Theorem nninfomnilem 15917

Description: Lemma for nninfomni 15918. (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 7222	. . 3 |- ℕ∞ e. _V
2		isomnimap 7238	. . 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 6756	. . . . . 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 15912	. . . . . . 7 |- E : ( 2o ^m ℕ∞ ) -->ℕ∞
7	6	ffvelcdmi 5713	. . . . . 6 |- ( r e. ( 2o ^m ℕ∞ ) -> ( E ` r ) e. ℕ∞ )
8	4, 7	ffvelcdmd 5715	. . . . 5 |- ( r e. ( 2o ^m ℕ∞ ) -> ( r ` ( E ` r ) ) e. 2o )
9		df2o3 6515	. . . . 5 |- 2o = { (/) , 1o }
10	8, 9	eleqtrdi 2297	. . . 4 |- ( r e. ( 2o ^m ℕ∞ ) -> ( r ` ( E ` r ) ) e. { (/) , 1o } )
11		elpri 3655	. . . 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 5584	. . . . . . 7 |- ( p = ( E ` r ) -> ( ( r ` p ) = (/) <-> ( r ` ( E ` r ) ) = (/) ) )
14	13	rspcev 2876	. . . . . 6 |- ( ( ( E ` r ) e. ℕ∞ /\ ( r ` ( E ` r ) ) = (/) ) -> E. p e. ℕ∞ ( r ` p ) = (/) )
15	14	ex 115	. . . . 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 109	. . . . . 6 |- ( ( r e. ( 2o ^m ℕ∞ ) /\ ( r ` ( E ` r ) ) = 1o ) -> r e. ( 2o ^m ℕ∞ ) )
18		simpr 110	. . . . . 6 |- ( ( r e. ( 2o ^m ℕ∞ ) /\ ( r ` ( E ` r ) ) = 1o ) -> ( r ` ( E ` r ) ) = 1o )
19	5, 17, 18	nninfsel 15916	. . . . 5 |- ( ( r e. ( 2o ^m ℕ∞ ) /\ ( r ` ( E ` r ) ) = 1o ) -> A. p e. ℕ∞ ( r ` p ) = 1o )
20	19	ex 115	. . . 4 |- ( r e. ( 2o ^m ℕ∞ ) -> ( ( r ` ( E ` r ) ) = 1o -> A. p e. ℕ∞ ( r ` p ) = 1o ) )
21	16, 20	orim12d 787	. . 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 2563	1 |- ℕ∞ e. Omni

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709   = wceq 1372   e. wcel 2175   A.wral 2483   E.wrex 2484   _Vcvv 2771   (/)c0 3459   ifcif 3570   {cpr 3633   |-> cmpt 4104   suc csuc 4411   omcom 4637   ` cfv 5270  (class class class)co 5943   1oc1o 6494   2oc2o 6495   ^m cmap 6734  ℕ_∞xnninf 7220  Omnicomni 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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-nul 4169  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-iinf 4635

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-ral 2488  df-rex 2489  df-reu 2490  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-if 3571  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-tr 4142  df-id 4339  df-iord 4412  df-on 4414  df-suc 4417  df-iom 4638  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-ov 5946  df-oprab 5947  df-mpo 5948  df-1o 6501  df-2o 6502  df-map 6736  df-nninf 7221  df-omni 7236

This theorem is referenced by:  nninfomni  15918