Theorem nninfwlporlemd 7231

Description: Given two countably infinite sequences of zeroes and ones, they are equal if and only if a sequence formed by pointwise comparing them is all ones. (Contributed by Jim Kingdon, 6-Dec-2024.)

Hypotheses

Ref	Expression
nninfwlporlem.x	⊢ (𝜑 → 𝑋:ω⟶2o)
nninfwlporlem.y	⊢ (𝜑 → 𝑌:ω⟶2o)
nninfwlporlem.d	⊢ 𝐷 = (𝑖 ∈ ω ↦ if((𝑋‘𝑖) = (𝑌‘𝑖), 1o, ∅))

Assertion

Ref	Expression
nninfwlporlemd	⊢ (𝜑 → (𝑋 = 𝑌 ↔ 𝐷 = (𝑖 ∈ ω ↦ 1o)))

Distinct variable groups:   𝐷,𝑖   𝑖,𝑋   𝑖,𝑌   𝜑,𝑖

Proof of Theorem nninfwlporlemd

Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		1n0 6485	. . . . . . . . 9 ⊢ 1o ≠ ∅
2	1	neii 2366	. . . . . . . 8 ⊢ ¬ 1o = ∅
3	2	intnan 930	. . . . . . 7 ⊢ ¬ (¬ (𝑋‘𝑖) = (𝑌‘𝑖) ∧ 1o = ∅)
4	3	biorfi 747	. . . . . 6 ⊢ ((𝑋‘𝑖) = (𝑌‘𝑖) ↔ ((𝑋‘𝑖) = (𝑌‘𝑖) ∨ (¬ (𝑋‘𝑖) = (𝑌‘𝑖) ∧ 1o = ∅)))
5		eqid 2193	. . . . . . . 8 ⊢ 1o = 1o
6	5	biantru 302	. . . . . . 7 ⊢ ((𝑋‘𝑖) = (𝑌‘𝑖) ↔ ((𝑋‘𝑖) = (𝑌‘𝑖) ∧ 1o = 1o))
7	6	orbi1i 764	. . . . . 6 ⊢ (((𝑋‘𝑖) = (𝑌‘𝑖) ∨ (¬ (𝑋‘𝑖) = (𝑌‘𝑖) ∧ 1o = ∅)) ↔ (((𝑋‘𝑖) = (𝑌‘𝑖) ∧ 1o = 1o) ∨ (¬ (𝑋‘𝑖) = (𝑌‘𝑖) ∧ 1o = ∅)))
8	4, 7	bitri 184	. . . . 5 ⊢ ((𝑋‘𝑖) = (𝑌‘𝑖) ↔ (((𝑋‘𝑖) = (𝑌‘𝑖) ∧ 1o = 1o) ∨ (¬ (𝑋‘𝑖) = (𝑌‘𝑖) ∧ 1o = ∅)))
9		eqcom 2195	. . . . . 6 ⊢ (1o = (𝐷‘𝑖) ↔ (𝐷‘𝑖) = 1o)
10		nninfwlporlem.d	. . . . . . . . . 10 ⊢ 𝐷 = (𝑖 ∈ ω ↦ if((𝑋‘𝑖) = (𝑌‘𝑖), 1o, ∅))
11		fveq2 5554	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑗 → (𝑋‘𝑖) = (𝑋‘𝑗))
12		fveq2 5554	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑗 → (𝑌‘𝑖) = (𝑌‘𝑗))
13	11, 12	eqeq12d 2208	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑗 → ((𝑋‘𝑖) = (𝑌‘𝑖) ↔ (𝑋‘𝑗) = (𝑌‘𝑗)))
14	13	ifbid 3578	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑗 → if((𝑋‘𝑖) = (𝑌‘𝑖), 1o, ∅) = if((𝑋‘𝑗) = (𝑌‘𝑗), 1o, ∅))
15	14	cbvmptv 4125	. . . . . . . . . 10 ⊢ (𝑖 ∈ ω ↦ if((𝑋‘𝑖) = (𝑌‘𝑖), 1o, ∅)) = (𝑗 ∈ ω ↦ if((𝑋‘𝑗) = (𝑌‘𝑗), 1o, ∅))
16	10, 15	eqtri 2214	. . . . . . . . 9 ⊢ 𝐷 = (𝑗 ∈ ω ↦ if((𝑋‘𝑗) = (𝑌‘𝑗), 1o, ∅))
17		fveq2 5554	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑖 → (𝑋‘𝑗) = (𝑋‘𝑖))
