Theorem dcapnconst 13594

Description: Decidability of real number apartness 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 trilpo 13577 for more discussion of decidability of real number apartness.

This is a weaker form of dceqnconst 13593 and in fact this theorem can be proved using dceqnconst 13593 as shown at dcapnconstALT 13595. (Contributed by BJ and Jim Kingdon, 24-Jun-2024.)

Assertion

Ref	Expression
dcapnconst	|- ( 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 dcapnconst

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

Step	Hyp	Ref	Expression
1		reex 7849	. . . 4 |- RR e. _V
2	1	mptex 5690	. . 3 |- ( y e. RR |-> if ( y # 0 , 1 , 0 ) ) e. _V
3	2	a1i 9	. 2 |- ( A. x e. RR DECID x # 0 -> ( y e. RR |-> if ( y # 0 , 1 , 0 ) ) e. _V )
4		1zzd 9177	. . . . 5 |- ( ( A. x e. RR DECID x # 0 /\ y e. RR ) -> 1 e. ZZ )
5		0zd 9162	. . . . 5 |- ( ( A. x e. RR DECID x # 0 /\ y e. RR ) -> 0 e. ZZ )
6		breq1 3968	. . . . . . 7 |- ( x = y -> ( x # 0 <-> y # 0 ) )
7	6	dcbid 824	. . . . . 6 |- ( x = y -> (DECID x # 0 <-> DECID y # 0 ) )
8	7	rspccva 2815	. . . . 5 |- ( ( A. x e. RR DECID x # 0 /\ y e. RR ) -> DECID y # 0 )
9	4, 5, 8	ifcldcd 3540	. . . 4 |- ( ( A. x e. RR DECID x # 0 /\ y e. RR ) -> if ( y # 0 , 1 , 0 ) e. ZZ )
10	9	fmpttd 5619	. . 3 |- ( A. x e. RR DECID x # 0 -> ( y e. RR |-> if ( y # 0 , 1 , 0 ) ) : RR --> ZZ )
11		0re 7861	. . . . . 6 |- 0 e. RR
12		1zzd 9177	. . . . . . . 8 |- ( T. -> 1 e. ZZ )
13		0zd 9162	. . . . . . . 8 |- ( T. -> 0 e. ZZ )
14		0cn 7853	. . . . . . . . . . . 12 |- 0 e. CC
15		apirr 8463	. . . . . . . . . . . 12 |- ( 0 e. CC -> -. 0 # 0 )
16	14, 15	ax-mp 5	. . . . . . . . . . 11 |- -. 0 # 0
17	16	olci 722	. . . . . . . . . 10 |- ( 0 # 0 \/ -. 0 # 0 )
18		df-dc 821	. . . . . . . . . 10 |- (DECID 0 # 0 <-> ( 0 # 0 \/ -. 0 # 0 ) )
19	17, 18	mpbir 145	. . . . . . . . 9 |- DECID 0 # 0
20	19	a1i 9	. . . . . . . 8 |- ( T. -> DECID 0 # 0 )
21	12, 13, 20	ifcldcd 3540	. . . . . . 7 |- ( T. -> if ( 0 # 0 , 1 , 0 ) e. ZZ )
22	21	mptru 1344	. . . . . 6 |- if ( 0 # 0 , 1 , 0 ) e. ZZ
23		breq1 3968	. . . . . . . 8 |- ( y = 0 -> ( y # 0 <-> 0 # 0 ) )
24	23	ifbid 3526	. . . . . . 7 |- ( y = 0 -> if ( y # 0 , 1 , 0 ) = if ( 0 # 0 , 1 , 0 ) )
25		eqid 2157	. . . . . . 7 |- ( y e. RR |-> if ( y # 0 , 1 , 0 ) ) = ( y e. RR |-> if ( y # 0 , 1 , 0 ) )
26	24, 25	fvmptg 5541	. . . . . 6 |- ( ( 0 e. RR /\ if ( 0 # 0 , 1 , 0 ) e. ZZ ) -> ( ( y e. RR |-> if ( y # 0 , 1 , 0 ) ) ` 0 ) = if ( 0 # 0 , 1 , 0 ) )
27	11, 22, 26	mp2an 423	. . . . 5 |- ( ( y e. RR |-> if ( y # 0 , 1 , 0 ) ) ` 0 ) = if ( 0 # 0 , 1 , 0 )
28	16	iffalsei 3514	. . . . 5 |- if ( 0 # 0 , 1 , 0 ) = 0
29	27, 28	eqtri 2178	. . . 4 |- ( ( y e. RR |-> if ( y # 0 , 1 , 0 ) ) ` 0 ) = 0
30	29	a1i 9	. . 3 |- ( A. x e. RR DECID x # 0 -> ( ( y e. RR |-> if ( y # 0 , 1 , 0 ) ) ` 0 ) = 0 )
31		1ne0 8884	. . . . . 6 |- 1 =/= 0
32		breq1 3968	. . . . . . . . . 10 |- ( y = z -> ( y # 0 <-> z # 0 ) )
33	32	ifbid 3526	. . . . . . . . 9 |- ( y = z -> if ( y # 0 , 1 , 0 ) = if ( z # 0 , 1 , 0 ) )
34		rpre 9549	. . . . . . . . . 10 |- ( z e. RR+ -> z e. RR )
35	34	adantl 275	. . . . . . . . 9 |- ( ( A. x e. RR DECID x # 0 /\ z e. RR+ ) -> z e. RR )
36		1zzd 9177	. . . . . . . . . 10 |- ( ( A. x e. RR DECID x # 0 /\ z e. RR+ ) -> 1 e. ZZ )
37		0zd 9162	. . . . . . . . . 10 |- ( ( A. x e. RR DECID x # 0 /\ z e. RR+ ) -> 0 e. ZZ )
38		breq1 3968	. . . . . . . . . . . 12 |- ( x = z -> ( x # 0 <-> z # 0 ) )
39	38	dcbid 824	. . . . . . . . . . 11 |- ( x = z -> (DECID x # 0 <-> DECID z # 0 ) )
40		simpl 108	. . . . . . . . . . 11 |- ( ( A. x e. RR DECID x # 0 /\ z e. RR+ ) -> A. x e. RR DECID x # 0 )
