Theorem nninfdcinf 7462

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 5669	. . . . . 6 |- ( f = N -> ( f ` x ) = ( N ` x ) )
2	1	eqeq1d 2241	. . . . 5 |- ( f = N -> ( ( f ` x ) = 1o <-> ( N ` x ) = 1o ) )
3	2	ralbidv 2542	. . . 4 |- ( f = N -> ( A. x e. om ( f ` x ) = 1o <-> A. x e. om ( N ` x ) = 1o ) )
4	3	dcbid 846	. . 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 2827	. . . . 5 |- ( ph -> om e. _V )
7		iswomnimap 7457	. . . . 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 7413	. . . . 5 |- ( N e. ℕ∞ -> N : om --> 2o )
12	10, 11	syl 14	. . . 4 |- ( ph -> N : om --> 2o )
13		2onn 6754	. . . . . 6 |- 2o e. om
14	13	elexi 2826	. . . . 5 |- 2o e. _V
15		omex 4715	. . . . 5 |- om e. _V
16	14, 15	elmap 6911	. . . 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 2926	. 2 |- ( ph -> DECID A. x e. om ( N ` x ) = 1o )
19	12	ffnd 5509	. . . 4 |- ( ph -> N Fn om )
20		eqidd 2233	. . . 4 |- ( x = i -> 1o = 1o )
21		1onn 6753	. . . . 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 5782	. . 3 |- ( ph -> ( N = ( i e. om |-> 1o ) <-> A. x e. om ( N ` x ) = 1o ) )
25	24	dcbid 846	. 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 842   = wceq 1398   e. wcel 2203   A.wral 2520   _Vcvv 2813   |-> cmpt 4171   omcom 4712   -->wf 5348   ` cfv 5352  (class class class)co 6050   1oc1o 6640   2oc2o 6641   ^m cmap 6882  ℕ_∞xnninf 7410  WOmnicwomni 7454

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-fv 5360  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1o 6647  df-2o 6648  df-map 6884  df-nninf 7411  df-womni 7455

This theorem is referenced by:  nninfinfwlpo  7471