Theorem fzennn 14019

Description: The cardinality of a finite set of sequential integers. (See om2uz0i 13998 for a description of the hypothesis.) (Contributed by Mario Carneiro, 12-Feb-2013.) (Revised by Mario Carneiro, 7-Mar-2014.)

Hypothesis

Ref	Expression
fzennn.1	⊢ 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)

Assertion

Ref	Expression
fzennn	⊢ (𝑁 ∈ ℕ0 → (1...𝑁) ≈ (◡𝐺‘𝑁))

Proof of Theorem fzennn

Dummy variables 𝑚 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 7456	. . 3 ⊢ (𝑛 = 0 → (1...𝑛) = (1...0))
2		fveq2 6920	. . 3 ⊢ (𝑛 = 0 → (◡𝐺‘𝑛) = (◡𝐺‘0))
3	1, 2	breq12d 5179	. 2 ⊢ (𝑛 = 0 → ((1...𝑛) ≈ (◡𝐺‘𝑛) ↔ (1...0) ≈ (◡𝐺‘0)))
4		oveq2 7456	. . 3 ⊢ (𝑛 = 𝑚 → (1...𝑛) = (1...𝑚))
5		fveq2 6920	. . 3 ⊢ (𝑛 = 𝑚 → (◡𝐺‘𝑛) = (◡𝐺‘𝑚))
6	4, 5	breq12d 5179	. 2 ⊢ (𝑛 = 𝑚 → ((1...𝑛) ≈ (◡𝐺‘𝑛) ↔ (1...𝑚) ≈ (◡𝐺‘𝑚)))
7		oveq2 7456	. . 3 ⊢ (𝑛 = (𝑚 + 1) → (1...𝑛) = (1...(𝑚 + 1)))
8		fveq2 6920	. . 3 ⊢ (𝑛 = (𝑚 + 1) → (◡𝐺‘𝑛) = (◡𝐺‘(𝑚 + 1)))
9	7, 8	breq12d 5179	. 2 ⊢ (𝑛 = (𝑚 + 1) → ((1...𝑛) ≈ (◡𝐺‘𝑛) ↔ (1...(𝑚 + 1)) ≈ (◡𝐺‘(𝑚 + 1))))
10		oveq2 7456	. . 3 ⊢ (𝑛 = 𝑁 → (1...𝑛) = (1...𝑁))
11		fveq2 6920	. . 3 ⊢ (𝑛 = 𝑁 → (◡𝐺‘𝑛) = (◡𝐺‘𝑁))
12	10, 11	breq12d 5179	. 2 ⊢ (𝑛 = 𝑁 → ((1...𝑛) ≈ (◡𝐺‘𝑛) ↔ (1...𝑁) ≈ (◡𝐺‘𝑁)))
13		0ex 5325	. . . 4 ⊢ ∅ ∈ V
14	13	enref 9045	. . 3 ⊢ ∅ ≈ ∅
15		fz10 13605	. . 3 ⊢ (1...0) = ∅
16		0z 12650	. . . . . 6 ⊢ 0 ∈ ℤ
17		fzennn.1	. . . . . 6 ⊢ 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)
18	16, 17	om2uzf1oi 14004	. . . . 5 ⊢ 𝐺:ω–1-1-onto→(ℤ≥‘0)
19		peano1 7927	. . . . 5 ⊢ ∅ ∈ ω
