Theorem cardfz 7719

Description: The cardinality of a finite set of sequential integers. (See om2uz0i 7706 for a description of the antecedent.)

Hypothesis

Ref	Expression
cardfz.1	|- G = (rec({<.j, k>. | k = (j + 1)}, 0) |` om)

Assertion

Ref	Expression
cardfz	|- (N e. NN0 -> (card` (1...N)) = (`'G` N))

Distinct variable group:   j,k

Proof of Theorem cardfz

Step	Hyp	Ref	Expression
1		nn0z 7363	. . . . . . . 8 |- (m e. NN0 -> m e. ZZ)
2		nn0ge0 7326	. . . . . . . . . 10 |- (m e. NN0 -> 0 <_ m)
3		nn0re 7317	. . . . . . . . . . 11 |- (m e. NN0 -> m e. RR)
4		0re 6603	. . . . . . . . . . . 12 |- 0 e. RR
5		1re 6598	. . . . . . . . . . . 12 |- 1 e. RR
6		leadd1 6808	. . . . . . . . . . . 12 |- ((0 e. RR /\ m e. RR /\ 1 e. RR) -> (0 <_ m <-> (0 + 1) <_ (m + 1)))
7	4, 5, 6	mp3an13 1182	. . . . . . . . . . 11 |- (m e. RR -> (0 <_ m <-> (0 + 1) <_ (m + 1)))
8	3, 7	syl 12	. . . . . . . . . 10 |- (m e. NN0 -> (0 <_ m <-> (0 + 1) <_ (m + 1)))
9	2, 8	mpbid 212	. . . . . . . . 9 |- (m e. NN0 -> (0 + 1) <_ (m + 1))
10		ax1cn 6422	. . . . . . . . . 10 |- 1 e. CC
11	10	addid2i 6484	. . . . . . . . 9 |- (0 + 1) = 1
12	9, 11	syl5eqbrr 3371	. . . . . . . 8 |- (m e. NN0 -> 1 <_ (m + 1))
13		1z 7368	. . . . . . . . 9 |- 1 e. ZZ
14		fzsuc 7678	. . . . . . . . 9 |- ((1 e. ZZ /\ m e. ZZ /\ 1 <_ (m + 1)) -> (1...(m + 1)) = ((1...m) u. {(m + 1)}))
15	13, 14	mp3an1 1178	. . . . . . . 8 |- ((m e. ZZ /\ 1 <_ (m + 1)) -> (1...(m + 1)) = ((1...m) u. {(m + 1)}))
16	1, 12, 15	syl11anc 524	. . . . . . 7 |- (m e. NN0 -> (1...(m + 1)) = ((1...m) u. {(m + 1)}))
17	16	fveq2d 4685	. . . . . 6 |- (m e. NN0 -> (card` (1...(m + 1))) = (card` ((1...m) u. {(m + 1)})))
18		ltp1 6989	. . . . . . . . . . 11 |- (m e. RR -> m < (m + 1))
19		peano2re 6599	. . . . . . . . . . . 12 |- (m e. RR -> (m + 1) e. RR)
20		ltnle 6680	. . . . . . . . . . . 12 |- ((m e. RR /\ (m + 1) e. RR) -> (m < (m + 1) <-> -. (m + 1) <_ m))
21	19, 20	mpdan 768	. . . . . . . . . . 11 |- (m e. RR -> (m < (m + 1) <-> -. (m + 1) <_ m))
22	18, 21	mpbid 212	. . . . . . . . . 10 |- (m e. RR -> -. (m + 1) <_ m)
23	3, 22	syl 12	. . . . . . . . 9 |- (m e. NN0 -> -. (m + 1) <_ m)
24		elfzle2 7654	. . . . . . . . 9 |- ((m + 1) e. (1...m) -> (m + 1) <_ m)
25	23, 24	nsyl 131	. . . . . . . 8 |- (m e. NN0 -> -. (m + 1) e. (1...m))
26		oprex 4907	. . . . . . . . 9 |- (1...m) e. _V
27		oprex 4907	. . . . . . . . 9 |- (m + 1) e. _V
28	26, 27	unsnen 5985	. . . . . . . 8 |- (-. (m + 1) e. (1...m) -> ((1...m) u. {(m + 1)}) ~~ suc (card` (1...m)))
29	25, 28	syl 12	. . . . . . 7 |- (m e. NN0 -> ((1...m) u. {(m + 1)}) ~~ suc (card` (1...m)))
30		snex 3492	. . . . . . . . 9 |- {(m + 1)} e. _V
31	26, 30	unex 3796	. . . . . . . 8 |- ((1...m) u. {(m + 1)}) e. _V
32		fvex 4689	. . . . . . . . 9 |- (card` (1...m)) e. _V
33	32	sucex 3892	. . . . . . . 8 |- suc (card` (1...m)) e. _V
34		carden 5981	. . . . . . . 8 |- ((((1...m) u. {(m + 1)}) e. _V /\ suc (card` (1...m)) e. _V) -> ((card` ((1...m) u. {(m + 1)})) = (card` suc (card` (1...m))) <-> ((1...m) u. {(m + 1)}) ~~ suc (card` (1...m))))
