Theorem dceqnconst 15791

Description: Decidability of real number equality 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 redcwlpo 15786 for more discussion of decidability of real number equality. (Contributed by BJ and Jim Kingdon, 24-Jun-2024.) (Revised by Jim Kingdon, 23-Jul-2024.)

Assertion

Ref	Expression
dceqnconst	|- ( 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 dceqnconst

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

Step	Hyp	Ref	Expression
1		reex 8030	. . . 4 |- RR e. _V
2	1	mptex 5791	. . 3 |- ( y e. RR |-> if ( y = 0 , 0 , 1 ) ) e. _V
3	2	a1i 9	. 2 |- ( A. x e. RR DECID x = 0 -> ( y e. RR |-> if ( y = 0 , 0 , 1 ) ) e. _V )
4		0zd 9355	. . . . 5 |- ( ( A. x e. RR DECID x = 0 /\ y e. RR ) -> 0 e. ZZ )
5		1zzd 9370	. . . . 5 |- ( ( A. x e. RR DECID x = 0 /\ y e. RR ) -> 1 e. ZZ )
6		eqeq1 2203	. . . . . . 7 |- ( x = y -> ( x = 0 <-> y = 0 ) )
7	6	dcbid 839	. . . . . 6 |- ( x = y -> (DECID x = 0 <-> DECID y = 0 ) )
8	7	rspccva 2867	. . . . 5 |- ( ( A. x e. RR DECID x = 0 /\ y e. RR ) -> DECID y = 0 )
9	4, 5, 8	ifcldcd 3598	. . . 4 |- ( ( A. x e. RR DECID x = 0 /\ y e. RR ) -> if ( y = 0 , 0 , 1 ) e. ZZ )
10	9	fmpttd 5720	. . 3 |- ( A. x e. RR DECID x = 0 -> ( y e. RR |-> if ( y = 0 , 0 , 1 ) ) : RR --> ZZ )
11		0re 8043	. . . . . 6 |- 0 e. RR
12		0zd 9355	. . . . . . . 8 |- ( T. -> 0 e. ZZ )
13		1zzd 9370	. . . . . . . 8 |- ( T. -> 1 e. ZZ )
14		eqid 2196	. . . . . . . . . . 11 |- 0 = 0
15	14	orci 732	. . . . . . . . . 10 |- ( 0 = 0 \/ -. 0 = 0 )
16		df-dc 836	. . . . . . . . . 10 |- (DECID 0 = 0 <-> ( 0 = 0 \/ -. 0 = 0 ) )
17	15, 16	mpbir 146	. . . . . . . . 9 |- DECID 0 = 0
18	17	a1i 9	. . . . . . . 8 |- ( T. -> DECID 0 = 0 )
19	12, 13, 18	ifcldcd 3598	. . . . . . 7 |- ( T. -> if ( 0 = 0 , 0 , 1 ) e. ZZ )
20	19	mptru 1373	. . . . . 6 |- if ( 0 = 0 , 0 , 1 ) e. ZZ
21		eqeq1 2203	. . . . . . . 8 |- ( y = 0 -> ( y = 0 <-> 0 = 0 ) )
22	21	ifbid 3583	. . . . . . 7 |- ( y = 0 -> if ( y = 0 , 0 , 1 ) = if ( 0 = 0 , 0 , 1 ) )
23		eqid 2196	. . . . . . 7 |- ( y e. RR |-> if ( y = 0 , 0 , 1 ) ) = ( y e. RR |-> if ( y = 0 , 0 , 1 ) )
24	22, 23	fvmptg 5640	. . . . . 6 |- ( ( 0 e. RR /\ if ( 0 = 0 , 0 , 1 ) e. ZZ ) -> ( ( y e. RR |-> if ( y = 0 , 0 , 1 ) ) ` 0 ) = if ( 0 = 0 , 0 , 1 ) )
25	11, 20, 24	mp2an 426	. . . . 5 |- ( ( y e. RR |-> if ( y = 0 , 0 , 1 ) ) ` 0 ) = if ( 0 = 0 , 0 , 1 )
26	14	iftruei 3568	. . . . 5 |- if ( 0 = 0 , 0 , 1 ) = 0
27	25, 26	eqtri 2217	. . . 4 |- ( ( y e. RR |-> if ( y = 0 , 0 , 1 ) ) ` 0 ) = 0