20	18, 19	pm3.2i 470	. . . 4 ⊢ (𝐺:ω–1-1-onto→(ℤ≥‘0) ∧ ∅ ∈ ω)
21	16, 17	om2uz0i 13998	. . . 4 ⊢ (𝐺‘∅) = 0
22		f1ocnvfv 7314	. . . 4 ⊢ ((𝐺:ω–1-1-onto→(ℤ≥‘0) ∧ ∅ ∈ ω) → ((𝐺‘∅) = 0 → (◡𝐺‘0) = ∅))
23	20, 21, 22	mp2 9	. . 3 ⊢ (◡𝐺‘0) = ∅
24	14, 15, 23	3brtr4i 5196	. 2 ⊢ (1...0) ≈ (◡𝐺‘0)
25		simpr 484	. . . . 5 ⊢ ((𝑚 ∈ ℕ0 ∧ (1...𝑚) ≈ (◡𝐺‘𝑚)) → (1...𝑚) ≈ (◡𝐺‘𝑚))
26		ovex 7481	. . . . . . 7 ⊢ (𝑚 + 1) ∈ V
27		fvex 6933	. . . . . . 7 ⊢ (◡𝐺‘𝑚) ∈ V
28		en2sn 9106	. . . . . . 7 ⊢ (((𝑚 + 1) ∈ V ∧ (◡𝐺‘𝑚) ∈ V) → {(𝑚 + 1)} ≈ {(◡𝐺‘𝑚)})
29	26, 27, 28	mp2an 691	. . . . . 6 ⊢ {(𝑚 + 1)} ≈ {(◡𝐺‘𝑚)}
30	29	a1i 11	. . . . 5 ⊢ ((𝑚 ∈ ℕ0 ∧ (1...𝑚) ≈ (◡𝐺‘𝑚)) → {(𝑚 + 1)} ≈ {(◡𝐺‘𝑚)})
31		fzp1disj 13643	. . . . . 6 ⊢ ((1...𝑚) ∩ {(𝑚 + 1)}) = ∅
32	31	a1i 11	. . . . 5 ⊢ ((𝑚 ∈ ℕ0 ∧ (1...𝑚) ≈ (◡𝐺‘𝑚)) → ((1...𝑚) ∩ {(𝑚 + 1)}) = ∅)
33		f1ocnvdm 7321	. . . . . . . . . 10 ⊢ ((𝐺:ω–1-1-onto→(ℤ≥‘0) ∧ 𝑚 ∈ (ℤ≥‘0)) → (◡𝐺‘𝑚) ∈ ω)
34	18, 33	mpan 689	. . . . . . . . 9 ⊢ (𝑚 ∈ (ℤ≥‘0) → (◡𝐺‘𝑚) ∈ ω)
35		nn0uz 12945	. . . . . . . . 9 ⊢ ℕ0 = (ℤ≥‘0)
36	34, 35	eleq2s 2862	. . . . . . . 8 ⊢ (𝑚 ∈ ℕ0 → (◡𝐺‘𝑚) ∈ ω)
37		nnord 7911	. . . . . . . 8 ⊢ ((◡𝐺‘𝑚) ∈ ω → Ord (◡𝐺‘𝑚))
38		ordirr 6413	. . . . . . . 8 ⊢ (Ord (◡𝐺‘𝑚) → ¬ (◡𝐺‘𝑚) ∈ (◡𝐺‘𝑚))
39	36, 37, 38	3syl 18	. . . . . . 7 ⊢ (𝑚 ∈ ℕ0 → ¬ (◡𝐺‘𝑚) ∈ (◡𝐺‘𝑚))
40	39	adantr 480	. . . . . 6 ⊢ ((𝑚 ∈ ℕ0 ∧ (1...𝑚) ≈ (◡𝐺‘𝑚)) → ¬ (◡𝐺‘𝑚) ∈ (◡𝐺‘𝑚))
41		disjsn 4736	. . . . . 6 ⊢ (((◡𝐺‘𝑚) ∩ {(◡𝐺‘𝑚)}) = ∅ ↔ ¬ (◡𝐺‘𝑚) ∈ (◡𝐺‘𝑚))
42	40, 41	sylibr 234	. . . . 5 ⊢ ((𝑚 ∈ ℕ0 ∧ (1...𝑚) ≈ (◡𝐺‘𝑚)) → ((◡𝐺‘𝑚) ∩ {(◡𝐺‘𝑚)}) = ∅)
