Theorem nninfall 16787

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 6665	. . . . 5 |- 1o =/= (/)
2	1	nesymi 2458	. . . 4 |- -. (/) = 1o
3		simplr 529	. . . . . . . . . . 11 |- ( ( ( ph /\ p e. ℕ∞ ) /\ ( Q ` p ) = (/) ) -> p e. ℕ∞ )
4		nninff 7413	. . . . . . . . . . . 12 |- ( p e. ℕ∞ -> p : om --> 2o )
5	4	ffnd 5509	. . . . . . . . . . 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 488	. . . . . . . . . . . . . 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 488	. . . . . . . . . . . . . 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 488	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ p e. ℕ∞ ) /\ ( Q ` p ) = (/) ) -> A. n e. om ( Q ` ( i e. om |-> if ( i e. n , 1o , (/) ) ) ) = 1o )
13		simpr 110	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ p e. ℕ∞ ) /\ ( Q ` p ) = (/) ) -> ( Q ` p ) = (/) )
14	8, 10, 12, 3, 13	nninfalllem1 16786	. . . . . . . . . . . . 13 |- ( ( ( ph /\ p e. ℕ∞ ) /\ ( Q ` p ) = (/) ) -> A. n e. om ( p ` n ) = 1o )
15		eqeq1 2239	. . . . . . . . . . . . . . 15 |- ( a = ( p ` n ) -> ( a = 1o <-> ( p ` n ) = 1o ) )
16	15	ralrn 5815	. . . . . . . . . . . . . 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 167	. . . . . . . . . . . 12 |- ( ( ( ph /\ p e. ℕ∞ ) /\ ( Q ` p ) = (/) ) -> A. a e. ran p a = 1o )
19		peano1 4716	. . . . . . . . . . . . . . . 16 |- (/) e. om
20		elex2 2830	. . . . . . . . . . . . . . . 16 |- ( (/) e. om -> E. b b e. om )
21	19, 20	ax-mp 5	. . . . . . . . . . . . . . 15 |- E. b b e. om
22		fdm 5514	. . . . . . . . . . . . . . . . 17 |- ( p : om --> 2o -> dom p = om )
23	22	eleq2d 2302	. . . . . . . . . . . . . . . 16 |- ( p : om --> 2o -> ( b e. dom p <-> b e. om ) )
24	23	exbidv 1874	. . . . . . . . . . . . . . 15 |- ( p : om --> 2o -> ( E. b b e. dom p <-> E. b b e. om ) )
25	21, 24	mpbiri 168	. . . . . . . . . . . . . 14 |- ( p : om --> 2o -> E. b b e. dom p )
26		dmmrnm 4976	. . . . . . . . . . . . . . 15 |- ( E. b b e. dom p <-> E. a a e. ran p )
27		eqsnm 3859	. . . . . . . . . . . . . . 15 |- ( E. a a e. ran p -> ( ran p = { 1o } <-> A. a e. ran p a = 1o ) )
28	26, 27	sylbi 121	. . . . . . . . . . . . . 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 167	. . . . . . . . . . 11 |- ( ( ( ph /\ p e. ℕ∞ ) /\ ( Q ` p ) = (/) ) -> ran p = { 1o } )
32		eqimss 3292	. . . . . . . . . . 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 5356	. . . . . . . . . 10 |- ( p : om --> { 1o } <-> ( p Fn om /\ ran p C_ { 1o } ) )
35	6, 33, 34	sylanbrc 417	. . . . . . . . 9 |- ( ( ( ph /\ p e. ℕ∞ ) /\ ( Q ` p ) = (/) ) -> p : om --> { 1o } )
36		1onn 6753	. . . . . . . . . 10 |- 1o e. om
37		fconst2g 5899	. . . . . . . . . 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 122	. . . . . . . 8 |- ( ( ( ph /\ p e. ℕ∞ ) /\ ( Q ` p ) = (/) ) -> p = ( om X. { 1o } ) )
40		fconstmpt 4797	. . . . . . . 8 |- ( om X. { 1o } ) = ( x e. om |-> 1o )
41	39, 40	eqtrdi 2281	. . . . . . 7 |- ( ( ( ph /\ p e. ℕ∞ ) /\ ( Q ` p ) = (/) ) -> p = ( x e. om |-> 1o ) )
42	41	fveq2d 5674	. . . . . 6 |- ( ( ( ph /\ p e. ℕ∞ ) /\ ( Q ` p ) = (/) ) -> ( Q ` p ) = ( Q ` ( x e. om |-> 1o ) ) )
43	42, 13, 10	3eqtr3d 2273	. . . . 5 |- ( ( ( ph /\ p e. ℕ∞ ) /\ ( Q ` p ) = (/) ) -> (/) = 1o )
44	43	ex 115	. . . 4 |- ( ( ph /\ p e. ℕ∞ ) -> ( ( Q ` p ) = (/) -> (/) = 1o ) )
45	2, 44	mtoi 670	. . 3 |- ( ( ph /\ p e. ℕ∞ ) -> -. ( Q ` p ) = (/) )
46		elmapi 6904	. . . . . . 7 |- ( Q e. ( 2o ^m ℕ∞ ) -> Q :ℕ∞ --> 2o )
47	7, 46	syl 14	. . . . . 6 |- ( ph -> Q :ℕ∞ --> 2o )
48	47	ffvelcdmda 5812	. . . . 5 |- ( ( ph /\ p e. ℕ∞ ) -> ( Q ` p ) e. 2o )
49		elpri 3712	. . . . . 6 |- ( ( Q ` p ) e. { (/) , 1o } -> ( ( Q ` p ) = (/) \/ ( Q ` p ) = 1o ) )
50		df2o3 6662	. . . . . 6 |- 2o = { (/) , 1o }
51	49, 50	eleq2s 2327	. . . . 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 737	. . 3 |- ( ( ph /\ p e. ℕ∞ ) -> ( ( Q ` p ) = 1o \/ ( Q ` p ) = (/) ) )
54	45, 53	ecased 1386	. 2 |- ( ( ph /\ p e. ℕ∞ ) -> ( Q ` p ) = 1o )
55	54	ralrimiva 2615	1 |- ( ph -> A. p e. ℕ∞ ( Q ` p ) = 1o )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 716   = wceq 1398   E.wex 1541   e. wcel 2203   A.wral 2520   C_ wss 3211   (/)c0 3508   ifcif 3620   {csn 3689   {cpr 3690   |-> cmpt 4171   omcom 4712   X. cxp 4747   dom cdm 4749   ran crn 4750   Fn wfn 5347   -->wf 5348   ` cfv 5352  (class class class)co 6050   1oc1o 6640   2oc2o 6641   ^m cmap 6882  ℕ_∞xnninf 7410

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-3or 1006  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-if 3621  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-tr 4209  df-id 4414  df-iord 4487  df-on 4489  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

This theorem is referenced by:  nninfsel  16795  nninffeq  16798