Theorem nninfwlporlemd 7370

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 6599	. . . . . . . . 9 ⊢ 1o ≠ ∅
2	1	neii 2404	. . . . . . . 8 ⊢ ¬ 1o = ∅
3	2	intnan 936	. . . . . . 7 ⊢ ¬ (¬ (𝑋‘𝑖) = (𝑌‘𝑖) ∧ 1o = ∅)
4	3	biorfi 753	. . . . . 6 ⊢ ((𝑋‘𝑖) = (𝑌‘𝑖) ↔ ((𝑋‘𝑖) = (𝑌‘𝑖) ∨ (¬ (𝑋‘𝑖) = (𝑌‘𝑖) ∧ 1o = ∅)))
5		eqid 2231	. . . . . . . 8 ⊢ 1o = 1o
6	5	biantru 302	. . . . . . 7 ⊢ ((𝑋‘𝑖) = (𝑌‘𝑖) ↔ ((𝑋‘𝑖) = (𝑌‘𝑖) ∧ 1o = 1o))
7	6	orbi1i 770	. . . . . 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 2233	. . . . . 6 ⊢ (1o = (𝐷‘𝑖) ↔ (𝐷‘𝑖) = 1o)
10		nninfwlporlem.d	. . . . . . . . . 10 ⊢ 𝐷 = (𝑖 ∈ ω ↦ if((𝑋‘𝑖) = (𝑌‘𝑖), 1o, ∅))
11		fveq2 5639	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑗 → (𝑋‘𝑖) = (𝑋‘𝑗))
12		fveq2 5639	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑗 → (𝑌‘𝑖) = (𝑌‘𝑗))
13	11, 12	eqeq12d 2246	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑗 → ((𝑋‘𝑖) = (𝑌‘𝑖) ↔ (𝑋‘𝑗) = (𝑌‘𝑗)))
14	13	ifbid 3627	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑗 → if((𝑋‘𝑖) = (𝑌‘𝑖), 1o, ∅) = if((𝑋‘𝑗) = (𝑌‘𝑗), 1o, ∅))
15	14	cbvmptv 4185	. . . . . . . . . 10 ⊢ (𝑖 ∈ ω ↦ if((𝑋‘𝑖) = (𝑌‘𝑖), 1o, ∅)) = (𝑗 ∈ ω ↦ if((𝑋‘𝑗) = (𝑌‘𝑗), 1o, ∅))
16	10, 15	eqtri 2252	. . . . . . . . 9 ⊢ 𝐷 = (𝑗 ∈ ω ↦ if((𝑋‘𝑗) = (𝑌‘𝑗), 1o, ∅))
17		fveq2 5639	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑖 → (𝑋‘𝑗) = (𝑋‘𝑖))
18		fveq2 5639	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑖 → (𝑌‘𝑗) = (𝑌‘𝑖))
19	17, 18	eqeq12d 2246	. . . . . . . . . 10 ⊢ (𝑗 = 𝑖 → ((𝑋‘𝑗) = (𝑌‘𝑗) ↔ (𝑋‘𝑖) = (𝑌‘𝑖)))
20	19	ifbid 3627	. . . . . . . . 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 6609	. . . . . . . . . . 11 ⊢ 1o ∈ 2o
23	22	a1i 9	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ ω) → 1o ∈ 2o)
24		0lt2o 6608	. . . . . . . . . . 11 ⊢ ∅ ∈ 2o
25	24	a1i 9	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ ω) → ∅ ∈ 2o)
26		2ssom 6691	. . . . . . . . . . . 12 ⊢ 2o ⊆ ω
27		nninfwlporlem.x	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑋:ω⟶2o)
28	27	ffvelcdmda 5782	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ ω) → (𝑋‘𝑖) ∈ 2o)
29	26, 28	sselid 3225	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ ω) → (𝑋‘𝑖) ∈ ω)
30		nninfwlporlem.y	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑌:ω⟶2o)
31	30	ffvelcdmda 5782	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ ω) → (𝑌‘𝑖) ∈ 2o)
32	26, 31	sselid 3225	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ ω) → (𝑌‘𝑖) ∈ ω)
33		nndceq 6666	. . . . . . . . . . 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 3643	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ ω) → if((𝑋‘𝑖) = (𝑌‘𝑖), 1o, ∅) ∈ 2o)
36	16, 20, 21, 35	fvmptd3 5740	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ ω) → (𝐷‘𝑖) = if((𝑋‘𝑖) = (𝑌‘𝑖), 1o, ∅))
37	36	eqeq2d 2243	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ ω) → (1o = (𝐷‘𝑖) ↔ 1o = if((𝑋‘𝑖) = (𝑌‘𝑖), 1o, ∅)))
38		eqifdc 3642	. . . . . . . 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 2528	. . 3 ⊢ (𝜑 → (∀𝑖 ∈ ω (𝑋‘𝑖) = (𝑌‘𝑖) ↔ ∀𝑖 ∈ ω (𝐷‘𝑖) = 1o))
44		fveqeq2 5648	. . . 4 ⊢ (𝑖 = 𝑗 → ((𝐷‘𝑖) = 1o ↔ (𝐷‘𝑗) = 1o))
45	44	cbvralv 2767	. . 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 5483	. . 3 ⊢ (𝜑 → 𝑋 Fn ω)
48	30	ffnd 5483	. . 3 ⊢ (𝜑 → 𝑌 Fn ω)
49		eqfnfv 5744	. . 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 2605	. . . 4 ⊢ (𝜑 → ∀𝑖 ∈ ω if((𝑋‘𝑖) = (𝑌‘𝑖), 1o, ∅) ∈ 2o)
52	10	fnmpt 5459	. . . 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 2232	. . 3 ⊢ (𝑗 = 𝑖 → 1o = 1o)
55		1onn 6687	. . . 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 5751	. 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 715  DECID wdc 841   = wceq 1397   ∈ wcel 2202  ∀wral 2510  ∅c0 3494  ifcif 3605   ↦ cmpt 4150  ωcom 4688   Fn wfn 5321  ⟶wf 5322  ‘cfv 5326  1_oc1o 6574  2_oc2o 6575

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fv 5334  df-1o 6581  df-2o 6582

This theorem is referenced by:  nninfwlporlem  7371