Theorem nninfall 13379

Description: Given a decidable predicate on ℕ_∞, showing it holds for natural numbers and the point at infinity suffices to show it holds everywhere. The sense in which Q is a decidable predicate is that it assigns a value of either (/) or 1o (which can be thought of as false and true) to every element of ℕ_∞. Lemma 3.5 of [PradicBrown2022], p. 5. (Contributed by Jim Kingdon, 1-Aug-2022.)

Hypotheses

Ref	Expression
nninfall.q	|- ( ph -> Q e. ( 2o ^m ℕ∞ ) )
nninfall.inf	|- ( ph -> ( Q ` ( x e. om |-> 1o ) ) = 1o )
nninfall.n	|- ( ph -> A. n e. om ( Q ` ( i e. om |-> if ( i e. n , 1o , (/) ) ) ) = 1o )

Assertion

Ref	Expression
nninfall	|- ( ph -> A. p e. ℕ∞ ( Q ` p ) = 1o )

Distinct variable groups:   Q, n, i   n, p, i, ph

Allowed substitution hints:   ph( x)   Q( x, p)

Proof of Theorem nninfall

Dummy variables a b are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		1n0 6337	. . . . 5 |- 1o =/= (/)
2	1	nesymi 2355	. . . 4 |- -. (/) = 1o
3		simplr 520	. . . . . . . . . . 11 |- ( ( ( ph /\ p e. ℕ∞ ) /\ ( Q ` p ) = (/) ) -> p e. ℕ∞ )
4		nninff 13373	. . . . . . . . . . . 12 |- ( p e. ℕ∞ -> p : om --> 2o )
5	4	ffnd 5281	. . . . . . . . . . 11 |- ( p e. ℕ∞ -> p Fn om )
6	3, 5	syl 14	. . . . . . . . . 10 |- ( ( ( ph /\ p e. ℕ∞ ) /\ ( Q ` p ) = (/) ) -> p Fn om )
7		nninfall.q	. . . . . . . . . . . . . . 15 |- ( ph -> Q e. ( 2o ^m ℕ∞ ) )
8	7	ad2antrr 480	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ p e. ℕ∞ ) /\ ( Q ` p ) = (/) ) -> Q e. ( 2o ^m ℕ∞ ) )
9		nninfall.inf	. . . . . . . . . . . . . . 15 |- ( ph -> ( Q ` ( x e. om |-> 1o ) ) = 1o )
10	9	ad2antrr 480	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ p e. ℕ∞ ) /\ ( Q ` p ) = (/) ) -> ( Q ` ( x e. om |-> 1o ) ) = 1o )
11		nninfall.n	. . . . . . . . . . . . . . 15 |- ( ph -> A. n e. om ( Q ` ( i e. om |-> if ( i e. n , 1o , (/) ) ) ) = 1o )
12	11	ad2antrr 480	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ p e. ℕ∞ ) /\ ( Q ` p ) = (/) ) -> A. n e. om ( Q ` ( i e. om |-> if ( i e. n , 1o , (/) ) ) ) = 1o )
13		simpr 109	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ p e. ℕ∞ ) /\ ( Q ` p ) = (/) ) -> ( Q ` p ) = (/) )
14	8, 10, 12, 3, 13	nninfalllem1 13378	. . . . . . . . . . . . 13 |- ( ( ( ph /\ p e. ℕ∞ ) /\ ( Q ` p ) = (/) ) -> A. n e. om ( p ` n ) = 1o )
15		eqeq1 2147	. . . . . . . . . . . . . . 15 |- ( a = ( p ` n ) -> ( a = 1o <-> ( p ` n ) = 1o ) )
16	15	ralrn 5566	. . . . . . . . . . . . . 14 |- ( p Fn om -> ( A. a e. ran p a = 1o <-> A. n e. om ( p ` n ) = 1o ) )
17	3, 5, 16	3syl 17	. . . . . . . . . . . . 13 |- ( ( ( ph /\ p e. ℕ∞ ) /\ ( Q ` p ) = (/) ) -> ( A. a e. ran p a = 1o <-> A. n e. om ( p ` n ) = 1o ) )
18	14, 17	mpbird 166	. . . . . . . . . . . 12 |- ( ( ( ph /\ p e. ℕ∞ ) /\ ( Q ` p ) = (/) ) -> A. a e. ran p a = 1o )
19		peano1 4516	. . . . . . . . . . . . . . . 16 |- (/) e. om
20		elex2 2705	. . . . . . . . . . . . . . . 16 |- ( (/) e. om -> E. b b e. om )
21	19, 20	ax-mp 5	. . . . . . . . . . . . . . 15 |- E. b b e. om
22		fdm 5286	. . . . . . . . . . . . . . . . 17 |- ( p : om --> 2o -> dom p = om )
23	22	eleq2d 2210	. . . . . . . . . . . . . . . 16 |- ( p : om --> 2o -> ( b e. dom p <-> b e. om ) )
24	23	exbidv 1798	. . . . . . . . . . . . . . 15 |- ( p : om --> 2o -> ( E. b b e. dom p <-> E. b b e. om ) )
25	21, 24	mpbiri 167	. . . . . . . . . . . . . 14 |- ( p : om --> 2o -> E. b b e. dom p )
26		dmmrnm 4766	. . . . . . . . . . . . . . 15 |- ( E. b b e. dom p <-> E. a a e. ran p )