18		fveq2 5554	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑖 → (𝑌‘𝑗) = (𝑌‘𝑖))
19	17, 18	eqeq12d 2208	. . . . . . . . . 10 ⊢ (𝑗 = 𝑖 → ((𝑋‘𝑗) = (𝑌‘𝑗) ↔ (𝑋‘𝑖) = (𝑌‘𝑖)))
20	19	ifbid 3578	. . . . . . . . 9 ⊢ (𝑗 = 𝑖 → if((𝑋‘𝑗) = (𝑌‘𝑗), 1o, ∅) = if((𝑋‘𝑖) = (𝑌‘𝑖), 1o, ∅))
21		simpr 110	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ ω) → 𝑖 ∈ ω)
22		1lt2o 6495	. . . . . . . . . . 11 ⊢ 1o ∈ 2o
23	22	a1i 9	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ ω) → 1o ∈ 2o)
24		0lt2o 6494	. . . . . . . . . . 11 ⊢ ∅ ∈ 2o
25	24	a1i 9	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ ω) → ∅ ∈ 2o)
26		2ssom 6577	. . . . . . . . . . . 12 ⊢ 2o ⊆ ω
27		nninfwlporlem.x	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑋:ω⟶2o)
28	27	ffvelcdmda 5693	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ ω) → (𝑋‘𝑖) ∈ 2o)
29	26, 28	sselid 3177	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ ω) → (𝑋‘𝑖) ∈ ω)
30		nninfwlporlem.y	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑌:ω⟶2o)
31	30	ffvelcdmda 5693	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ ω) → (𝑌‘𝑖) ∈ 2o)
32	26, 31	sselid 3177	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ ω) → (𝑌‘𝑖) ∈ ω)
33		nndceq 6552	. . . . . . . . . . 11 ⊢ (((𝑋‘𝑖) ∈ ω ∧ (𝑌‘𝑖) ∈ ω) → DECID (𝑋‘𝑖) = (𝑌‘𝑖))
34	29, 32, 33	syl2anc 411	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ ω) → DECID (𝑋‘𝑖) = (𝑌‘𝑖))
35	23, 25, 34	ifcldcd 3593	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ ω) → if((𝑋‘𝑖) = (𝑌‘𝑖), 1o, ∅) ∈ 2o)
36	16, 20, 21, 35	fvmptd3 5651	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ ω) → (𝐷‘𝑖) = if((𝑋‘𝑖) = (𝑌‘𝑖), 1o, ∅))
37	36	eqeq2d 2205	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ ω) → (1o = (𝐷‘𝑖) ↔ 1o = if((𝑋‘𝑖) = (𝑌‘𝑖), 1o, ∅)))
38		eqifdc 3592	. . . . . . . 8 ⊢ (DECID (𝑋‘𝑖) = (𝑌‘𝑖) → (1o = if((𝑋‘𝑖) = (𝑌‘𝑖), 1o, ∅) ↔ (((𝑋‘𝑖) = (𝑌‘𝑖) ∧ 1o = 1o) ∨ (¬ (𝑋‘𝑖) = (𝑌‘𝑖) ∧ 1o = ∅))))
