Theorem 1tonninf 10440

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 5517	. . . 4 ⊢ (𝐼‘1) = (((𝐹 ∘ ◡𝐺) ∪ {⟨+∞, (ω × {1o})⟩})‘1)
3		1nn0 9192	. . . . . 6 ⊢ 1 ∈ ℕ0
4		nn0xnn0 9243	. . . . . 6 ⊢ (1 ∈ ℕ0 → 1 ∈ ℕ0*)
5	3, 4	ax-mp 5	. . . . 5 ⊢ 1 ∈ ℕ0*
6		nn0nepnf 9247	. . . . . . 7 ⊢ (1 ∈ ℕ0 → 1 ≠ +∞)
7	3, 6	ax-mp 5	. . . . . 6 ⊢ 1 ≠ +∞
8	7	necomi 2432	. . . . 5 ⊢ +∞ ≠ 1
9		fvunsng 5711	. . . . 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 10428	. . . . . . 7 ⊢ 𝐺:ω–1-1-onto→ℕ0
13		f1ocnv 5475	. . . . . . 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 5462	. . . . . 6 ⊢ (◡𝐺:ℕ0–1-1-onto→ω → ◡𝐺:ℕ0⟶ω)
16	14, 15	ax-mp 5	. . . . 5 ⊢ ◡𝐺:ℕ0⟶ω
17		fvco3 5588	. . . . 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 2202	. . 3 ⊢ (𝐼‘1) = (𝐹‘(◡𝐺‘1))
20		df-1o 6417	. . . . . . 7 ⊢ 1o = suc ∅
21	20	fveq2i 5519	. . . . . 6 ⊢ (𝐺‘1o) = (𝐺‘suc ∅)
22		0zd 9265	. . . . . . . . 9 ⊢ (⊤ → 0 ∈ ℤ)
23		peano1 4594	. . . . . . . . . 10 ⊢ ∅ ∈ ω
24	23	a1i 9	. . . . . . . . 9 ⊢ (⊤ → ∅ ∈ ω)
25	22, 11, 24	frec2uzsucd 10401	. . . . . . . 8 ⊢ (⊤ → (𝐺‘suc ∅) = ((𝐺‘∅) + 1))
26	25	mptru 1362	. . . . . . 7 ⊢ (𝐺‘suc ∅) = ((𝐺‘∅) + 1)
27	22, 11	frec2uz0d 10399	. . . . . . . . 9 ⊢ (⊤ → (𝐺‘∅) = 0)
28	27	mptru 1362	. . . . . . . 8 ⊢ (𝐺‘∅) = 0
29	28	oveq1i 5885	. . . . . . 7 ⊢ ((𝐺‘∅) + 1) = (0 + 1)
30	26, 29	eqtri 2198	. . . . . 6 ⊢ (𝐺‘suc ∅) = (0 + 1)
31		0p1e1 9033	. . . . . 6 ⊢ (0 + 1) = 1
32	21, 30, 31	3eqtri 2202	. . . . 5 ⊢ (𝐺‘1o) = 1
33		1onn 6521	. . . . . 6 ⊢ 1o ∈ ω
34		f1ocnvfv 5780	. . . . . 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 5519	. . 3 ⊢ (𝐹‘(◡𝐺‘1)) = (𝐹‘1o)
38		eleq2 2241	. . . . . . 7 ⊢ (𝑛 = 1o → (𝑖 ∈ 𝑛 ↔ 𝑖 ∈ 1o))
39	38	ifbid 3556	. . . . . 6 ⊢ (𝑛 = 1o → if(𝑖 ∈ 𝑛, 1o, ∅) = if(𝑖 ∈ 1o, 1o, ∅))
40	39	mpteq2dv 4095	. . . . 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 4593	. . . . . 6 ⊢ ω ∈ V
43	42	mptex 5743	. . . . 5 ⊢ (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅)) ∈ V
44	40, 41, 43	fvmpt3i 5597	. . . 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 2202	. 2 ⊢ (𝐼‘1) = (𝑖 ∈ ω ↦ if(𝑖 ∈ 1o, 1o, ∅))
47		el1o 6438	. . . 4 ⊢ (𝑖 ∈ 1o ↔ 𝑖 = ∅)
48		ifbi 3555	. . . 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 4091	. 2 ⊢ (𝑖 ∈ ω ↦ if(𝑖 ∈ 1o, 1o, ∅)) = (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, ∅))
51		eqeq1 2184	. . . 4 ⊢ (𝑖 = 𝑥 → (𝑖 = ∅ ↔ 𝑥 = ∅))
52	51	ifbid 3556	. . 3 ⊢ (𝑖 = 𝑥 → if(𝑖 = ∅, 1o, ∅) = if(𝑥 = ∅, 1o, ∅))
53	52	cbvmptv 4100	. 2 ⊢ (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, ∅)) = (𝑥 ∈ ω ↦ if(𝑥 = ∅, 1o, ∅))
54	46, 50, 53	3eqtri 2202	1 ⊢ (𝐼‘1) = (𝑥 ∈ ω ↦ if(𝑥 = ∅, 1o, ∅))

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1353  ⊤wtru 1354   ∈ wcel 2148   ≠ wne 2347   ∪ cun 3128  ∅c0 3423  ifcif 3535  {csn 3593  ⟨cop 3596   ↦ cmpt 4065  suc csuc 4366  ωcom 4590   × cxp 4625  ^◡ccnv 4626   ∘ ccom 4631  ⟶wf 5213  –1-1-onto→wf1o 5216  ‘cfv 5217  (class class class)co 5875  freccfrec 6391  1_oc1o 6410  0cc0 7811  1c1 7812   + caddc 7814  +∞cpnf 7989  ℕ₀cn0 9176  ℕ₀^*cxnn0 9239  ℤcz 9253

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588  ax-cnex 7902  ax-resscn 7903  ax-1cn 7904  ax-1re 7905  ax-icn 7906  ax-addcl 7907  ax-addrcl 7908  ax-mulcl 7909  ax-addcom 7911  ax-addass 7913  ax-distr 7915  ax-i2m1 7916  ax-0lt1 7917  ax-0id 7919  ax-rnegex 7920  ax-cnre 7922  ax-pre-ltirr 7923  ax-pre-ltwlin 7924  ax-pre-lttrn 7925  ax-pre-ltadd 7927

This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-if 3536  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-id 4294  df-iord 4367  df-on 4369  df-ilim 4370  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-riota 5831  df-ov 5878  df-oprab 5879  df-mpo 5880  df-recs 6306  df-frec 6392  df-1o 6417  df-pnf 7994  df-mnf 7995  df-xr 7996  df-ltxr 7997  df-le 7998  df-sub 8130  df-neg 8131  df-inn 8920  df-n0 9177  df-xnn0 9240  df-z 9254  df-uz 9529

This theorem is referenced by: (None)