43		unen 9112	. . . . 5 ⊢ ((((1...𝑚) ≈ (◡𝐺‘𝑚) ∧ {(𝑚 + 1)} ≈ {(◡𝐺‘𝑚)}) ∧ (((1...𝑚) ∩ {(𝑚 + 1)}) = ∅ ∧ ((◡𝐺‘𝑚) ∩ {(◡𝐺‘𝑚)}) = ∅)) → ((1...𝑚) ∪ {(𝑚 + 1)}) ≈ ((◡𝐺‘𝑚) ∪ {(◡𝐺‘𝑚)}))
44	25, 30, 32, 42, 43	syl22anc 838	. . . 4 ⊢ ((𝑚 ∈ ℕ0 ∧ (1...𝑚) ≈ (◡𝐺‘𝑚)) → ((1...𝑚) ∪ {(𝑚 + 1)}) ≈ ((◡𝐺‘𝑚) ∪ {(◡𝐺‘𝑚)}))
45		1z 12673	. . . . . 6 ⊢ 1 ∈ ℤ
46		1m1e0 12365	. . . . . . . . . 10 ⊢ (1 − 1) = 0
47	46	fveq2i 6923	. . . . . . . . 9 ⊢ (ℤ≥‘(1 − 1)) = (ℤ≥‘0)
48	35, 47	eqtr4i 2771	. . . . . . . 8 ⊢ ℕ0 = (ℤ≥‘(1 − 1))
49	48	eleq2i 2836	. . . . . . 7 ⊢ (𝑚 ∈ ℕ0 ↔ 𝑚 ∈ (ℤ≥‘(1 − 1)))
50	49	biimpi 216	. . . . . 6 ⊢ (𝑚 ∈ ℕ0 → 𝑚 ∈ (ℤ≥‘(1 − 1)))
51		fzsuc2 13642	. . . . . 6 ⊢ ((1 ∈ ℤ ∧ 𝑚 ∈ (ℤ≥‘(1 − 1))) → (1...(𝑚 + 1)) = ((1...𝑚) ∪ {(𝑚 + 1)}))
52	45, 50, 51	sylancr 586	. . . . 5 ⊢ (𝑚 ∈ ℕ0 → (1...(𝑚 + 1)) = ((1...𝑚) ∪ {(𝑚 + 1)}))
53	52	adantr 480	. . . 4 ⊢ ((𝑚 ∈ ℕ0 ∧ (1...𝑚) ≈ (◡𝐺‘𝑚)) → (1...(𝑚 + 1)) = ((1...𝑚) ∪ {(𝑚 + 1)}))
54		peano2 7929	. . . . . . . . 9 ⊢ ((◡𝐺‘𝑚) ∈ ω → suc (◡𝐺‘𝑚) ∈ ω)
55	36, 54	syl 17	. . . . . . . 8 ⊢ (𝑚 ∈ ℕ0 → suc (◡𝐺‘𝑚) ∈ ω)
56	55, 18	jctil 519	. . . . . . 7 ⊢ (𝑚 ∈ ℕ0 → (𝐺:ω–1-1-onto→(ℤ≥‘0) ∧ suc (◡𝐺‘𝑚) ∈ ω))
57	16, 17	om2uzsuci 13999	. . . . . . . . 9 ⊢ ((◡𝐺‘𝑚) ∈ ω → (𝐺‘suc (◡𝐺‘𝑚)) = ((𝐺‘(◡𝐺‘𝑚)) + 1))
58	36, 57	syl 17	. . . . . . . 8 ⊢ (𝑚 ∈ ℕ0 → (𝐺‘suc (◡𝐺‘𝑚)) = ((𝐺‘(◡𝐺‘𝑚)) + 1))
59	35	eleq2i 2836	. . . . . . . . . . 11 ⊢ (𝑚 ∈ ℕ0 ↔ 𝑚 ∈ (ℤ≥‘0))
60	59	biimpi 216	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ0 → 𝑚 ∈ (ℤ≥‘0))
61		f1ocnvfv2 7313	. . . . . . . . . 10 ⊢ ((𝐺:ω–1-1-onto→(ℤ≥‘0) ∧ 𝑚 ∈ (ℤ≥‘0)) → (𝐺‘(◡𝐺‘𝑚)) = 𝑚)
62	18, 60, 61	sylancr 586	. . . . . . . . 9 ⊢ (𝑚 ∈ ℕ0 → (𝐺‘(◡𝐺‘𝑚)) = 𝑚)
