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Theorem enq0ref 7446
Description: The equivalence relation for nonnegative fractions is reflexive. Lemma for enq0er 7448. (Contributed by Jim Kingdon, 14-Nov-2019.)
Assertion
Ref Expression
enq0ref (𝑓 ∈ (ω × N) ↔ 𝑓 ~Q0 𝑓)

Proof of Theorem enq0ref
Dummy variables 𝑢 𝑣 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elxpi 4654 . . . . . 6 (𝑓 ∈ (ω × N) → ∃𝑧𝑤(𝑓 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ ω ∧ 𝑤N)))
2 elxpi 4654 . . . . . 6 (𝑓 ∈ (ω × N) → ∃𝑣𝑢(𝑓 = ⟨𝑣, 𝑢⟩ ∧ (𝑣 ∈ ω ∧ 𝑢N)))
3 ee4anv 1944 . . . . . 6 (∃𝑧𝑤𝑣𝑢((𝑓 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ ω ∧ 𝑤N)) ∧ (𝑓 = ⟨𝑣, 𝑢⟩ ∧ (𝑣 ∈ ω ∧ 𝑢N))) ↔ (∃𝑧𝑤(𝑓 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ ω ∧ 𝑤N)) ∧ ∃𝑣𝑢(𝑓 = ⟨𝑣, 𝑢⟩ ∧ (𝑣 ∈ ω ∧ 𝑢N))))
41, 2, 3sylanbrc 417 . . . . 5 (𝑓 ∈ (ω × N) → ∃𝑧𝑤𝑣𝑢((𝑓 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ ω ∧ 𝑤N)) ∧ (𝑓 = ⟨𝑣, 𝑢⟩ ∧ (𝑣 ∈ ω ∧ 𝑢N))))
5 eqtr2 2206 . . . . . . . . . . . 12 ((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑓 = ⟨𝑣, 𝑢⟩) → ⟨𝑧, 𝑤⟩ = ⟨𝑣, 𝑢⟩)
6 vex 2752 . . . . . . . . . . . . 13 𝑧 ∈ V
7 vex 2752 . . . . . . . . . . . . 13 𝑤 ∈ V
86, 7opth 4249 . . . . . . . . . . . 12 (⟨𝑧, 𝑤⟩ = ⟨𝑣, 𝑢⟩ ↔ (𝑧 = 𝑣𝑤 = 𝑢))
95, 8sylib 122 . . . . . . . . . . 11 ((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑓 = ⟨𝑣, 𝑢⟩) → (𝑧 = 𝑣𝑤 = 𝑢))
10 oveq1 5895 . . . . . . . . . . . 12 (𝑧 = 𝑣 → (𝑧 ·o 𝑢) = (𝑣 ·o 𝑢))
11 oveq2 5896 . . . . . . . . . . . . 13 (𝑢 = 𝑤 → (𝑣 ·o 𝑢) = (𝑣 ·o 𝑤))
1211equcoms 1718 . . . . . . . . . . . 12 (𝑤 = 𝑢 → (𝑣 ·o 𝑢) = (𝑣 ·o 𝑤))
1310, 12sylan9eq 2240 . . . . . . . . . . 11 ((𝑧 = 𝑣𝑤 = 𝑢) → (𝑧 ·o 𝑢) = (𝑣 ·o 𝑤))
149, 13syl 14 . . . . . . . . . 10 ((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑓 = ⟨𝑣, 𝑢⟩) → (𝑧 ·o 𝑢) = (𝑣 ·o 𝑤))
1514ancli 323 . . . . . . . . 9 ((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑓 = ⟨𝑣, 𝑢⟩) → ((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑓 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑣 ·o 𝑤)))
1615ad2ant2r 509 . . . . . . . 8 (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ ω ∧ 𝑤N)) ∧ (𝑓 = ⟨𝑣, 𝑢⟩ ∧ (𝑣 ∈ ω ∧ 𝑢N))) → ((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑓 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑣 ·o 𝑤)))
17 pinn 7322 . . . . . . . . . . . . . 14 (𝑤N𝑤 ∈ ω)
18 nnmcom 6504 . . . . . . . . . . . . . 14 ((𝑣 ∈ ω ∧ 𝑤 ∈ ω) → (𝑣 ·o 𝑤) = (𝑤 ·o 𝑣))
1917, 18sylan2 286 . . . . . . . . . . . . 13 ((𝑣 ∈ ω ∧ 𝑤N) → (𝑣 ·o 𝑤) = (𝑤 ·o 𝑣))
2019eqeq2d 2199 . . . . . . . . . . . 12 ((𝑣 ∈ ω ∧ 𝑤N) → ((𝑧 ·o 𝑢) = (𝑣 ·o 𝑤) ↔ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)))
2120ancoms 268 . . . . . . . . . . 11 ((𝑤N𝑣 ∈ ω) → ((𝑧 ·o 𝑢) = (𝑣 ·o 𝑤) ↔ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)))
2221ad2ant2lr 510 . . . . . . . . . 10 (((𝑧 ∈ ω ∧ 𝑤N) ∧ (𝑣 ∈ ω ∧ 𝑢N)) → ((𝑧 ·o 𝑢) = (𝑣 ·o 𝑤) ↔ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)))
