Theorem nninfninc 7182

Description: All values beyond a zero in an ℕ_∞ sequence are zero. This is another way of stating that elements of ℕ_∞ are nonincreasing. (Contributed by Jim Kingdon, 12-Jul-2025.)

Hypotheses

Ref	Expression
nninfninc.a	|- ( ph -> A e. ℕ∞ )
nninfninc.x	|- ( ph -> X e. om )
nninfninc.y	|- ( ph -> Y e. om )
nninfninc.le	|- ( ph -> X C_ Y )
nninfninc.z	|- ( ph -> ( A ` X ) = (/) )

Assertion

Ref	Expression
nninfninc	|- ( ph -> ( A ` Y ) = (/) )

Proof of Theorem nninfninc

Dummy variables f i n k w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nninfninc.le	. 2 |- ( ph -> X C_ Y )
2		sseq2 3203	. . . 4 |- ( n = Y -> ( X C_ n <-> X C_ Y ) )
3		fveqeq2 5563	. . . 4 |- ( n = Y -> ( ( A ` n ) = (/) <-> ( A ` Y ) = (/) ) )
4	2, 3	imbi12d 234	. . 3 |- ( n = Y -> ( ( X C_ n -> ( A ` n ) = (/) ) <-> ( X C_ Y -> ( A ` Y ) = (/) ) ) )
5		sseq2 3203	. . . . . . . . 9 |- ( w = (/) -> ( X C_ w <-> X C_ (/) ) )
6	5	anbi2d 464	. . . . . . . 8 |- ( w = (/) -> ( ( ph /\ X C_ w ) <-> ( ph /\ X C_ (/) ) ) )
7		fveqeq2 5563	. . . . . . . 8 |- ( w = (/) -> ( ( A ` w ) = (/) <-> ( A ` (/) ) = (/) ) )
8	6, 7	imbi12d 234	. . . . . . 7 |- ( w = (/) -> ( ( ( ph /\ X C_ w ) -> ( A ` w ) = (/) ) <-> ( ( ph /\ X C_ (/) ) -> ( A ` (/) ) = (/) ) ) )
9		sseq2 3203	. . . . . . . . 9 |- ( w = k -> ( X C_ w <-> X C_ k ) )
10	9	anbi2d 464	. . . . . . . 8 |- ( w = k -> ( ( ph /\ X C_ w ) <-> ( ph /\ X C_ k ) ) )
11		fveqeq2 5563	. . . . . . . 8 |- ( w = k -> ( ( A ` w ) = (/) <-> ( A ` k ) = (/) ) )
12	10, 11	imbi12d 234	. . . . . . 7 |- ( w = k -> ( ( ( ph /\ X C_ w ) -> ( A ` w ) = (/) ) <-> ( ( ph /\ X C_ k ) -> ( A ` k ) = (/) ) ) )
13		sseq2 3203	. . . . . . . . 9 |- ( w = suc k -> ( X C_ w <-> X C_ suc k ) )
14	13	anbi2d 464	. . . . . . . 8 |- ( w = suc k -> ( ( ph /\ X C_ w ) <-> ( ph /\ X C_ suc k ) ) )
15		fveqeq2 5563	. . . . . . . 8 |- ( w = suc k -> ( ( A ` w ) = (/) <-> ( A ` suc k ) = (/) ) )
16	14, 15	imbi12d 234	. . . . . . 7 |- ( w = suc k -> ( ( ( ph /\ X C_ w ) -> ( A ` w ) = (/) ) <-> ( ( ph /\ X C_ suc k ) -> ( A ` suc k ) = (/) ) ) )
17		sseq2 3203	. . . . . . . . 9 |- ( w = n -> ( X C_ w <-> X C_ n ) )
18	17	anbi2d 464	. . . . . . . 8 |- ( w = n -> ( ( ph /\ X C_ w ) <-> ( ph /\ X C_ n ) ) )
19		fveqeq2 5563	. . . . . . . 8 |- ( w = n -> ( ( A ` w ) = (/) <-> ( A ` n ) = (/) ) )
20	18, 19	imbi12d 234	. . . . . . 7 |- ( w = n -> ( ( ( ph /\ X C_ w ) -> ( A ` w ) = (/) ) <-> ( ( ph /\ X C_ n ) -> ( A ` n ) = (/) ) ) )
21		ss0 3487	. . . . . . . . . 10 |- ( X C_ (/) -> X = (/) )
