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Theorem nninfwlporlemd 7164
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 (𝜑𝑋:ω⟶2o)
nninfwlporlem.y (𝜑𝑌:ω⟶2o)
nninfwlporlem.d 𝐷 = (𝑖 ∈ ω ↦ if((𝑋𝑖) = (𝑌𝑖), 1o, ∅))
Assertion
Ref Expression
nninfwlporlemd (𝜑 → (𝑋 = 𝑌𝐷 = (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.
StepHypRef Expression
1 1n0 6427 . . . . . . . . 9 1o ≠ ∅
21neii 2349 . . . . . . . 8 ¬ 1o = ∅
32intnan 929 . . . . . . 7 ¬ (¬ (𝑋𝑖) = (𝑌𝑖) ∧ 1o = ∅)
43biorfi 746 . . . . . 6 ((𝑋𝑖) = (𝑌𝑖) ↔ ((𝑋𝑖) = (𝑌𝑖) ∨ (¬ (𝑋𝑖) = (𝑌𝑖) ∧ 1o = ∅)))
5 eqid 2177 . . . . . . . 8 1o = 1o
65biantru 302 . . . . . . 7 ((𝑋𝑖) = (𝑌𝑖) ↔ ((𝑋𝑖) = (𝑌𝑖) ∧ 1o = 1o))
76orbi1i 763 . . . . . 6 (((𝑋𝑖) = (𝑌𝑖) ∨ (¬ (𝑋𝑖) = (𝑌𝑖) ∧ 1o = ∅)) ↔ (((𝑋𝑖) = (𝑌𝑖) ∧ 1o = 1o) ∨ (¬ (𝑋𝑖) = (𝑌𝑖) ∧ 1o = ∅)))
84, 7bitri 184 . . . . 5 ((𝑋𝑖) = (𝑌𝑖) ↔ (((𝑋𝑖) = (𝑌𝑖) ∧ 1o = 1o) ∨ (¬ (𝑋𝑖) = (𝑌𝑖) ∧ 1o = ∅)))
9 eqcom 2179 . . . . . 6 (1o = (𝐷𝑖) ↔ (𝐷𝑖) = 1o)
10 nninfwlporlem.d . . . . . . . . . 10 𝐷 = (𝑖 ∈ ω ↦ if((𝑋𝑖) = (𝑌𝑖), 1o, ∅))
11 fveq2 5511 . . . . . . . . . . . . 13 (𝑖 = 𝑗 → (𝑋𝑖) = (𝑋𝑗))
12 fveq2 5511 . . . . . . . . . . . . 13 (𝑖 = 𝑗 → (𝑌𝑖) = (𝑌𝑗))
1311, 12eqeq12d 2192 . . . . . . . . . . . 12 (𝑖 = 𝑗 → ((𝑋𝑖) = (𝑌𝑖) ↔ (𝑋𝑗) = (𝑌𝑗)))
1413ifbid 3555 . . . . . . . . . . 11 (𝑖 = 𝑗 → if((𝑋𝑖) = (𝑌𝑖), 1o, ∅) = if((𝑋𝑗) = (𝑌𝑗), 1o, ∅))
1514cbvmptv 4096 . . . . . . . . . 10 (𝑖 ∈ ω ↦ if((𝑋𝑖) = (𝑌𝑖), 1o, ∅)) = (𝑗 ∈ ω ↦ if((𝑋𝑗) = (𝑌𝑗), 1o, ∅))
1610, 15eqtri 2198 . . . . . . . . 9 𝐷 = (𝑗 ∈ ω ↦ if((𝑋𝑗) = (𝑌𝑗), 1o, ∅))
17 fveq2 5511 . . . . . . . . . . 11 (𝑗 = 𝑖 → (𝑋𝑗) = (𝑋𝑖))
18 fveq2 5511 . . . . . . . . . . 11 (𝑗 = 𝑖 → (𝑌𝑗) = (𝑌𝑖))
1917, 18eqeq12d 2192 . . . . . . . . . 10 (𝑗 = 𝑖 → ((𝑋𝑗) = (𝑌𝑗) ↔ (𝑋𝑖) = (𝑌𝑖)))
