Theorem cardfz 6671

Description: The cardinality of a finite set of sequential integers. (See om2uz0i 6658 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		1z 6327	. . . . . . . . 9 |- 1 e. ZZ
2		fzsuc 6636	. . . . . . . . 9 |- ((1 e. ZZ /\ m e. ZZ /\ 1 <_ (m + 1)) -> (1...(m + 1)) = ((1...m) u. {(m + 1)}))
3	1, 2	mp3an1 909	. . . . . . . 8 |- ((m e. ZZ /\ 1 <_ (m + 1)) -> (1...(m + 1)) = ((1...m) u. {(m + 1)}))
4		nn0z 6322	. . . . . . . 8 |- (m e. NN0 -> m e. ZZ)
5		nn0ge0 6285	. . . . . . . . . 10 |- (m e. NN0 -> 0 <_ m)
6		nn0re 6276	. . . . . . . . . . 11 |- (m e. NN0 -> m e. RR)
7		0re 5594	. . . . . . . . . . . 12 |- 0 e. RR
8		1re 5589	. . . . . . . . . . . 12 |- 1 e. RR
9		leadd1 5779	. . . . . . . . . . . 12 |- ((0 e. RR /\ m e. RR /\ 1 e. RR) -> (0 <_ m <-> (0 + 1) <_ (m + 1)))
10	7, 8, 9	mp3an13 913	. . . . . . . . . . 11 |- (m e. RR -> (0 <_ m <-> (0 + 1) <_ (m + 1)))
11	6, 10	syl 10	. . . . . . . . . 10 |- (m e. NN0 -> (0 <_ m <-> (0 + 1) <_ (m + 1)))
12	5, 11	mpbid 193	. . . . . . . . 9 |- (m e. NN0 -> (0 + 1) <_ (m + 1))
13		ax1cn 5423	. . . . . . . . . 10 |- 1 e. CC
14	13	addid2i 5485	. . . . . . . . 9 |- (0 + 1) = 1
15	12, 14	syl5eqbrr 2722	. . . . . . . 8 |- (m e. NN0 -> 1 <_ (m + 1))
16	3, 4, 15	sylanc 473	. . . . . . 7 |- (m e. NN0 -> (1...(m + 1)) = ((1...m) u. {(m + 1)}))
17	16	fveq2d 3839	. . . . . 6 |- (m e. NN0 -> (card` (1...(m + 1))) = (card` ((1...m) u. {(m + 1)})))
18		ltp1 5951	. . . . . . . . . . 11 |- (m e. RR -> m < (m + 1))
19		peano2re 5590	. . . . . . . . . . . 12 |- (m e. RR -> (m + 1) e. RR)
20		ltnle 5665	. . . . . . . . . . . 12 |- ((m e. RR /\ (m + 1) e. RR) -> (m < (m + 1) <-> -. (m + 1) <_ m))
21	19, 20	mpdan 708	. . . . . . . . . . 11 |- (m e. RR -> (m < (m + 1) <-> -. (m + 1) <_ m))
22	18, 21	mpbid 193	. . . . . . . . . 10 |- (m e. RR -> -. (m + 1) <_ m)
23	6, 22	syl 10	. . . . . . . . 9 |- (m e. NN0 -> -. (m + 1) <_ m)
24		elfzle2 6612	. . . . . . . . 9 |- ((m + 1) e. (1...m) -> (m + 1) <_ m)
25	23, 24	nsyl 115	. . . . . . . 8 |- (m e. NN0 -> -. (m + 1) e. (1...m))
26		oprex 4041	. . . . . . . . 9 |- (1...m) e. V
27		oprex 4041	. . . . . . . . 9 |- (m + 1) e. V
28	26, 27	unsnen 4983	. . . . . . . 8 |- (-. (m + 1) e. (1...m) -> ((1...m) u. {(m + 1)}) ~~ suc (card` (1...m)))
29	25, 28	syl 10	. . . . . . 7 |- (m e. NN0 -> ((1...m) u. {(m + 1)}) ~~ suc (card` (1...m)))
30		snex 2826	. . . . . . . . 9 |- {(m + 1)} e. V
31	26, 30	unex 3095	. . . . . . . 8 |- ((1...m) u. {(m + 1)}) e. V
32		fvex 3843	. . . . . . . . 9 |- (card` (1...m)) e. V
