Theorem frecfzennn 9203

Description: The cardinality of a finite set of sequential integers. (See frec2uz0d 9185 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 5520	. . 3 |- ( n = 0 -> ( 1 ... n ) = ( 1 ... 0 ) )
2		fveq2 5178	. . 3 |- ( n = 0 -> ( `' G ` n ) = ( `' G ` 0 ) )
3	1, 2	breq12d 3777	. 2 |- ( n = 0 -> ( ( 1 ... n ) ~~ ( `' G ` n ) <-> ( 1 ... 0 ) ~~ ( `' G ` 0 ) ) )
4		oveq2 5520	. . 3 |- ( n = m -> ( 1 ... n ) = ( 1 ... m ) )
5		fveq2 5178	. . 3 |- ( n = m -> ( `' G ` n ) = ( `' G ` m ) )
6	4, 5	breq12d 3777	. 2 |- ( n = m -> ( ( 1 ... n ) ~~ ( `' G ` n ) <-> ( 1 ... m ) ~~ ( `' G ` m ) ) )
7		oveq2 5520	. . 3 |- ( n = ( m + 1 ) -> ( 1 ... n ) = ( 1 ... ( m + 1 ) ) )
8		fveq2 5178	. . 3 |- ( n = ( m + 1 ) -> ( `' G ` n ) = ( `' G ` ( m + 1 ) ) )
9	7, 8	breq12d 3777	. 2 |- ( n = ( m + 1 ) -> ( ( 1 ... n ) ~~ ( `' G ` n ) <-> ( 1 ... ( m + 1 ) ) ~~ ( `' G ` ( m + 1 ) ) ) )
10		oveq2 5520	. . 3 |- ( n = N -> ( 1 ... n ) = ( 1 ... N ) )
11		fveq2 5178	. . 3 |- ( n = N -> ( `' G ` n ) = ( `' G ` N ) )
12	10, 11	breq12d 3777	. 2 |- ( n = N -> ( ( 1 ... n ) ~~ ( `' G ` n ) <-> ( 1 ... N ) ~~ ( `' G ` N ) ) )
13		0ex 3884	. . . 4 |- (/) e. _V
14	13	enref 6245	. . 3 |- (/) ~~ (/)
15		fz10 8910	. . 3 |- ( 1 ... 0 ) = (/)
16		0zd 8257	. . . . . . 7 |- ( T. -> 0 e. ZZ )
17		frecfzennn.1	. . . . . . 7 |- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )
18	16, 17	frec2uzf1od 9192	. . . . . 6 |- ( T. -> G : om -1-1-onto-> ( ZZ>= ` 0 ) )
19	18	trud 1252	. . . . 5 |- G : om -1-1-onto-> ( ZZ>= ` 0 )
20		peano1 4317	. . . . 5 |- (/) e. om
21	19, 20	pm3.2i 257	. . . 4 |- ( G : om -1-1-onto-> ( ZZ>= ` 0 ) /\ (/) e. om )
22	16, 17	frec2uz0d 9185	. . . . 5 |- ( T. -> ( G ` (/) ) = 0 )
23	22	trud 1252	. . . 4 |- ( G ` (/) ) = 0
24		f1ocnvfv 5419	. . . 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 3792	. 2 |- ( 1 ... 0 ) ~~ ( `' G ` 0 )
27		simpr 103	. . . . 5 |- ( ( m e. NN0 /\ ( 1 ... m ) ~~ ( `' G ` m ) ) -> ( 1 ... m ) ~~ ( `' G ` m ) )
28		peano2nn0 8222	. . . . . . 7 |- ( m e. NN0 -> ( m + 1 ) e. NN0 )
29		zex 8254	. . . . . . . . . . . . . . 15 |- ZZ e. _V
30	29	mptex 5387	. . . . . . . . . . . . . 14 |- ( x e. ZZ |-> ( x + 1 ) ) e. _V
31		vex 2560	. . . . . . . . . . . . . 14 |- z e. _V
32	30, 31	fvex 5195	. . . . . . . . . . . . 13 |- ( ( x e. ZZ |-> ( x + 1 ) ) ` z ) e. _V
33	32	ax-gen 1338	. . . . . . . . . . . 12 |- A. z ( ( x e. ZZ |-> ( x + 1 ) ) ` z ) e. _V
34		0z 8256	. . . . . . . . . . . 12 |- 0 e. ZZ
35		frecfnom 5986	. . . . . . . . . . . 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 402	. . . . . . . . . . 11 |- frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 ) Fn om
37	17	fneq1i 4993	. . . . . . . . . . 11 |- ( G Fn om <-> frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 ) Fn om )
38	36, 37	mpbir 134	. . . . . . . . . 10 |- G Fn om
39		omex 4316	. . . . . . . . . 10 |- om e. _V
40		fnex 5383	. . . . . . . . . 10 |- ( ( G Fn om /\ om e. _V ) -> G e. _V )
