Theorem nninfdcinf 7182

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 5526	. . . . . 6 |- ( f = N -> ( f ` x ) = ( N ` x ) )
2	1	eqeq1d 2196	. . . . 5 |- ( f = N -> ( ( f ` x ) = 1o <-> ( N ` x ) = 1o ) )
3	2	ralbidv 2487	. . . 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 2762	. . . . 5 |- ( ph -> om e. _V )
7		iswomnimap 7177	. . . . 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 7134	. . . . 5 |- ( N e. ℕ∞ -> N : om --> 2o )
12	10, 11	syl 14	. . . 4 |- ( ph -> N : om --> 2o )
13		2onn 6535	. . . . . 6 |- 2o e. om
14	13	elexi 2761	. . . . 5 |- 2o e. _V
15		omex 4604	. . . . 5 |- om e. _V
16	14, 15	elmap 6690	. . . 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 2858	. 2 |- ( ph -> DECID A. x e. om ( N ` x ) = 1o )
19	12	ffnd 5378	. . . 4 |- ( ph -> N Fn om )
20		eqidd 2188	. . . 4 |- ( x = i -> 1o = 1o )
21		1onn 6534	. . . . 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 5633	. . 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 1363   e. wcel 2158   A.wral 2465   _Vcvv 2749   |-> cmpt 4076   omcom 4601   -->wf 5224   ` cfv 5228  (class class class)co 5888   1oc1o 6423   2oc2o 6424   ^m cmap 6661  ℕ_∞xnninf 7131  WOmnicwomni 7174

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-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599

This theorem depends on definitions:  df-bi 117  df-dc 836  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-ral 2470  df-rex 2471  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-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-fv 5236  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1o 6430  df-2o 6431  df-map 6663  df-nninf 7132  df-womni 7175

This theorem is referenced by: (None)