Theorem 1tonninf 10396

Description: The mapping of one into ℕ_∞ is a sequence which is a one followed by zeroes. (Contributed by Jim Kingdon, 17-Jul-2022.)

Hypotheses

Ref	Expression
fxnn0nninf.g	⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)
fxnn0nninf.f	⊢ 𝐹 = (𝑛 ∈ ω ↦ (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅)))
fxnn0nninf.i	⊢ 𝐼 = ((𝐹 ∘ ◡𝐺) ∪ {⟨+∞, (ω × {1o})⟩})

Assertion

Ref	Expression
1tonninf	⊢ (𝐼‘1) = (𝑥 ∈ ω ↦ if(𝑥 = ∅, 1o, ∅))

Distinct variable groups:   𝑖,𝑛   𝑥,𝑖

Allowed substitution hints:   𝐹(𝑥,𝑖,𝑛)   𝐺(𝑥,𝑖,𝑛)   𝐼(𝑥,𝑖,𝑛)

Proof of Theorem 1tonninf

Step	Hyp	Ref	Expression
1		fxnn0nninf.i	. . . . 5 ⊢ 𝐼 = ((𝐹 ∘ ◡𝐺) ∪ {⟨+∞, (ω × {1o})⟩})
2	1	fveq1i 5497	. . . 4 ⊢ (𝐼‘1) = (((𝐹 ∘ ◡𝐺) ∪ {⟨+∞, (ω × {1o})⟩})‘1)
3		1nn0 9151	. . . . . 6 ⊢ 1 ∈ ℕ0
4		nn0xnn0 9202	. . . . . 6 ⊢ (1 ∈ ℕ0 → 1 ∈ ℕ0*)
5	3, 4	ax-mp 5	. . . . 5 ⊢ 1 ∈ ℕ0*
6		nn0nepnf 9206	. . . . . . 7 ⊢ (1 ∈ ℕ0 → 1 ≠ +∞)
7	3, 6	ax-mp 5	. . . . . 6 ⊢ 1 ≠ +∞
8	7	necomi 2425	. . . . 5 ⊢ +∞ ≠ 1
9		fvunsng 5690	. . . . 5 ⊢ ((1 ∈ ℕ0* ∧ +∞ ≠ 1) → (((𝐹 ∘ ◡𝐺) ∪ {⟨+∞, (ω × {1o})⟩})‘1) = ((𝐹 ∘ ◡𝐺)‘1))
10	5, 8, 9	mp2an 424	. . . 4 ⊢ (((𝐹 ∘ ◡𝐺) ∪ {⟨+∞, (ω × {1o})⟩})‘1) = ((𝐹 ∘ ◡𝐺)‘1)
11		fxnn0nninf.g	. . . . . . . 8 ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)
12	11	frechashgf1o 10384	. . . . . . 7 ⊢ 𝐺:ω–1-1-onto→ℕ0
13		f1ocnv 5455	. . . . . . 7 ⊢ (𝐺:ω–1-1-onto→ℕ0 → ◡𝐺:ℕ0–1-1-onto→ω)
14	12, 13	ax-mp 5	. . . . . 6 ⊢ ◡𝐺:ℕ0–1-1-onto→ω
15		f1of 5442	. . . . . 6 ⊢ (◡𝐺:ℕ0–1-1-onto→ω → ◡𝐺:ℕ0⟶ω)
16	14, 15	ax-mp 5	. . . . 5 ⊢ ◡𝐺:ℕ0⟶ω
17		fvco3 5567	. . . . 5 ⊢ ((◡𝐺:ℕ0⟶ω ∧ 1 ∈ ℕ0) → ((𝐹 ∘ ◡𝐺)‘1) = (𝐹‘(◡𝐺‘1)))
18	16, 3, 17	mp2an 424	. . . 4 ⊢ ((𝐹 ∘ ◡𝐺)‘1) = (𝐹‘(◡𝐺‘1))
19	2, 10, 18	3eqtri 2195	. . 3 ⊢ (𝐼‘1) = (𝐹‘(◡𝐺‘1))
20		df-1o 6395	. . . . . . 7 ⊢ 1o = suc ∅
21	20	fveq2i 5499	. . . . . 6 ⊢ (𝐺‘1o) = (𝐺‘suc ∅)
22		0zd 9224	. . . . . . . . 9 ⊢ (⊤ → 0 ∈ ℤ)
23		peano1 4578	. . . . . . . . . 10 ⊢ ∅ ∈ ω
24	23	a1i 9	. . . . . . . . 9 ⊢ (⊤ → ∅ ∈ ω)
25	22, 11, 24	frec2uzsucd 10357	. . . . . . . 8 ⊢ (⊤ → (𝐺‘suc ∅) = ((𝐺‘∅) + 1))
26	25	mptru 1357	. . . . . . 7 ⊢ (𝐺‘suc ∅) = ((𝐺‘∅) + 1)
27	22, 11	frec2uz0d 10355	. . . . . . . . 9 ⊢ (⊤ → (𝐺‘∅) = 0)
28	27	mptru 1357	. . . . . . . 8 ⊢ (𝐺‘∅) = 0
29	28	oveq1i 5863	. . . . . . 7 ⊢ ((𝐺‘∅) + 1) = (0 + 1)
30	26, 29	eqtri 2191	. . . . . 6 ⊢ (𝐺‘suc ∅) = (0 + 1)
31		0p1e1 8992	. . . . . 6 ⊢ (0 + 1) = 1
32	21, 30, 31	3eqtri 2195	. . . . 5 ⊢ (𝐺‘1o) = 1
33		1onn 6499	. . . . . 6 ⊢ 1o ∈ ω