41	38, 39, 40	mp2an 402	. . . . . . . . 9 |- G e. _V
42	41	cnvex 4856	. . . . . . . 8 |- `' G e. _V
43		vex 2560	. . . . . . . 8 |- m e. _V
44	42, 43	fvex 5195	. . . . . . 7 |- ( `' G ` m ) e. _V
45		en2sn 6290	. . . . . . 7 |- ( ( ( m + 1 ) e. NN0 /\ ( `' G ` m ) e. _V ) -> { ( m + 1 ) } ~~ { ( `' G ` m ) } )
46	28, 44, 45	sylancl 392	. . . . . 6 |- ( m e. NN0 -> { ( m + 1 ) } ~~ { ( `' G ` m ) } )
47	46	adantr 261	. . . . 5 |- ( ( m e. NN0 /\ ( 1 ... m ) ~~ ( `' G ` m ) ) -> { ( m + 1 ) } ~~ { ( `' G ` m ) } )
48		fzp1disj 8942	. . . . . 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 5421	. . . . . . . . . 10 |- ( ( G : om -1-1-onto-> ( ZZ>= ` 0 ) /\ m e. ( ZZ>= ` 0 ) ) -> ( `' G ` m ) e. om )
51	19, 50	mpan 400	. . . . . . . . 9 |- ( m e. ( ZZ>= ` 0 ) -> ( `' G ` m ) e. om )
52		nn0uz 8507	. . . . . . . . 9 |- NN0 = ( ZZ>= ` 0 )
53	51, 52	eleq2s 2132	. . . . . . . 8 |- ( m e. NN0 -> ( `' G ` m ) e. om )
54		nnord 4334	. . . . . . . 8 |- ( ( `' G ` m ) e. om -> Ord ( `' G ` m ) )
55		ordirr 4267	. . . . . . . 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 261	. . . . . 6 |- ( ( m e. NN0 /\ ( 1 ... m ) ~~ ( `' G ` m ) ) -> -. ( `' G ` m ) e. ( `' G ` m ) )
58		disjsn 3432	. . . . . 6 |- ( ( ( `' G ` m ) i^i { ( `' G ` m ) } ) = (/) <-> -. ( `' G ` m ) e. ( `' G ` m ) )
59	57, 58	sylibr 137	. . . . 5 |- ( ( m e. NN0 /\ ( 1 ... m ) ~~ ( `' G ` m ) ) -> ( ( `' G ` m ) i^i { ( `' G ` m ) } ) = (/) )
60		unen 6293	. . . . 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 1136	. . . 4 |- ( ( m e. NN0 /\ ( 1 ... m ) ~~ ( `' G ` m ) ) -> ( ( 1 ... m ) u. { ( m + 1 ) } ) ~~ ( ( `' G ` m ) u. { ( `' G ` m ) } ) )
62		1z 8271	. . . . . 6 |- 1 e. ZZ
63		1m1e0 7984	. . . . . . . . . 10 |- ( 1 - 1 ) = 0
64	63	fveq2i 5181	. . . . . . . . 9 |- ( ZZ>= ` ( 1 - 1 ) ) = ( ZZ>= ` 0 )
65	52, 64	eqtr4i 2063	. . . . . . . 8 |- NN0 = ( ZZ>= ` ( 1 - 1 ) )
66	65	eleq2i 2104	. . . . . . 7 |- ( m e. NN0 <-> m e. ( ZZ>= ` ( 1 - 1 ) ) )
67	66	biimpi 113	. . . . . 6 |- ( m e. NN0 -> m e. ( ZZ>= ` ( 1 - 1 ) ) )
68		fzsuc2 8941	. . . . . 6 |- ( ( 1 e. ZZ /\ m e. ( ZZ>= ` ( 1 - 1 ) ) ) -> ( 1 ... ( m + 1 ) ) = ( ( 1 ... m ) u. { ( m + 1 ) } ) )
69	62, 67, 68	sylancr 393	. . . . 5 |- ( m e. NN0 -> ( 1 ... ( m + 1 ) ) = ( ( 1 ... m ) u. { ( m + 1 ) } ) )
70	69	adantr 261	. . . 4 |- ( ( m e. NN0 /\ ( 1 ... m ) ~~ ( `' G ` m ) ) -> ( 1 ... ( m + 1 ) ) = ( ( 1 ... m ) u. { ( m + 1 ) } ) )
71		peano2 4318	. . . . . . . . 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 295	. . . . . . 7 |- ( m e. NN0 -> ( G : om -1-1-onto-> ( ZZ>= ` 0 ) /\ suc ( `' G ` m ) e. om ) )
74		0zd 8257	. . . . . . . . . 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 9187	. . . . . . . . 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 2104	. . . . . . . . . . 11 |- ( m e. NN0 <-> m e. ( ZZ>= ` 0 ) )
79	78	biimpi 113	. . . . . . . . . 10 |- ( m e. NN0 -> m e. ( ZZ>= ` 0 ) )
