Theorem dcapnconst 14464

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 14447 for more discussion of decidability of real number apartness.

This is a weaker form of dceqnconst 14463 and in fact this theorem can be proved using dceqnconst 14463 as shown at dcapnconstALT 14465. (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 7936	. . . 4 |- RR e. _V
2	1	mptex 5738	. . 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 9269	. . . . 5 |- ( ( A. x e. RR DECID x # 0 /\ y e. RR ) -> 1 e. ZZ )
5		0zd 9254	. . . . 5 |- ( ( A. x e. RR DECID x # 0 /\ y e. RR ) -> 0 e. ZZ )
6		breq1 4003	. . . . . . 7 |- ( x = y -> ( x # 0 <-> y # 0 ) )
7	6	dcbid 838	. . . . . 6 |- ( x = y -> (DECID x # 0 <-> DECID y # 0 ) )
8	7	rspccva 2840	. . . . 5 |- ( ( A. x e. RR DECID x # 0 /\ y e. RR ) -> DECID y # 0 )
9	4, 5, 8	ifcldcd 3569	. . . 4 |- ( ( A. x e. RR DECID x # 0 /\ y e. RR ) -> if ( y # 0 , 1 , 0 ) e. ZZ )
10	9	fmpttd 5667	. . 3 |- ( A. x e. RR DECID x # 0 -> ( y e. RR |-> if ( y # 0 , 1 , 0 ) ) : RR --> ZZ )
11		0re 7948	. . . . . 6 |- 0 e. RR
12		1zzd 9269	. . . . . . . 8 |- ( T. -> 1 e. ZZ )
13		0zd 9254	. . . . . . . 8 |- ( T. -> 0 e. ZZ )
14		0cn 7940	. . . . . . . . . . . 12 |- 0 e. CC
15		apirr 8552	. . . . . . . . . . . 12 |- ( 0 e. CC -> -. 0 # 0 )
16	14, 15	ax-mp 5	. . . . . . . . . . 11 |- -. 0 # 0
17	16	olci 732	. . . . . . . . . 10 |- ( 0 # 0 \/ -. 0 # 0 )
18		df-dc 835	. . . . . . . . . 10 |- (DECID 0 # 0 <-> ( 0 # 0 \/ -. 0 # 0 ) )
19	17, 18	mpbir 146	. . . . . . . . 9 |- DECID 0 # 0
20	19	a1i 9	. . . . . . . 8 |- ( T. -> DECID 0 # 0 )
21	12, 13, 20	ifcldcd 3569	. . . . . . 7 |- ( T. -> if ( 0 # 0 , 1 , 0 ) e. ZZ )
22	21	mptru 1362	. . . . . 6 |- if ( 0 # 0 , 1 , 0 ) e. ZZ
23		breq1 4003	. . . . . . . 8 |- ( y = 0 -> ( y # 0 <-> 0 # 0 ) )
24	23	ifbid 3555	. . . . . . 7 |- ( y = 0 -> if ( y # 0 , 1 , 0 ) = if ( 0 # 0 , 1 , 0 ) )
25		eqid 2177	. . . . . . 7 |- ( y e. RR |-> if ( y # 0 , 1 , 0 ) ) = ( y e. RR |-> if ( y # 0 , 1 , 0 ) )
26	24, 25	fvmptg 5588	. . . . . 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 426	. . . . 5 |- ( ( y e. RR |-> if ( y # 0 , 1 , 0 ) ) ` 0 ) = if ( 0 # 0 , 1 , 0 )
28	16	iffalsei 3543	. . . . 5 |- if ( 0 # 0 , 1 , 0 ) = 0
29	27, 28	eqtri 2198	. . . 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 8976	. . . . . 6 |- 1 =/= 0
32		breq1 4003	. . . . . . . . . 10 |- ( y = z -> ( y # 0 <-> z # 0 ) )
33	32	ifbid 3555	. . . . . . . . 9 |- ( y = z -> if ( y # 0 , 1 , 0 ) = if ( z # 0 , 1 , 0 ) )
34		rpre 9647	. . . . . . . . . 10 |- ( z e. RR+ -> z e. RR )
35	34	adantl 277	. . . . . . . . 9 |- ( ( A. x e. RR DECID x # 0 /\ z e. RR+ ) -> z e. RR )
36		1zzd 9269	. . . . . . . . . 10 |- ( ( A. x e. RR DECID x # 0 /\ z e. RR+ ) -> 1 e. ZZ )
37		0zd 9254	. . . . . . . . . 10 |- ( ( A. x e. RR DECID x # 0 /\ z e. RR+ ) -> 0 e. ZZ )
38		breq1 4003	. . . . . . . . . . . 12 |- ( x = z -> ( x # 0 <-> z # 0 ) )
39	38	dcbid 838	. . . . . . . . . . 11 |- ( x = z -> (DECID x # 0 <-> DECID z # 0 ) )
