Theorem dcapnconst 17085

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 17066 for more discussion of decidability of real number apartness.

This is a weaker form of dceqnconst 17084 and in fact this theorem can be proved using dceqnconst 17084 as shown at dcapnconstALT 17086. (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 8307	. . . 4 |- RR e. _V
2	1	mptex 5937	. . 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 9654	. . . . 5 |- ( ( A. x e. RR DECID x # 0 /\ y e. RR ) -> 1 e. ZZ )
5		0zd 9639	. . . . 5 |- ( ( A. x e. RR DECID x # 0 /\ y e. RR ) -> 0 e. ZZ )
6		breq1 4131	. . . . . . 7 |- ( x = y -> ( x # 0 <-> y # 0 ) )
7	6	dcbid 850	. . . . . 6 |- ( x = y -> (DECID x # 0 <-> DECID y # 0 ) )
8	7	rspccva 2928	. . . . 5 |- ( ( A. x e. RR DECID x # 0 /\ y e. RR ) -> DECID y # 0 )
9	4, 5, 8	ifcldcd 3678	. . . 4 |- ( ( A. x e. RR DECID x # 0 /\ y e. RR ) -> if ( y # 0 , 1 , 0 ) e. ZZ )
10	9	fmpttd 5857	. . 3 |- ( A. x e. RR DECID x # 0 -> ( y e. RR |-> if ( y # 0 , 1 , 0 ) ) : RR --> ZZ )
11		0re 8320	. . . . . 6 |- 0 e. RR
12		1zzd 9654	. . . . . . . 8 |- ( T. -> 1 e. ZZ )
13		0zd 9639	. . . . . . . 8 |- ( T. -> 0 e. ZZ )
14		0cn 8312	. . . . . . . . . . . 12 |- 0 e. CC
15		apirr 8927	. . . . . . . . . . . 12 |- ( 0 e. CC -> -. 0 # 0 )
16	14, 15	ax-mp 5	. . . . . . . . . . 11 |- -. 0 # 0
17	16	olci 744	. . . . . . . . . 10 |- ( 0 # 0 \/ -. 0 # 0 )
18		df-dc 847	. . . . . . . . . 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 3678	. . . . . . 7 |- ( T. -> if ( 0 # 0 , 1 , 0 ) e. ZZ )
22	21	mptru 1411	. . . . . 6 |- if ( 0 # 0 , 1 , 0 ) e. ZZ
23		breq1 4131	. . . . . . . 8 |- ( y = 0 -> ( y # 0 <-> 0 # 0 ) )
24	23	ifbid 3662	. . . . . . 7 |- ( y = 0 -> if ( y # 0 , 1 , 0 ) = if ( 0 # 0 , 1 , 0 ) )
25		eqid 2238	. . . . . . 7 |- ( y e. RR |-> if ( y # 0 , 1 , 0 ) ) = ( y e. RR |-> if ( y # 0 , 1 , 0 ) )
26	24, 25	fvmptg 5778	. . . . . 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 430	. . . . 5 |- ( ( y e. RR |-> if ( y # 0 , 1 , 0 ) ) ` 0 ) = if ( 0 # 0 , 1 , 0 )
28	16	iffalsei 3649	. . . . 5 |- if ( 0 # 0 , 1 , 0 ) = 0
29	27, 28	eqtri 2259	. . . 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 9355	. . . . . 6 |- 1 =/= 0
32		breq1 4131	. . . . . . . . . 10 |- ( y = z -> ( y # 0 <-> z # 0 ) )
33	32	ifbid 3662	. . . . . . . . 9 |- ( y = z -> if ( y # 0 , 1 , 0 ) = if ( z # 0 , 1 , 0 ) )
34		rpre 10044	. . . . . . . . . 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 9654	. . . . . . . . . 10 |- ( ( A. x e. RR DECID x # 0 /\ z e. RR+ ) -> 1 e. ZZ )
37		0zd 9639	. . . . . . . . . 10 |- ( ( A. x e. RR DECID x # 0 /\ z e. RR+ ) -> 0 e. ZZ )
38		breq1 4131	. . . . . . . . . . . 12 |- ( x = z -> ( x # 0 <-> z # 0 ) )
39	38	dcbid 850	. . . . . . . . . . 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 2934	. . . . . . . . . 10 |- ( ( A. x e. RR DECID x # 0 /\ z e. RR+ ) -> DECID z # 0 )
42	36, 37, 41	ifcldcd 3678	. . . . . . . . 9 |- ( ( A. x e. RR DECID x # 0 /\ z e. RR+ ) -> if ( z # 0 , 1 , 0 ) e. ZZ )
43	25, 33, 35, 42	fvmptd3 5796	. . . . . . . 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 10054	. . . . . . . . . 10 |- ( z e. RR+ -> z # 0 )
45	44	iftrued 3647	. . . . . . . . 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 2271	. . . . . . 7 |- ( ( A. x e. RR DECID x # 0 /\ z e. RR+ ) -> ( ( y e. RR |-> if ( y # 0 , 1 , 0 ) ) ` z ) = 1 )
48	47	neeq1d 2438	. . . . . 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 2623	. . . 4 |- ( A. x e. RR DECID x # 0 -> A. z e. RR+ ( ( y e. RR |-> if ( y # 0 , 1 , 0 ) ) ` z ) =/= 0 )
51		fveq2 5693	. . . . . 6 |- ( z = x -> ( ( y e. RR |-> if ( y # 0 , 1 , 0 ) ) ` z ) = ( ( y e. RR |-> if ( y # 0 , 1 , 0 ) ) ` x ) )
52	51	neeq1d 2438	. . . . 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 2786	. . . 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 1208	. 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 5514	. . 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 5692	. . . 4 |- ( f = ( y e. RR |-> if ( y # 0 , 1 , 0 ) ) -> ( f ` 0 ) = ( ( y e. RR |-> if ( y # 0 , 1 , 0 ) ) ` 0 ) )
58	57	eqeq1d 2247	. . 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 5692	. . . . 5 |- ( f = ( y e. RR |-> if ( y # 0 , 1 , 0 ) ) -> ( f ` x ) = ( ( y e. RR |-> if ( y # 0 , 1 , 0 ) ) ` x ) )
60	59	neeq1d 2438	. . . 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 2550	. . 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 1353	. 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 2971	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 720  DECID wdc 846   /\ w3a 1009   = wceq 1402   T. wtru 1403   E.wex 1545   e. wcel 2209   =/= wne 2420   A.wral 2528   _Vcvv 2821   ifcif 3638   class class class wbr 4128   |-> cmpt 4190   -->wf 5371   ` cfv 5375   CCcc 8171   RRcr 8172   0cc0 8173   1c1 8174   # cap 8903   ZZcz 9627   RR+crp 10037

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-mulrcl 8272  ax-addcom 8273  ax-mulcom 8274  ax-addass 8275  ax-mulass 8276  ax-distr 8277  ax-i2m1 8278  ax-0lt1 8279  ax-1rid 8280  ax-0id 8281  ax-rnegex 8282  ax-precex 8283  ax-cnre 8284  ax-pre-ltirr 8285  ax-pre-ltwlin 8286  ax-pre-lttrn 8287  ax-pre-apti 8288  ax-pre-ltadd 8289  ax-pre-mulgt0 8290

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-if 3639  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-pnf 8356  df-mnf 8357  df-xr 8358  df-ltxr 8359  df-le 8360  df-sub 8493  df-neg 8494  df-reap 8897  df-ap 8904  df-inn 9288  df-z 9628  df-rp 10038

This theorem is referenced by: (None)