33	32	sucex 3168	. . . . . . . 8 |- suc (card` (1...m)) e. V
34		carden 4979	. . . . . . . 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 701	. . . . . . 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 198	. . . . . 6 |- (m e. NN0 -> (card` ((1...m) u. {(m + 1)})) = (card` suc (card` (1...m))))
37	17, 36	eqtrd 1550	. . . . 5 |- (m e. NN0 -> (card` (1...(m + 1))) = (card` suc (card` (1...m))))
38	37	adantr 389	. . . 4 |- ((m e. NN0 /\ (card` (1...m)) = (`'G` m)) -> (card` (1...(m + 1))) = (card` suc (card` (1...m))))
39		suceq 3038	. . . . . 6 |- ((card` (1...m)) = (`'G` m) -> suc (card` (1...m)) = suc (`'G` m))
40	39	fveq2d 3839	. . . . 5 |- ((card` (1...m)) = (`'G` m) -> (card` suc (card` (1...m))) = (card` suc (`'G` m)))
41		nn0zrab 6326	. . . . . . . . . 10 |- NN0 = {n e. ZZ | 0 <_ n}
42	41	eleq2i 1581	. . . . . . . . 9 |- (m e. NN0 <-> m e. {n e. ZZ | 0 <_ n})
43		0z 6314	. . . . . . . . . . 11 |- 0 e. ZZ
44		cardfz.1	. . . . . . . . . . 11 |- G = (rec({<.j, k>. | k = (j + 1)}, 0) |` om)
45	43, 44	om2uzf1oi 6664	. . . . . . . . . 10 |- G:om-1-1-onto->{n e. ZZ | 0 <_ n}
46		f1ocnvdm 3998	. . . . . . . . . 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 699	. . . . . . . . 9 |- (m e. {n e. ZZ | 0 <_ n} -> (`'G` m) e. om)
48	42, 47	sylbi 197	. . . . . . . 8 |- (m e. NN0 -> (`'G` m) e. om)
49		peano2 3238	. . . . . . . 8 |- ((`'G` m) e. om -> suc (`'G` m) e. om)
50	48, 49	syl 10	. . . . . . 7 |- (m e. NN0 -> suc (`'G` m) e. om)
51		cardnn 4970	. . . . . . 7 |- (suc (`'G` m) e. om -> (card` suc (`'G` m)) = suc (`'G` m))
52	50, 51	syl 10	. . . . . 6 |- (m e. NN0 -> (card` suc (`'G` m)) = suc (`'G` m))
53		f1ocnvfv 3994	. . . . . . 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)))
54	50, 45	jctil 290	. . . . . . 7 |- (m e. NN0 -> (G:om-1-1-onto->{n e. ZZ | 0 <_ n} /\ suc (`'G` m) e. om))
55	43, 44	om2uzsuci 6659	. . . . . . . . 9 |- ((`'G` m) e. om -> (G` suc (`'G` m)) = ((G` (`'G` m)) + 1))
56	48, 55	syl 10	. . . . . . . 8 |- (m e. NN0 -> (G` suc (`'G` m)) = ((G` (`'G` m)) + 1))
57	42	biimpi 149	. . . . . . . . . . 11 |- (m e. NN0 -> m e. {n e. ZZ | 0 <_ n})
58	57, 45	jctil 290	. . . . . . . . . 10 |- (m e. NN0 -> (G:om-1-1-onto->{n e. ZZ | 0 <_ n} /\ m e. {n e. ZZ | 0 <_ n}))
59		f1ocnvfv2 3993	. . . . . . . . . 10 |- ((G:om-1-1-onto->{n e. ZZ | 0 <_ n} /\ m e. {n e. ZZ | 0 <_ n}) -> (G` (`'G` m)) = m)
60	58, 59	syl 10	. . . . . . . . 9 |- (m e. NN0 -> (G` (`'G` m)) = m)
61	60	opreq1d 4033	. . . . . . . 8 |- (m e. NN0 -> ((G` (`'G` m)) + 1) = (m + 1))