40		simpl 109	. . . . . . . . . . 11 |- ( ( A. x e. RR DECID x # 0 /\ z e. RR+ ) -> A. x e. RR DECID x # 0 )
41	39, 40, 35	rspcdva 2846	. . . . . . . . . 10 |- ( ( A. x e. RR DECID x # 0 /\ z e. RR+ ) -> DECID z # 0 )
42	36, 37, 41	ifcldcd 3569	. . . . . . . . 9 |- ( ( A. x e. RR DECID x # 0 /\ z e. RR+ ) -> if ( z # 0 , 1 , 0 ) e. ZZ )
43	25, 33, 35, 42	fvmptd3 5605	. . . . . . . 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 9657	. . . . . . . . . 10 |- ( z e. RR+ -> z # 0 )
45	44	iftrued 3541	. . . . . . . . 9 |- ( z e. RR+ -> if ( z # 0 , 1 , 0 ) = 1 )
46	45	adantl 277	. . . . . . . 8 |- ( ( A. x e. RR DECID x # 0 /\ z e. RR+ ) -> if ( z # 0 , 1 , 0 ) = 1 )
47	43, 46	eqtrd 2210	. . . . . . 7 |- ( ( A. x e. RR DECID x # 0 /\ z e. RR+ ) -> ( ( y e. RR |-> if ( y # 0 , 1 , 0 ) ) ` z ) = 1 )
48	47	neeq1d 2365	. . . . . 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 168	. . . . 5 |- ( ( A. x e. RR DECID x # 0 /\ z e. RR+ ) -> ( ( y e. RR |-> if ( y # 0 , 1 , 0 ) ) ` z ) =/= 0 )
50	49	ralrimiva 2550	. . . 4 |- ( A. x e. RR DECID x # 0 -> A. z e. RR+ ( ( y e. RR |-> if ( y # 0 , 1 , 0 ) ) ` z ) =/= 0 )
51		fveq2 5511	. . . . . 6 |- ( z = x -> ( ( y e. RR |-> if ( y # 0 , 1 , 0 ) ) ` z ) = ( ( y e. RR |-> if ( y # 0 , 1 , 0 ) ) ` x ) )
52	51	neeq1d 2365	. . . . 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 2703	. . . 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 122	. . 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 1177	. 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 5344	. . 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 5510	. . . 4 |- ( f = ( y e. RR |-> if ( y # 0 , 1 , 0 ) ) -> ( f ` 0 ) = ( ( y e. RR |-> if ( y # 0 , 1 , 0 ) ) ` 0 ) )
58	57	eqeq1d 2186	. . 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 5510	. . . . 5 |- ( f = ( y e. RR |-> if ( y # 0 , 1 , 0 ) ) -> ( f ` x ) = ( ( y e. RR |-> if ( y # 0 , 1 , 0 ) ) ` x ) )
60	59	neeq1d 2365	. . . 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 2477	. . 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 1312	. 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 2882	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 708  DECID wdc 834   /\ w3a 978   = wceq 1353   T. wtru 1354   E.wex 1492   e. wcel 2148   =/= wne 2347   A.wral 2455   _Vcvv 2737   ifcif 3534   class class class wbr 4000   |-> cmpt 4061   -->wf 5208   ` cfv 5212   CCcc 7800   RRcr 7801   0cc0 7802   1c1 7803   # cap 8528   ZZcz 9242   RR+crp 9640

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-mulrcl 7901  ax-addcom 7902  ax-mulcom 7903  ax-addass 7904  ax-mulass 7905  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-1rid 7909  ax-0id 7910  ax-rnegex 7911  ax-precex 7912  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918  ax-pre-mulgt0 7919

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-reap 8522  df-ap 8529  df-inn 8909  df-z 9243  df-rp 9641

This theorem is referenced by: (None)