Theorem frecfzennn 10382

Description: The cardinality of a finite set of sequential integers. (See frec2uz0d 10355 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 5861	. . 3 |- ( n = 0 -> ( 1 ... n ) = ( 1 ... 0 ) )
2		fveq2 5496	. . 3 |- ( n = 0 -> ( `' G ` n ) = ( `' G ` 0 ) )
3	1, 2	breq12d 4002	. 2 |- ( n = 0 -> ( ( 1 ... n ) ~~ ( `' G ` n ) <-> ( 1 ... 0 ) ~~ ( `' G ` 0 ) ) )
4		oveq2 5861	. . 3 |- ( n = m -> ( 1 ... n ) = ( 1 ... m ) )
5		fveq2 5496	. . 3 |- ( n = m -> ( `' G ` n ) = ( `' G ` m ) )
6	4, 5	breq12d 4002	. 2 |- ( n = m -> ( ( 1 ... n ) ~~ ( `' G ` n ) <-> ( 1 ... m ) ~~ ( `' G ` m ) ) )
7		oveq2 5861	. . 3 |- ( n = ( m + 1 ) -> ( 1 ... n ) = ( 1 ... ( m + 1 ) ) )
8		fveq2 5496	. . 3 |- ( n = ( m + 1 ) -> ( `' G ` n ) = ( `' G ` ( m + 1 ) ) )
9	7, 8	breq12d 4002	. 2 |- ( n = ( m + 1 ) -> ( ( 1 ... n ) ~~ ( `' G ` n ) <-> ( 1 ... ( m + 1 ) ) ~~ ( `' G ` ( m + 1 ) ) ) )
10		oveq2 5861	. . 3 |- ( n = N -> ( 1 ... n ) = ( 1 ... N ) )
11		fveq2 5496	. . 3 |- ( n = N -> ( `' G ` n ) = ( `' G ` N ) )
12	10, 11	breq12d 4002	. 2 |- ( n = N -> ( ( 1 ... n ) ~~ ( `' G ` n ) <-> ( 1 ... N ) ~~ ( `' G ` N ) ) )
13		0ex 4116	. . . 4 |- (/) e. _V
14	13	enref 6743	. . 3 |- (/) ~~ (/)
15		fz10 10002	. . 3 |- ( 1 ... 0 ) = (/)
16		0zd 9224	. . . . . . 7 |- ( T. -> 0 e. ZZ )
17		frecfzennn.1	. . . . . . 7 |- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )
18	16, 17	frec2uzf1od 10362	. . . . . 6 |- ( T. -> G : om -1-1-onto-> ( ZZ>= ` 0 ) )
19	18	mptru 1357	. . . . 5 |- G : om -1-1-onto-> ( ZZ>= ` 0 )
20		peano1 4578	. . . . 5 |- (/) e. om
21	19, 20	pm3.2i 270	. . . 4 |- ( G : om -1-1-onto-> ( ZZ>= ` 0 ) /\ (/) e. om )
22	16, 17	frec2uz0d 10355	. . . . 5 |- ( T. -> ( G ` (/) ) = 0 )
23	22	mptru 1357	. . . 4 |- ( G ` (/) ) = 0
24		f1ocnvfv 5758	. . . 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 4019	. 2 |- ( 1 ... 0 ) ~~ ( `' G ` 0 )
27		simpr 109	. . . . 5 |- ( ( m e. NN0 /\ ( 1 ... m ) ~~ ( `' G ` m ) ) -> ( 1 ... m ) ~~ ( `' G ` m ) )
28		peano2nn0 9175	. . . . . . 7 |- ( m e. NN0 -> ( m + 1 ) e. NN0 )
29		zex 9221	. . . . . . . . . . . . . . 15 |- ZZ e. _V
30	29	mptex 5722	. . . . . . . . . . . . . 14 |- ( x e. ZZ |-> ( x + 1 ) ) e. _V
31		vex 2733	. . . . . . . . . . . . . 14 |- z e. _V
32	30, 31	fvex 5516	. . . . . . . . . . . . 13 |- ( ( x e. ZZ |-> ( x + 1 ) ) ` z ) e. _V
33	32	ax-gen 1442	. . . . . . . . . . . 12 |- A. z ( ( x e. ZZ |-> ( x + 1 ) ) ` z ) e. _V
34		0z 9223	. . . . . . . . . . . 12 |- 0 e. ZZ
35		frecfnom 6380	. . . . . . . . . . . 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 424	. . . . . . . . . . 11 |- frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 ) Fn om