2019ifbid 3555 . . . . . . . . 9 (𝑗 = 𝑖 → if((𝑋𝑗) = (𝑌𝑗), 1o, ∅) = if((𝑋𝑖) = (𝑌𝑖), 1o, ∅))
21 simpr 110 . . . . . . . . 9 ((𝜑𝑖 ∈ ω) → 𝑖 ∈ ω)
22 1lt2o 6437 . . . . . . . . . . 11 1o ∈ 2o
2322a1i 9 . . . . . . . . . 10 ((𝜑𝑖 ∈ ω) → 1o ∈ 2o)
24 0lt2o 6436 . . . . . . . . . . 11 ∅ ∈ 2o
2524a1i 9 . . . . . . . . . 10 ((𝜑𝑖 ∈ ω) → ∅ ∈ 2o)
26 2ssom 6519 . . . . . . . . . . . 12 2o ⊆ ω
27 nninfwlporlem.x . . . . . . . . . . . . 13 (𝜑𝑋:ω⟶2o)
2827ffvelcdmda 5647 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ ω) → (𝑋𝑖) ∈ 2o)
2926, 28sselid 3153 . . . . . . . . . . 11 ((𝜑𝑖 ∈ ω) → (𝑋𝑖) ∈ ω)
30 nninfwlporlem.y . . . . . . . . . . . . 13 (𝜑𝑌:ω⟶2o)
3130ffvelcdmda 5647 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ ω) → (𝑌𝑖) ∈ 2o)
3226, 31sselid 3153 . . . . . . . . . . 11 ((𝜑𝑖 ∈ ω) → (𝑌𝑖) ∈ ω)
33 nndceq 6494 . . . . . . . . . . 11 (((𝑋𝑖) ∈ ω ∧ (𝑌𝑖) ∈ ω) → DECID (𝑋𝑖) = (𝑌𝑖))
3429, 32, 33syl2anc 411 . . . . . . . . . 10 ((𝜑𝑖 ∈ ω) → DECID (𝑋𝑖) = (𝑌𝑖))
3523, 25, 34ifcldcd 3569 . . . . . . . . 9 ((𝜑𝑖 ∈ ω) → if((𝑋𝑖) = (𝑌𝑖), 1o, ∅) ∈ 2o)
3616, 20, 21, 35fvmptd3 5605 . . . . . . . 8 ((𝜑𝑖 ∈ ω) → (𝐷𝑖) = if((𝑋𝑖) = (𝑌𝑖), 1o, ∅))
3736eqeq2d 2189 . . . . . . 7 ((𝜑𝑖 ∈ ω) → (1o = (𝐷𝑖) ↔ 1o = if((𝑋𝑖) = (𝑌𝑖), 1o, ∅)))
38 eqifdc 3568 . . . . . . . 8 (DECID (𝑋𝑖) = (𝑌𝑖) → (1o = if((𝑋𝑖) = (𝑌𝑖), 1o, ∅) ↔ (((𝑋𝑖) = (𝑌𝑖) ∧ 1o = 1o) ∨ (¬ (𝑋𝑖) = (𝑌𝑖) ∧ 1o = ∅))))
3934, 38syl 14 . . . . . . 7 ((𝜑𝑖 ∈ ω) → (1o = if((𝑋𝑖) = (𝑌𝑖), 1o, ∅) ↔ (((𝑋𝑖) = (𝑌𝑖) ∧ 1o = 1o) ∨ (¬ (𝑋𝑖) = (𝑌𝑖) ∧ 1o = ∅))))
4037, 39bitrd 188 . . . . . 6 ((𝜑𝑖 ∈ ω) → (1o = (𝐷𝑖) ↔ (((𝑋𝑖) = (𝑌𝑖) ∧ 1o = 1o) ∨ (¬ (𝑋𝑖) = (𝑌𝑖) ∧ 1o = ∅))))