22	21	adantl 277	. . . . . . . . 9 |- ( ( ph /\ X C_ (/) ) -> X = (/) )
23	22	fveq2d 5558	. . . . . . . 8 |- ( ( ph /\ X C_ (/) ) -> ( A ` X ) = ( A ` (/) ) )
24		nninfninc.z	. . . . . . . . 9 |- ( ph -> ( A ` X ) = (/) )
25	24	adantr 276	. . . . . . . 8 |- ( ( ph /\ X C_ (/) ) -> ( A ` X ) = (/) )
26	23, 25	eqtr3d 2228	. . . . . . 7 |- ( ( ph /\ X C_ (/) ) -> ( A ` (/) ) = (/) )
27		suceq 4433	. . . . . . . . . . . . . 14 |- ( i = k -> suc i = suc k )
28	27	fveq2d 5558	. . . . . . . . . . . . 13 |- ( i = k -> ( A ` suc i ) = ( A ` suc k ) )
29		fveq2 5554	. . . . . . . . . . . . 13 |- ( i = k -> ( A ` i ) = ( A ` k ) )
30	28, 29	sseq12d 3210	. . . . . . . . . . . 12 |- ( i = k -> ( ( A ` suc i ) C_ ( A ` i ) <-> ( A ` suc k ) C_ ( A ` k ) ) )
31		simplrl 535	. . . . . . . . . . . . 13 |- ( ( ( ( k e. om /\ ( ( ph /\ X C_ k ) -> ( A ` k ) = (/) ) ) /\ ( ph /\ X C_ suc k ) ) /\ X e. suc k ) -> ph )
32		nninfninc.a	. . . . . . . . . . . . . . 15 |- ( ph -> A e. ℕ∞ )
33		fveq1 5553	. . . . . . . . . . . . . . . . . 18 |- ( f = A -> ( f ` suc i ) = ( A ` suc i ) )
34		fveq1 5553	. . . . . . . . . . . . . . . . . 18 |- ( f = A -> ( f ` i ) = ( A ` i ) )
35	33, 34	sseq12d 3210	. . . . . . . . . . . . . . . . 17 |- ( f = A -> ( ( f ` suc i ) C_ ( f ` i ) <-> ( A ` suc i ) C_ ( A ` i ) ) )
36	35	ralbidv 2494	. . . . . . . . . . . . . . . 16 |- ( f = A -> ( A. i e. om ( f ` suc i ) C_ ( f ` i ) <-> A. i e. om ( A ` suc i ) C_ ( A ` i ) ) )
37		df-nninf 7179	. . . . . . . . . . . . . . . 16 |- ℕ∞ = { f e. ( 2o ^m om ) | A. i e. om ( f ` suc i ) C_ ( f ` i ) }
38	36, 37	elrab2 2919	. . . . . . . . . . . . . . 15 |- ( A e. ℕ∞ <-> ( A e. ( 2o ^m om ) /\ A. i e. om ( A ` suc i ) C_ ( A ` i ) ) )
39	32, 38	sylib 122	. . . . . . . . . . . . . 14 |- ( ph -> ( A e. ( 2o ^m om ) /\ A. i e. om ( A ` suc i ) C_ ( A ` i ) ) )
40	39	simprd 114	. . . . . . . . . . . . 13 |- ( ph -> A. i e. om ( A ` suc i ) C_ ( A ` i ) )
41	31, 40	syl 14	. . . . . . . . . . . 12 |- ( ( ( ( k e. om /\ ( ( ph /\ X C_ k ) -> ( A ` k ) = (/) ) ) /\ ( ph /\ X C_ suc k ) ) /\ X e. suc k ) -> A. i e. om ( A ` suc i ) C_ ( A ` i ) )
42		simpll 527	. . . . . . . . . . . . 13 |- ( ( ( k e. om /\ ( ( ph /\ X C_ k ) -> ( A ` k ) = (/) ) ) /\ ( ph /\ X C_ suc k ) ) -> k e. om )
43	42	adantr 276	. . . . . . . . . . . 12 |- ( ( ( ( k e. om /\ ( ( ph /\ X C_ k ) -> ( A ` k ) = (/) ) ) /\ ( ph /\ X C_ suc k ) ) /\ X e. suc k ) -> k e. om )