80		f1ocnvfv2 5418	. . . . . . . . . 10 |- ( ( G : om -1-1-onto-> ( ZZ>= ` 0 ) /\ m e. ( ZZ>= ` 0 ) ) -> ( G ` ( `' G ` m ) ) = m )
81	19, 79, 80	sylancr 393	. . . . . . . . 9 |- ( m e. NN0 -> ( G ` ( `' G ` m ) ) = m )
82	81	oveq1d 5527	. . . . . . . 8 |- ( m e. NN0 -> ( ( G ` ( `' G ` m ) ) + 1 ) = ( m + 1 ) )
83	77, 82	eqtrd 2072	. . . . . . 7 |- ( m e. NN0 -> ( G ` suc ( `' G ` m ) ) = ( m + 1 ) )
84		f1ocnvfv 5419	. . . . . . 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 56	. . . . . 6 |- ( m e. NN0 -> ( `' G ` ( m + 1 ) ) = suc ( `' G ` m ) )
86	85	adantr 261	. . . . 5 |- ( ( m e. NN0 /\ ( 1 ... m ) ~~ ( `' G ` m ) ) -> ( `' G ` ( m + 1 ) ) = suc ( `' G ` m ) )
87		df-suc 4108	. . . . 5 |- suc ( `' G ` m ) = ( ( `' G ` m ) u. { ( `' G ` m ) } )
88	86, 87	syl6eq 2088	. . . 4 |- ( ( m e. NN0 /\ ( 1 ... m ) ~~ ( `' G ` m ) ) -> ( `' G ` ( m + 1 ) ) = ( ( `' G ` m ) u. { ( `' G ` m ) } ) )
89	61, 70, 88	3brtr4d 3794	. . 3 |- ( ( m e. NN0 /\ ( 1 ... m ) ~~ ( `' G ` m ) ) -> ( 1 ... ( m + 1 ) ) ~~ ( `' G ` ( m + 1 ) ) )
90	89	ex 108	. 2 |- ( m e. NN0 -> ( ( 1 ... m ) ~~ ( `' G ` m ) -> ( 1 ... ( m + 1 ) ) ~~ ( `' G ` ( m + 1 ) ) ) )
91	3, 6, 9, 12, 26, 90	nn0ind 8352	1 |- ( N e. NN0 -> ( 1 ... N ) ~~ ( `' G ` N ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 97   A.wal 1241   = wceq 1243   T. wtru 1244   e. wcel 1393   _Vcvv 2557   u. cun 2915   i^i cin 2916   (/)c0 3224   {csn 3375   class class class wbr 3764   |-> cmpt 3818   Ord word 4099   suc csuc 4102   omcom 4313   `'ccnv 4344   Fn wfn 4897   -1-1-onto->wf1o 4901   ` cfv 4902  (class class class)co 5512  freccfrec 5977   ~~ cen 6219   0cc0 6889   1c1 6890   + caddc 6892   - cmin 7182   NN0cn0 8181   ZZcz 8245   ZZ>=cuz 8473   ...cfz 8874

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311  ax-cnex 6975  ax-resscn 6976  ax-1cn 6977  ax-1re 6978  ax-icn 6979  ax-addcl 6980  ax-addrcl 6981  ax-mulcl 6982  ax-addcom 6984  ax-addass 6986  ax-distr 6988  ax-i2m1 6989  ax-0id 6992  ax-rnegex 6993  ax-cnre 6995  ax-pre-ltirr 6996  ax-pre-ltwlin 6997  ax-pre-lttrn 6998  ax-pre-apti 6999  ax-pre-ltadd 7000

This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-nel 2207  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-riota 5468  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-frec 5978  df-1o 6001  df-2o 6002  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-en 6222  df-ni 6402  df-pli 6403  df-mi 6404  df-lti 6405  df-plpq 6442  df-mpq 6443  df-enq 6445  df-nqqs 6446  df-plqqs 6447  df-mqqs 6448  df-1nqqs 6449  df-rq 6450  df-ltnqqs 6451  df-enq0 6522  df-nq0 6523  df-0nq0 6524  df-plq0 6525  df-mq0 6526  df-inp 6564  df-i1p 6565  df-iplp 6566  df-iltp 6568  df-enr 6811  df-nr 6812  df-ltr 6815  df-0r 6816  df-1r 6817  df-0 6896  df-1 6897  df-r 6899  df-lt 6902  df-pnf 7062  df-mnf 7063  df-xr 7064  df-ltxr 7065  df-le 7066  df-sub 7184  df-neg 7185  df-inn 7915  df-n0 8182  df-z 8246  df-uz 8474  df-fz 8875

This theorem is referenced by:  frecfzen2  9204