Theorem frecfzennn 10841

Description: The cardinality of a finite set of sequential integers. (See frec2uz0d 10814 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 6083	. . 3 |- ( n = 0 -> ( 1 ... n ) = ( 1 ... 0 ) )
2		fveq2 5690	. . 3 |- ( n = 0 -> ( `' G ` n ) = ( `' G ` 0 ) )
3	1, 2	breq12d 4138	. 2 |- ( n = 0 -> ( ( 1 ... n ) ~~ ( `' G ` n ) <-> ( 1 ... 0 ) ~~ ( `' G ` 0 ) ) )
4		oveq2 6083	. . 3 |- ( n = m -> ( 1 ... n ) = ( 1 ... m ) )
5		fveq2 5690	. . 3 |- ( n = m -> ( `' G ` n ) = ( `' G ` m ) )
6	4, 5	breq12d 4138	. 2 |- ( n = m -> ( ( 1 ... n ) ~~ ( `' G ` n ) <-> ( 1 ... m ) ~~ ( `' G ` m ) ) )
7		oveq2 6083	. . 3 |- ( n = ( m + 1 ) -> ( 1 ... n ) = ( 1 ... ( m + 1 ) ) )
8		fveq2 5690	. . 3 |- ( n = ( m + 1 ) -> ( `' G ` n ) = ( `' G ` ( m + 1 ) ) )
9	7, 8	breq12d 4138	. 2 |- ( n = ( m + 1 ) -> ( ( 1 ... n ) ~~ ( `' G ` n ) <-> ( 1 ... ( m + 1 ) ) ~~ ( `' G ` ( m + 1 ) ) ) )
10		oveq2 6083	. . 3 |- ( n = N -> ( 1 ... n ) = ( 1 ... N ) )
11		fveq2 5690	. . 3 |- ( n = N -> ( `' G ` n ) = ( `' G ` N ) )
12	10, 11	breq12d 4138	. 2 |- ( n = N -> ( ( 1 ... n ) ~~ ( `' G ` n ) <-> ( 1 ... N ) ~~ ( `' G ` N ) ) )
13		0ex 4255	. . . 4 |- (/) e. _V
14	13	enref 7041	. . 3 |- (/) ~~ (/)
15		fz10 10429	. . 3 |- ( 1 ... 0 ) = (/)
16		0zd 9635	. . . . . . 7 |- ( T. -> 0 e. ZZ )
17		frecfzennn.1	. . . . . . 7 |- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )
18	16, 17	frec2uzf1od 10821	. . . . . 6 |- ( T. -> G : om -1-1-onto-> ( ZZ>= ` 0 ) )
19	18	mptru 1411	. . . . 5 |- G : om -1-1-onto-> ( ZZ>= ` 0 )
20		peano1 4736	. . . . 5 |- (/) e. om
21	19, 20	pm3.2i 272	. . . 4 |- ( G : om -1-1-onto-> ( ZZ>= ` 0 ) /\ (/) e. om )
22	16, 17	frec2uz0d 10814	. . . . 5 |- ( T. -> ( G ` (/) ) = 0 )
23	22	mptru 1411	. . . 4 |- ( G ` (/) ) = 0
24		f1ocnvfv 5975	. . . 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 4155	. 2 |- ( 1 ... 0 ) ~~ ( `' G ` 0 )
27		simpr 110	. . . . 5 |- ( ( m e. NN0 /\ ( 1 ... m ) ~~ ( `' G ` m ) ) -> ( 1 ... m ) ~~ ( `' G ` m ) )
28		peano2nn0 9582	. . . . . . 7 |- ( m e. NN0 -> ( m + 1 ) e. NN0 )
29		zex 9632	. . . . . . . . . . . . . . 15 |- ZZ e. _V
30	29	mptex 5934	. . . . . . . . . . . . . 14 |- ( x e. ZZ |-> ( x + 1 ) ) e. _V
31		vex 2824	. . . . . . . . . . . . . 14 |- z e. _V
32	30, 31	fvex 5710	. . . . . . . . . . . . 13 |- ( ( x e. ZZ |-> ( x + 1 ) ) ` z ) e. _V
33	32	ax-gen 1502	. . . . . . . . . . . 12 |- A. z ( ( x e. ZZ |-> ( x + 1 ) ) ` z ) e. _V
34		0z 9634	. . . . . . . . . . . 12 |- 0 e. ZZ
35		frecfnom 6662	. . . . . . . . . . . 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 430	. . . . . . . . . . 11 |- frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 ) Fn om
37	17	fneq1i 5470	. . . . . . . . . . 11 |- ( G Fn om <-> frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 ) Fn om )
38	36, 37	mpbir 146	. . . . . . . . . 10 |- G Fn om
39		omex 4735	. . . . . . . . . 10 |- om e. _V
40		fnex 5928	. . . . . . . . . 10 |- ( ( G Fn om /\ om e. _V ) -> G e. _V )
41	38, 39, 40	mp2an 430	. . . . . . . . 9 |- G e. _V
42	41	cnvex 5321	. . . . . . . 8 |- `' G e. _V
43		vex 2824	. . . . . . . 8 |- m e. _V
44	42, 43	fvex 5710	. . . . . . 7 |- ( `' G ` m ) e. _V
45		en2sn 7092	. . . . . . 7 |- ( ( ( m + 1 ) e. NN0 /\ ( `' G ` m ) e. _V ) -> { ( m + 1 ) } ~~ { ( `' G ` m ) } )
46	28, 44, 45	sylancl 417	. . . . . 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 10465	. . . . . 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 5977	. . . . . . . . . 10 |- ( ( G : om -1-1-onto-> ( ZZ>= ` 0 ) /\ m e. ( ZZ>= ` 0 ) ) -> ( `' G ` m ) e. om )