35	31, 33, 34	mp2an 761	. . . . . . 7 |- ((card` ((1...m) u. {(m + 1)})) = (card` suc (card` (1...m))) <-> ((1...m) u. {(m + 1)}) ~~ suc (card` (1...m)))
36	29, 35	sylibr 217	. . . . . 6 |- (m e. NN0 -> (card` ((1...m) u. {(m + 1)})) = (card` suc (card` (1...m))))
37	17, 36	eqtrd 1925	. . . . 5 |- (m e. NN0 -> (card` (1...(m + 1))) = (card` suc (card` (1...m))))
38	37	adantr 425	. . . 4 |- ((m e. NN0 /\ (card` (1...m)) = (`'G` m)) -> (card` (1...(m + 1))) = (card` suc (card` (1...m))))
39		suceq 3729	. . . . . 6 |- ((card` (1...m)) = (`'G` m) -> suc (card` (1...m)) = suc (`'G` m))
40	39	fveq2d 4685	. . . . 5 |- ((card` (1...m)) = (`'G` m) -> (card` suc (card` (1...m))) = (card` suc (`'G` m)))
41		nn0zrab 7367	. . . . . . . . . 10 |- NN0 = {n e. ZZ | 0 <_ n}
42	41	eleq2i 1961	. . . . . . . . 9 |- (m e. NN0 <-> m e. {n e. ZZ | 0 <_ n})
43		0z 7355	. . . . . . . . . . 11 |- 0 e. ZZ
44		cardfz.1	. . . . . . . . . . 11 |- G = (rec({<.j, k>. | k = (j + 1)}, 0) |` om)
45	43, 44	om2uzf1oi 7712	. . . . . . . . . 10 |- G:om-1-1-onto->{n e. ZZ | 0 <_ n}
46		f1ocnvdm 4860	. . . . . . . . . 10 |- ((G:om-1-1-onto->{n e. ZZ | 0 <_ n} /\ m e. {n e. ZZ | 0 <_ n}) -> (`'G` m) e. om)
47	45, 46	mpan 759	. . . . . . . . 9 |- (m e. {n e. ZZ | 0 <_ n} -> (`'G` m) e. om)
48	42, 47	sylbi 216	. . . . . . . 8 |- (m e. NN0 -> (`'G` m) e. om)
49		peano2 3972	. . . . . . . 8 |- ((`'G` m) e. om -> suc (`'G` m) e. om)
50	48, 49	syl 12	. . . . . . 7 |- (m e. NN0 -> suc (`'G` m) e. om)
51		cardnn 5870	. . . . . . 7 |- (suc (`'G` m) e. om -> (card` suc (`'G` m)) = suc (`'G` m))
52	50, 51	syl 12	. . . . . 6 |- (m e. NN0 -> (card` suc (`'G` m)) = suc (`'G` m))
53	50, 45	jctil 316	. . . . . . 7 |- (m e. NN0 -> (G:om-1-1-onto->{n e. ZZ | 0 <_ n} /\ suc (`'G` m) e. om))
54	43, 44	om2uzsuci 7707	. . . . . . . . 9 |- ((`'G` m) e. om -> (G` suc (`'G` m)) = ((G` (`'G` m)) + 1))
55	48, 54	syl 12	. . . . . . . 8 |- (m e. NN0 -> (G` suc (`'G` m)) = ((G` (`'G` m)) + 1))
56		f1ocnvfv2 4855	. . . . . . . . . 10 |- ((G:om-1-1-onto->{n e. ZZ | 0 <_ n} /\ m e. {n e. ZZ | 0 <_ n}) -> (G` (`'G` m)) = m)
57	42	biimpi 168	. . . . . . . . . 10 |- (m e. NN0 -> m e. {n e. ZZ | 0 <_ n})
58	56, 45, 57	sylancr 526	. . . . . . . . 9 |- (m e. NN0 -> (G` (`'G` m)) = m)
59	58	opreq1d 4897	. . . . . . . 8 |- (m e. NN0 -> ((G` (`'G` m)) + 1) = (m + 1))
60	55, 59	eqtrd 1925	. . . . . . 7 |- (m e. NN0 -> (G` suc (`'G` m)) = (m + 1))
61		f1ocnvfv 4856	. . . . . . 7 |- ((G:om-1-1-onto->{n e. ZZ | 0 <_ n} /\ suc (`'G` m) e. om) -> ((G` suc (`'G` m)) = (m + 1) -> (`'G` (m + 1)) = suc (`'G` m)))
62	53, 60, 61	sylc 83	. . . . . 6 |- (m e. NN0 -> (`'G` (m + 1)) = suc (`'G` m))
63	52, 62	eqtr4d 1928	. . . . 5 |- (m e. NN0 -> (card` suc (`'G` m)) = (`'G` (m + 1)))
64	40, 63	sylan9eqr 1951	. . . 4 |- ((m e. NN0 /\ (card` (1...m)) = (`'G` m)) -> (card` suc (card` (1...m))) = (`'G` (m + 1)))
