Theorem 1tonninf 10533

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 5559	. . . 4 ⊢ (𝐼‘1) = (((𝐹 ∘ ◡𝐺) ∪ {⟨+∞, (ω × {1o})⟩})‘1)
3		1nn0 9265	. . . . . 6 ⊢ 1 ∈ ℕ0
4		nn0xnn0 9316	. . . . . 6 ⊢ (1 ∈ ℕ0 → 1 ∈ ℕ0*)
5	3, 4	ax-mp 5	. . . . 5 ⊢ 1 ∈ ℕ0*
6		nn0nepnf 9320	. . . . . . 7 ⊢ (1 ∈ ℕ0 → 1 ≠ +∞)
7	3, 6	ax-mp 5	. . . . . 6 ⊢ 1 ≠ +∞
8	7	necomi 2452	. . . . 5 ⊢ +∞ ≠ 1
9		fvunsng 5756	. . . . 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 10520	. . . . . . 7 ⊢ 𝐺:ω–1-1-onto→ℕ0
13		f1ocnv 5517	. . . . . . 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 5504	. . . . . 6 ⊢ (◡𝐺:ℕ0–1-1-onto→ω → ◡𝐺:ℕ0⟶ω)
16	14, 15	ax-mp 5	. . . . 5 ⊢ ◡𝐺:ℕ0⟶ω
17		fvco3 5632	. . . . 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 2221	. . 3 ⊢ (𝐼‘1) = (𝐹‘(◡𝐺‘1))
20		df-1o 6474	. . . . . . 7 ⊢ 1o = suc ∅
21	20	fveq2i 5561	. . . . . 6 ⊢ (𝐺‘1o) = (𝐺‘suc ∅)
22		0zd 9338	. . . . . . . . 9 ⊢ (⊤ → 0 ∈ ℤ)
23		peano1 4630	. . . . . . . . . 10 ⊢ ∅ ∈ ω
24	23	a1i 9	. . . . . . . . 9 ⊢ (⊤ → ∅ ∈ ω)
25	22, 11, 24	frec2uzsucd 10493	. . . . . . . 8 ⊢ (⊤ → (𝐺‘suc ∅) = ((𝐺‘∅) + 1))
26	25	mptru 1373	. . . . . . 7 ⊢ (𝐺‘suc ∅) = ((𝐺‘∅) + 1)
27	22, 11	frec2uz0d 10491	. . . . . . . . 9 ⊢ (⊤ → (𝐺‘∅) = 0)
28	27	mptru 1373	. . . . . . . 8 ⊢ (𝐺‘∅) = 0
29	28	oveq1i 5932	. . . . . . 7 ⊢ ((𝐺‘∅) + 1) = (0 + 1)
30	26, 29	eqtri 2217	. . . . . 6 ⊢ (𝐺‘suc ∅) = (0 + 1)
31		0p1e1 9104	. . . . . 6 ⊢ (0 + 1) = 1
32	21, 30, 31	3eqtri 2221	. . . . 5 ⊢ (𝐺‘1o) = 1
33		1onn 6578	. . . . . 6 ⊢ 1o ∈ ω
34		f1ocnvfv 5826	. . . . . 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 5561	. . 3 ⊢ (𝐹‘(◡𝐺‘1)) = (𝐹‘1o)
38		eleq2 2260	. . . . . . 7 ⊢ (𝑛 = 1o → (𝑖 ∈ 𝑛 ↔ 𝑖 ∈ 1o))
39	38	ifbid 3582	. . . . . 6 ⊢ (𝑛 = 1o → if(𝑖 ∈ 𝑛, 1o, ∅) = if(𝑖 ∈ 1o, 1o, ∅))
40	39	mpteq2dv 4124	. . . . 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 4629	. . . . . 6 ⊢ ω ∈ V
43	42	mptex 5788	. . . . 5 ⊢ (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅)) ∈ V
44	40, 41, 43	fvmpt3i 5641	. . . 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 2221	. 2 ⊢ (𝐼‘1) = (𝑖 ∈ ω ↦ if(𝑖 ∈ 1o, 1o, ∅))
47		el1o 6495	. . . 4 ⊢ (𝑖 ∈ 1o ↔ 𝑖 = ∅)
48		ifbi 3581	. . . 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 4120	. 2 ⊢ (𝑖 ∈ ω ↦ if(𝑖 ∈ 1o, 1o, ∅)) = (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, ∅))
51		eqeq1 2203	. . . 4 ⊢ (𝑖 = 𝑥 → (𝑖 = ∅ ↔ 𝑥 = ∅))
52	51	ifbid 3582	. . 3 ⊢ (𝑖 = 𝑥 → if(𝑖 = ∅, 1o, ∅) = if(𝑥 = ∅, 1o, ∅))
53	52	cbvmptv 4129	. 2 ⊢ (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, ∅)) = (𝑥 ∈ ω ↦ if(𝑥 = ∅, 1o, ∅))
54	46, 50, 53	3eqtri 2221	1 ⊢ (𝐼‘1) = (𝑥 ∈ ω ↦ if(𝑥 = ∅, 1o, ∅))

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1364  ⊤wtru 1365   ∈ wcel 2167   ≠ wne 2367   ∪ cun 3155  ∅c0 3450  ifcif 3561  {csn 3622  ⟨cop 3625   ↦ cmpt 4094  suc csuc 4400  ωcom 4626   × cxp 4661  ^◡ccnv 4662   ∘ ccom 4667  ⟶wf 5254  –1-1-onto→wf1o 5257  ‘cfv 5258  (class class class)co 5922  freccfrec 6448  1_oc1o 6467  0cc0 7879  1c1 7880   + caddc 7882  +∞cpnf 8058  ℕ₀cn0 9249  ℕ₀^*cxnn0 9312  ℤcz 9326

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-0id 7987  ax-rnegex 7988  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-ltadd 7995

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-iord 4401  df-on 4403  df-ilim 4404  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-recs 6363  df-frec 6449  df-1o 6474  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-inn 8991  df-n0 9250  df-xnn0 9313  df-z 9327  df-uz 9602

This theorem is referenced by: (None)