62	56, 61	eqtrd 1550	. . . . . . 7 |- (m e. NN0 -> (G` suc (`'G` m)) = (m + 1))
63	53, 54, 62	sylc 68	. . . . . 6 |- (m e. NN0 -> (`'G` (m + 1)) = suc (`'G` m))
64	52, 63	eqtr4d 1553	. . . . 5 |- (m e. NN0 -> (card` suc (`'G` m)) = (`'G` (m + 1)))
65	40, 64	sylan9eqr 1572	. . . 4 |- ((m e. NN0 /\ (card` (1...m)) = (`'G` m)) -> (card` suc (card` (1...m))) = (`'G` (m + 1)))
66	38, 65	eqtrd 1550	. . 3 |- ((m e. NN0 /\ (card` (1...m)) = (`'G` m)) -> (card` (1...(m + 1))) = (`'G` (m + 1)))
67	66	ex 371	. 2 |- (m e. NN0 -> ((card` (1...m)) = (`'G` m) -> (card` (1...(m + 1))) = (`'G` (m + 1))))
68		card0 4969	. . 3 |- (card` (/)) = (/)
69		lt01 5836	. . . . 5 |- 0 < 1
70		fzn 6621	. . . . . 6 |- ((1 e. ZZ /\ 0 e. ZZ) -> (0 < 1 <-> (1...0) = (/)))
71	1, 43, 70	mp2an 701	. . . . 5 |- (0 < 1 <-> (1...0) = (/))
72	69, 71	mpbi 187	. . . 4 |- (1...0) = (/)
73	72	fveq2i 3838	. . 3 |- (card` (1...0)) = (card` (/))
74		peano1 3237	. . . . 5 |- (/) e. om
75	45, 74	pm3.2i 283	. . . 4 |- (G:om-1-1-onto->{n e. ZZ | 0 <_ n} /\ (/) e. om)
76	43, 44	om2uz0i 6658	. . . 4 |- (G` (/)) = 0
77		f1ocnvfv 3994	. . . 4 |- ((G:om-1-1-onto->{n e. ZZ | 0 <_ n} /\ (/) e. om) -> ((G` (/)) = 0 -> (`'G` 0) = (/)))
78	75, 76, 77	mp2 43	. . 3 |- (`'G` 0) = (/)
79	68, 73, 78	3eqtr4i 1548	. 2 |- (card` (1...0)) = (`'G` 0)
80		opreq2 4027	. . . 4 |- (n = 0 -> (1...n) = (1...0))
81	80	fveq2d 3839	. . 3 |- (n = 0 -> (card` (1...n)) = (card` (1...0)))
82		fveq2 3835	. . 3 |- (n = 0 -> (`'G` n) = (`'G` 0))
83	81, 82	eqeq12d 1532	. 2 |- (n = 0 -> ((card` (1...n)) = (`'G` n) <-> (card` (1...0)) = (`'G` 0)))
84		opreq2 4027	. . . 4 |- (n = m -> (1...n) = (1...m))
85	84	fveq2d 3839	. . 3 |- (n = m -> (card` (1...n)) = (card` (1...m)))
86		fveq2 3835	. . 3 |- (n = m -> (`'G` n) = (`'G` m))
87	85, 86	eqeq12d 1532	. 2 |- (n = m -> ((card` (1...n)) = (`'G` n) <-> (card` (1...m)) = (`'G` m)))
88		opreq2 4027	. . . 4 |- (n = (m + 1) -> (1...n) = (1...(m + 1)))
89	88	fveq2d 3839	. . 3 |- (n = (m + 1) -> (card` (1...n)) = (card` (1...(m + 1))))
90		fveq2 3835	. . 3 |- (n = (m + 1) -> (`'G` n) = (`'G` (m + 1)))
91	89, 90	eqeq12d 1532	. 2 |- (n = (m + 1) -> ((card` (1...n)) = (`'G` n) <-> (card` (1...(m + 1))) = (`'G` (m + 1))))
92		opreq2 4027	. . . 4 |- (n = N -> (1...n) = (1...N))
93	92	fveq2d 3839	. . 3 |- (n = N -> (card` (1...n)) = (card` (1...N)))
94		fveq2 3835	. . 3 |- (n = N -> (`'G` n) = (`'G` N))
95	93, 94	eqeq12d 1532	. 2 |- (n = N -> ((card` (1...n)) = (`'G` n) <-> (card` (1...N)) = (`'G` N)))