65	38, 64	eqtrd 1925	. . 3 |- ((m e. NN0 /\ (card` (1...m)) = (`'G` m)) -> (card` (1...(m + 1))) = (`'G` (m + 1)))
66	65	ex 402	. 2 |- (m e. NN0 -> ((card` (1...m)) = (`'G` m) -> (card` (1...(m + 1))) = (`'G` (m + 1))))
67		card0 5869	. . 3 |- (card` (/)) = (/)
68		lt01 6871	. . . . 5 |- 0 < 1
69		fzn 7663	. . . . . 6 |- ((1 e. ZZ /\ 0 e. ZZ) -> (0 < 1 <-> (1...0) = (/)))
70	13, 43, 69	mp2an 761	. . . . 5 |- (0 < 1 <-> (1...0) = (/))
71	68, 70	mpbi 206	. . . 4 |- (1...0) = (/)
72	71	fveq2i 4684	. . 3 |- (card` (1...0)) = (card` (/))
73		peano1 3971	. . . . 5 |- (/) e. om
74	45, 73	pm3.2i 307	. . . 4 |- (G:om-1-1-onto->{n e. ZZ | 0 <_ n} /\ (/) e. om)
75	43, 44	om2uz0i 7706	. . . 4 |- (G` (/)) = 0
76		f1ocnvfv 4856	. . . 4 |- ((G:om-1-1-onto->{n e. ZZ | 0 <_ n} /\ (/) e. om) -> ((G` (/)) = 0 -> (`'G` 0) = (/)))
77	74, 75, 76	mp2 54	. . 3 |- (`'G` 0) = (/)
78	67, 72, 77	3eqtr4i 1921	. 2 |- (card` (1...0)) = (`'G` 0)
79		opreq2 4890	. . . 4 |- (n = 0 -> (1...n) = (1...0))
80	79	fveq2d 4685	. . 3 |- (n = 0 -> (card` (1...n)) = (card` (1...0)))
81		fveq2 4681	. . 3 |- (n = 0 -> (`'G` n) = (`'G` 0))
82	80, 81	eqeq12d 1899	. 2 |- (n = 0 -> ((card` (1...n)) = (`'G` n) <-> (card` (1...0)) = (`'G` 0)))
83		opreq2 4890	. . . 4 |- (n = m -> (1...n) = (1...m))
84	83	fveq2d 4685	. . 3 |- (n = m -> (card` (1...n)) = (card` (1...m)))
85		fveq2 4681	. . 3 |- (n = m -> (`'G` n) = (`'G` m))
86	84, 85	eqeq12d 1899	. 2 |- (n = m -> ((card` (1...n)) = (`'G` n) <-> (card` (1...m)) = (`'G` m)))
87		opreq2 4890	. . . 4 |- (n = (m + 1) -> (1...n) = (1...(m + 1)))
88	87	fveq2d 4685	. . 3 |- (n = (m + 1) -> (card` (1...n)) = (card` (1...(m + 1))))
89		fveq2 4681	. . 3 |- (n = (m + 1) -> (`'G` n) = (`'G` (m + 1)))
90	88, 89	eqeq12d 1899	. 2 |- (n = (m + 1) -> ((card` (1...n)) = (`'G` n) <-> (card` (1...(m + 1))) = (`'G` (m + 1))))
91		opreq2 4890	. . . 4 |- (n = N -> (1...n) = (1...N))
92	91	fveq2d 4685	. . 3 |- (n = N -> (card` (1...n)) = (card` (1...N)))
93		fveq2 4681	. . 3 |- (n = N -> (`'G` n) = (`'G` N))
94	92, 93	eqeq12d 1899	. 2 |- (n = N -> ((card` (1...n)) = (`'G` n) <-> (card` (1...N)) = (`'G` N)))
95	66, 78, 82, 86, 90, 94	nn0indALT 7425	1 |- (N e. NN0 -> (card` (1...N)) = (`'G` N))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  {crab 2108  _Vcvv 2292   u. cun 2591  (/)c0 2875  {csn 3044   class class class wbr 3338  {copab 3395  suc csuc 3659  omcom 3949  `'ccnv 3985   |` cres 3988  -1-1-onto->wf1o 3997  ` cfv 3998  (class class class)co 4884  reccrdg 5139   ~~ cen 5423  cardccrd 5859  RRcr 6385  0cc0 6386  1c1 6387   + caddc 6389   <_ cle 6448  NN0cn0 6450  ZZcz 6451   < clt 6653  ...cfz 7637

This theorem is referenced by:  fz1fi 8229  hashfz1 8232

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731  ax-ac 5906

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-card 5862  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-n 7108  df-n0 7309  df-z 7345  df-uz 7587  df-fz 7638

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