Theorem frecfzennn 10784

Description: The cardinality of a finite set of sequential integers. (See frec2uz0d 10757 for a description of the hypothesis.) (Contributed by Jim Kingdon, 18-May-2020.)

Hypothesis

Ref	Expression
frecfzennn.1	⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)

Assertion

Ref	Expression
frecfzennn	⊢ (𝑁 ∈ ℕ0 → (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 6057	. . 3 ⊢ (𝑛 = 0 → (1...𝑛) = (1...0))
2		fveq2 5669	. . 3 ⊢ (𝑛 = 0 → (◡𝐺‘𝑛) = (◡𝐺‘0))
3	1, 2	breq12d 4121	. 2 ⊢ (𝑛 = 0 → ((1...𝑛) ≈ (◡𝐺‘𝑛) ↔ (1...0) ≈ (◡𝐺‘0)))
4		oveq2 6057	. . 3 ⊢ (𝑛 = 𝑚 → (1...𝑛) = (1...𝑚))
5		fveq2 5669	. . 3 ⊢ (𝑛 = 𝑚 → (◡𝐺‘𝑛) = (◡𝐺‘𝑚))
6	4, 5	breq12d 4121	. 2 ⊢ (𝑛 = 𝑚 → ((1...𝑛) ≈ (◡𝐺‘𝑛) ↔ (1...𝑚) ≈ (◡𝐺‘𝑚)))
7		oveq2 6057	. . 3 ⊢ (𝑛 = (𝑚 + 1) → (1...𝑛) = (1...(𝑚 + 1)))
8		fveq2 5669	. . 3 ⊢ (𝑛 = (𝑚 + 1) → (◡𝐺‘𝑛) = (◡𝐺‘(𝑚 + 1)))
9	7, 8	breq12d 4121	. 2 ⊢ (𝑛 = (𝑚 + 1) → ((1...𝑛) ≈ (◡𝐺‘𝑛) ↔ (1...(𝑚 + 1)) ≈ (◡𝐺‘(𝑚 + 1))))
10		oveq2 6057	. . 3 ⊢ (𝑛 = 𝑁 → (1...𝑛) = (1...𝑁))
11		fveq2 5669	. . 3 ⊢ (𝑛 = 𝑁 → (◡𝐺‘𝑛) = (◡𝐺‘𝑁))
12	10, 11	breq12d 4121	. 2 ⊢ (𝑛 = 𝑁 → ((1...𝑛) ≈ (◡𝐺‘𝑛) ↔ (1...𝑁) ≈ (◡𝐺‘𝑁)))
13		0ex 4236	. . . 4 ⊢ ∅ ∈ V
14	13	enref 7003	. . 3 ⊢ ∅ ≈ ∅
15		fz10 10376	. . 3 ⊢ (1...0) = ∅
16		0zd 9585	. . . . . . 7 ⊢ (⊤ → 0 ∈ ℤ)
17		frecfzennn.1	. . . . . . 7 ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)
18	16, 17	frec2uzf1od 10764	. . . . . 6 ⊢ (⊤ → 𝐺:ω–1-1-onto→(ℤ≥‘0))
19	18	mptru 1407	. . . . 5 ⊢ 𝐺:ω–1-1-onto→(ℤ≥‘0)
20		peano1 4715	. . . . 5 ⊢ ∅ ∈ ω
21	19, 20	pm3.2i 272	. . . 4 ⊢ (𝐺:ω–1-1-onto→(ℤ≥‘0) ∧ ∅ ∈ ω)
22	16, 17	frec2uz0d 10757	. . . . 5 ⊢ (⊤ → (𝐺‘∅) = 0)
23	22	mptru 1407	. . . 4 ⊢ (𝐺‘∅) = 0
24		f1ocnvfv 5951	. . . 4 ⊢ ((𝐺:ω–1-1-onto→(ℤ≥‘0) ∧ ∅ ∈ ω) → ((𝐺‘∅) = 0 → (◡𝐺‘0) = ∅))
25	21, 23, 24	mp2 16	. . 3 ⊢ (◡𝐺‘0) = ∅
26	14, 15, 25	3brtr4i 4138	. 2 ⊢ (1...0) ≈ (◡𝐺‘0)
27		simpr 110	. . . . 5 ⊢ ((𝑚 ∈ ℕ0 ∧ (1...𝑚) ≈ (◡𝐺‘𝑚)) → (1...𝑚) ≈ (◡𝐺‘𝑚))
28		peano2nn0 9532	. . . . . . 7 ⊢ (𝑚 ∈ ℕ0 → (𝑚 + 1) ∈ ℕ0)
29		zex 9582	. . . . . . . . . . . . . . 15 ⊢ ℤ ∈ V
