Theorem dceqnconst 16837

Description: Decidability of real number equality implies the existence of a certain non-constant function from real numbers to integers. Variation of Exercise 11.6(i) of [HoTT], p. (varies). See redcwlpo 16832 for more discussion of decidability of real number equality. (Contributed by BJ and Jim Kingdon, 24-Jun-2024.) (Revised by Jim Kingdon, 23-Jul-2024.)

Assertion

Ref	Expression
dceqnconst	|- ( A. x e. RR DECID x = 0 -> E. f ( f : RR --> ZZ /\ ( f ` 0 ) = 0 /\ A. x e. RR+ ( f ` x ) =/= 0 ) )

Distinct variable group:   x, f

Proof of Theorem dceqnconst

Dummy variables y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		reex 8260	. . . 4 |- RR e. _V
2	1	mptex 5911	. . 3 |- ( y e. RR |-> if ( y = 0 , 0 , 1 ) ) e. _V
3	2	a1i 9	. 2 |- ( A. x e. RR DECID x = 0 -> ( y e. RR |-> if ( y = 0 , 0 , 1 ) ) e. _V )
4		0zd 9588	. . . . 5 |- ( ( A. x e. RR DECID x = 0 /\ y e. RR ) -> 0 e. ZZ )
5		1zzd 9603	. . . . 5 |- ( ( A. x e. RR DECID x = 0 /\ y e. RR ) -> 1 e. ZZ )
6		eqeq1 2239	. . . . . . 7 |- ( x = y -> ( x = 0 <-> y = 0 ) )
7	6	dcbid 846	. . . . . 6 |- ( x = y -> (DECID x = 0 <-> DECID y = 0 ) )
8	7	rspccva 2919	. . . . 5 |- ( ( A. x e. RR DECID x = 0 /\ y e. RR ) -> DECID y = 0 )
9	4, 5, 8	ifcldcd 3659	. . . 4 |- ( ( A. x e. RR DECID x = 0 /\ y e. RR ) -> if ( y = 0 , 0 , 1 ) e. ZZ )
10	9	fmpttd 5831	. . 3 |- ( A. x e. RR DECID x = 0 -> ( y e. RR |-> if ( y = 0 , 0 , 1 ) ) : RR --> ZZ )
11		0re 8273	. . . . . 6 |- 0 e. RR
12		0zd 9588	. . . . . . . 8 |- ( T. -> 0 e. ZZ )
13		1zzd 9603	. . . . . . . 8 |- ( T. -> 1 e. ZZ )
14		eqid 2232	. . . . . . . . . . 11 |- 0 = 0
15	14	orci 739	. . . . . . . . . 10 |- ( 0 = 0 \/ -. 0 = 0 )
16		df-dc 843	. . . . . . . . . 10 |- (DECID 0 = 0 <-> ( 0 = 0 \/ -. 0 = 0 ) )
17	15, 16	mpbir 146	. . . . . . . . 9 |- DECID 0 = 0
18	17	a1i 9	. . . . . . . 8 |- ( T. -> DECID 0 = 0 )
19	12, 13, 18	ifcldcd 3659	. . . . . . 7 |- ( T. -> if ( 0 = 0 , 0 , 1 ) e. ZZ )
20	19	mptru 1407	. . . . . 6 |- if ( 0 = 0 , 0 , 1 ) e. ZZ
21		eqeq1 2239	. . . . . . . 8 |- ( y = 0 -> ( y = 0 <-> 0 = 0 ) )
22	21	ifbid 3643	. . . . . . 7 |- ( y = 0 -> if ( y = 0 , 0 , 1 ) = if ( 0 = 0 , 0 , 1 ) )
23		eqid 2232	. . . . . . 7 |- ( y e. RR |-> if ( y = 0 , 0 , 1 ) ) = ( y e. RR |-> if ( y = 0 , 0 , 1 ) )
24	22, 23	fvmptg 5752	. . . . . 6 |- ( ( 0 e. RR /\ if ( 0 = 0 , 0 , 1 ) e. ZZ ) -> ( ( y e. RR |-> if ( y = 0 , 0 , 1 ) ) ` 0 ) = if ( 0 = 0 , 0 , 1 ) )
25	11, 20, 24	mp2an 426	. . . . 5 |- ( ( y e. RR |-> if ( y = 0 , 0 , 1 ) ) ` 0 ) = if ( 0 = 0 , 0 , 1 )
26	14	iftruei 3627	. . . . 5 |- if ( 0 = 0 , 0 , 1 ) = 0
27	25, 26	eqtri 2253	. . . 4 |- ( ( y e. RR |-> if ( y = 0 , 0 , 1 ) ) ` 0 ) = 0
