Theorem 1tonninf 10747

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 5649	. . . 4 ⊢ (𝐼‘1) = (((𝐹 ∘ ◡𝐺) ∪ {⟨+∞, (ω × {1o})⟩})‘1)
3		1nn0 9461	. . . . . 6 ⊢ 1 ∈ ℕ0
4		nn0xnn0 9512	. . . . . 6 ⊢ (1 ∈ ℕ0 → 1 ∈ ℕ0*)
5	3, 4	ax-mp 5	. . . . 5 ⊢ 1 ∈ ℕ0*
6		nn0nepnf 9516	. . . . . . 7 ⊢ (1 ∈ ℕ0 → 1 ≠ +∞)
7	3, 6	ax-mp 5	. . . . . 6 ⊢ 1 ≠ +∞
8	7	necomi 2488	. . . . 5 ⊢ +∞ ≠ 1
9		fvunsng 5856	. . . . 5 ⊢ ((1 ∈ ℕ0* ∧ +∞ ≠ 1) → (((𝐹 ∘ ◡𝐺) ∪ {⟨+∞, (ω × {1o})⟩})‘1) = ((𝐹 ∘ ◡𝐺)‘1))
10	5, 8, 9	mp2an 426	. . . 4 ⊢ (((𝐹 ∘ ◡𝐺) ∪ {⟨+∞, (ω × {1o})⟩})‘1) = ((𝐹 ∘ ◡𝐺)‘1)
11		fxnn0nninf.g	. . . . . . . 8 ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)
12	11	frechashgf1o 10734	. . . . . . 7 ⊢ 𝐺:ω–1-1-onto→ℕ0
13		f1ocnv 5605	. . . . . . 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 5592	. . . . . 6 ⊢ (◡𝐺:ℕ0–1-1-onto→ω → ◡𝐺:ℕ0⟶ω)
16	14, 15	ax-mp 5	. . . . 5 ⊢ ◡𝐺:ℕ0⟶ω
17		fvco3 5726	. . . . 5 ⊢ ((◡𝐺:ℕ0⟶ω ∧ 1 ∈ ℕ0) → ((𝐹 ∘ ◡𝐺)‘1) = (𝐹‘(◡𝐺‘1)))
18	16, 3, 17	mp2an 426	. . . 4 ⊢ ((𝐹 ∘ ◡𝐺)‘1) = (𝐹‘(◡𝐺‘1))
19	2, 10, 18	3eqtri 2256	. . 3 ⊢ (𝐼‘1) = (𝐹‘(◡𝐺‘1))
20		df-1o 6625	. . . . . . 7 ⊢ 1o = suc ∅
21	20	fveq2i 5651	. . . . . 6 ⊢ (𝐺‘1o) = (𝐺‘suc ∅)
22		0zd 9534	. . . . . . . . 9 ⊢ (⊤ → 0 ∈ ℤ)
23		peano1 4698	. . . . . . . . . 10 ⊢ ∅ ∈ ω
24	23	a1i 9	. . . . . . . . 9 ⊢ (⊤ → ∅ ∈ ω)
25	22, 11, 24	frec2uzsucd 10707	. . . . . . . 8 ⊢ (⊤ → (𝐺‘suc ∅) = ((𝐺‘∅) + 1))
26	25	mptru 1407	. . . . . . 7 ⊢ (𝐺‘suc ∅) = ((𝐺‘∅) + 1)
27	22, 11	frec2uz0d 10705	. . . . . . . . 9 ⊢ (⊤ → (𝐺‘∅) = 0)
28	27	mptru 1407	. . . . . . . 8 ⊢ (𝐺‘∅) = 0
29	28	oveq1i 6038	. . . . . . 7 ⊢ ((𝐺‘∅) + 1) = (0 + 1)
30	26, 29	eqtri 2252	. . . . . 6 ⊢ (𝐺‘suc ∅) = (0 + 1)
31		0p1e1 9300	. . . . . 6 ⊢ (0 + 1) = 1
32	21, 30, 31	3eqtri 2256	. . . . 5 ⊢ (𝐺‘1o) = 1
33		1onn 6731	. . . . . 6 ⊢ 1o ∈ ω
34		f1ocnvfv 5930	. . . . . 6 ⊢ ((𝐺:ω–1-1-onto→ℕ0 ∧ 1o ∈ ω) → ((𝐺‘1o) = 1 → (◡𝐺‘1) = 1o))
35	12, 33, 34	mp2an 426	. . . . 5 ⊢ ((𝐺‘1o) = 1 → (◡𝐺‘1) = 1o)
36	32, 35	ax-mp 5	. . . 4 ⊢ (◡𝐺‘1) = 1o
37	36	fveq2i 5651	. . 3 ⊢ (𝐹‘(◡𝐺‘1)) = (𝐹‘1o)
38		eleq2 2295	. . . . . . 7 ⊢ (𝑛 = 1o → (𝑖 ∈ 𝑛 ↔ 𝑖 ∈ 1o))
39	38	ifbid 3631	. . . . . 6 ⊢ (𝑛 = 1o → if(𝑖 ∈ 𝑛, 1o, ∅) = if(𝑖 ∈ 1o, 1o, ∅))
40	39	mpteq2dv 4185	. . . . 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 4697	. . . . . 6 ⊢ ω ∈ V
43	42	mptex 5890	. . . . 5 ⊢ (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅)) ∈ V
44	40, 41, 43	fvmpt3i 5735	. . . 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 2256	. 2 ⊢ (𝐼‘1) = (𝑖 ∈ ω ↦ if(𝑖 ∈ 1o, 1o, ∅))
47		el1o 6648	. . . 4 ⊢ (𝑖 ∈ 1o ↔ 𝑖 = ∅)
48		ifbi 3630	. . . 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 4181	. 2 ⊢ (𝑖 ∈ ω ↦ if(𝑖 ∈ 1o, 1o, ∅)) = (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, ∅))
51		eqeq1 2238	. . . 4 ⊢ (𝑖 = 𝑥 → (𝑖 = ∅ ↔ 𝑥 = ∅))
52	51	ifbid 3631	. . 3 ⊢ (𝑖 = 𝑥 → if(𝑖 = ∅, 1o, ∅) = if(𝑥 = ∅, 1o, ∅))
53	52	cbvmptv 4190	. 2 ⊢ (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, ∅)) = (𝑥 ∈ ω ↦ if(𝑥 = ∅, 1o, ∅))
54	46, 50, 53	3eqtri 2256	1 ⊢ (𝐼‘1) = (𝑥 ∈ ω ↦ if(𝑥 = ∅, 1o, ∅))

