Theorem dcapnconst 16074

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 16056 for more discussion of decidability of real number apartness.

This is a weaker form of dceqnconst 16073 and in fact this theorem can be proved using dceqnconst 16073 as shown at dcapnconstALT 16075. (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 8066	. . . 4 |- RR e. _V
2	1	mptex 5817	. . 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 9406	. . . . 5 |- ( ( A. x e. RR DECID x # 0 /\ y e. RR ) -> 1 e. ZZ )
5		0zd 9391	. . . . 5 |- ( ( A. x e. RR DECID x # 0 /\ y e. RR ) -> 0 e. ZZ )
6		breq1 4050	. . . . . . 7 |- ( x = y -> ( x # 0 <-> y # 0 ) )
7	6	dcbid 840	. . . . . 6 |- ( x = y -> (DECID x # 0 <-> DECID y # 0 ) )
8	7	rspccva 2877	. . . . 5 |- ( ( A. x e. RR DECID x # 0 /\ y e. RR ) -> DECID y # 0 )
9	4, 5, 8	ifcldcd 3609	. . . 4 |- ( ( A. x e. RR DECID x # 0 /\ y e. RR ) -> if ( y # 0 , 1 , 0 ) e. ZZ )
10	9	fmpttd 5742	. . 3 |- ( A. x e. RR DECID x # 0 -> ( y e. RR |-> if ( y # 0 , 1 , 0 ) ) : RR --> ZZ )
11		0re 8079	. . . . . 6 |- 0 e. RR
12		1zzd 9406	. . . . . . . 8 |- ( T. -> 1 e. ZZ )
13		0zd 9391	. . . . . . . 8 |- ( T. -> 0 e. ZZ )
14		0cn 8071	. . . . . . . . . . . 12 |- 0 e. CC
15		apirr 8685	. . . . . . . . . . . 12 |- ( 0 e. CC -> -. 0 # 0 )
16	14, 15	ax-mp 5	. . . . . . . . . . 11 |- -. 0 # 0
17	16	olci 734	. . . . . . . . . 10 |- ( 0 # 0 \/ -. 0 # 0 )
18		df-dc 837	. . . . . . . . . 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 3609	. . . . . . 7 |- ( T. -> if ( 0 # 0 , 1 , 0 ) e. ZZ )
22	21	mptru 1382	. . . . . 6 |- if ( 0 # 0 , 1 , 0 ) e. ZZ
23		breq1 4050	. . . . . . . 8 |- ( y = 0 -> ( y # 0 <-> 0 # 0 ) )
24	23	ifbid 3593	. . . . . . 7 |- ( y = 0 -> if ( y # 0 , 1 , 0 ) = if ( 0 # 0 , 1 , 0 ) )
25		eqid 2206	. . . . . . 7 |- ( y e. RR |-> if ( y # 0 , 1 , 0 ) ) = ( y e. RR |-> if ( y # 0 , 1 , 0 ) )
26	24, 25	fvmptg 5662	. . . . . 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 3581	. . . . 5 |- if ( 0 # 0 , 1 , 0 ) = 0
29	27, 28	eqtri 2227	. . . 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 9111	. . . . . 6 |- 1 =/= 0
32		breq1 4050	. . . . . . . . . 10 |- ( y = z -> ( y # 0 <-> z # 0 ) )
33	32	ifbid 3593	. . . . . . . . 9 |- ( y = z -> if ( y # 0 , 1 , 0 ) = if ( z # 0 , 1 , 0 ) )
34		rpre 9789	. . . . . . . . . 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 9406	. . . . . . . . . 10 |- ( ( A. x e. RR DECID x # 0 /\ z e. RR+ ) -> 1 e. ZZ )
37		0zd 9391	. . . . . . . . . 10 |- ( ( A. x e. RR DECID x # 0 /\ z e. RR+ ) -> 0 e. ZZ )
38		breq1 4050	. . . . . . . . . . . 12 |- ( x = z -> ( x # 0 <-> z # 0 ) )
39	38	dcbid 840	. . . . . . . . . . 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 2883	. . . . . . . . . 10 |- ( ( A. x e. RR DECID x # 0 /\ z e. RR+ ) -> DECID z # 0 )
42	36, 37, 41	ifcldcd 3609	. . . . . . . . 9 |- ( ( A. x e. RR DECID x # 0 /\ z e. RR+ ) -> if ( z # 0 , 1 , 0 ) e. ZZ )
43	25, 33, 35, 42	fvmptd3 5680	. . . . . . . 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 9799	. . . . . . . . . 10 |- ( z e. RR+ -> z # 0 )
45	44	iftrued 3579	. . . . . . . . 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 2239	. . . . . . 7 |- ( ( A. x e. RR DECID x # 0 /\ z e. RR+ ) -> ( ( y e. RR |-> if ( y # 0 , 1 , 0 ) ) ` z ) = 1 )
48	47	neeq1d 2395	. . . . . 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 2580	. . . 4 |- ( A. x e. RR DECID x # 0 -> A. z e. RR+ ( ( y e. RR |-> if ( y # 0 , 1 , 0 ) ) ` z ) =/= 0 )
51		fveq2 5583	. . . . . 6 |- ( z = x -> ( ( y e. RR |-> if ( y # 0 , 1 , 0 ) ) ` z ) = ( ( y e. RR |-> if ( y # 0 , 1 , 0 ) ) ` x ) )
52	51	neeq1d 2395	. . . . 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 2739	. . . 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 1180	. 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 5414	. . 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 5582	. . . 4 |- ( f = ( y e. RR |-> if ( y # 0 , 1 , 0 ) ) -> ( f ` 0 ) = ( ( y e. RR |-> if ( y # 0 , 1 , 0 ) ) ` 0 ) )
58	57	eqeq1d 2215	. . 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 5582	. . . . 5 |- ( f = ( y e. RR |-> if ( y # 0 , 1 , 0 ) ) -> ( f ` x ) = ( ( y e. RR |-> if ( y # 0 , 1 , 0 ) ) ` x ) )
60	59	neeq1d 2395	. . . 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 2507	. . 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 1325	. 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 2919	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 710  DECID wdc 836   /\ w3a 981   = wceq 1373   T. wtru 1374   E.wex 1516   e. wcel 2177   =/= wne 2377   A.wral 2485   _Vcvv 2773   ifcif 3572   class class class wbr 4047   |-> cmpt 4109   -->wf 5272   ` cfv 5276   CCcc 7930   RRcr 7931   0cc0 7932   1c1 7933   # cap 8661   ZZcz 9379   RR+crp 9782

