . . . . 6 |- ( x = z -> (DECID x = y <-> DECID z = y ) )
4		eqeq2 2180	. . . . . . 7 |- ( y = 0 -> ( z = y <-> z = 0 ) )
5	4	dcbid 833	. . . . . 6 |- ( y = 0 -> (DECID z = y <-> DECID z = 0 ) )
6	3, 5	rspc2v 2847	. . . . 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 423	. . . 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 2543	. 2 |- ( A. x e. RR A. y e. RR DECID x = y -> A. z e. RR DECID z = 0 )
10		eqeq1 2177	. . . . . 6 |- ( z = ( x - y ) -> ( z = 0 <-> ( x - y ) = 0 ) )
11	10	dcbid 833	. . . . 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 8183	. . . . . 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 2839	. . . 4 |- ( ( A. z e. RR DECID z = 0 /\ ( x e. RR /\ y e. RR ) ) -> DECID ( x - y ) = 0 )
16		simprl 526	. . . . . . 7 |- ( ( A. z e. RR DECID z = 0 /\ ( x e. RR /\ y e. RR ) ) -> x e. RR )
17	16	recnd 7948	. . . . . 6 |- ( ( A. z e. RR DECID z = 0 /\ ( x e. RR /\ y e. RR ) ) -> x e. CC )
18		simprr 527	. . . . . . 7 |- ( ( A. z e. RR DECID z = 0 /\ ( x e. RR /\ y e. RR ) ) -> y e. RR )
19	18	recnd 7948	. . . . . 6 |- ( ( A. z e. RR DECID z = 0 /\ ( x e. RR /\ y e. RR ) ) -> y e. CC )
20	17, 19	subeq0ad 8240	. . . . 5 |- ( ( A. z e. RR DECID z = 0 /\ ( x e. RR /\ y e. RR ) ) -> ( ( x - y ) = 0 <-> x = y ) )
21	20	dcbid 833	. . . 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 2552	. 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 829   = wceq 1348   e. wcel 2141   A.wral 2448  (class class class)co 5853   RRcr 7773   0cc0 7774   - cmin 8090

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-setind 4521  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-addcom 7874  ax-addass 7876  ax-distr 7878  ax-i2m1 7879  ax-0id 7882  ax-rnegex 7883  ax-cnre 7885

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-sub 8092  df-neg 8093

This theorem is referenced by:  dcapnconstALT  14093