30	29	mptex 5911	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℤ ↦ (𝑥 + 1)) ∈ V
31		vex 2815	. . . . . . . . . . . . . 14 ⊢ 𝑧 ∈ V
32	30, 31	fvex 5689	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℤ ↦ (𝑥 + 1))‘𝑧) ∈ V
33	32	ax-gen 1498	. . . . . . . . . . . 12 ⊢ ∀𝑧((𝑥 ∈ ℤ ↦ (𝑥 + 1))‘𝑧) ∈ V
34		0z 9584	. . . . . . . . . . . 12 ⊢ 0 ∈ ℤ
35		frecfnom 6631	. . . . . . . . . . . 12 ⊢ ((∀𝑧((𝑥 ∈ ℤ ↦ (𝑥 + 1))‘𝑧) ∈ V ∧ 0 ∈ ℤ) → frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) Fn ω)
36	33, 34, 35	mp2an 426	. . . . . . . . . . 11 ⊢ frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) Fn ω
37	17	fneq1i 5449	. . . . . . . . . . 11 ⊢ (𝐺 Fn ω ↔ frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) Fn ω)
38	36, 37	mpbir 146	. . . . . . . . . 10 ⊢ 𝐺 Fn ω
39		omex 4714	. . . . . . . . . 10 ⊢ ω ∈ V
40		fnex 5905	. . . . . . . . . 10 ⊢ ((𝐺 Fn ω ∧ ω ∈ V) → 𝐺 ∈ V)
41	38, 39, 40	mp2an 426	. . . . . . . . 9 ⊢ 𝐺 ∈ V
42	41	cnvex 5300	. . . . . . . 8 ⊢ ◡𝐺 ∈ V
43		vex 2815	. . . . . . . 8 ⊢ 𝑚 ∈ V
44	42, 43	fvex 5689	. . . . . . 7 ⊢ (◡𝐺‘𝑚) ∈ V
45		en2sn 7054	. . . . . . 7 ⊢ (((𝑚 + 1) ∈ ℕ0 ∧ (◡𝐺‘𝑚) ∈ V) → {(𝑚 + 1)} ≈ {(◡𝐺‘𝑚)})
46	28, 44, 45	sylancl 413	. . . . . 6 ⊢ (𝑚 ∈ ℕ0 → {(𝑚 + 1)} ≈ {(◡𝐺‘𝑚)})
47	46	adantr 276	. . . . 5 ⊢ ((𝑚 ∈ ℕ0 ∧ (1...𝑚) ≈ (◡𝐺‘𝑚)) → {(𝑚 + 1)} ≈ {(◡𝐺‘𝑚)})
48		fzp1disj 10410	. . . . . 6 ⊢ ((1...𝑚) ∩ {(𝑚 + 1)}) = ∅
49	48	a1i 9	. . . . 5 ⊢ ((𝑚 ∈ ℕ0 ∧ (1...𝑚) ≈ (◡𝐺‘𝑚)) → ((1...𝑚) ∩ {(𝑚 + 1)}) = ∅)
50		f1ocnvdm 5953	. . . . . . . . . 10 ⊢ ((𝐺:ω–1-1-onto→(ℤ≥‘0) ∧ 𝑚 ∈ (ℤ≥‘0)) → (◡𝐺‘𝑚) ∈ ω)
51	19, 50	mpan 424	. . . . . . . . 9 ⊢ (𝑚 ∈ (ℤ≥‘0) → (◡𝐺‘𝑚) ∈ ω)
52		nn0uz 9885	. . . . . . . . 9 ⊢ ℕ0 = (ℤ≥‘0)
53	51, 52	eleq2s 2327	. . . . . . . 8 ⊢ (𝑚 ∈ ℕ0 → (◡𝐺‘𝑚) ∈ ω)
54		nnord 4733	. . . . . . . 8 ⊢ ((◡𝐺‘𝑚) ∈ ω → Ord (◡𝐺‘𝑚))
55		ordirr 4663	. . . . . . . 8 ⊢ (Ord (◡𝐺‘𝑚) → ¬ (◡𝐺‘𝑚) ∈ (◡𝐺‘𝑚))
56	53, 54, 55	3syl 17	. . . . . . 7 ⊢ (𝑚 ∈ ℕ0 → ¬ (◡𝐺‘𝑚) ∈ (◡𝐺‘𝑚))
57	56	adantr 276	. . . . . 6 ⊢ ((𝑚 ∈ ℕ0 ∧ (1...𝑚) ≈ (◡𝐺‘𝑚)) → ¬ (◡𝐺‘𝑚) ∈ (◡𝐺‘𝑚))
