Theorem dcapnconst 16543

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 16525 for more discussion of decidability of real number apartness.

This is a weaker form of dceqnconst 16542 and in fact this theorem can be proved using dceqnconst 16542 as shown at dcapnconstALT 16544. (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 8149	. . . 4 |- RR e. _V
2	1	mptex 5872	. . 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 9489	. . . . 5 |- ( ( A. x e. RR DECID x # 0 /\ y e. RR ) -> 1 e. ZZ )
5		0zd 9474	. . . . 5 |- ( ( A. x e. RR DECID x # 0 /\ y e. RR ) -> 0 e. ZZ )
6		breq1 4086	. . . . . . 7 |- ( x = y -> ( x # 0 <-> y # 0 ) )
7	6	dcbid 843	. . . . . 6 |- ( x = y -> (DECID x # 0 <-> DECID y # 0 ) )
8	7	rspccva 2906	. . . . 5 |- ( ( A. x e. RR DECID x # 0 /\ y e. RR ) -> DECID y # 0 )
9	4, 5, 8	ifcldcd 3640	. . . 4 |- ( ( A. x e. RR DECID x # 0 /\ y e. RR ) -> if ( y # 0 , 1 , 0 ) e. ZZ )
10	9	fmpttd 5795	. . 3 |- ( A. x e. RR DECID x # 0 -> ( y e. RR |-> if ( y # 0 , 1 , 0 ) ) : RR --> ZZ )
11		0re 8162	. . . . . 6 |- 0 e. RR
12		1zzd 9489	. . . . . . . 8 |- ( T. -> 1 e. ZZ )
13		0zd 9474	. . . . . . . 8 |- ( T. -> 0 e. ZZ )
14		0cn 8154	. . . . . . . . . . . 12 |- 0 e. CC
15		apirr 8768	. . . . . . . . . . . 12 |- ( 0 e. CC -> -. 0 # 0 )
16	14, 15	ax-mp 5	. . . . . . . . . . 11 |- -. 0 # 0
17	16	olci 737	. . . . . . . . . 10 |- ( 0 # 0 \/ -. 0 # 0 )
18		df-dc 840	. . . . . . . . . 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 3640	. . . . . . 7 |- ( T. -> if ( 0 # 0 , 1 , 0 ) e. ZZ )
22	21	mptru 1404	. . . . . 6 |- if ( 0 # 0 , 1 , 0 ) e. ZZ
23		breq1 4086	. . . . . . . 8 |- ( y = 0 -> ( y # 0 <-> 0 # 0 ) )
24	23	ifbid 3624	. . . . . . 7 |- ( y = 0 -> if ( y # 0 , 1 , 0 ) = if ( 0 # 0 , 1 , 0 ) )
25		eqid 2229	. . . . . . 7 |- ( y e. RR |-> if ( y # 0 , 1 , 0 ) ) = ( y e. RR |-> if ( y # 0 , 1 , 0 ) )
26	24, 25	fvmptg 5715	. . . . . 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 3611	. . . . 5 |- if ( 0 # 0 , 1 , 0 ) = 0
29	27, 28	eqtri 2250	. . . 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 9194	. . . . . 6 |- 1 =/= 0
32		breq1 4086	. . . . . . . . . 10 |- ( y = z -> ( y # 0 <-> z # 0 ) )
33	32	ifbid 3624	. . . . . . . . 9 |- ( y = z -> if ( y # 0 , 1 , 0 ) = if ( z # 0 , 1 , 0 ) )
34		rpre 9873	. . . . . . . . . 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 9489	. . . . . . . . . 10 |- ( ( A. x e. RR DECID x # 0 /\ z e. RR+ ) -> 1 e. ZZ )
37		0zd 9474	. . . . . . . . . 10 |- ( ( A. x e. RR DECID x # 0 /\ z e. RR+ ) -> 0 e. ZZ )
38		breq1 4086	. . . . . . . . . . . 12 |- ( x = z -> ( x # 0 <-> z # 0 ) )
39	38	dcbid 843	. . . . . . . . . . 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 2912	. . . . . . . . . 10 |- ( ( A. x e. RR DECID x # 0 /\ z e. RR+ ) -> DECID z # 0 )
42	36, 37, 41	ifcldcd 3640	. . . . . . . . 9 |- ( ( A. x e. RR DECID x # 0 /\ z e. RR+ ) -> if ( z # 0 , 1 , 0 ) e. ZZ )
43	25, 33, 35, 42	fvmptd3 5733	. . . . . . . 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 9883	. . . . . . . . . 10 |- ( z e. RR+ -> z # 0 )
45	44	iftrued 3609	. . . . . . . . 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 2262	. . . . . . 7 |- ( ( A. x e. RR DECID x # 0 /\ z e. RR+ ) -> ( ( y e. RR |-> if ( y # 0 , 1 , 0 ) ) ` z ) = 1 )
48	47	neeq1d 2418	. . . . . 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 2603	. . . 4 |- ( A. x e. RR DECID x # 0 -> A. z e. RR+ ( ( y e. RR |-> if ( y # 0 , 1 , 0 ) ) ` z ) =/= 0 )
51		fveq2 5632	. . . . . 6 |- ( z = x -> ( ( y e. RR |-> if ( y # 0 , 1 , 0 ) ) ` z ) = ( ( y e. RR |-> if ( y # 0 , 1 , 0 ) ) ` x ) )
52	51	neeq1d 2418	. . . . 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 2765	. . . 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 1201	. 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 5459	. . 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 5631	. . . 4 |- ( f = ( y e. RR |-> if ( y # 0 , 1 , 0 ) ) -> ( f ` 0 ) = ( ( y e. RR |-> if ( y # 0 , 1 , 0 ) ) ` 0 ) )
58	57	eqeq1d 2238	. . 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 5631	. . . . 5 |- ( f = ( y e. RR |-> if ( y # 0 , 1 , 0 ) ) -> ( f ` x ) = ( ( y e. RR |-> if ( y # 0 , 1 , 0 ) ) ` x ) )
60	59	neeq1d 2418	. . . 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 2530	. . 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 1346	. 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 2948	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 713  DECID wdc 839   /\ w3a 1002   = wceq 1395   T. wtru 1396   E.wex 1538   e. wcel 2200   =/= wne 2400   A.wral 2508   _Vcvv 2799   ifcif 3602   class class class wbr 4083   |-> cmpt 4145   -->wf 5317   ` cfv 5321   CCcc 8013   RRcr 8014   0cc0 8015   1c1 8016   # cap 8744   ZZcz 9462   RR+crp 9866