27		eqsnm 3690	. . . . . . . . . . . . . . 15 |- ( E. a a e. ran p -> ( ran p = { 1o } <-> A. a e. ran p a = 1o ) )
28	26, 27	sylbi 120	. . . . . . . . . . . . . 14 |- ( E. b b e. dom p -> ( ran p = { 1o } <-> A. a e. ran p a = 1o ) )
29	25, 28	syl 14	. . . . . . . . . . . . 13 |- ( p : om --> 2o -> ( ran p = { 1o } <-> A. a e. ran p a = 1o ) )
30	3, 4, 29	3syl 17	. . . . . . . . . . . 12 |- ( ( ( ph /\ p e. ℕ∞ ) /\ ( Q ` p ) = (/) ) -> ( ran p = { 1o } <-> A. a e. ran p a = 1o ) )
31	18, 30	mpbird 166	. . . . . . . . . . 11 |- ( ( ( ph /\ p e. ℕ∞ ) /\ ( Q ` p ) = (/) ) -> ran p = { 1o } )
32		eqimss 3156	. . . . . . . . . . 11 |- ( ran p = { 1o } -> ran p C_ { 1o } )
33	31, 32	syl 14	. . . . . . . . . 10 |- ( ( ( ph /\ p e. ℕ∞ ) /\ ( Q ` p ) = (/) ) -> ran p C_ { 1o } )
34		df-f 5135	. . . . . . . . . 10 |- ( p : om --> { 1o } <-> ( p Fn om /\ ran p C_ { 1o } ) )
35	6, 33, 34	sylanbrc 414	. . . . . . . . 9 |- ( ( ( ph /\ p e. ℕ∞ ) /\ ( Q ` p ) = (/) ) -> p : om --> { 1o } )
36		1onn 6424	. . . . . . . . . 10 |- 1o e. om
37		fconst2g 5643	. . . . . . . . . 10 |- ( 1o e. om -> ( p : om --> { 1o } <-> p = ( om X. { 1o } ) ) )
38	36, 37	ax-mp 5	. . . . . . . . 9 |- ( p : om --> { 1o } <-> p = ( om X. { 1o } ) )
39	35, 38	sylib 121	. . . . . . . 8 |- ( ( ( ph /\ p e. ℕ∞ ) /\ ( Q ` p ) = (/) ) -> p = ( om X. { 1o } ) )
40		fconstmpt 4594	. . . . . . . 8 |- ( om X. { 1o } ) = ( x e. om |-> 1o )
41	39, 40	eqtrdi 2189	. . . . . . 7 |- ( ( ( ph /\ p e. ℕ∞ ) /\ ( Q ` p ) = (/) ) -> p = ( x e. om |-> 1o ) )
42	41	fveq2d 5433	. . . . . 6 |- ( ( ( ph /\ p e. ℕ∞ ) /\ ( Q ` p ) = (/) ) -> ( Q ` p ) = ( Q ` ( x e. om |-> 1o ) ) )
43	42, 13, 10	3eqtr3d 2181	. . . . 5 |- ( ( ( ph /\ p e. ℕ∞ ) /\ ( Q ` p ) = (/) ) -> (/) = 1o )
44	43	ex 114	. . . 4 |- ( ( ph /\ p e. ℕ∞ ) -> ( ( Q ` p ) = (/) -> (/) = 1o ) )
45	2, 44	mtoi 654	. . 3 |- ( ( ph /\ p e. ℕ∞ ) -> -. ( Q ` p ) = (/) )
46		elmapi 6572	. . . . . . 7 |- ( Q e. ( 2o ^m ℕ∞ ) -> Q :ℕ∞ --> 2o )
47	7, 46	syl 14	. . . . . 6 |- ( ph -> Q :ℕ∞ --> 2o )
48	47	ffvelrnda 5563	. . . . 5 |- ( ( ph /\ p e. ℕ∞ ) -> ( Q ` p ) e. 2o )
49		elpri 3555	. . . . . 6 |- ( ( Q ` p ) e. { (/) , 1o } -> ( ( Q ` p ) = (/) \/ ( Q ` p ) = 1o ) )
50		df2o3 6335	. . . . . 6 |- 2o = { (/) , 1o }
51	49, 50	eleq2s 2235	. . . . 5 |- ( ( Q ` p ) e. 2o -> ( ( Q ` p ) = (/) \/ ( Q ` p ) = 1o ) )
52	48, 51	syl 14	. . . 4 |- ( ( ph /\ p e. ℕ∞ ) -> ( ( Q ` p ) = (/) \/ ( Q ` p ) = 1o ) )
53	52	orcomd 719	. . 3 |- ( ( ph /\ p e. ℕ∞ ) -> ( ( Q ` p ) = 1o \/ ( Q ` p ) = (/) ) )
54	45, 53	ecased 1328	. 2 |- ( ( ph /\ p e. ℕ∞ ) -> ( Q ` p ) = 1o )
55	54	ralrimiva 2508	1 |- ( ph -> A. p e. ℕ∞ ( Q ` p ) = 1o )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 698   = wceq 1332   E.wex 1469   e. wcel 1481   A.wral 2417   C_ wss 3076   (/)c0 3368   ifcif 3479   {csn 3532   {cpr 3533   |-> cmpt 3997   omcom 4512   X. cxp 4545   dom cdm 4547   ran crn 4548   Fn wfn 5126   -->wf 5127   ` cfv 5131  (class class class)co 5782   1oc1o 6314   2oc2o 6315   ^m cmap 6550  ℕ_∞xnninf 7013

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-iinf 4510

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-if 3480  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-id 4223  df-iord 4296  df-on 4298  df-suc 4301  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-fv 5139  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1o 6321  df-2o 6322  df-map 6552  df-nninf 7015

This theorem is referenced by:  nninfsel  13388  nninffeq  13391