Theorem frecfzennn 10643

Description: The cardinality of a finite set of sequential integers. (See frec2uz0d 10616 for a description of the hypothesis.) (Contributed by Jim Kingdon, 18-May-2020.)

Hypothesis

Ref	Expression
frecfzennn.1	|- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )

Assertion

Ref	Expression
frecfzennn	|- ( N e. NN0 -> ( 1 ... N ) ~~ ( `' G ` N ) )

Proof of Theorem frecfzennn

Dummy variables m n z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 6008	. . 3 |- ( n = 0 -> ( 1 ... n ) = ( 1 ... 0 ) )
2		fveq2 5626	. . 3 |- ( n = 0 -> ( `' G ` n ) = ( `' G ` 0 ) )
3	1, 2	breq12d 4095	. 2 |- ( n = 0 -> ( ( 1 ... n ) ~~ ( `' G ` n ) <-> ( 1 ... 0 ) ~~ ( `' G ` 0 ) ) )
4		oveq2 6008	. . 3 |- ( n = m -> ( 1 ... n ) = ( 1 ... m ) )
5		fveq2 5626	. . 3 |- ( n = m -> ( `' G ` n ) = ( `' G ` m ) )
6	4, 5	breq12d 4095	. 2 |- ( n = m -> ( ( 1 ... n ) ~~ ( `' G ` n ) <-> ( 1 ... m ) ~~ ( `' G ` m ) ) )
7		oveq2 6008	. . 3 |- ( n = ( m + 1 ) -> ( 1 ... n ) = ( 1 ... ( m + 1 ) ) )
8		fveq2 5626	. . 3 |- ( n = ( m + 1 ) -> ( `' G ` n ) = ( `' G ` ( m + 1 ) ) )
9	7, 8	breq12d 4095	. 2 |- ( n = ( m + 1 ) -> ( ( 1 ... n ) ~~ ( `' G ` n ) <-> ( 1 ... ( m + 1 ) ) ~~ ( `' G ` ( m + 1 ) ) ) )
10		oveq2 6008	. . 3 |- ( n = N -> ( 1 ... n ) = ( 1 ... N ) )
11		fveq2 5626	. . 3 |- ( n = N -> ( `' G ` n ) = ( `' G ` N ) )
12	10, 11	breq12d 4095	. 2 |- ( n = N -> ( ( 1 ... n ) ~~ ( `' G ` n ) <-> ( 1 ... N ) ~~ ( `' G ` N ) ) )
13		0ex 4210	. . . 4 |- (/) e. _V
14	13	enref 6914	. . 3 |- (/) ~~ (/)
15		fz10 10238	. . 3 |- ( 1 ... 0 ) = (/)
16		0zd 9454	. . . . . . 7 |- ( T. -> 0 e. ZZ )
17		frecfzennn.1	. . . . . . 7 |- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )
18	16, 17	frec2uzf1od 10623	. . . . . 6 |- ( T. -> G : om -1-1-onto-> ( ZZ>= ` 0 ) )
19	18	mptru 1404	. . . . 5 |- G : om -1-1-onto-> ( ZZ>= ` 0 )
20		peano1 4685	. . . . 5 |- (/) e. om
21	19, 20	pm3.2i 272	. . . 4 |- ( G : om -1-1-onto-> ( ZZ>= ` 0 ) /\ (/) e. om )
22	16, 17	frec2uz0d 10616	. . . . 5 |- ( T. -> ( G ` (/) ) = 0 )
23	22	mptru 1404	. . . 4 |- ( G ` (/) ) = 0
24		f1ocnvfv 5902	. . . 4 |- ( ( G : om -1-1-onto-> ( ZZ>= ` 0 ) /\ (/) e. om ) -> ( ( G ` (/) ) = 0 -> ( `' G ` 0 ) = (/) ) )
25	21, 23, 24	mp2 16	. . 3 |- ( `' G ` 0 ) = (/)
26	14, 15, 25	3brtr4i 4112	. 2 |- ( 1 ... 0 ) ~~ ( `' G ` 0 )
27		simpr 110	. . . . 5 |- ( ( m e. NN0 /\ ( 1 ... m ) ~~ ( `' G ` m ) ) -> ( 1 ... m ) ~~ ( `' G ` m ) )