28	27	a1i 9	. . 3 |- ( A. x e. RR DECID x = 0 -> ( ( y e. RR |-> if ( y = 0 , 0 , 1 ) ) ` 0 ) = 0 )
29		1ne0 9304	. . . . . 6 |- 1 =/= 0
30		eqeq1 2239	. . . . . . . . . 10 |- ( y = z -> ( y = 0 <-> z = 0 ) )
31	30	ifbid 3643	. . . . . . . . 9 |- ( y = z -> if ( y = 0 , 0 , 1 ) = if ( z = 0 , 0 , 1 ) )
32		rpre 9992	. . . . . . . . . 10 |- ( z e. RR+ -> z e. RR )
33	32	adantl 277	. . . . . . . . 9 |- ( ( A. x e. RR DECID x = 0 /\ z e. RR+ ) -> z e. RR )
34		0zd 9588	. . . . . . . . . 10 |- ( ( A. x e. RR DECID x = 0 /\ z e. RR+ ) -> 0 e. ZZ )
35		1zzd 9603	. . . . . . . . . 10 |- ( ( A. x e. RR DECID x = 0 /\ z e. RR+ ) -> 1 e. ZZ )
36		eqeq1 2239	. . . . . . . . . . . 12 |- ( x = z -> ( x = 0 <-> z = 0 ) )
37	36	dcbid 846	. . . . . . . . . . 11 |- ( x = z -> (DECID x = 0 <-> DECID z = 0 ) )
38		simpl 109	. . . . . . . . . . 11 |- ( ( A. x e. RR DECID x = 0 /\ z e. RR+ ) -> A. x e. RR DECID x = 0 )
39	37, 38, 33	rspcdva 2925	. . . . . . . . . 10 |- ( ( A. x e. RR DECID x = 0 /\ z e. RR+ ) -> DECID z = 0 )
40	34, 35, 39	ifcldcd 3659	. . . . . . . . 9 |- ( ( A. x e. RR DECID x = 0 /\ z e. RR+ ) -> if ( z = 0 , 0 , 1 ) e. ZZ )
41	23, 31, 33, 40	fvmptd3 5770	. . . . . . . 8 |- ( ( A. x e. RR DECID x = 0 /\ z e. RR+ ) -> ( ( y e. RR |-> if ( y = 0 , 0 , 1 ) ) ` z ) = if ( z = 0 , 0 , 1 ) )
42		rpne0 10001	. . . . . . . . . . 11 |- ( z e. RR+ -> z =/= 0 )
43	42	neneqd 2433	. . . . . . . . . 10 |- ( z e. RR+ -> -. z = 0 )
44	43	iffalsed 3631	. . . . . . . . 9 |- ( z e. RR+ -> if ( z = 0 , 0 , 1 ) = 1 )
45	44	adantl 277	. . . . . . . 8 |- ( ( A. x e. RR DECID x = 0 /\ z e. RR+ ) -> if ( z = 0 , 0 , 1 ) = 1 )
46	41, 45	eqtrd 2265	. . . . . . 7 |- ( ( A. x e. RR DECID x = 0 /\ z e. RR+ ) -> ( ( y e. RR |-> if ( y = 0 , 0 , 1 ) ) ` z ) = 1 )
47	46	neeq1d 2430	. . . . . 6 |- ( ( A. x e. RR DECID x = 0 /\ z e. RR+ ) -> ( ( ( y e. RR |-> if ( y = 0 , 0 , 1 ) ) ` z ) =/= 0 <-> 1 =/= 0 ) )
48	29, 47	mpbiri 168	. . . . 5 |- ( ( A. x e. RR DECID x = 0 /\ z e. RR+ ) -> ( ( y e. RR |-> if ( y = 0 , 0 , 1 ) ) ` z ) =/= 0 )
49	48	ralrimiva 2615	. . . 4 |- ( A. x e. RR DECID x = 0 -> A. z e. RR+ ( ( y e. RR |-> if ( y = 0 , 0 , 1 ) ) ` z ) =/= 0 )
50		fveq2 5669	. . . . . 6 |- ( z = x -> ( ( y e. RR |-> if ( y = 0 , 0 , 1 ) ) ` z ) = ( ( y e. RR |-> if ( y = 0 , 0 , 1 ) ) ` x ) )
51	50	neeq1d 2430	. . . . 5 |- ( z = x -> ( ( ( y e. RR |-> if ( y = 0 , 0 , 1 ) ) ` z ) =/= 0 <-> ( ( y e. RR |-> if ( y = 0 , 0 , 1 ) ) ` x ) =/= 0 ) )
52	51	cbvralv 2777	. . . 4 |- ( A. z e. RR+ ( ( y e. RR |-> if ( y = 0 , 0 , 1 ) ) ` z ) =/= 0 <-> A. x e. RR+ ( ( y e. RR |-> if ( y = 0 , 0 , 1 ) ) ` x ) =/= 0 )