37	17	fneq1i 5292	. . . . . . . . . . 11 |- ( G Fn om <-> frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 ) Fn om )
38	36, 37	mpbir 145	. . . . . . . . . 10 |- G Fn om
39		omex 4577	. . . . . . . . . 10 |- om e. _V
40		fnex 5718	. . . . . . . . . 10 |- ( ( G Fn om /\ om e. _V ) -> G e. _V )
41	38, 39, 40	mp2an 424	. . . . . . . . 9 |- G e. _V
42	41	cnvex 5149	. . . . . . . 8 |- `' G e. _V
43		vex 2733	. . . . . . . 8 |- m e. _V
44	42, 43	fvex 5516	. . . . . . 7 |- ( `' G ` m ) e. _V
45		en2sn 6791	. . . . . . 7 |- ( ( ( m + 1 ) e. NN0 /\ ( `' G ` m ) e. _V ) -> { ( m + 1 ) } ~~ { ( `' G ` m ) } )
46	28, 44, 45	sylancl 411	. . . . . 6 |- ( m e. NN0 -> { ( m + 1 ) } ~~ { ( `' G ` m ) } )
47	46	adantr 274	. . . . 5 |- ( ( m e. NN0 /\ ( 1 ... m ) ~~ ( `' G ` m ) ) -> { ( m + 1 ) } ~~ { ( `' G ` m ) } )
48		fzp1disj 10036	. . . . . 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 5760	. . . . . . . . . 10 |- ( ( G : om -1-1-onto-> ( ZZ>= ` 0 ) /\ m e. ( ZZ>= ` 0 ) ) -> ( `' G ` m ) e. om )
51	19, 50	mpan 422	. . . . . . . . 9 |- ( m e. ( ZZ>= ` 0 ) -> ( `' G ` m ) e. om )
52		nn0uz 9521	. . . . . . . . 9 |- NN0 = ( ZZ>= ` 0 )
53	51, 52	eleq2s 2265	. . . . . . . 8 |- ( m e. NN0 -> ( `' G ` m ) e. om )
54		nnord 4596	. . . . . . . 8 |- ( ( `' G ` m ) e. om -> Ord ( `' G ` m ) )
55		ordirr 4526	. . . . . . . 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 274	. . . . . 6 |- ( ( m e. NN0 /\ ( 1 ... m ) ~~ ( `' G ` m ) ) -> -. ( `' G ` m ) e. ( `' G ` m ) )
58		disjsn 3645	. . . . . 6 |- ( ( ( `' G ` m ) i^i { ( `' G ` m ) } ) = (/) <-> -. ( `' G ` m ) e. ( `' G ` m ) )
59	57, 58	sylibr 133	. . . . 5 |- ( ( m e. NN0 /\ ( 1 ... m ) ~~ ( `' G ` m ) ) -> ( ( `' G ` m ) i^i { ( `' G ` m ) } ) = (/) )
60		unen 6794	. . . . 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 1234	. . . 4 |- ( ( m e. NN0 /\ ( 1 ... m ) ~~ ( `' G ` m ) ) -> ( ( 1 ... m ) u. { ( m + 1 ) } ) ~~ ( ( `' G ` m ) u. { ( `' G ` m ) } ) )
62		1z 9238	. . . . . 6 |- 1 e. ZZ
63		1m1e0 8947	. . . . . . . . . 10 |- ( 1 - 1 ) = 0
64	63	fveq2i 5499	. . . . . . . . 9 |- ( ZZ>= ` ( 1 - 1 ) ) = ( ZZ>= ` 0 )
65	52, 64	eqtr4i 2194	. . . . . . . 8 |- NN0 = ( ZZ>= ` ( 1 - 1 ) )
66	65	eleq2i 2237	. . . . . . 7 |- ( m e. NN0 <-> m e. ( ZZ>= ` ( 1 - 1 ) ) )
67	66	biimpi 119	. . . . . 6 |- ( m e. NN0 -> m e. ( ZZ>= ` ( 1 - 1 ) ) )
68		fzsuc2 10035	. . . . . 6 |- ( ( 1 e. ZZ /\ m e. ( ZZ>= ` ( 1 - 1 ) ) ) -> ( 1 ... ( m + 1 ) ) = ( ( 1 ... m ) u. { ( m + 1 ) } ) )
69	62, 67, 68	sylancr 412	. . . . 5 |- ( m e. NN0 -> ( 1 ... ( m + 1 ) ) = ( ( 1 ... m ) u. { ( m + 1 ) } ) )
70	69	adantr 274	. . . 4 |- ( ( m e. NN0 /\ ( 1 ... m ) ~~ ( `' G ` m ) ) -> ( 1 ... ( m + 1 ) ) = ( ( 1 ... m ) u. { ( m + 1 ) } ) )