28		peano2nn0 9405	. . . . . . 7 |- ( m e. NN0 -> ( m + 1 ) e. NN0 )
29		zex 9451	. . . . . . . . . . . . . . 15 |- ZZ e. _V
30	29	mptex 5864	. . . . . . . . . . . . . 14 |- ( x e. ZZ |-> ( x + 1 ) ) e. _V
31		vex 2802	. . . . . . . . . . . . . 14 |- z e. _V
32	30, 31	fvex 5646	. . . . . . . . . . . . 13 |- ( ( x e. ZZ |-> ( x + 1 ) ) ` z ) e. _V
33	32	ax-gen 1495	. . . . . . . . . . . 12 |- A. z ( ( x e. ZZ |-> ( x + 1 ) ) ` z ) e. _V
34		0z 9453	. . . . . . . . . . . 12 |- 0 e. ZZ
35		frecfnom 6545	. . . . . . . . . . . 12 |- ( ( A. z ( ( x e. ZZ |-> ( x + 1 ) ) ` z ) e. _V /\ 0 e. ZZ ) -> frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 ) Fn om )
36	33, 34, 35	mp2an 426	. . . . . . . . . . 11 |- frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 ) Fn om
37	17	fneq1i 5414	. . . . . . . . . . 11 |- ( G Fn om <-> frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 ) Fn om )
38	36, 37	mpbir 146	. . . . . . . . . 10 |- G Fn om
39		omex 4684	. . . . . . . . . 10 |- om e. _V
40		fnex 5860	. . . . . . . . . 10 |- ( ( G Fn om /\ om e. _V ) -> G e. _V )
41	38, 39, 40	mp2an 426	. . . . . . . . 9 |- G e. _V
42	41	cnvex 5266	. . . . . . . 8 |- `' G e. _V
43		vex 2802	. . . . . . . 8 |- m e. _V
44	42, 43	fvex 5646	. . . . . . 7 |- ( `' G ` m ) e. _V
45		en2sn 6964	. . . . . . 7 |- ( ( ( m + 1 ) e. NN0 /\ ( `' G ` m ) e. _V ) -> { ( m + 1 ) } ~~ { ( `' G ` m ) } )
46	28, 44, 45	sylancl 413	. . . . . 6 |- ( m e. NN0 -> { ( m + 1 ) } ~~ { ( `' G ` m ) } )
47	46	adantr 276	. . . . 5 |- ( ( m e. NN0 /\ ( 1 ... m ) ~~ ( `' G ` m ) ) -> { ( m + 1 ) } ~~ { ( `' G ` m ) } )
48		fzp1disj 10272	. . . . . 6 |- ( ( 1 ... m ) i^i { ( m + 1 ) } ) = (/)
49	48	a1i 9	. . . . 5 |- ( ( m e. NN0 /\ ( 1 ... m ) ~~ ( `' G ` m ) ) -> ( ( 1 ... m ) i^i { ( m + 1 ) } ) = (/) )
50		f1ocnvdm 5904	. . . . . . . . . 10 |- ( ( G : om -1-1-onto-> ( ZZ>= ` 0 ) /\ m e. ( ZZ>= ` 0 ) ) -> ( `' G ` m ) e. om )
51	19, 50	mpan 424	. . . . . . . . 9 |- ( m e. ( ZZ>= ` 0 ) -> ( `' G ` m ) e. om )
52		nn0uz 9753	. . . . . . . . 9 |- NN0 = ( ZZ>= ` 0 )
53	51, 52	eleq2s 2324	. . . . . . . 8 |- ( m e. NN0 -> ( `' G ` m ) e. om )
54		nnord 4703	. . . . . . . 8 |- ( ( `' G ` m ) e. om -> Ord ( `' G ` m ) )
55		ordirr 4633	. . . . . . . 8 |- ( Ord ( `' G ` m ) -> -. ( `' G ` m ) e. ( `' G ` m ) )
56	53, 54, 55	3syl 17	. . . . . . 7 |- ( m e. NN0 -> -. ( `' G ` m ) e. ( `' G ` m ) )
57	56	adantr 276	. . . . . 6 |- ( ( m e. NN0 /\ ( 1 ... m ) ~~ ( `' G ` m ) ) -> -. ( `' G ` m ) e. ( `' G ` m ) )
