Theorem nninfall 12788

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 6259	. . . . 5 |- 1o =/= (/)
2	1	nesymi 2313	. . . 4 |- -. (/) = 1o
3		simplr 500	. . . . . . . . . . 11 |- ( ( ( ph /\ p e. ℕ∞ ) /\ ( Q ` p ) = (/) ) -> p e. ℕ∞ )
4		nninff 12782	. . . . . . . . . . . 12 |- ( p e. ℕ∞ -> p : om --> 2o )
5	4	ffnd 5209	. . . . . . . . . . 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 475	. . . . . . . . . . . . . 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 475	. . . . . . . . . . . . . 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 475	. . . . . . . . . . . . . 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 12787	. . . . . . . . . . . . 13 |- ( ( ( ph /\ p e. ℕ∞ ) /\ ( Q ` p ) = (/) ) -> A. n e. om ( p ` n ) = 1o )
15		eqeq1 2106	. . . . . . . . . . . . . . 15 |- ( a = ( p ` n ) -> ( a = 1o <-> ( p ` n ) = 1o ) )
16	15	ralrn 5490	. . . . . . . . . . . . . 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 4446	. . . . . . . . . . . . . . . 16 |- (/) e. om
20		elex2 2657	. . . . . . . . . . . . . . . 16 |- ( (/) e. om -> E. b b e. om )
21	19, 20	ax-mp 7	. . . . . . . . . . . . . . 15 |- E. b b e. om
22		fdm 5214	. . . . . . . . . . . . . . . . 17 |- ( p : om --> 2o -> dom p = om )
23	22	eleq2d 2169	. . . . . . . . . . . . . . . 16 |- ( p : om --> 2o -> ( b e. dom p <-> b e. om ) )
24	23	exbidv 1764	. . . . . . . . . . . . . . 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 4696	. . . . . . . . . . . . . . 15 |- ( E. b b e. dom p <-> E. a a e. ran p )
27		eqsnm 3629	. . . . . . . . . . . . . . 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 3101	. . . . . . . . . . 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 5063	. . . . . . . . . 10 |- ( p : om --> { 1o } <-> ( p Fn om /\ ran p C_ { 1o } ) )
35	6, 33, 34	sylanbrc 411	. . . . . . . . 9 |- ( ( ( ph /\ p e. ℕ∞ ) /\ ( Q ` p ) = (/) ) -> p : om --> { 1o } )
36		1onn 6346	. . . . . . . . . 10 |- 1o e. om
37		fconst2g 5567	. . . . . . . . . 10 |- ( 1o e. om -> ( p : om --> { 1o } <-> p = ( om X. { 1o } ) ) )
38	36, 37	ax-mp 7	. . . . . . . . 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 4524	. . . . . . . 8 |- ( om X. { 1o } ) = ( x e. om |-> 1o )
41	39, 40	syl6eq 2148	. . . . . . 7 |- ( ( ( ph /\ p e. ℕ∞ ) /\ ( Q ` p ) = (/) ) -> p = ( x e. om |-> 1o ) )
42	41	fveq2d 5357	. . . . . 6 |- ( ( ( ph /\ p e. ℕ∞ ) /\ ( Q ` p ) = (/) ) -> ( Q ` p ) = ( Q ` ( x e. om |-> 1o ) ) )
43	42, 13, 10	3eqtr3d 2140	. . . . 5 |- ( ( ( ph /\ p e. ℕ∞ ) /\ ( Q ` p ) = (/) ) -> (/) = 1o )
44	43	ex 114	. . . 4 |- ( ( ph /\ p e. ℕ∞ ) -> ( ( Q ` p ) = (/) -> (/) = 1o ) )
45	2, 44	mtoi 631	. . 3 |- ( ( ph /\ p e. ℕ∞ ) -> -. ( Q ` p ) = (/) )
46		elmapi 6494	. . . . . . 7 |- ( Q e. ( 2o ^m ℕ∞ ) -> Q :ℕ∞ --> 2o )
47	7, 46	syl 14	. . . . . 6 |- ( ph -> Q :ℕ∞ --> 2o )
48	47	ffvelrnda 5487	. . . . 5 |- ( ( ph /\ p e. ℕ∞ ) -> ( Q ` p ) e. 2o )
49		elpri 3497	. . . . . 6 |- ( ( Q ` p ) e. { (/) , 1o } -> ( ( Q ` p ) = (/) \/ ( Q ` p ) = 1o ) )
50		df2o3 6257	. . . . . 6 |- 2o = { (/) , 1o }
51	49, 50	eleq2s 2194	. . . . 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 689	. . 3 |- ( ( ph /\ p e. ℕ∞ ) -> ( ( Q ` p ) = 1o \/ ( Q ` p ) = (/) ) )
54	45, 53	ecased 1295	. 2 |- ( ( ph /\ p e. ℕ∞ ) -> ( Q ` p ) = 1o )
55	54	ralrimiva 2464	1 |- ( ph -> A. p e. ℕ∞ ( Q ` p ) = 1o )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 670   = wceq 1299   E.wex 1436   e. wcel 1448   A.wral 2375   C_ wss 3021   (/)c0 3310   ifcif 3421   {csn 3474   {cpr 3475   |-> cmpt 3929   omcom 4442   X. cxp 4475   dom cdm 4477   ran crn 4478   Fn wfn 5054   -->wf 5055   ` cfv 5059  (class class class)co 5706   1oc1o 6236   2oc2o 6237   ^m cmap 6472  ℕ_∞xnninf 6917

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-iinf 4440

This theorem depends on definitions:  df-bi 116  df-dc 787  df-3or 931  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-ral 2380  df-rex 2381  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-if 3422  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-br 3876  df-opab 3930  df-mpt 3931  df-tr 3967  df-id 4153  df-iord 4226  df-on 4228  df-suc 4231  df-iom 4443  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-fv 5067  df-ov 5709  df-oprab 5710  df-mpo 5711  df-1o 6243  df-2o 6244  df-map 6474  df-nninf 6919

This theorem is referenced by:  nninfsel  12797  nninffeq  12800