53	49, 52	sylib 122	. . 3 |- ( A. x e. RR DECID x = 0 -> A. x e. RR+ ( ( y e. RR |-> if ( y = 0 , 0 , 1 ) ) ` x ) =/= 0 )
54	10, 28, 53	3jca 1204	. 2 |- ( A. x e. RR DECID x = 0 -> ( ( y e. RR |-> if ( y = 0 , 0 , 1 ) ) : RR --> ZZ /\ ( ( y e. RR |-> if ( y = 0 , 0 , 1 ) ) ` 0 ) = 0 /\ A. x e. RR+ ( ( y e. RR |-> if ( y = 0 , 0 , 1 ) ) ` x ) =/= 0 ) )
55		feq1 5490	. . 3 |- ( f = ( y e. RR |-> if ( y = 0 , 0 , 1 ) ) -> ( f : RR --> ZZ <-> ( y e. RR |-> if ( y = 0 , 0 , 1 ) ) : RR --> ZZ ) )
56		fveq1 5668	. . . 4 |- ( f = ( y e. RR |-> if ( y = 0 , 0 , 1 ) ) -> ( f ` 0 ) = ( ( y e. RR |-> if ( y = 0 , 0 , 1 ) ) ` 0 ) )
57	56	eqeq1d 2241	. . 3 |- ( f = ( y e. RR |-> if ( y = 0 , 0 , 1 ) ) -> ( ( f ` 0 ) = 0 <-> ( ( y e. RR |-> if ( y = 0 , 0 , 1 ) ) ` 0 ) = 0 ) )
58		fveq1 5668	. . . . 5 |- ( f = ( y e. RR |-> if ( y = 0 , 0 , 1 ) ) -> ( f ` x ) = ( ( y e. RR |-> if ( y = 0 , 0 , 1 ) ) ` x ) )
59	58	neeq1d 2430	. . . 4 |- ( f = ( y e. RR |-> if ( y = 0 , 0 , 1 ) ) -> ( ( f ` x ) =/= 0 <-> ( ( y e. RR |-> if ( y = 0 , 0 , 1 ) ) ` x ) =/= 0 ) )
60	59	ralbidv 2542	. . 3 |- ( f = ( y e. RR |-> if ( y = 0 , 0 , 1 ) ) -> ( A. x e. RR+ ( f ` x ) =/= 0 <-> A. x e. RR+ ( ( y e. RR |-> if ( y = 0 , 0 , 1 ) ) ` x ) =/= 0 ) )
61	55, 57, 60	3anbi123d 1349	. 2 |- ( f = ( y e. RR |-> if ( y = 0 , 0 , 1 ) ) -> ( ( f : RR --> ZZ /\ ( f ` 0 ) = 0 /\ A. x e. RR+ ( f ` x ) =/= 0 ) <-> ( ( y e. RR |-> if ( y = 0 , 0 , 1 ) ) : RR --> ZZ /\ ( ( y e. RR |-> if ( y = 0 , 0 , 1 ) ) ` 0 ) = 0 /\ A. x e. RR+ ( ( y e. RR |-> if ( y = 0 , 0 , 1 ) ) ` x ) =/= 0 ) ) )
62	3, 54, 61	elabd 2961	1 |- ( A. x e. RR DECID x = 0 -> E. f ( f : RR --> ZZ /\ ( f ` 0 ) = 0 /\ A. x e. RR+ ( f ` x ) =/= 0 ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 716  DECID wdc 842   /\ w3a 1005   = wceq 1398   T. wtru 1399   E.wex 1541   e. wcel 2203   =/= wne 2412   A.wral 2520   _Vcvv 2812   ifcif 3619   |-> cmpt 4170   -->wf 5347   ` cfv 5351   RRcr 8125   0cc0 8126   1c1 8127   ZZcz 9576   RR+crp 9985

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-cnex 8217  ax-resscn 8218  ax-1cn 8219  ax-1re 8220  ax-icn 8221  ax-addcl 8222  ax-addrcl 8223  ax-mulcl 8224  ax-addcom 8226  ax-addass 8228  ax-distr 8230  ax-i2m1 8231  ax-0lt1 8232  ax-0id 8234  ax-rnegex 8235  ax-cnre 8237  ax-pre-ltirr 8238  ax-pre-ltwlin 8239  ax-pre-lttrn 8240  ax-pre-ltadd 8242

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-if 3620  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-pnf 8309  df-mnf 8310  df-xr 8311  df-ltxr 8312  df-le 8313  df-sub 8445  df-neg 8446  df-inn 9237  df-z 9577  df-rp 9986

This theorem is referenced by:  dcapnconstALT  16839