58		disjsn 3728	. . . . . 6 |- ( ( ( `' G ` m ) i^i { ( `' G ` m ) } ) = (/) <-> -. ( `' G ` m ) e. ( `' G ` m ) )
59	57, 58	sylibr 134	. . . . 5 |- ( ( m e. NN0 /\ ( 1 ... m ) ~~ ( `' G ` m ) ) -> ( ( `' G ` m ) i^i { ( `' G ` m ) } ) = (/) )
60		unen 6967	. . . . 5 |- ( ( ( ( 1 ... m ) ~~ ( `' G ` m ) /\ { ( m + 1 ) } ~~ { ( `' G ` m ) } ) /\ ( ( ( 1 ... m ) i^i { ( m + 1 ) } ) = (/) /\ ( ( `' G ` m ) i^i { ( `' G ` m ) } ) = (/) ) ) -> ( ( 1 ... m ) u. { ( m + 1 ) } ) ~~ ( ( `' G ` m ) u. { ( `' G ` m ) } ) )
61	27, 47, 49, 59, 60	syl22anc 1272	. . . 4 |- ( ( m e. NN0 /\ ( 1 ... m ) ~~ ( `' G ` m ) ) -> ( ( 1 ... m ) u. { ( m + 1 ) } ) ~~ ( ( `' G ` m ) u. { ( `' G ` m ) } ) )
62		1z 9468	. . . . . 6 |- 1 e. ZZ
63		1m1e0 9175	. . . . . . . . . 10 |- ( 1 - 1 ) = 0
64	63	fveq2i 5629	. . . . . . . . 9 |- ( ZZ>= ` ( 1 - 1 ) ) = ( ZZ>= ` 0 )
65	52, 64	eqtr4i 2253	. . . . . . . 8 |- NN0 = ( ZZ>= ` ( 1 - 1 ) )
66	65	eleq2i 2296	. . . . . . 7 |- ( m e. NN0 <-> m e. ( ZZ>= ` ( 1 - 1 ) ) )
67	66	biimpi 120	. . . . . 6 |- ( m e. NN0 -> m e. ( ZZ>= ` ( 1 - 1 ) ) )
68		fzsuc2 10271	. . . . . 6 |- ( ( 1 e. ZZ /\ m e. ( ZZ>= ` ( 1 - 1 ) ) ) -> ( 1 ... ( m + 1 ) ) = ( ( 1 ... m ) u. { ( m + 1 ) } ) )
69	62, 67, 68	sylancr 414	. . . . 5 |- ( m e. NN0 -> ( 1 ... ( m + 1 ) ) = ( ( 1 ... m ) u. { ( m + 1 ) } ) )
70	69	adantr 276	. . . 4 |- ( ( m e. NN0 /\ ( 1 ... m ) ~~ ( `' G ` m ) ) -> ( 1 ... ( m + 1 ) ) = ( ( 1 ... m ) u. { ( m + 1 ) } ) )
71		peano2 4686	. . . . . . . . 9 |- ( ( `' G ` m ) e. om -> suc ( `' G ` m ) e. om )
72	53, 71	syl 14	. . . . . . . 8 |- ( m e. NN0 -> suc ( `' G ` m ) e. om )
73	72, 19	jctil 312	. . . . . . 7 |- ( m e. NN0 -> ( G : om -1-1-onto-> ( ZZ>= ` 0 ) /\ suc ( `' G ` m ) e. om ) )
74		0zd 9454	. . . . . . . . . 10 |- ( ( `' G ` m ) e. om -> 0 e. ZZ )
75		id 19	. . . . . . . . . 10 |- ( ( `' G ` m ) e. om -> ( `' G ` m ) e. om )
76	74, 17, 75	frec2uzsucd 10618	. . . . . . . . 9 |- ( ( `' G ` m ) e. om -> ( G ` suc ( `' G ` m ) ) = ( ( G ` ( `' G ` m ) ) + 1 ) )
77	53, 76	syl 14	. . . . . . . 8 |- ( m e. NN0 -> ( G ` suc ( `' G ` m ) ) = ( ( G ` ( `' G ` m ) ) + 1 ) )
78	52	eleq2i 2296	. . . . . . . . . . 11 |- ( m e. NN0 <-> m e. ( ZZ>= ` 0 ) )
79	78	biimpi 120	. . . . . . . . . 10 |- ( m e. NN0 -> m e. ( ZZ>= ` 0 ) )
80		f1ocnvfv2 5901	. . . . . . . . . 10 |- ( ( G : om -1-1-onto-> ( ZZ>= ` 0 ) /\ m e. ( ZZ>= ` 0 ) ) -> ( G ` ( `' G ` m ) ) = m )
81	19, 79, 80	sylancr 414	. . . . . . . . 9 |- ( m e. NN0 -> ( G ` ( `' G ` m ) ) = m )