44	30, 41, 43	rspcdva 2869	. . . . . . . . . . 11 |- ( ( ( ( k e. om /\ ( ( ph /\ X C_ k ) -> ( A ` k ) = (/) ) ) /\ ( ph /\ X C_ suc k ) ) /\ X e. suc k ) -> ( A ` suc k ) C_ ( A ` k ) )
45		simpr 110	. . . . . . . . . . . . 13 |- ( ( ( ( k e. om /\ ( ( ph /\ X C_ k ) -> ( A ` k ) = (/) ) ) /\ ( ph /\ X C_ suc k ) ) /\ X e. suc k ) -> X e. suc k )
46		nninfninc.x	. . . . . . . . . . . . . . . 16 |- ( ph -> X e. om )
47	46	ad2antrl 490	. . . . . . . . . . . . . . 15 |- ( ( ( k e. om /\ ( ( ph /\ X C_ k ) -> ( A ` k ) = (/) ) ) /\ ( ph /\ X C_ suc k ) ) -> X e. om )
48	47	adantr 276	. . . . . . . . . . . . . 14 |- ( ( ( ( k e. om /\ ( ( ph /\ X C_ k ) -> ( A ` k ) = (/) ) ) /\ ( ph /\ X C_ suc k ) ) /\ X e. suc k ) -> X e. om )
49		nnsssuc 6555	. . . . . . . . . . . . . 14 |- ( ( X e. om /\ k e. om ) -> ( X C_ k <-> X e. suc k ) )
50	48, 43, 49	syl2anc 411	. . . . . . . . . . . . 13 |- ( ( ( ( k e. om /\ ( ( ph /\ X C_ k ) -> ( A ` k ) = (/) ) ) /\ ( ph /\ X C_ suc k ) ) /\ X e. suc k ) -> ( X C_ k <-> X e. suc k ) )
51	45, 50	mpbird 167	. . . . . . . . . . . 12 |- ( ( ( ( k e. om /\ ( ( ph /\ X C_ k ) -> ( A ` k ) = (/) ) ) /\ ( ph /\ X C_ suc k ) ) /\ X e. suc k ) -> X C_ k )
52		simpllr 534	. . . . . . . . . . . 12 |- ( ( ( ( k e. om /\ ( ( ph /\ X C_ k ) -> ( A ` k ) = (/) ) ) /\ ( ph /\ X C_ suc k ) ) /\ X e. suc k ) -> ( ( ph /\ X C_ k ) -> ( A ` k ) = (/) ) )
53	31, 51, 52	mp2and 433	. . . . . . . . . . 11 |- ( ( ( ( k e. om /\ ( ( ph /\ X C_ k ) -> ( A ` k ) = (/) ) ) /\ ( ph /\ X C_ suc k ) ) /\ X e. suc k ) -> ( A ` k ) = (/) )
54	44, 53	sseqtrd 3217	. . . . . . . . . 10 |- ( ( ( ( k e. om /\ ( ( ph /\ X C_ k ) -> ( A ` k ) = (/) ) ) /\ ( ph /\ X C_ suc k ) ) /\ X e. suc k ) -> ( A ` suc k ) C_ (/) )
55		ss0 3487	. . . . . . . . . 10 |- ( ( A ` suc k ) C_ (/) -> ( A ` suc k ) = (/) )
56	54, 55	syl 14	. . . . . . . . 9 |- ( ( ( ( k e. om /\ ( ( ph /\ X C_ k ) -> ( A ` k ) = (/) ) ) /\ ( ph /\ X C_ suc k ) ) /\ X e. suc k ) -> ( A ` suc k ) = (/) )
57		simpr 110	. . . . . . . . . . 11 |- ( ( ( ( k e. om /\ ( ( ph /\ X C_ k ) -> ( A ` k ) = (/) ) ) /\ ( ph /\ X C_ suc k ) ) /\ X = suc k ) -> X = suc k )
58	57	fveq2d 5558	. . . . . . . . . 10 |- ( ( ( ( k e. om /\ ( ( ph /\ X C_ k ) -> ( A ` k ) = (/) ) ) /\ ( ph /\ X C_ suc k ) ) /\ X = suc k ) -> ( A ` X ) = ( A ` suc k ) )
59		simplrl 535	. . . . . . . . . . 11 |- ( ( ( ( k e. om /\ ( ( ph /\ X C_ k ) -> ( A ` k ) = (/) ) ) /\ ( ph /\ X C_ suc k ) ) /\ X = suc k ) -> ph )
60	59, 24	syl 14	. . . . . . . . . 10 |- ( ( ( ( k e. om /\ ( ( ph /\ X C_ k ) -> ( A ` k ) = (/) ) ) /\ ( ph /\ X C_ suc k ) ) /\ X = suc k ) -> ( A ` X ) = (/) )