28	27	a1i 9	. . 3 |- ( A. x e. RR DECID x = 0 -> ( ( y e. RR |-> if ( y = 0 , 0 , 1 ) ) ` 0 ) = 0 )
29		1ne0 9075	. . . . . 6 |- 1 =/= 0
30		eqeq1 2203	. . . . . . . . . 10 |- ( y = z -> ( y = 0 <-> z = 0 ) )
31	30	ifbid 3583	. . . . . . . . 9 |- ( y = z -> if ( y = 0 , 0 , 1 ) = if ( z = 0 , 0 , 1 ) )
32		rpre 9752	. . . . . . . . . 10 |- ( z e. RR+ -> z e. RR )
33	32	adantl 277	. . . . . . . . 9 |- ( ( A. x e. RR DECID x = 0 /\ z e. RR+ ) -> z e. RR )
34		0zd 9355	. . . . . . . . . 10 |- ( ( A. x e. RR DECID x = 0 /\ z e. RR+ ) -> 0 e. ZZ )
35		1zzd 9370	. . . . . . . . . 10 |- ( ( A. x e. RR DECID x = 0 /\ z e. RR+ ) -> 1 e. ZZ )
36		eqeq1 2203	. . . . . . . . . . . 12 |- ( x = z -> ( x = 0 <-> z = 0 ) )
37	36	dcbid 839	. . . . . . . . . . 11 |- ( x = z -> (DECID x = 0 <-> DECID z = 0 ) )
38		simpl 109	. . . . . . . . . . 11 |- ( ( A. x e. RR DECID x = 0 /\ z e. RR+ ) -> A. x e. RR DECID x = 0 )
39	37, 38, 33	rspcdva 2873	. . . . . . . . . 10 |- ( ( A. x e. RR DECID x = 0 /\ z e. RR+ ) -> DECID z = 0 )
40	34, 35, 39	ifcldcd 3598	. . . . . . . . 9 |- ( ( A. x e. RR DECID x = 0 /\ z e. RR+ ) -> if ( z = 0 , 0 , 1 ) e. ZZ )
41	23, 31, 33, 40	fvmptd3 5658	. . . . . . . 8 |- ( ( A. x e. RR DECID x = 0 /\ z e. RR+ ) -> ( ( y e. RR |-> if ( y = 0 , 0 , 1 ) ) ` z ) = if ( z = 0 , 0 , 1 ) )
42		rpne0 9761	. . . . . . . . . . 11 |- ( z e. RR+ -> z =/= 0 )
43	42	neneqd 2388	. . . . . . . . . 10 |- ( z e. RR+ -> -. z = 0 )
44	43	iffalsed 3572	. . . . . . . . 9 |- ( z e. RR+ -> if ( z = 0 , 0 , 1 ) = 1 )
45	44	adantl 277	. . . . . . . 8 |- ( ( A. x e. RR DECID x = 0 /\ z e. RR+ ) -> if ( z = 0 , 0 , 1 ) = 1 )
46	41, 45	eqtrd 2229	. . . . . . 7 |- ( ( A. x e. RR DECID x = 0 /\ z e. RR+ ) -> ( ( y e. RR |-> if ( y = 0 , 0 , 1 ) ) ` z ) = 1 )
47	46	neeq1d 2385	. . . . . 6 |- ( ( A. x e. RR DECID x = 0 /\ z e. RR+ ) -> ( ( ( y e. RR |-> if ( y = 0 , 0 , 1 ) ) ` z ) =/= 0 <-> 1 =/= 0 ) )
48	29, 47	mpbiri 168	. . . . 5 |- ( ( A. x e. RR DECID x = 0 /\ z e. RR+ ) -> ( ( y e. RR |-> if ( y = 0 , 0 , 1 ) ) ` z ) =/= 0 )
49	48	ralrimiva 2570	. . . 4 |- ( A. x e. RR DECID x = 0 -> A. z e. RR+ ( ( y e. RR |-> if ( y = 0 , 0 , 1 ) ) ` z ) =/= 0 )
50		fveq2 5561	. . . . . 6 |- ( z = x -> ( ( y e. RR |-> if ( y = 0 , 0 , 1 ) ) ` z ) = ( ( y e. RR |-> if ( y = 0 , 0 , 1 ) ) ` x ) )
51	50	neeq1d 2385	. . . . 5 |- ( z = x -> ( ( ( y e. RR |-> if ( y = 0 , 0 , 1 ) ) ` z ) =/= 0 <-> ( ( y e. RR |-> if ( y = 0 , 0 , 1 ) ) ` x ) =/= 0 ) )