41	39, 40, 35	rspcdva 2821	. . . . . . . . . 10 |- ( ( A. x e. RR DECID x # 0 /\ z e. RR+ ) -> DECID z # 0 )
42	36, 37, 41	ifcldcd 3540	. . . . . . . . 9 |- ( ( A. x e. RR DECID x # 0 /\ z e. RR+ ) -> if ( z # 0 , 1 , 0 ) e. ZZ )
43	25, 33, 35, 42	fvmptd3 5558	. . . . . . . 8 |- ( ( A. x e. RR DECID x # 0 /\ z e. RR+ ) -> ( ( y e. RR |-> if ( y # 0 , 1 , 0 ) ) ` z ) = if ( z # 0 , 1 , 0 ) )
44		rpap0 9559	. . . . . . . . . 10 |- ( z e. RR+ -> z # 0 )
45	44	iftrued 3512	. . . . . . . . 9 |- ( z e. RR+ -> if ( z # 0 , 1 , 0 ) = 1 )
46	45	adantl 275	. . . . . . . 8 |- ( ( A. x e. RR DECID x # 0 /\ z e. RR+ ) -> if ( z # 0 , 1 , 0 ) = 1 )
47	43, 46	eqtrd 2190	. . . . . . 7 |- ( ( A. x e. RR DECID x # 0 /\ z e. RR+ ) -> ( ( y e. RR |-> if ( y # 0 , 1 , 0 ) ) ` z ) = 1 )
48	47	neeq1d 2345	. . . . . 6 |- ( ( A. x e. RR DECID x # 0 /\ z e. RR+ ) -> ( ( ( y e. RR |-> if ( y # 0 , 1 , 0 ) ) ` z ) =/= 0 <-> 1 =/= 0 ) )
49	31, 48	mpbiri 167	. . . . 5 |- ( ( A. x e. RR DECID x # 0 /\ z e. RR+ ) -> ( ( y e. RR |-> if ( y # 0 , 1 , 0 ) ) ` z ) =/= 0 )
50	49	ralrimiva 2530	. . . 4 |- ( A. x e. RR DECID x # 0 -> A. z e. RR+ ( ( y e. RR |-> if ( y # 0 , 1 , 0 ) ) ` z ) =/= 0 )
51		fveq2 5465	. . . . . 6 |- ( z = x -> ( ( y e. RR |-> if ( y # 0 , 1 , 0 ) ) ` z ) = ( ( y e. RR |-> if ( y # 0 , 1 , 0 ) ) ` x ) )
52	51	neeq1d 2345	. . . . 5 |- ( z = x -> ( ( ( y e. RR |-> if ( y # 0 , 1 , 0 ) ) ` z ) =/= 0 <-> ( ( y e. RR |-> if ( y # 0 , 1 , 0 ) ) ` x ) =/= 0 ) )
53	52	cbvralv 2680	. . . 4 |- ( A. z e. RR+ ( ( y e. RR |-> if ( y # 0 , 1 , 0 ) ) ` z ) =/= 0 <-> A. x e. RR+ ( ( y e. RR |-> if ( y # 0 , 1 , 0 ) ) ` x ) =/= 0 )
54	50, 53	sylib 121	. . 3 |- ( A. x e. RR DECID x # 0 -> A. x e. RR+ ( ( y e. RR |-> if ( y # 0 , 1 , 0 ) ) ` x ) =/= 0 )
55	10, 30, 54	3jca 1162	. 2 |- ( A. x e. RR DECID x # 0 -> ( ( y e. RR |-> if ( y # 0 , 1 , 0 ) ) : RR --> ZZ /\ ( ( y e. RR |-> if ( y # 0 , 1 , 0 ) ) ` 0 ) = 0 /\ A. x e. RR+ ( ( y e. RR |-> if ( y # 0 , 1 , 0 ) ) ` x ) =/= 0 ) )
56		feq1 5299	. . 3 |- ( f = ( y e. RR |-> if ( y # 0 , 1 , 0 ) ) -> ( f : RR --> ZZ <-> ( y e. RR |-> if ( y # 0 , 1 , 0 ) ) : RR --> ZZ ) )
57		fveq1 5464	. . . 4 |- ( f = ( y e. RR |-> if ( y # 0 , 1 , 0 ) ) -> ( f ` 0 ) = ( ( y e. RR |-> if ( y # 0 , 1 , 0 ) ) ` 0 ) )
58	57	eqeq1d 2166	. . 3 |- ( f = ( y e. RR |-> if ( y # 0 , 1 , 0 ) ) -> ( ( f ` 0 ) = 0 <-> ( ( y e. RR |-> if ( y # 0 , 1 , 0 ) ) ` 0 ) = 0 ) )
59		fveq1 5464	. . . . 5 |- ( f = ( y e. RR |-> if ( y # 0 , 1 , 0 ) ) -> ( f ` x ) = ( ( y e. RR |-> if ( y # 0 , 1 , 0 ) ) ` x ) )
60	59	neeq1d 2345	. . . 4 |- ( f = ( y e. RR |-> if ( y # 0 , 1 , 0 ) ) -> ( ( f ` x ) =/= 0 <-> ( ( y e. RR |-> if ( y # 0 , 1 , 0 ) ) ` x ) =/= 0 ) )
61	60	ralbidv 2457	. . 3 |- ( f = ( y e. RR |-> if ( y # 0 , 1 , 0 ) ) -> ( A. x e. RR+ ( f ` x ) =/= 0 <-> A. x e. RR+ ( ( y e. RR |-> if ( y # 0 , 1 , 0 ) ) ` x ) =/= 0 ) )
62	56, 58, 61	3anbi123d 1294	. 2 |- ( f = ( y e. RR |-> if ( y # 0 , 1 , 0 ) ) -> ( ( f : RR --> ZZ /\ ( f ` 0 ) = 0 /\ A. x e. RR+ ( f ` x ) =/= 0 ) <-> ( ( y e. RR |-> if ( y # 0 , 1 , 0 ) ) : RR --> ZZ /\ ( ( y e. RR |-> if ( y # 0 , 1 , 0 ) ) ` 0 ) = 0 /\ A. x e. RR+ ( ( y e. RR |-> if ( y # 0 , 1 , 0 ) ) ` x ) =/= 0 ) ) )
63	3, 55, 62	elabd 2857	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 103   \/ wo 698  DECID wdc 820   /\ w3a 963   = wceq 1335   T. wtru 1336   E.wex 1472   e. wcel 2128   =/= wne 2327   A.wral 2435   _Vcvv 2712   ifcif 3505   class class class wbr 3965   |-> cmpt 4025   -->wf 5163   ` cfv 5167   CCcc 7713   RRcr 7714   0cc0 7715   1c1 7716   # cap 8439   ZZcz 9150   RR+crp 9542