419, 40bitr3id 194 . . . . 5 ((𝜑𝑖 ∈ ω) → ((𝐷𝑖) = 1o ↔ (((𝑋𝑖) = (𝑌𝑖) ∧ 1o = 1o) ∨ (¬ (𝑋𝑖) = (𝑌𝑖) ∧ 1o = ∅))))
428, 41bitr4id 199 . . . 4 ((𝜑𝑖 ∈ ω) → ((𝑋𝑖) = (𝑌𝑖) ↔ (𝐷𝑖) = 1o))
4342ralbidva 2473 . . 3 (𝜑 → (∀𝑖 ∈ ω (𝑋𝑖) = (𝑌𝑖) ↔ ∀𝑖 ∈ ω (𝐷𝑖) = 1o))
44 fveqeq2 5520 . . . 4 (𝑖 = 𝑗 → ((𝐷𝑖) = 1o ↔ (𝐷𝑗) = 1o))
4544cbvralv 2703 . . 3 (∀𝑖 ∈ ω (𝐷𝑖) = 1o ↔ ∀𝑗 ∈ ω (𝐷𝑗) = 1o)
4643, 45bitrdi 196 . 2 (𝜑 → (∀𝑖 ∈ ω (𝑋𝑖) = (𝑌𝑖) ↔ ∀𝑗 ∈ ω (𝐷𝑗) = 1o))
4727ffnd 5362 . . 3 (𝜑𝑋 Fn ω)
4830ffnd 5362 . . 3 (𝜑𝑌 Fn ω)
49 eqfnfv 5609 . . 3 ((𝑋 Fn ω ∧ 𝑌 Fn ω) → (𝑋 = 𝑌 ↔ ∀𝑖 ∈ ω (𝑋𝑖) = (𝑌𝑖)))
5047, 48, 49syl2anc 411 . 2 (𝜑 → (𝑋 = 𝑌 ↔ ∀𝑖 ∈ ω (𝑋𝑖) = (𝑌𝑖)))
5135ralrimiva 2550 . . . 4 (𝜑 → ∀𝑖 ∈ ω if((𝑋𝑖) = (𝑌𝑖), 1o, ∅) ∈ 2o)
5210fnmpt 5338 . . . 4 (∀𝑖 ∈ ω if((𝑋𝑖) = (𝑌𝑖), 1o, ∅) ∈ 2o𝐷 Fn ω)
5351, 52syl 14 . . 3 (𝜑𝐷 Fn ω)
54 eqidd 2178 . . 3 (𝑗 = 𝑖 → 1o = 1o)
55 1onn 6515 . . . 4 1o ∈ ω
5655a1i 9 . . 3 ((𝜑𝑗 ∈ ω) → 1o ∈ ω)
5755a1i 9 . . 3 ((𝜑𝑖 ∈ ω) → 1o ∈ ω)
5853, 54, 56, 57fnmptfvd 5616 . 2 (𝜑 → (𝐷 = (𝑖 ∈ ω ↦ 1o) ↔ ∀𝑗 ∈ ω (𝐷𝑗) = 1o))
5946, 50, 583bitr4d 220 1 (𝜑 → (𝑋 = 𝑌𝐷 = (𝑖 ∈ ω ↦ 1o)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wb 105  wo 708  DECID wdc 834   = wceq 1353  wcel 2148  wral 2455  c0 3422  ifcif 3534  cmpt 4061  ωcom 4586   Fn wfn 5207  wf 5208  cfv 5212  1oc1o 6404  2oc2o 6405
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-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  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-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-iord 4363  df-on 4365  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-fv 5220  df-1o 6411  df-2o 6412
This theorem is referenced by:  nninfwlporlem  7165
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