2322ad2ant2l 508 . . . . . . . . 9 (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ ω ∧ 𝑤N)) ∧ (𝑓 = ⟨𝑣, 𝑢⟩ ∧ (𝑣 ∈ ω ∧ 𝑢N))) → ((𝑧 ·o 𝑢) = (𝑣 ·o 𝑤) ↔ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)))
2423anbi2d 464 . . . . . . . 8 (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ ω ∧ 𝑤N)) ∧ (𝑓 = ⟨𝑣, 𝑢⟩ ∧ (𝑣 ∈ ω ∧ 𝑢N))) → (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑓 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑣 ·o 𝑤)) ↔ ((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑓 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣))))
2516, 24mpbid 147 . . . . . . 7 (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ ω ∧ 𝑤N)) ∧ (𝑓 = ⟨𝑣, 𝑢⟩ ∧ (𝑣 ∈ ω ∧ 𝑢N))) → ((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑓 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)))
26252eximi 1611 . . . . . 6 (∃𝑣𝑢((𝑓 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ ω ∧ 𝑤N)) ∧ (𝑓 = ⟨𝑣, 𝑢⟩ ∧ (𝑣 ∈ ω ∧ 𝑢N))) → ∃𝑣𝑢((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑓 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)))
27262eximi 1611 . . . . 5 (∃𝑧𝑤𝑣𝑢((𝑓 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ ω ∧ 𝑤N)) ∧ (𝑓 = ⟨𝑣, 𝑢⟩ ∧ (𝑣 ∈ ω ∧ 𝑢N))) → ∃𝑧𝑤𝑣𝑢((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑓 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)))
284, 27syl 14 . . . 4 (𝑓 ∈ (ω × N) → ∃𝑧𝑤𝑣𝑢((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑓 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)))
2928ancli 323 . . 3 (𝑓 ∈ (ω × N) → (𝑓 ∈ (ω × N) ∧ ∃𝑧𝑤𝑣𝑢((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑓 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣))))
30 vex 2752 . . . . 5 𝑓 ∈ V
31 eleq1 2250 . . . . . . 7 (𝑥 = 𝑓 → (𝑥 ∈ (ω × N) ↔ 𝑓 ∈ (ω × N)))
3231anbi1d 465 . . . . . 6 (𝑥 = 𝑓 → ((𝑥 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ↔ (𝑓 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N))))
33 eqeq1 2194 . . . . . . . . 9 (𝑥 = 𝑓 → (𝑥 = ⟨𝑧, 𝑤⟩ ↔ 𝑓 = ⟨𝑧, 𝑤⟩))
3433anbi1d 465 . . . . . . . 8 (𝑥 = 𝑓 → ((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ↔ (𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩)))
3534anbi1d 465 . . . . . . 7 (𝑥 = 𝑓 → (((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ↔ ((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣))))
36354exbidv 1880 . . . . . 6 (𝑥 = 𝑓 → (∃𝑧𝑤𝑣𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ↔ ∃𝑧𝑤𝑣𝑢((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣))))
3732, 36anbi12d 473 . . . . 5 (𝑥 = 𝑓 → (((𝑥 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣))) ↔ ((𝑓 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ∧ ∃𝑧𝑤𝑣𝑢((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)))))
38 eleq1 2250 . . . . . . 7 (𝑦 = 𝑓 → (𝑦 ∈ (ω × N) ↔ 𝑓 ∈ (ω × N)))