51	19, 50	mpan 428	. . . . . . . . 9 |- ( m e. ( ZZ>= ` 0 ) -> ( `' G ` m ) e. om )
52		nn0uz 9936	. . . . . . . . 9 |- NN0 = ( ZZ>= ` 0 )
53	51, 52	eleq2s 2333	. . . . . . . 8 |- ( m e. NN0 -> ( `' G ` m ) e. om )
54		nnord 4754	. . . . . . . 8 |- ( ( `' G ` m ) e. om -> Ord ( `' G ` m ) )
55		ordirr 4684	. . . . . . . 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 3767	. . . . . 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 7095	. . . . 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 1279	. . . 4 |- ( ( m e. NN0 /\ ( 1 ... m ) ~~ ( `' G ` m ) ) -> ( ( 1 ... m ) u. { ( m + 1 ) } ) ~~ ( ( `' G ` m ) u. { ( `' G ` m ) } ) )
62		1z 9649	. . . . . 6 |- 1 e. ZZ
63		1m1e0 9352	. . . . . . . . . 10 |- ( 1 - 1 ) = 0
64	63	fveq2i 5693	. . . . . . . . 9 |- ( ZZ>= ` ( 1 - 1 ) ) = ( ZZ>= ` 0 )
65	52, 64	eqtr4i 2262	. . . . . . . 8 |- NN0 = ( ZZ>= ` ( 1 - 1 ) )
66	65	eleq2i 2305	. . . . . . 7 |- ( m e. NN0 <-> m e. ( ZZ>= ` ( 1 - 1 ) ) )
67	66	biimpi 120	. . . . . 6 |- ( m e. NN0 -> m e. ( ZZ>= ` ( 1 - 1 ) ) )
68		fzsuc2 10464	. . . . . 6 |- ( ( 1 e. ZZ /\ m e. ( ZZ>= ` ( 1 - 1 ) ) ) -> ( 1 ... ( m + 1 ) ) = ( ( 1 ... m ) u. { ( m + 1 ) } ) )
69	62, 67, 68	sylancr 418	. . . . 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 4737	. . . . . . . . 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 9635	. . . . . . . . . 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 10816	. . . . . . . . 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 2305	. . . . . . . . . . 11 |- ( m e. NN0 <-> m e. ( ZZ>= ` 0 ) )
79	78	biimpi 120	. . . . . . . . . 10 |- ( m e. NN0 -> m e. ( ZZ>= ` 0 ) )
80		f1ocnvfv2 5974	. . . . . . . . . 10 |- ( ( G : om -1-1-onto-> ( ZZ>= ` 0 ) /\ m e. ( ZZ>= ` 0 ) ) -> ( G ` ( `' G ` m ) ) = m )
81	19, 79, 80	sylancr 418	. . . . . . . . 9 |- ( m e. NN0 -> ( G ` ( `' G ` m ) ) = m )
82	81	oveq1d 6090	. . . . . . . 8 |- ( m e. NN0 -> ( ( G ` ( `' G ` m ) ) + 1 ) = ( m + 1 ) )
83	77, 82	eqtrd 2271	. . . . . . 7 |- ( m e. NN0 -> ( G ` suc ( `' G ` m ) ) = ( m + 1 ) )
84		f1ocnvfv 5975	. . . . . . 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 4511	. . . . 5 |- suc ( `' G ` m ) = ( ( `' G ` m ) u. { ( `' G ` m ) } )
88	86, 87	eqtrdi 2287	. . . 4 |- ( ( m e. NN0 /\ ( 1 ... m ) ~~ ( `' G ` m ) ) -> ( `' G ` ( m + 1 ) ) = ( ( `' G ` m ) u. { ( `' G ` m ) } ) )
89	61, 70, 88	3brtr4d 4157	. . 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 9739	1 |- ( N e. NN0 -> ( 1 ... N ) ~~ ( `' G ` N ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   A.wal 1400   = wceq 1402   T. wtru 1403   e. wcel 2209   _Vcvv 2821   u. cun 3218   i^i cin 3219   (/)c0 3520   {csn 3705   class class class wbr 4125   |-> cmpt 4187   Ord word 4502   suc csuc 4505   omcom 4732   `'ccnv 4768   Fn wfn 5367   -1-1-onto->wf1o 5371   ` cfv 5372  (class class class)co 6075  freccfrec 6651   ~~ cen 7010   0cc0 8169   1c1 8170   + caddc 8172   - cmin 8487   NN0cn0 9542   ZZcz 9623   ZZ>=cuz 9900   ...cfz 10390

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285

This theorem depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-recs 6566  df-frec 6652  df-1o 6677  df-er 6797  df-en 7013  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-inn 9284  df-n0 9543  df-z 9624  df-uz 9901  df-fz 10391

This theorem is referenced by:  frecfzen2  10842  hashfz1  11200