Theorem nninfomnilem 13732

Description: Lemma for nninfomni 13733. (Contributed by Jim Kingdon, 10-Aug-2022.)

Hypothesis

Ref	Expression
nninfsel.e	|- E = ( q e. ( 2o ^m ℕ∞ ) |-> ( n e. om |-> if ( A. k e. suc n ( q ` ( i e. om |-> if ( i e. k , 1o , (/) ) ) ) = 1o , 1o , (/) ) ) )

Assertion

Ref	Expression
nninfomnilem	|- ℕ∞ e. Omni

Distinct variable groups:   i, E, k, n   i, q, k, n

Allowed substitution hint:   E( q)

Proof of Theorem nninfomnilem

Dummy variables p r are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nninfex 7077	. . 3 |- ℕ∞ e. _V
2		isomnimap 7092	. . 3 |- (ℕ∞ e. _V -> (ℕ∞ e. Omni <-> A. r e. ( 2o ^m ℕ∞ ) ( E. p e. ℕ∞ ( r ` p ) = (/) \/ A. p e. ℕ∞ ( r ` p ) = 1o ) ) )
3	1, 2	ax-mp 5	. 2 |- (ℕ∞ e. Omni <-> A. r e. ( 2o ^m ℕ∞ ) ( E. p e. ℕ∞ ( r ` p ) = (/) \/ A. p e. ℕ∞ ( r ` p ) = 1o ) )
4		elmapi 6627	. . . . . 6 |- ( r e. ( 2o ^m ℕ∞ ) -> r :ℕ∞ --> 2o )
5		nninfsel.e	. . . . . . . 8 |- E = ( q e. ( 2o ^m ℕ∞ ) |-> ( n e. om |-> if ( A. k e. suc n ( q ` ( i e. om |-> if ( i e. k , 1o , (/) ) ) ) = 1o , 1o , (/) ) ) )
6	5	nninfself 13727	. . . . . . 7 |- E : ( 2o ^m ℕ∞ ) -->ℕ∞
7	6	ffvelrni 5613	. . . . . 6 |- ( r e. ( 2o ^m ℕ∞ ) -> ( E ` r ) e. ℕ∞ )
8	4, 7	ffvelrnd 5615	. . . . 5 |- ( r e. ( 2o ^m ℕ∞ ) -> ( r ` ( E ` r ) ) e. 2o )
9		df2o3 6389	. . . . 5 |- 2o = { (/) , 1o }
10	8, 9	eleqtrdi 2257	. . . 4 |- ( r e. ( 2o ^m ℕ∞ ) -> ( r ` ( E ` r ) ) e. { (/) , 1o } )
11		elpri 3593	. . . 4 |- ( ( r ` ( E ` r ) ) e. { (/) , 1o } -> ( ( r ` ( E ` r ) ) = (/) \/ ( r ` ( E ` r ) ) = 1o ) )
12	10, 11	syl 14	. . 3 |- ( r e. ( 2o ^m ℕ∞ ) -> ( ( r ` ( E ` r ) ) = (/) \/ ( r ` ( E ` r ) ) = 1o ) )
13		fveqeq2 5489	. . . . . . 7 |- ( p = ( E ` r ) -> ( ( r ` p ) = (/) <-> ( r ` ( E ` r ) ) = (/) ) )
14	13	rspcev 2825	. . . . . 6 |- ( ( ( E ` r ) e. ℕ∞ /\ ( r ` ( E ` r ) ) = (/) ) -> E. p e. ℕ∞ ( r ` p ) = (/) )
15	14	ex 114	. . . . 5 |- ( ( E ` r ) e. ℕ∞ -> ( ( r ` ( E ` r ) ) = (/) -> E. p e. ℕ∞ ( r ` p ) = (/) ) )
16	7, 15	syl 14	. . . 4 |- ( r e. ( 2o ^m ℕ∞ ) -> ( ( r ` ( E ` r ) ) = (/) -> E. p e. ℕ∞ ( r ` p ) = (/) ) )
17		simpl 108	. . . . . 6 |- ( ( r e. ( 2o ^m ℕ∞ ) /\ ( r ` ( E ` r ) ) = 1o ) -> r e. ( 2o ^m ℕ∞ ) )
18		simpr 109	. . . . . 6 |- ( ( r e. ( 2o ^m ℕ∞ ) /\ ( r ` ( E ` r ) ) = 1o ) -> ( r ` ( E ` r ) ) = 1o )
19	5, 17, 18	nninfsel 13731	. . . . 5 |- ( ( r e. ( 2o ^m ℕ∞ ) /\ ( r ` ( E ` r ) ) = 1o ) -> A. p e. ℕ∞ ( r ` p ) = 1o )
20	19	ex 114	. . . 4 |- ( r e. ( 2o ^m ℕ∞ ) -> ( ( r ` ( E ` r ) ) = 1o -> A. p e. ℕ∞ ( r ` p ) = 1o ) )
21	16, 20	orim12d 776	. . 3 |- ( r e. ( 2o ^m ℕ∞ ) -> ( ( ( r ` ( E ` r ) ) = (/) \/ ( r ` ( E ` r ) ) = 1o ) -> ( E. p e. ℕ∞ ( r ` p ) = (/) \/ A. p e. ℕ∞ ( r ` p ) = 1o ) ) )
22	12, 21	mpd 13	. 2 |- ( r e. ( 2o ^m ℕ∞ ) -> ( E. p e. ℕ∞ ( r ` p ) = (/) \/ A. p e. ℕ∞ ( r ` p ) = 1o ) )
23	3, 22	mprgbir 2522	1 |- ℕ∞ e. Omni

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 698   = wceq 1342   e. wcel 2135   A.wral 2442   E.wrex 2443   _Vcvv 2721   (/)c0 3404   ifcif 3515   {cpr 3571   |-> cmpt 4037   suc csuc 4337   omcom 4561   ` cfv 5182  (class class class)co 5836   1oc1o 6368   2oc2o 6369   ^m cmap 6605  ℕ_∞xnninf 7075  Omnicomni 7089

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 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-coll 4091  ax-sep 4094  ax-nul 4102  ax-pow 4147  ax-pr 4181  ax-un 4405  ax-setind 4508  ax-iinf 4559

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 968  df-3an 969  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ne 2335  df-ral 2447  df-rex 2448  df-reu 2449  df-rab 2451  df-v 2723  df-sbc 2947  df-csb 3041  df-dif 3113  df-un 3115  df-in 3117  df-ss 3124  df-nul 3405  df-if 3516  df-pw 3555  df-sn 3576  df-pr 3577  df-op 3579  df-uni 3784  df-int 3819  df-iun 3862  df-br 3977  df-opab 4038  df-mpt 4039  df-tr 4075  df-id 4265  df-iord 4338  df-on 4340  df-suc 4343  df-iom 4562  df-xp 4604  df-rel 4605  df-cnv 4606  df-co 4607  df-dm 4608  df-rn 4609  df-res 4610  df-ima 4611  df-iota 5147  df-fun 5184  df-fn 5185  df-f 5186  df-f1 5187  df-fo 5188  df-f1o 5189  df-fv 5190  df-ov 5839  df-oprab 5840  df-mpo 5841  df-1o 6375  df-2o 6376  df-map 6607  df-nninf 7076  df-omni 7090

This theorem is referenced by:  nninfomni  13733