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-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-pow 4134  ax-pr 4168  ax-un 4392  ax-setind 4494  ax-cnex 7806  ax-resscn 7807  ax-1cn 7808  ax-1re 7809  ax-icn 7810  ax-addcl 7811  ax-addrcl 7812  ax-mulcl 7813  ax-mulrcl 7814  ax-addcom 7815  ax-mulcom 7816  ax-addass 7817  ax-mulass 7818  ax-distr 7819  ax-i2m1 7820  ax-0lt1 7821  ax-1rid 7822  ax-0id 7823  ax-rnegex 7824  ax-precex 7825  ax-cnre 7826  ax-pre-ltirr 7827  ax-pre-ltwlin 7828  ax-pre-lttrn 7829  ax-pre-apti 7830  ax-pre-ltadd 7831  ax-pre-mulgt0 7832

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  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-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-if 3506  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-id 4252  df-xp 4589  df-rel 4590  df-cnv 4591  df-co 4592  df-dm 4593  df-rn 4594  df-res 4595  df-ima 4596  df-iota 5132  df-fun 5169  df-fn 5170  df-f 5171  df-f1 5172  df-fo 5173  df-f1o 5174  df-fv 5175  df-riota 5774  df-ov 5821  df-oprab 5822  df-mpo 5823  df-pnf 7897  df-mnf 7898  df-xr 7899  df-ltxr 7900  df-le 7901  df-sub 8031  df-neg 8032  df-reap 8433  df-ap 8440  df-inn 8817  df-z 9151  df-rp 9543

This theorem is referenced by: (None)