3938anbi2d 464 . . . . . 6 (𝑦 = 𝑓 → ((𝑓 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ↔ (𝑓 ∈ (ω × N) ∧ 𝑓 ∈ (ω × N))))
40 eqeq1 2194 . . . . . . . . 9 (𝑦 = 𝑓 → (𝑦 = ⟨𝑣, 𝑢⟩ ↔ 𝑓 = ⟨𝑣, 𝑢⟩))
4140anbi2d 464 . . . . . . . 8 (𝑦 = 𝑓 → ((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ↔ (𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑓 = ⟨𝑣, 𝑢⟩)))
4241anbi1d 465 . . . . . . 7 (𝑦 = 𝑓 → (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ↔ ((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑓 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣))))
43424exbidv 1880 . . . . . 6 (𝑦 = 𝑓 → (∃𝑧𝑤𝑣𝑢((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ↔ ∃𝑧𝑤𝑣𝑢((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑓 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣))))
4439, 43anbi12d 473 . . . . 5 (𝑦 = 𝑓 → (((𝑓 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ∧ ∃𝑧𝑤𝑣𝑢((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣))) ↔ ((𝑓 ∈ (ω × N) ∧ 𝑓 ∈ (ω × N)) ∧ ∃𝑧𝑤𝑣𝑢((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑓 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)))))
45 df-enq0 7437 . . . . 5 ~Q0 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)))}
4630, 30, 37, 44, 45brab 4284 . . . 4 (𝑓 ~Q0 𝑓 ↔ ((𝑓 ∈ (ω × N) ∧ 𝑓 ∈ (ω × N)) ∧ ∃𝑧𝑤𝑣𝑢((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑓 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣))))
47 anidm 396 . . . . 5 ((𝑓 ∈ (ω × N) ∧ 𝑓 ∈ (ω × N)) ↔ 𝑓 ∈ (ω × N))
4847anbi1i 458 . . . 4 (((𝑓 ∈ (ω × N) ∧ 𝑓 ∈ (ω × N)) ∧ ∃𝑧𝑤𝑣𝑢((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑓 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣))) ↔ (𝑓 ∈ (ω × N) ∧ ∃𝑧𝑤𝑣𝑢((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑓 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣))))
4946, 48bitri 184 . . 3 (𝑓 ~Q0 𝑓 ↔ (𝑓 ∈ (ω × N) ∧ ∃𝑧𝑤𝑣𝑢((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑓 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣))))
5029, 49sylibr 134 . 2 (𝑓 ∈ (ω × N) → 𝑓 ~Q0 𝑓)
5149simplbi 274 . 2 (𝑓 ~Q0 𝑓𝑓 ∈ (ω × N))
5250, 51impbii 126 1 (𝑓 ∈ (ω × N) ↔ 𝑓 ~Q0 𝑓)
Colors of variables: wff set class
Syntax hints:  wa 104  wb 105   = wceq 1363  wex 1502  wcel 2158  cop 3607   class class class wbr 4015  ωcom 4601   × cxp 4636  (class class class)co 5888   ·o comu 6429  Ncnpi 7285   ~Q0 ceq0 7299
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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-ral 2470  df-rex 2471  df-reu 2472  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-id 4305  df-iord 4378  df-on 4380  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6155  df-2nd 6156  df-recs 6320  df-irdg 6385  df-oadd 6435  df-omul 6436  df-ni 7317  df-enq0 7437
This theorem is referenced by:  enq0er  7448
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