Theorem nninfall 16081

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 6530	. . . . 5 |- 1o =/= (/)
2	1	nesymi 2423	. . . 4 |- -. (/) = 1o
3		simplr 528	. . . . . . . . . . 11 |- ( ( ( ph /\ p e. ℕ∞ ) /\ ( Q ` p ) = (/) ) -> p e. ℕ∞ )
4		nninff 7238	. . . . . . . . . . . 12 |- ( p e. ℕ∞ -> p : om --> 2o )
5	4	ffnd 5435	. . . . . . . . . . 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 16080	. . . . . . . . . . . . 13 |- ( ( ( ph /\ p e. ℕ∞ ) /\ ( Q ` p ) = (/) ) -> A. n e. om ( p ` n ) = 1o )
15		eqeq1 2213	. . . . . . . . . . . . . . 15 |- ( a = ( p ` n ) -> ( a = 1o <-> ( p ` n ) = 1o ) )
16	15	ralrn 5730	. . . . . . . . . . . . . 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 4649	. . . . . . . . . . . . . . . 16 |- (/) e. om
20		elex2 2790	. . . . . . . . . . . . . . . 16 |- ( (/) e. om -> E. b b e. om )
21	19, 20	ax-mp 5	. . . . . . . . . . . . . . 15 |- E. b b e. om
22		fdm 5440	. . . . . . . . . . . . . . . . 17 |- ( p : om --> 2o -> dom p = om )
23	22	eleq2d 2276	. . . . . . . . . . . . . . . 16 |- ( p : om --> 2o -> ( b e. dom p <-> b e. om ) )
24	23	exbidv 1849	. . . . . . . . . . . . . . 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 4905	. . . . . . . . . . . . . . 15 |- ( E. b b e. dom p <-> E. a a e. ran p )
27		eqsnm 3801	. . . . . . . . . . . . . . 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 3251	. . . . . . . . . . 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 5283	. . . . . . . . . 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 6618	. . . . . . . . . 10 |- 1o e. om
37		fconst2g 5811	. . . . . . . . . 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 4729	. . . . . . . 8 |- ( om X. { 1o } ) = ( x e. om |-> 1o )
41	39, 40	eqtrdi 2255	. . . . . . 7 |- ( ( ( ph /\ p e. ℕ∞ ) /\ ( Q ` p ) = (/) ) -> p = ( x e. om |-> 1o ) )
42	41	fveq2d 5592	. . . . . 6 |- ( ( ( ph /\ p e. ℕ∞ ) /\ ( Q ` p ) = (/) ) -> ( Q ` p ) = ( Q ` ( x e. om |-> 1o ) ) )
43	42, 13, 10	3eqtr3d 2247	. . . . 5 |- ( ( ( ph /\ p e. ℕ∞ ) /\ ( Q ` p ) = (/) ) -> (/) = 1o )
44	43	ex 115	. . . 4 |- ( ( ph /\ p e. ℕ∞ ) -> ( ( Q ` p ) = (/) -> (/) = 1o ) )
45	2, 44	mtoi 666	. . 3 |- ( ( ph /\ p e. ℕ∞ ) -> -. ( Q ` p ) = (/) )
46		elmapi 6769	. . . . . . 7 |- ( Q e. ( 2o ^m ℕ∞ ) -> Q :ℕ∞ --> 2o )
47	7, 46	syl 14	. . . . . 6 |- ( ph -> Q :ℕ∞ --> 2o )
48	47	ffvelcdmda 5727	. . . . 5 |- ( ( ph /\ p e. ℕ∞ ) -> ( Q ` p ) e. 2o )
49		elpri 3660	. . . . . 6 |- ( ( Q ` p ) e. { (/) , 1o } -> ( ( Q ` p ) = (/) \/ ( Q ` p ) = 1o ) )
50		df2o3 6528	. . . . . 6 |- 2o = { (/) , 1o }
51	49, 50	eleq2s 2301	. . . . 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 731	. . 3 |- ( ( ph /\ p e. ℕ∞ ) -> ( ( Q ` p ) = 1o \/ ( Q ` p ) = (/) ) )
54	45, 53	ecased 1362	. 2 |- ( ( ph /\ p e. ℕ∞ ) -> ( Q ` p ) = 1o )
55	54	ralrimiva 2580	1 |- ( ph -> A. p e. ℕ∞ ( Q ` p ) = 1o )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 710   = wceq 1373   E.wex 1516   e. wcel 2177   A.wral 2485   C_ wss 3170   (/)c0 3464   ifcif 3575   {csn 3637   {cpr 3638   |-> cmpt 4112   omcom 4645   X. cxp 4680   dom cdm 4682   ran crn 4683   Fn wfn 5274   -->wf 5275   ` cfv 5279  (class class class)co 5956   1oc1o 6507   2oc2o 6508   ^m cmap 6747  ℕ_∞xnninf 7235

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4169  ax-nul 4177  ax-pow 4225  ax-pr 4260  ax-un 4487  ax-setind 4592  ax-iinf 4643

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-if 3576  df-pw 3622  df-sn 3643  df-pr 3644  df-op 3646  df-uni 3856  df-int 3891  df-br 4051  df-opab 4113  df-mpt 4114  df-tr 4150  df-id 4347  df-iord 4420  df-on 4422  df-suc 4425  df-iom 4646  df-xp 4688  df-rel 4689  df-cnv 4690  df-co 4691  df-dm 4692  df-rn 4693  df-iota 5240  df-fun 5281  df-fn 5282  df-f 5283  df-fv 5287  df-ov 5959  df-oprab 5960  df-mpo 5961  df-1o 6514  df-2o 6515  df-map 6749  df-nninf 7236

This theorem is referenced by:  nninfsel  16089  nninffeq  16092