Theorem nninfwlporlemd 7502

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 6695	. . . . . . . . 9 ⊢ 1o ≠ ∅
2	1	neii 2422	. . . . . . . 8 ⊢ ¬ 1o = ∅
3	2	intnan 941	. . . . . . 7 ⊢ ¬ (¬ (𝑋‘𝑖) = (𝑌‘𝑖) ∧ 1o = ∅)
4	3	biorfi 758	. . . . . 6 ⊢ ((𝑋‘𝑖) = (𝑌‘𝑖) ↔ ((𝑋‘𝑖) = (𝑌‘𝑖) ∨ (¬ (𝑋‘𝑖) = (𝑌‘𝑖) ∧ 1o = ∅)))
5		eqid 2238	. . . . . . . 8 ⊢ 1o = 1o
6	5	biantru 302	. . . . . . 7 ⊢ ((𝑋‘𝑖) = (𝑌‘𝑖) ↔ ((𝑋‘𝑖) = (𝑌‘𝑖) ∧ 1o = 1o))
7	6	orbi1i 775	. . . . . 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 2240	. . . . . 6 ⊢ (1o = (𝐷‘𝑖) ↔ (𝐷‘𝑖) = 1o)
10		nninfwlporlem.d	. . . . . . . . . 10 ⊢ 𝐷 = (𝑖 ∈ ω ↦ if((𝑋‘𝑖) = (𝑌‘𝑖), 1o, ∅))
11		fveq2 5690	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑗 → (𝑋‘𝑖) = (𝑋‘𝑗))
12		fveq2 5690	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑗 → (𝑌‘𝑖) = (𝑌‘𝑗))
13	11, 12	eqeq12d 2253	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑗 → ((𝑋‘𝑖) = (𝑌‘𝑖) ↔ (𝑋‘𝑗) = (𝑌‘𝑗)))
14	13	ifbid 3659	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑗 → if((𝑋‘𝑖) = (𝑌‘𝑖), 1o, ∅) = if((𝑋‘𝑗) = (𝑌‘𝑗), 1o, ∅))
15	14	cbvmptv 4222	. . . . . . . . . 10 ⊢ (𝑖 ∈ ω ↦ if((𝑋‘𝑖) = (𝑌‘𝑖), 1o, ∅)) = (𝑗 ∈ ω ↦ if((𝑋‘𝑗) = (𝑌‘𝑗), 1o, ∅))
16	10, 15	eqtri 2259	. . . . . . . . 9 ⊢ 𝐷 = (𝑗 ∈ ω ↦ if((𝑋‘𝑗) = (𝑌‘𝑗), 1o, ∅))
17		fveq2 5690	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑖 → (𝑋‘𝑗) = (𝑋‘𝑖))
18		fveq2 5690	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑖 → (𝑌‘𝑗) = (𝑌‘𝑖))
19	17, 18	eqeq12d 2253	. . . . . . . . . 10 ⊢ (𝑗 = 𝑖 → ((𝑋‘𝑗) = (𝑌‘𝑗) ↔ (𝑋‘𝑖) = (𝑌‘𝑖)))
20	19	ifbid 3659	. . . . . . . . 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 6705	. . . . . . . . . . 11 ⊢ 1o ∈ 2o
23	22	a1i 9	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ ω) → 1o ∈ 2o)
24		0lt2o 6704	. . . . . . . . . . 11 ⊢ ∅ ∈ 2o
25	24	a1i 9	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ ω) → ∅ ∈ 2o)
26		2ssom 6787	. . . . . . . . . . . 12 ⊢ 2o ⊆ ω
27		nninfwlporlem.x	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑋:ω⟶2o)
28	27	ffvelcdmda 5834	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ ω) → (𝑋‘𝑖) ∈ 2o)
29	26, 28	sselid 3246	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ ω) → (𝑋‘𝑖) ∈ ω)
30		nninfwlporlem.y	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑌:ω⟶2o)
31	30	ffvelcdmda 5834	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ ω) → (𝑌‘𝑖) ∈ 2o)
32	26, 31	sselid 3246	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ ω) → (𝑌‘𝑖) ∈ ω)
33		nndceq 6762	. . . . . . . . . . 11 ⊢ (((𝑋‘𝑖) ∈ ω ∧ (𝑌‘𝑖) ∈ ω) → DECID (𝑋‘𝑖) = (𝑌‘𝑖))
34	29, 32, 33	syl2anc 415	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ ω) → DECID (𝑋‘𝑖) = (𝑌‘𝑖))
35	23, 25, 34	ifcldcd 3675	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ ω) → if((𝑋‘𝑖) = (𝑌‘𝑖), 1o, ∅) ∈ 2o)
36	16, 20, 21, 35	fvmptd3 5793	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ ω) → (𝐷‘𝑖) = if((𝑋‘𝑖) = (𝑌‘𝑖), 1o, ∅))
37	36	eqeq2d 2250	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ ω) → (1o = (𝐷‘𝑖) ↔ 1o = if((𝑋‘𝑖) = (𝑌‘𝑖), 1o, ∅)))
38		eqifdc 3674	. . . . . . . 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 2546	. . 3 ⊢ (𝜑 → (∀𝑖 ∈ ω (𝑋‘𝑖) = (𝑌‘𝑖) ↔ ∀𝑖 ∈ ω (𝐷‘𝑖) = 1o))
44		fveqeq2 5699	. . . 4 ⊢ (𝑖 = 𝑗 → ((𝐷‘𝑖) = 1o ↔ (𝐷‘𝑗) = 1o))
45	44	cbvralv 2786	. . 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 5529	. . 3 ⊢ (𝜑 → 𝑋 Fn ω)
48	30	ffnd 5529	. . 3 ⊢ (𝜑 → 𝑌 Fn ω)
49		eqfnfv 5797	. . 3 ⊢ ((𝑋 Fn ω ∧ 𝑌 Fn ω) → (𝑋 = 𝑌 ↔ ∀𝑖 ∈ ω (𝑋‘𝑖) = (𝑌‘𝑖)))
50	47, 48, 49	syl2anc 415	. 2 ⊢ (𝜑 → (𝑋 = 𝑌 ↔ ∀𝑖 ∈ ω (𝑋‘𝑖) = (𝑌‘𝑖)))
51	35	ralrimiva 2623	. . . 4 ⊢ (𝜑 → ∀𝑖 ∈ ω if((𝑋‘𝑖) = (𝑌‘𝑖), 1o, ∅) ∈ 2o)
52	10	fnmpt 5505	. . . 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 2239	. . 3 ⊢ (𝑗 = 𝑖 → 1o = 1o)
55		1onn 6783	. . . 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 5804	. 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 720  DECID wdc 846   = wceq 1402   ∈ wcel 2209  ∀wral 2528  ∅c0 3520  ifcif 3635   ↦ cmpt 4187  ωcom 4732   Fn wfn 5367  ⟶wf 5368  ‘cfv 5372  1_oc1o 6670  2_oc2o 6671

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-fv 5380  df-1o 6677  df-2o 6678

This theorem is referenced by:  nninfwlporlem  7503