82	81	oveq1d 6015	. . . . . . . 8 |- ( m e. NN0 -> ( ( G ` ( `' G ` m ) ) + 1 ) = ( m + 1 ) )
83	77, 82	eqtrd 2262	. . . . . . 7 |- ( m e. NN0 -> ( G ` suc ( `' G ` m ) ) = ( m + 1 ) )
84		f1ocnvfv 5902	. . . . . . 7 |- ( ( G : om -1-1-onto-> ( ZZ>= ` 0 ) /\ suc ( `' G ` m ) e. om ) -> ( ( G ` suc ( `' G ` m ) ) = ( m + 1 ) -> ( `' G ` ( m + 1 ) ) = suc ( `' G ` m ) ) )
85	73, 83, 84	sylc 62	. . . . . 6 |- ( m e. NN0 -> ( `' G ` ( m + 1 ) ) = suc ( `' G ` m ) )
86	85	adantr 276	. . . . 5 |- ( ( m e. NN0 /\ ( 1 ... m ) ~~ ( `' G ` m ) ) -> ( `' G ` ( m + 1 ) ) = suc ( `' G ` m ) )
87		df-suc 4461	. . . . 5 |- suc ( `' G ` m ) = ( ( `' G ` m ) u. { ( `' G ` m ) } )
88	86, 87	eqtrdi 2278	. . . 4 |- ( ( m e. NN0 /\ ( 1 ... m ) ~~ ( `' G ` m ) ) -> ( `' G ` ( m + 1 ) ) = ( ( `' G ` m ) u. { ( `' G ` m ) } ) )
89	61, 70, 88	3brtr4d 4114	. . 3 |- ( ( m e. NN0 /\ ( 1 ... m ) ~~ ( `' G ` m ) ) -> ( 1 ... ( m + 1 ) ) ~~ ( `' G ` ( m + 1 ) ) )
90	89	ex 115	. 2 |- ( m e. NN0 -> ( ( 1 ... m ) ~~ ( `' G ` m ) -> ( 1 ... ( m + 1 ) ) ~~ ( `' G ` ( m + 1 ) ) ) )
91	3, 6, 9, 12, 26, 90	nn0ind 9557	1 |- ( N e. NN0 -> ( 1 ... N ) ~~ ( `' G ` N ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   A.wal 1393   = wceq 1395   T. wtru 1396   e. wcel 2200   _Vcvv 2799   u. cun 3195   i^i cin 3196   (/)c0 3491   {csn 3666   class class class wbr 4082   |-> cmpt 4144   Ord word 4452   suc csuc 4455   omcom 4681   `'ccnv 4717   Fn wfn 5312   -1-1-onto->wf1o 5316   ` cfv 5317  (class class class)co 6000  freccfrec 6534   ~~ cen 6883   0cc0 7995   1c1 7996   + caddc 7998   - cmin 8313   NN0cn0 9365   ZZcz 9442   ZZ>=cuz 9718   ...cfz 10200

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-iinf 4679  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-addcom 8095  ax-addass 8097  ax-distr 8099  ax-i2m1 8100  ax-0lt1 8101  ax-0id 8103  ax-rnegex 8104  ax-cnre 8106  ax-pre-ltirr 8107  ax-pre-ltwlin 8108  ax-pre-lttrn 8109  ax-pre-apti 8110  ax-pre-ltadd 8111

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4383  df-iord 4456  df-on 4458  df-ilim 4459  df-suc 4461  df-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-recs 6449  df-frec 6535  df-1o 6560  df-er 6678  df-en 6886  df-pnf 8179  df-mnf 8180  df-xr 8181  df-ltxr 8182  df-le 8183  df-sub 8315  df-neg 8316  df-inn 9107  df-n0 9366  df-z 9443  df-uz 9719  df-fz 10201

This theorem is referenced by:  frecfzen2  10644  hashfz1  11000