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1398  ⊤wtru 1399   ∈ wcel 2202   ≠ wne 2403   ∪ cun 3199  ∅c0 3496  ifcif 3607  {csn 3673  ⟨cop 3676   ↦ cmpt 4155  suc csuc 4468  ωcom 4694   × cxp 4729  ^◡ccnv 4730   ∘ ccom 4735  ⟶wf 5329  –1-1-onto→wf1o 5332  ‘cfv 5333  (class class class)co 6028  freccfrec 6599  1_oc1o 6618  0cc0 8075  1c1 8076   + caddc 8078  +∞cpnf 8254  ℕ₀cn0 9445  ℕ₀^*cxnn0 9508  ℤcz 9522

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-addcom 8175  ax-addass 8177  ax-distr 8179  ax-i2m1 8180  ax-0lt1 8181  ax-0id 8183  ax-rnegex 8184  ax-cnre 8186  ax-pre-ltirr 8187  ax-pre-ltwlin 8188  ax-pre-lttrn 8189  ax-pre-ltadd 8191

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 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-recs 6514  df-frec 6600  df-1o 6625  df-pnf 8259  df-mnf 8260  df-xr 8261  df-ltxr 8262  df-le 8263  df-sub 8395  df-neg 8396  df-inn 9187  df-n0 9446  df-xnn0 9509  df-z 9523  df-uz 9799

This theorem is referenced by: (None)