34		f1ocnvfv 5758	. . . . . 6 ⊢ ((𝐺:ω–1-1-onto→ℕ0 ∧ 1o ∈ ω) → ((𝐺‘1o) = 1 → (◡𝐺‘1) = 1o))
35	12, 33, 34	mp2an 424	. . . . 5 ⊢ ((𝐺‘1o) = 1 → (◡𝐺‘1) = 1o)
36	32, 35	ax-mp 5	. . . 4 ⊢ (◡𝐺‘1) = 1o
37	36	fveq2i 5499	. . 3 ⊢ (𝐹‘(◡𝐺‘1)) = (𝐹‘1o)
38		eleq2 2234	. . . . . . 7 ⊢ (𝑛 = 1o → (𝑖 ∈ 𝑛 ↔ 𝑖 ∈ 1o))
39	38	ifbid 3547	. . . . . 6 ⊢ (𝑛 = 1o → if(𝑖 ∈ 𝑛, 1o, ∅) = if(𝑖 ∈ 1o, 1o, ∅))
40	39	mpteq2dv 4080	. . . . 5 ⊢ (𝑛 = 1o → (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅)) = (𝑖 ∈ ω ↦ if(𝑖 ∈ 1o, 1o, ∅)))
41		fxnn0nninf.f	. . . . 5 ⊢ 𝐹 = (𝑛 ∈ ω ↦ (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅)))
42		omex 4577	. . . . . 6 ⊢ ω ∈ V
43	42	mptex 5722	. . . . 5 ⊢ (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅)) ∈ V
44	40, 41, 43	fvmpt3i 5576	. . . 4 ⊢ (1o ∈ ω → (𝐹‘1o) = (𝑖 ∈ ω ↦ if(𝑖 ∈ 1o, 1o, ∅)))
45	33, 44	ax-mp 5	. . 3 ⊢ (𝐹‘1o) = (𝑖 ∈ ω ↦ if(𝑖 ∈ 1o, 1o, ∅))
46	19, 37, 45	3eqtri 2195	. 2 ⊢ (𝐼‘1) = (𝑖 ∈ ω ↦ if(𝑖 ∈ 1o, 1o, ∅))
47		el1o 6416	. . . 4 ⊢ (𝑖 ∈ 1o ↔ 𝑖 = ∅)
48		ifbi 3546	. . . 4 ⊢ ((𝑖 ∈ 1o ↔ 𝑖 = ∅) → if(𝑖 ∈ 1o, 1o, ∅) = if(𝑖 = ∅, 1o, ∅))
49	47, 48	ax-mp 5	. . 3 ⊢ if(𝑖 ∈ 1o, 1o, ∅) = if(𝑖 = ∅, 1o, ∅)
50	49	mpteq2i 4076	. 2 ⊢ (𝑖 ∈ ω ↦ if(𝑖 ∈ 1o, 1o, ∅)) = (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, ∅))
51		eqeq1 2177	. . . 4 ⊢ (𝑖 = 𝑥 → (𝑖 = ∅ ↔ 𝑥 = ∅))
52	51	ifbid 3547	. . 3 ⊢ (𝑖 = 𝑥 → if(𝑖 = ∅, 1o, ∅) = if(𝑥 = ∅, 1o, ∅))
53	52	cbvmptv 4085	. 2 ⊢ (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, ∅)) = (𝑥 ∈ ω ↦ if(𝑥 = ∅, 1o, ∅))
54	46, 50, 53	3eqtri 2195	1 ⊢ (𝐼‘1) = (𝑥 ∈ ω ↦ if(𝑥 = ∅, 1o, ∅))

Syntax hints:   → wi 4   ↔ wb 104   = wceq 1348  ⊤wtru 1349   ∈ wcel 2141   ≠ wne 2340   ∪ cun 3119  ∅c0 3414  ifcif 3526  {csn 3583  ⟨cop 3586   ↦ cmpt 4050  suc csuc 4350  ωcom 4574   × cxp 4609  ^◡ccnv 4610   ∘ ccom 4615  ⟶wf 5194  –1-1-onto→wf1o 5197  ‘cfv 5198  (class class class)co 5853  freccfrec 6369  1_oc1o 6388  0cc0 7774  1c1 7775   + caddc 7777  +∞cpnf 7951  ℕ₀cn0 9135  ℕ₀^*cxnn0 9198  ℤcz 9212

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-addcom 7874  ax-addass 7876  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-0id 7882  ax-rnegex 7883  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-ltadd 7890

This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-recs 6284  df-frec 6370  df-1o 6395  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-inn 8879  df-n0 9136  df-xnn0 9199  df-z 9213  df-uz 9488

This theorem is referenced by: (None)