Theorem nninfdcinf 7246

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 5560	. . . . . 6 |- ( f = N -> ( f ` x ) = ( N ` x ) )
2	1	eqeq1d 2205	. . . . 5 |- ( f = N -> ( ( f ` x ) = 1o <-> ( N ` x ) = 1o ) )
3	2	ralbidv 2497	. . . 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 2776	. . . . 5 |- ( ph -> om e. _V )
7		iswomnimap 7241	. . . . 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 7197	. . . . 5 |- ( N e. ℕ∞ -> N : om --> 2o )
12	10, 11	syl 14	. . . 4 |- ( ph -> N : om --> 2o )
13		2onn 6588	. . . . . 6 |- 2o e. om
14	13	elexi 2775	. . . . 5 |- 2o e. _V
15		omex 4630	. . . . 5 |- om e. _V
16	14, 15	elmap 6745	. . . 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 2873	. 2 |- ( ph -> DECID A. x e. om ( N ` x ) = 1o )
19	12	ffnd 5411	. . . 4 |- ( ph -> N Fn om )
20		eqidd 2197	. . . 4 |- ( x = i -> 1o = 1o )
21		1onn 6587	. . . . 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 5669	. . 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 2167   A.wral 2475   _Vcvv 2763   |-> cmpt 4095   omcom 4627   -->wf 5255   ` cfv 5259  (class class class)co 5925   1oc1o 6476   2oc2o 6477   ^m cmap 6716  ℕ_∞xnninf 7194  WOmnicwomni 7238

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-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625

This theorem depends on definitions:  df-bi 117  df-dc 836  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-ral 2480  df-rex 2481  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-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-fv 5267  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1o 6483  df-2o 6484  df-map 6718  df-nninf 7195  df-womni 7239

This theorem is referenced by: (None)