58		disjsn 3750	. . . . . 6 ⊢ (((◡𝐺‘𝑚) ∩ {(◡𝐺‘𝑚)}) = ∅ ↔ ¬ (◡𝐺‘𝑚) ∈ (◡𝐺‘𝑚))
59	57, 58	sylibr 134	. . . . 5 ⊢ ((𝑚 ∈ ℕ0 ∧ (1...𝑚) ≈ (◡𝐺‘𝑚)) → ((◡𝐺‘𝑚) ∩ {(◡𝐺‘𝑚)}) = ∅)
60		unen 7057	. . . . 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)}))
61	27, 47, 49, 59, 60	syl22anc 1275	. . . 4 ⊢ ((𝑚 ∈ ℕ0 ∧ (1...𝑚) ≈ (◡𝐺‘𝑚)) → ((1...𝑚) ∪ {(𝑚 + 1)}) ≈ ((◡𝐺‘𝑚) ∪ {(◡𝐺‘𝑚)}))
62		1z 9599	. . . . . 6 ⊢ 1 ∈ ℤ
63		1m1e0 9302	. . . . . . . . . 10 ⊢ (1 − 1) = 0
64	63	fveq2i 5672	. . . . . . . . 9 ⊢ (ℤ≥‘(1 − 1)) = (ℤ≥‘0)
65	52, 64	eqtr4i 2256	. . . . . . . 8 ⊢ ℕ0 = (ℤ≥‘(1 − 1))
66	65	eleq2i 2299	. . . . . . 7 ⊢ (𝑚 ∈ ℕ0 ↔ 𝑚 ∈ (ℤ≥‘(1 − 1)))
67	66	biimpi 120	. . . . . 6 ⊢ (𝑚 ∈ ℕ0 → 𝑚 ∈ (ℤ≥‘(1 − 1)))
68		fzsuc2 10409	. . . . . 6 ⊢ ((1 ∈ ℤ ∧ 𝑚 ∈ (ℤ≥‘(1 − 1))) → (1...(𝑚 + 1)) = ((1...𝑚) ∪ {(𝑚 + 1)}))
69	62, 67, 68	sylancr 414	. . . . 5 ⊢ (𝑚 ∈ ℕ0 → (1...(𝑚 + 1)) = ((1...𝑚) ∪ {(𝑚 + 1)}))
70	69	adantr 276	. . . 4 ⊢ ((𝑚 ∈ ℕ0 ∧ (1...𝑚) ≈ (◡𝐺‘𝑚)) → (1...(𝑚 + 1)) = ((1...𝑚) ∪ {(𝑚 + 1)}))
71		peano2 4716	. . . . . . . . 9 ⊢ ((◡𝐺‘𝑚) ∈ ω → suc (◡𝐺‘𝑚) ∈ ω)
72	53, 71	syl 14	. . . . . . . 8 ⊢ (𝑚 ∈ ℕ0 → suc (◡𝐺‘𝑚) ∈ ω)
73	72, 19	jctil 312	. . . . . . 7 ⊢ (𝑚 ∈ ℕ0 → (𝐺:ω–1-1-onto→(ℤ≥‘0) ∧ suc (◡𝐺‘𝑚) ∈ ω))
74		0zd 9585	. . . . . . . . . 10 ⊢ ((◡𝐺‘𝑚) ∈ ω → 0 ∈ ℤ)
75		id 19	. . . . . . . . . 10 ⊢ ((◡𝐺‘𝑚) ∈ ω → (◡𝐺‘𝑚) ∈ ω)
76	74, 17, 75	frec2uzsucd 10759	. . . . . . . . 9 ⊢ ((◡𝐺‘𝑚) ∈ ω → (𝐺‘suc (◡𝐺‘𝑚)) = ((𝐺‘(◡𝐺‘𝑚)) + 1))
77	53, 76	syl 14	. . . . . . . 8 ⊢ (𝑚 ∈ ℕ0 → (𝐺‘suc (◡𝐺‘𝑚)) = ((𝐺‘(◡𝐺‘𝑚)) + 1))
78	52	eleq2i 2299	. . . . . . . . . . 11 ⊢ (𝑚 ∈ ℕ0 ↔ 𝑚 ∈ (ℤ≥‘0))
79	78	biimpi 120	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ0 → 𝑚 ∈ (ℤ≥‘0))
80		f1ocnvfv2 5950	. . . . . . . . . 10 ⊢ ((𝐺:ω–1-1-onto→(ℤ≥‘0) ∧ 𝑚 ∈ (ℤ≥‘0)) → (𝐺‘(◡𝐺‘𝑚)) = 𝑚)
81	19, 79, 80	sylancr 414	. . . . . . . . 9 ⊢ (𝑚 ∈ ℕ0 → (𝐺‘(◡𝐺‘𝑚)) = 𝑚)
82	81	oveq1d 6064	. . . . . . . 8 ⊢ (𝑚 ∈ ℕ0 → ((𝐺‘(◡𝐺‘𝑚)) + 1) = (𝑚 + 1))
83	77, 82	eqtrd 2265	. . . . . . 7 ⊢ (𝑚 ∈ ℕ0 → (𝐺‘suc (◡𝐺‘𝑚)) = (𝑚 + 1))