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-pow 4259  ax-pr 4294  ax-un 4525  ax-setind 4630  ax-cnex 8106  ax-resscn 8107  ax-1cn 8108  ax-1re 8109  ax-icn 8110  ax-addcl 8111  ax-addrcl 8112  ax-mulcl 8113  ax-mulrcl 8114  ax-addcom 8115  ax-mulcom 8116  ax-addass 8117  ax-mulass 8118  ax-distr 8119  ax-i2m1 8120  ax-0lt1 8121  ax-1rid 8122  ax-0id 8123  ax-rnegex 8124  ax-precex 8125  ax-cnre 8126  ax-pre-ltirr 8127  ax-pre-ltwlin 8128  ax-pre-lttrn 8129  ax-pre-apti 8130  ax-pre-ltadd 8131  ax-pre-mulgt0 8132

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4385  df-xp 4726  df-rel 4727  df-cnv 4728  df-co 4729  df-dm 4730  df-rn 4731  df-res 4732  df-ima 4733  df-iota 5281  df-fun 5323  df-fn 5324  df-f 5325  df-f1 5326  df-fo 5327  df-f1o 5328  df-fv 5329  df-riota 5963  df-ov 6013  df-oprab 6014  df-mpo 6015  df-pnf 8199  df-mnf 8200  df-xr 8201  df-ltxr 8202  df-le 8203  df-sub 8335  df-neg 8336  df-reap 8738  df-ap 8745  df-inn 9127  df-z 9463  df-rp 9867

This theorem is referenced by: (None)