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4163  ax-sep 4166  ax-pow 4222  ax-pr 4257  ax-un 4484  ax-setind 4589  ax-cnex 8023  ax-resscn 8024  ax-1cn 8025  ax-1re 8026  ax-icn 8027  ax-addcl 8028  ax-addrcl 8029  ax-mulcl 8030  ax-mulrcl 8031  ax-addcom 8032  ax-mulcom 8033  ax-addass 8034  ax-mulass 8035  ax-distr 8036  ax-i2m1 8037  ax-0lt1 8038  ax-1rid 8039  ax-0id 8040  ax-rnegex 8041  ax-precex 8042  ax-cnre 8043  ax-pre-ltirr 8044  ax-pre-ltwlin 8045  ax-pre-lttrn 8046  ax-pre-apti 8047  ax-pre-ltadd 8048  ax-pre-mulgt0 8049

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3000  df-csb 3095  df-dif 3169  df-un 3171  df-in 3173  df-ss 3180  df-if 3573  df-pw 3619  df-sn 3640  df-pr 3641  df-op 3643  df-uni 3853  df-int 3888  df-iun 3931  df-br 4048  df-opab 4110  df-mpt 4111  df-id 4344  df-xp 4685  df-rel 4686  df-cnv 4687  df-co 4688  df-dm 4689  df-rn 4690  df-res 4691  df-ima 4692  df-iota 5237  df-fun 5278  df-fn 5279  df-f 5280  df-f1 5281  df-fo 5282  df-f1o 5283  df-fv 5284  df-riota 5906  df-ov 5954  df-oprab 5955  df-mpo 5956  df-pnf 8116  df-mnf 8117  df-xr 8118  df-ltxr 8119  df-le 8120  df-sub 8252  df-neg 8253  df-reap 8655  df-ap 8662  df-inn 9044  df-z 9380  df-rp 9783

This theorem is referenced by: (None)