Theorem nninfdcinf 7188

Description: The Weak Limited Principle of Omniscience (WLPO) implies that it is decidable whether an element of ℕ_∞ equals the point at infinity. (Contributed by Jim Kingdon, 3-Dec-2024.)

Hypotheses

Ref	Expression
nninfdcinf.w	|- ( ph -> om e. WOmni )
nninfdcinf.n	|- ( ph -> N e. ℕ∞ )

Assertion

Ref	Expression
nninfdcinf	|- ( ph -> DECID N = ( i e. om |-> 1o ) )

Distinct variable groups:   i, N   ph, i

Proof of Theorem nninfdcinf

Dummy variables f x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq1 5529	. . . . . 6 |- ( f = N -> ( f ` x ) = ( N ` x ) )
2	1	eqeq1d 2198	. . . . 5 |- ( f = N -> ( ( f ` x ) = 1o <-> ( N ` x ) = 1o ) )
3	2	ralbidv 2490	. . . 4 |- ( f = N -> ( A. x e. om ( f ` x ) = 1o <-> A. x e. om ( N ` x ) = 1o ) )
4	3	dcbid 839	. . 3 |- ( f = N -> (DECID A. x e. om ( f ` x ) = 1o <-> DECID A. x e. om ( N ` x ) = 1o ) )
5		nninfdcinf.w	. . . 4 |- ( ph -> om e. WOmni )
6	5	elexd 2765	. . . . 5 |- ( ph -> om e. _V )
7		iswomnimap 7183	. . . . 5 |- ( om e. _V -> ( om e. WOmni <-> A. f e. ( 2o ^m om )DECID A. x e. om ( f ` x ) = 1o ) )
8	6, 7	syl 14	. . . 4 |- ( ph -> ( om e. WOmni <-> A. f e. ( 2o ^m om )DECID A. x e. om ( f ` x ) = 1o ) )
9	5, 8	mpbid 147	. . 3 |- ( ph -> A. f e. ( 2o ^m om )DECID A. x e. om ( f ` x ) = 1o )
10		nninfdcinf.n	. . . . 5 |- ( ph -> N e. ℕ∞ )
11		nninff 7140	. . . . 5 |- ( N e. ℕ∞ -> N : om --> 2o )
12	10, 11	syl 14	. . . 4 |- ( ph -> N : om --> 2o )
13		2onn 6540	. . . . . 6 |- 2o e. om
14	13	elexi 2764	. . . . 5 |- 2o e. _V
15		omex 4607	. . . . 5 |- om e. _V
16	14, 15	elmap 6695	. . . 4 |- ( N e. ( 2o ^m om ) <-> N : om --> 2o )
17	12, 16	sylibr 134	. . 3 |- ( ph -> N e. ( 2o ^m om ) )
18	4, 9, 17	rspcdva 2861	. 2 |- ( ph -> DECID A. x e. om ( N ` x ) = 1o )
19	12	ffnd 5381	. . . 4 |- ( ph -> N Fn om )
20		eqidd 2190	. . . 4 |- ( x = i -> 1o = 1o )
21		1onn 6539	. . . . 5 |- 1o e. om
22	21	a1i 9	. . . 4 |- ( ( ph /\ x e. om ) -> 1o e. om )
23	21	a1i 9	. . . 4 |- ( ( ph /\ i e. om ) -> 1o e. om )
24	19, 20, 22, 23	fnmptfvd 5636	. . 3 |- ( ph -> ( N = ( i e. om |-> 1o ) <-> A. x e. om ( N ` x ) = 1o ) )
25	24	dcbid 839	. 2 |- ( ph -> (DECID N = ( i e. om |-> 1o ) <-> DECID A. x e. om ( N ` x ) = 1o ) )
26	18, 25	mpbird 167	1 |- ( ph -> DECID N = ( i e. om |-> 1o ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105  DECID wdc 835   = wceq 1364   e. wcel 2160   A.wral 2468   _Vcvv 2752   |-> cmpt 4079   omcom 4604   -->wf 5227   ` cfv 5231  (class class class)co 5891   1oc1o 6428   2oc2o 6429   ^m cmap 6666  ℕ_∞xnninf 7137  WOmnicwomni 7180

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-nul 4144  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-iinf 4602

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4308  df-suc 4386  df-iom 4605  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-fv 5239  df-ov 5894  df-oprab 5895  df-mpo 5896  df-1o 6435  df-2o 6436  df-map 6668  df-nninf 7138  df-womni 7181

This theorem is referenced by: (None)