84		f1ocnvfv 5951	. . . . . . 7 ⊢ ((𝐺:ω–1-1-onto→(ℤ≥‘0) ∧ suc (◡𝐺‘𝑚) ∈ ω) → ((𝐺‘suc (◡𝐺‘𝑚)) = (𝑚 + 1) → (◡𝐺‘(𝑚 + 1)) = suc (◡𝐺‘𝑚)))
85	73, 83, 84	sylc 62	. . . . . 6 ⊢ (𝑚 ∈ ℕ0 → (◡𝐺‘(𝑚 + 1)) = suc (◡𝐺‘𝑚))
86	85	adantr 276	. . . . 5 ⊢ ((𝑚 ∈ ℕ0 ∧ (1...𝑚) ≈ (◡𝐺‘𝑚)) → (◡𝐺‘(𝑚 + 1)) = suc (◡𝐺‘𝑚))
87		df-suc 4491	. . . . 5 ⊢ suc (◡𝐺‘𝑚) = ((◡𝐺‘𝑚) ∪ {(◡𝐺‘𝑚)})
88	86, 87	eqtrdi 2281	. . . 4 ⊢ ((𝑚 ∈ ℕ0 ∧ (1...𝑚) ≈ (◡𝐺‘𝑚)) → (◡𝐺‘(𝑚 + 1)) = ((◡𝐺‘𝑚) ∪ {(◡𝐺‘𝑚)}))
89	61, 70, 88	3brtr4d 4140	. . 3 ⊢ ((𝑚 ∈ ℕ0 ∧ (1...𝑚) ≈ (◡𝐺‘𝑚)) → (1...(𝑚 + 1)) ≈ (◡𝐺‘(𝑚 + 1)))
90	89	ex 115	. 2 ⊢ (𝑚 ∈ ℕ0 → ((1...𝑚) ≈ (◡𝐺‘𝑚) → (1...(𝑚 + 1)) ≈ (◡𝐺‘(𝑚 + 1))))
91	3, 6, 9, 12, 26, 90	nn0ind 9688	1 ⊢ (𝑁 ∈ ℕ0 → (1...𝑁) ≈ (◡𝐺‘𝑁))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104  ∀wal 1396   = wceq 1398  ⊤wtru 1399   ∈ wcel 2203  Vcvv 2812   ∪ cun 3208   ∩ cin 3209  ∅c0 3507  {csn 3688   class class class wbr 4108   ↦ cmpt 4170  Ord word 4482  suc csuc 4485  ωcom 4711  ^◡ccnv 4747   Fn wfn 5346  –1-1-onto→wf1o 5350  ‘cfv 5351  (class class class)co 6049  freccfrec 6620   ≈ cen 6972  0cc0 8123  1c1 8124   + caddc 8126   − cmin 8440  ℕ₀cn0 9492  ℤcz 9573  ℤ_≥cuz 9849  ...cfz 10338

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-iinf 4709  ax-cnex 8214  ax-resscn 8215  ax-1cn 8216  ax-1re 8217  ax-icn 8218  ax-addcl 8219  ax-addrcl 8220  ax-mulcl 8221  ax-addcom 8223  ax-addass 8225  ax-distr 8227  ax-i2m1 8228  ax-0lt1 8229  ax-0id 8231  ax-rnegex 8232  ax-cnre 8234  ax-pre-ltirr 8235  ax-pre-ltwlin 8236  ax-pre-lttrn 8237  ax-pre-apti 8238  ax-pre-ltadd 8239

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-tr 4208  df-id 4413  df-iord 4486  df-on 4488  df-ilim 4489  df-suc 4491  df-iom 4712  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-recs 6535  df-frec 6621  df-1o 6646  df-er 6766  df-en 6975  df-pnf 8306  df-mnf 8307  df-xr 8308  df-ltxr 8309  df-le 8310  df-sub 8442  df-neg 8443  df-inn 9234  df-n0 9493  df-z 9574  df-uz 9850  df-fz 10339

This theorem is referenced by:  frecfzen2  10785  hashfz1  11141