Theorem dceqnconst 15080

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 15075 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 7958	. . . 4 |- RR e. _V
2	1	mptex 5755	. . 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 9278	. . . . 5 |- ( ( A. x e. RR DECID x = 0 /\ y e. RR ) -> 0 e. ZZ )
5		1zzd 9293	. . . . 5 |- ( ( A. x e. RR DECID x = 0 /\ y e. RR ) -> 1 e. ZZ )
6		eqeq1 2194	. . . . . . 7 |- ( x = y -> ( x = 0 <-> y = 0 ) )
7	6	dcbid 839	. . . . . 6 |- ( x = y -> (DECID x = 0 <-> DECID y = 0 ) )
8	7	rspccva 2852	. . . . 5 |- ( ( A. x e. RR DECID x = 0 /\ y e. RR ) -> DECID y = 0 )
9	4, 5, 8	ifcldcd 3582	. . . 4 |- ( ( A. x e. RR DECID x = 0 /\ y e. RR ) -> if ( y = 0 , 0 , 1 ) e. ZZ )
10	9	fmpttd 5684	. . 3 |- ( A. x e. RR DECID x = 0 -> ( y e. RR |-> if ( y = 0 , 0 , 1 ) ) : RR --> ZZ )
11		0re 7970	. . . . . 6 |- 0 e. RR
12		0zd 9278	. . . . . . . 8 |- ( T. -> 0 e. ZZ )
13		1zzd 9293	. . . . . . . 8 |- ( T. -> 1 e. ZZ )
14		eqid 2187	. . . . . . . . . . 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 3582	. . . . . . 7 |- ( T. -> if ( 0 = 0 , 0 , 1 ) e. ZZ )
20	19	mptru 1372	. . . . . 6 |- if ( 0 = 0 , 0 , 1 ) e. ZZ
21		eqeq1 2194	. . . . . . . 8 |- ( y = 0 -> ( y = 0 <-> 0 = 0 ) )
22	21	ifbid 3567	. . . . . . 7 |- ( y = 0 -> if ( y = 0 , 0 , 1 ) = if ( 0 = 0 , 0 , 1 ) )
23		eqid 2187	. . . . . . 7 |- ( y e. RR |-> if ( y = 0 , 0 , 1 ) ) = ( y e. RR |-> if ( y = 0 , 0 , 1 ) )
24	22, 23	fvmptg 5605	. . . . . 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 3552	. . . . 5 |- if ( 0 = 0 , 0 , 1 ) = 0
27	25, 26	eqtri 2208	. . . 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 9000	. . . . . 6 |- 1 =/= 0
30		eqeq1 2194	. . . . . . . . . 10 |- ( y = z -> ( y = 0 <-> z = 0 ) )
31	30	ifbid 3567	. . . . . . . . 9 |- ( y = z -> if ( y = 0 , 0 , 1 ) = if ( z = 0 , 0 , 1 ) )
32		rpre 9673	. . . . . . . . . 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 9278	. . . . . . . . . 10 |- ( ( A. x e. RR DECID x = 0 /\ z e. RR+ ) -> 0 e. ZZ )
35		1zzd 9293	. . . . . . . . . 10 |- ( ( A. x e. RR DECID x = 0 /\ z e. RR+ ) -> 1 e. ZZ )
36		eqeq1 2194	. . . . . . . . . . . 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 2858	. . . . . . . . . 10 |- ( ( A. x e. RR DECID x = 0 /\ z e. RR+ ) -> DECID z = 0 )
40	34, 35, 39	ifcldcd 3582	. . . . . . . . 9 |- ( ( A. x e. RR DECID x = 0 /\ z e. RR+ ) -> if ( z = 0 , 0 , 1 ) e. ZZ )
41	23, 31, 33, 40	fvmptd3 5622	. . . . . . . 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 9682	. . . . . . . . . . 11 |- ( z e. RR+ -> z =/= 0 )
43	42	neneqd 2378	. . . . . . . . . 10 |- ( z e. RR+ -> -. z = 0 )
44	43	iffalsed 3556	. . . . . . . . 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 2220	. . . . . . 7 |- ( ( A. x e. RR DECID x = 0 /\ z e. RR+ ) -> ( ( y e. RR |-> if ( y = 0 , 0 , 1 ) ) ` z ) = 1 )
47	46	neeq1d 2375	. . . . . 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 2560	. . . 4 |- ( A. x e. RR DECID x = 0 -> A. z e. RR+ ( ( y e. RR |-> if ( y = 0 , 0 , 1 ) ) ` z ) =/= 0 )
50		fveq2 5527	. . . . . 6 |- ( z = x -> ( ( y e. RR |-> if ( y = 0 , 0 , 1 ) ) ` z ) = ( ( y e. RR |-> if ( y = 0 , 0 , 1 ) ) ` x ) )
51	50	neeq1d 2375	. . . . 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 2715	. . . 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 1178	. 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 5360	. . 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 5526	. . . 4 |- ( f = ( y e. RR |-> if ( y = 0 , 0 , 1 ) ) -> ( f ` 0 ) = ( ( y e. RR |-> if ( y = 0 , 0 , 1 ) ) ` 0 ) )
57	56	eqeq1d 2196	. . 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 5526	. . . . 5 |- ( f = ( y e. RR |-> if ( y = 0 , 0 , 1 ) ) -> ( f ` x ) = ( ( y e. RR |-> if ( y = 0 , 0 , 1 ) ) ` x ) )
59	58	neeq1d 2375	. . . 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 2487	. . 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 1322	. 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 2894	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 979   = wceq 1363   T. wtru 1364   E.wex 1502   e. wcel 2158   =/= wne 2357   A.wral 2465   _Vcvv 2749   ifcif 3546   |-> cmpt 4076   -->wf 5224   ` cfv 5228   RRcr 7823   0cc0 7824   1c1 7825   ZZcz 9266   RR+crp 9666

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-cnex 7915  ax-resscn 7916  ax-1cn 7917  ax-1re 7918  ax-icn 7919  ax-addcl 7920  ax-addrcl 7921  ax-mulcl 7922  ax-addcom 7924  ax-addass 7926  ax-distr 7928  ax-i2m1 7929  ax-0lt1 7930  ax-0id 7932  ax-rnegex 7933  ax-cnre 7935  ax-pre-ltirr 7936  ax-pre-ltwlin 7937  ax-pre-lttrn 7938  ax-pre-ltadd 7940

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-if 3547  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-pnf 8007  df-mnf 8008  df-xr 8009  df-ltxr 8010  df-le 8011  df-sub 8143  df-neg 8144  df-inn 8933  df-z 9267  df-rp 9667

This theorem is referenced by:  dcapnconstALT  15082