96	67, 79, 83, 87, 91, 95	nn0indALT 6384	1 |- (N e. NN0 -> (card` (1...N)) = (`'G` N))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 144   /\ wa 221   = wceq 992   e. wcel 994  {crab 1694  Vcvv 1857   u. cun 2097  (/)c0 2332  {csn 2467   class class class wbr 2692  {copab 2740  suc csuc 2977  omcom 3218  `'ccnv 3250   |` cres 3253  -1-1-onto->wf1o 3262  ` cfv 3263  (class class class)co 4021  reccrdg 4232   ~~ cen 4505  cardccrd 4959  RRcr 5387  0cc0 5388  1c1 5389   + caddc 5391   <_ cle 5449  NN0cn0 5451  ZZcz 5452   < clt 5640  ...cfz 6595

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-9 1001  ax-10 1002  ax-11 1003  ax-12 1004  ax-13 1005  ax-14 1006  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255  ax-ext 1500  ax-rep 2767  ax-sep 2777  ax-nul 2784  ax-pow 2818  ax-pr 2855  ax-un 3089  ax-inf2 4770  ax-ac 4890

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-3or 782  df-3an 783  df-ex 1017  df-sb 1209  df-eu 1421  df-mo 1422  df-clab 1506  df-cleq 1511  df-clel 1514  df-ne 1630  df-nel 1631  df-ral 1695  df-rex 1696  df-reu 1697  df-rab 1698  df-v 1858  df-sbc 1987  df-csb 2052  df-dif 2101  df-un 2102  df-in 2103  df-ss 2105  df-pss 2107  df-nul 2333  df-if 2416  df-pw 2459  df-sn 2470  df-pr 2471  df-tp 2473  df-op 2474  df-uni 2570  df-int 2601  df-iun 2635  df-br 2693  df-opab 2741  df-tr 2755  df-eprel 2910  df-id 2913  df-po 2918  df-so 2929  df-fr 2947  df-we 2962  df-ord 2978  df-on 2979  df-lim 2980  df-suc 2981  df-om 3219  df-xp 3265  df-rel 3266  df-cnv 3267  df-co 3268  df-dm 3269  df-rn 3270  df-res 3271  df-ima 3272  df-fun 3273  df-fn 3274  df-f 3275  df-f1 3276  df-fo 3277  df-f1o 3278  df-fv 3279  df-opr 4023  df-oprab 4024  df-1st 4140  df-2nd 4141  df-rdg 4233  df-1o 4269  df-oadd 4271  df-omul 4272  df-er 4401  df-ec 4403  df-qs 4406  df-en 4509  df-dom 4510  df-sdom 4511  df-card 4962  df-ni 5154  df-pli 5155  df-mi 5156  df-lti 5157  df-plpq 5189  df-mpq 5190  df-enq 5191  df-nq 5192  df-plq 5193  df-mq 5194  df-rq 5195  df-ltq 5196  df-1q 5197  df-np 5240  df-1p 5241  df-plp 5242  df-mp 5243  df-ltp 5244  df-plpr 5318  df-mpr 5319  df-enr 5320  df-nr 5321  df-plr 5322  df-mr 5323  df-ltr 5324  df-0r 5325  df-1r 5326  df-m1r 5327  df-c 5394  df-0 5395  df-1 5396  df-i 5397  df-r 5398  df-plus 5399  df-mul 5400  df-lt 5401  df-sub 5510  df-neg 5512  df-pnf 5641  df-mnf 5642  df-xr 5643  df-ltxr 5644  df-le 5645  df-n 6070  df-n0 6268  df-z 6304  df-uz 6545  df-fz 6596

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