61	58, 60	eqtr3d 2228	. . . . . . . . 9 |- ( ( ( ( k e. om /\ ( ( ph /\ X C_ k ) -> ( A ` k ) = (/) ) ) /\ ( ph /\ X C_ suc k ) ) /\ X = suc k ) -> ( A ` suc k ) = (/) )
62		simprr 531	. . . . . . . . . . 11 |- ( ( ( k e. om /\ ( ( ph /\ X C_ k ) -> ( A ` k ) = (/) ) ) /\ ( ph /\ X C_ suc k ) ) -> X C_ suc k )
63		peano2 4627	. . . . . . . . . . . . 13 |- ( k e. om -> suc k e. om )
64	42, 63	syl 14	. . . . . . . . . . . 12 |- ( ( ( k e. om /\ ( ( ph /\ X C_ k ) -> ( A ` k ) = (/) ) ) /\ ( ph /\ X C_ suc k ) ) -> suc k e. om )
65		nnsssuc 6555	. . . . . . . . . . . 12 |- ( ( X e. om /\ suc k e. om ) -> ( X C_ suc k <-> X e. suc suc k ) )
66	47, 64, 65	syl2anc 411	. . . . . . . . . . 11 |- ( ( ( k e. om /\ ( ( ph /\ X C_ k ) -> ( A ` k ) = (/) ) ) /\ ( ph /\ X C_ suc k ) ) -> ( X C_ suc k <-> X e. suc suc k ) )
67	62, 66	mpbid 147	. . . . . . . . . 10 |- ( ( ( k e. om /\ ( ( ph /\ X C_ k ) -> ( A ` k ) = (/) ) ) /\ ( ph /\ X C_ suc k ) ) -> X e. suc suc k )
68		elsuci 4434	. . . . . . . . . 10 |- ( X e. suc suc k -> ( X e. suc k \/ X = suc k ) )
69	67, 68	syl 14	. . . . . . . . 9 |- ( ( ( k e. om /\ ( ( ph /\ X C_ k ) -> ( A ` k ) = (/) ) ) /\ ( ph /\ X C_ suc k ) ) -> ( X e. suc k \/ X = suc k ) )
70	56, 61, 69	mpjaodan 799	. . . . . . . 8 |- ( ( ( k e. om /\ ( ( ph /\ X C_ k ) -> ( A ` k ) = (/) ) ) /\ ( ph /\ X C_ suc k ) ) -> ( A ` suc k ) = (/) )
71	70	exp31 364	. . . . . . 7 |- ( k e. om -> ( ( ( ph /\ X C_ k ) -> ( A ` k ) = (/) ) -> ( ( ph /\ X C_ suc k ) -> ( A ` suc k ) = (/) ) ) )
72	8, 12, 16, 20, 26, 71	finds 4632	. . . . . 6 |- ( n e. om -> ( ( ph /\ X C_ n ) -> ( A ` n ) = (/) ) )
73	72	com12 30	. . . . 5 |- ( ( ph /\ X C_ n ) -> ( n e. om -> ( A ` n ) = (/) ) )
74	73	impancom 260	. . . 4 |- ( ( ph /\ n e. om ) -> ( X C_ n -> ( A ` n ) = (/) ) )
75	74	ralrimiva 2567	. . 3 |- ( ph -> A. n e. om ( X C_ n -> ( A ` n ) = (/) ) )
76		nninfninc.y	. . 3 |- ( ph -> Y e. om )
77	4, 75, 76	rspcdva 2869	. 2 |- ( ph -> ( X C_ Y -> ( A ` Y ) = (/) ) )
78	1, 77	mpd 13	1 |- ( ph -> ( A ` Y ) = (/) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709   = wceq 1364   e. wcel 2164   A.wral 2472   C_ wss 3153   (/)c0 3446   suc csuc 4396   omcom 4622   ` cfv 5254  (class class class)co 5918   2oc2o 6463   ^m cmap 6702  ℕ_∞xnninf 7178

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-br 4030  df-tr 4128  df-iord 4397  df-on 4399  df-suc 4402  df-iom 4623  df-iota 5215  df-fv 5262  df-nninf 7179

This theorem is referenced by:  nninfctlemfo  12177