63	62	oveq1d 7463	. . . . . . . 8 ⊢ (𝑚 ∈ ℕ0 → ((𝐺‘(◡𝐺‘𝑚)) + 1) = (𝑚 + 1))
64	58, 63	eqtrd 2780	. . . . . . 7 ⊢ (𝑚 ∈ ℕ0 → (𝐺‘suc (◡𝐺‘𝑚)) = (𝑚 + 1))
65		f1ocnvfv 7314	. . . . . . 7 ⊢ ((𝐺:ω–1-1-onto→(ℤ≥‘0) ∧ suc (◡𝐺‘𝑚) ∈ ω) → ((𝐺‘suc (◡𝐺‘𝑚)) = (𝑚 + 1) → (◡𝐺‘(𝑚 + 1)) = suc (◡𝐺‘𝑚)))
66	56, 64, 65	sylc 65	. . . . . 6 ⊢ (𝑚 ∈ ℕ0 → (◡𝐺‘(𝑚 + 1)) = suc (◡𝐺‘𝑚))
67	66	adantr 480	. . . . 5 ⊢ ((𝑚 ∈ ℕ0 ∧ (1...𝑚) ≈ (◡𝐺‘𝑚)) → (◡𝐺‘(𝑚 + 1)) = suc (◡𝐺‘𝑚))
68		df-suc 6401	. . . . 5 ⊢ suc (◡𝐺‘𝑚) = ((◡𝐺‘𝑚) ∪ {(◡𝐺‘𝑚)})
69	67, 68	eqtrdi 2796	. . . 4 ⊢ ((𝑚 ∈ ℕ0 ∧ (1...𝑚) ≈ (◡𝐺‘𝑚)) → (◡𝐺‘(𝑚 + 1)) = ((◡𝐺‘𝑚) ∪ {(◡𝐺‘𝑚)}))
70	44, 53, 69	3brtr4d 5198	. . 3 ⊢ ((𝑚 ∈ ℕ0 ∧ (1...𝑚) ≈ (◡𝐺‘𝑚)) → (1...(𝑚 + 1)) ≈ (◡𝐺‘(𝑚 + 1)))
71	70	ex 412	. 2 ⊢ (𝑚 ∈ ℕ0 → ((1...𝑚) ≈ (◡𝐺‘𝑚) → (1...(𝑚 + 1)) ≈ (◡𝐺‘(𝑚 + 1))))
72	3, 6, 9, 12, 24, 71	nn0ind 12738	1 ⊢ (𝑁 ∈ ℕ0 → (1...𝑁) ≈ (◡𝐺‘𝑁))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108  Vcvv 3488   ∪ cun 3974   ∩ cin 3975  ∅c0 4352  {csn 4648   class class class wbr 5166   ↦ cmpt 5249  ^◡ccnv 5699   ↾ cres 5702  Ord word 6394  suc csuc 6397  –1-1-onto→wf1o 6572  ‘cfv 6573  (class class class)co 7448  ωcom 7903  reccrdg 8465   ≈ cen 9000  0cc0 11184  1c1 11185   + caddc 11187   − cmin 11520  ℕ₀cn0 12553  ℤcz 12639  ℤ_≥cuz 12903  ...cfz 13567

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-er 8763  df-en 9004  df-dom 9005  df-sdom 9006  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-nn 12294  df-n0 12554  df-z 12640  df-uz 12904  df-fz 13568

This theorem is referenced by:  fzen2  14020  cardfz  14021