39	34, 38	syl 14	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ ω) → (1o = if((𝑋‘𝑖) = (𝑌‘𝑖), 1o, ∅) ↔ (((𝑋‘𝑖) = (𝑌‘𝑖) ∧ 1o = 1o) ∨ (¬ (𝑋‘𝑖) = (𝑌‘𝑖) ∧ 1o = ∅))))
40	37, 39	bitrd 188	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ ω) → (1o = (𝐷‘𝑖) ↔ (((𝑋‘𝑖) = (𝑌‘𝑖) ∧ 1o = 1o) ∨ (¬ (𝑋‘𝑖) = (𝑌‘𝑖) ∧ 1o = ∅))))
41	9, 40	bitr3id 194	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ ω) → ((𝐷‘𝑖) = 1o ↔ (((𝑋‘𝑖) = (𝑌‘𝑖) ∧ 1o = 1o) ∨ (¬ (𝑋‘𝑖) = (𝑌‘𝑖) ∧ 1o = ∅))))
42	8, 41	bitr4id 199	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ω) → ((𝑋‘𝑖) = (𝑌‘𝑖) ↔ (𝐷‘𝑖) = 1o))
43	42	ralbidva 2490	. . 3 ⊢ (𝜑 → (∀𝑖 ∈ ω (𝑋‘𝑖) = (𝑌‘𝑖) ↔ ∀𝑖 ∈ ω (𝐷‘𝑖) = 1o))
44		fveqeq2 5563	. . . 4 ⊢ (𝑖 = 𝑗 → ((𝐷‘𝑖) = 1o ↔ (𝐷‘𝑗) = 1o))
45	44	cbvralv 2726	. . 3 ⊢ (∀𝑖 ∈ ω (𝐷‘𝑖) = 1o ↔ ∀𝑗 ∈ ω (𝐷‘𝑗) = 1o)
46	43, 45	bitrdi 196	. 2 ⊢ (𝜑 → (∀𝑖 ∈ ω (𝑋‘𝑖) = (𝑌‘𝑖) ↔ ∀𝑗 ∈ ω (𝐷‘𝑗) = 1o))
47	27	ffnd 5404	. . 3 ⊢ (𝜑 → 𝑋 Fn ω)
48	30	ffnd 5404	. . 3 ⊢ (𝜑 → 𝑌 Fn ω)
49		eqfnfv 5655	. . 3 ⊢ ((𝑋 Fn ω ∧ 𝑌 Fn ω) → (𝑋 = 𝑌 ↔ ∀𝑖 ∈ ω (𝑋‘𝑖) = (𝑌‘𝑖)))
50	47, 48, 49	syl2anc 411	. 2 ⊢ (𝜑 → (𝑋 = 𝑌 ↔ ∀𝑖 ∈ ω (𝑋‘𝑖) = (𝑌‘𝑖)))
51	35	ralrimiva 2567	. . . 4 ⊢ (𝜑 → ∀𝑖 ∈ ω if((𝑋‘𝑖) = (𝑌‘𝑖), 1o, ∅) ∈ 2o)
52	10	fnmpt 5380	. . . 4 ⊢ (∀𝑖 ∈ ω if((𝑋‘𝑖) = (𝑌‘𝑖), 1o, ∅) ∈ 2o → 𝐷 Fn ω)
53	51, 52	syl 14	. . 3 ⊢ (𝜑 → 𝐷 Fn ω)
54		eqidd 2194	. . 3 ⊢ (𝑗 = 𝑖 → 1o = 1o)
55		1onn 6573	. . . 4 ⊢ 1o ∈ ω
56	55	a1i 9	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ ω) → 1o ∈ ω)
57	55	a1i 9	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ ω) → 1o ∈ ω)
58	53, 54, 56, 57	fnmptfvd 5662	. 2 ⊢ (𝜑 → (𝐷 = (𝑖 ∈ ω ↦ 1o) ↔ ∀𝑗 ∈ ω (𝐷‘𝑗) = 1o))
59	46, 50, 58	3bitr4d 220	1 ⊢ (𝜑 → (𝑋 = 𝑌 ↔ 𝐷 = (𝑖 ∈ ω ↦ 1o)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 709  DECID wdc 835   = wceq 1364   ∈ wcel 2164  ∀wral 2472  ∅c0 3446  ifcif 3557   ↦ cmpt 4090  ωcom 4622   Fn wfn 5249  ⟶wf 5250  ‘cfv 5254  1_oc1o 6462  2_oc2o 6463

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-iord 4397  df-on 4399  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-fv 5262  df-1o 6469  df-2o 6470

This theorem is referenced by:  nninfwlporlem  7232