71		peano2 4579	. . . . . . . . 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 310	. . . . . . 7 |- ( m e. NN0 -> ( G : om -1-1-onto-> ( ZZ>= ` 0 ) /\ suc ( `' G ` m ) e. om ) )
74		0zd 9224	. . . . . . . . . 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 10357	. . . . . . . . 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 2237	. . . . . . . . . . 11 |- ( m e. NN0 <-> m e. ( ZZ>= ` 0 ) )
79	78	biimpi 119	. . . . . . . . . 10 |- ( m e. NN0 -> m e. ( ZZ>= ` 0 ) )
80		f1ocnvfv2 5757	. . . . . . . . . 10 |- ( ( G : om -1-1-onto-> ( ZZ>= ` 0 ) /\ m e. ( ZZ>= ` 0 ) ) -> ( G ` ( `' G ` m ) ) = m )
81	19, 79, 80	sylancr 412	. . . . . . . . 9 |- ( m e. NN0 -> ( G ` ( `' G ` m ) ) = m )
82	81	oveq1d 5868	. . . . . . . 8 |- ( m e. NN0 -> ( ( G ` ( `' G ` m ) ) + 1 ) = ( m + 1 ) )
83	77, 82	eqtrd 2203	. . . . . . 7 |- ( m e. NN0 -> ( G ` suc ( `' G ` m ) ) = ( m + 1 ) )
84		f1ocnvfv 5758	. . . . . . 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 274	. . . . 5 |- ( ( m e. NN0 /\ ( 1 ... m ) ~~ ( `' G ` m ) ) -> ( `' G ` ( m + 1 ) ) = suc ( `' G ` m ) )
87		df-suc 4356	. . . . 5 |- suc ( `' G ` m ) = ( ( `' G ` m ) u. { ( `' G ` m ) } )
88	86, 87	eqtrdi 2219	. . . 4 |- ( ( m e. NN0 /\ ( 1 ... m ) ~~ ( `' G ` m ) ) -> ( `' G ` ( m + 1 ) ) = ( ( `' G ` m ) u. { ( `' G ` m ) } ) )
89	61, 70, 88	3brtr4d 4021	. . 3 |- ( ( m e. NN0 /\ ( 1 ... m ) ~~ ( `' G ` m ) ) -> ( 1 ... ( m + 1 ) ) ~~ ( `' G ` ( m + 1 ) ) )
90	89	ex 114	. 2 |- ( m e. NN0 -> ( ( 1 ... m ) ~~ ( `' G ` m ) -> ( 1 ... ( m + 1 ) ) ~~ ( `' G ` ( m + 1 ) ) ) )
91	3, 6, 9, 12, 26, 90	nn0ind 9326	1 |- ( N e. NN0 -> ( 1 ... N ) ~~ ( `' G ` N ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   A.wal 1346   = wceq 1348   T. wtru 1349   e. wcel 2141   _Vcvv 2730   u. cun 3119   i^i cin 3120   (/)c0 3414   {csn 3583   class class class wbr 3989   |-> cmpt 4050   Ord word 4347   suc csuc 4350   omcom 4574   `'ccnv 4610   Fn wfn 5193   -1-1-onto->wf1o 5197   ` cfv 5198  (class class class)co 5853  freccfrec 6369   ~~ cen 6716   0cc0 7774   1c1 7775   + caddc 7777   - cmin 8090   NN0cn0 9135   ZZcz 9212   ZZ>=cuz 9487   ...cfz 9965

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-addcom 7874  ax-addass 7876  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-0id 7882  ax-rnegex 7883  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890

This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-recs 6284  df-frec 6370  df-1o 6395  df-er 6513  df-en 6719  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-inn 8879  df-n0 9136  df-z 9213  df-uz 9488  df-fz 9966

This theorem is referenced by:  frecfzen2  10383  hashfz1  10717