52	51	cbvralv 2729	. . . 4 |- ( A. z e. RR+ ( ( y e. RR |-> if ( y = 0 , 0 , 1 ) ) ` z ) =/= 0 <-> A. x e. RR+ ( ( y e. RR |-> if ( y = 0 , 0 , 1 ) ) ` x ) =/= 0 )
53	49, 52	sylib 122	. . 3 |- ( A. x e. RR DECID x = 0 -> A. x e. RR+ ( ( y e. RR |-> if ( y = 0 , 0 , 1 ) ) ` x ) =/= 0 )
54	10, 28, 53	3jca 1179	. 2 |- ( A. x e. RR DECID x = 0 -> ( ( y e. RR |-> if ( y = 0 , 0 , 1 ) ) : RR --> ZZ /\ ( ( y e. RR |-> if ( y = 0 , 0 , 1 ) ) ` 0 ) = 0 /\ A. x e. RR+ ( ( y e. RR |-> if ( y = 0 , 0 , 1 ) ) ` x ) =/= 0 ) )
55		feq1 5393	. . 3 |- ( f = ( y e. RR |-> if ( y = 0 , 0 , 1 ) ) -> ( f : RR --> ZZ <-> ( y e. RR |-> if ( y = 0 , 0 , 1 ) ) : RR --> ZZ ) )
56		fveq1 5560	. . . 4 |- ( f = ( y e. RR |-> if ( y = 0 , 0 , 1 ) ) -> ( f ` 0 ) = ( ( y e. RR |-> if ( y = 0 , 0 , 1 ) ) ` 0 ) )
57	56	eqeq1d 2205	. . 3 |- ( f = ( y e. RR |-> if ( y = 0 , 0 , 1 ) ) -> ( ( f ` 0 ) = 0 <-> ( ( y e. RR |-> if ( y = 0 , 0 , 1 ) ) ` 0 ) = 0 ) )
58		fveq1 5560	. . . . 5 |- ( f = ( y e. RR |-> if ( y = 0 , 0 , 1 ) ) -> ( f ` x ) = ( ( y e. RR |-> if ( y = 0 , 0 , 1 ) ) ` x ) )
59	58	neeq1d 2385	. . . 4 |- ( f = ( y e. RR |-> if ( y = 0 , 0 , 1 ) ) -> ( ( f ` x ) =/= 0 <-> ( ( y e. RR |-> if ( y = 0 , 0 , 1 ) ) ` x ) =/= 0 ) )
60	59	ralbidv 2497	. . 3 |- ( f = ( y e. RR |-> if ( y = 0 , 0 , 1 ) ) -> ( A. x e. RR+ ( f ` x ) =/= 0 <-> A. x e. RR+ ( ( y e. RR |-> if ( y = 0 , 0 , 1 ) ) ` x ) =/= 0 ) )
61	55, 57, 60	3anbi123d 1323	. 2 |- ( f = ( y e. RR |-> if ( y = 0 , 0 , 1 ) ) -> ( ( f : RR --> ZZ /\ ( f ` 0 ) = 0 /\ A. x e. RR+ ( f ` x ) =/= 0 ) <-> ( ( y e. RR |-> if ( y = 0 , 0 , 1 ) ) : RR --> ZZ /\ ( ( y e. RR |-> if ( y = 0 , 0 , 1 ) ) ` 0 ) = 0 /\ A. x e. RR+ ( ( y e. RR |-> if ( y = 0 , 0 , 1 ) ) ` x ) =/= 0 ) ) )
62	3, 54, 61	elabd 2909	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 709  DECID wdc 835   /\ w3a 980   = wceq 1364   T. wtru 1365   E.wex 1506   e. wcel 2167   =/= wne 2367   A.wral 2475   _Vcvv 2763   ifcif 3562   |-> cmpt 4095   -->wf 5255   ` cfv 5259   RRcr 7895   0cc0 7896   1c1 7897   ZZcz 9343   RR+crp 9745

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-addcom 7996  ax-addass 7998  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-0id 8004  ax-rnegex 8005  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-ltadd 8012

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-inn 9008  df-z 9344  df-rp 9746

This theorem is referenced by:  dcapnconstALT  15793