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Theorem enq0sym 7182
Description: The equivalence relation for nonnegative fractions is symmetric. Lemma for enq0er 7185. (Contributed by Jim Kingdon, 14-Nov-2019.)
Assertion
Ref Expression
enq0sym (𝑓 ~Q0 𝑔𝑔 ~Q0 𝑓)

Proof of Theorem enq0sym
Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑢 𝑣 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2658 . . . . . . . 8 𝑓 ∈ V
2 vex 2658 . . . . . . . 8 𝑔 ∈ V
3 eleq1 2175 . . . . . . . . . 10 (𝑥 = 𝑓 → (𝑥 ∈ (ω × N) ↔ 𝑓 ∈ (ω × N)))
43anbi1d 458 . . . . . . . . 9 (𝑥 = 𝑓 → ((𝑥 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ↔ (𝑓 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N))))
5 eqeq1 2119 . . . . . . . . . . . 12 (𝑥 = 𝑓 → (𝑥 = ⟨𝑧, 𝑤⟩ ↔ 𝑓 = ⟨𝑧, 𝑤⟩))
65anbi1d 458 . . . . . . . . . . 11 (𝑥 = 𝑓 → ((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ↔ (𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩)))
76anbi1d 458 . . . . . . . . . 10 (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))))
874exbidv 1822 . . . . . . . . 9 (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))))
94, 8anbi12d 462 . . . . . . . 8 (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)))))
10 eleq1 2175 . . . . . . . . . 10 (𝑦 = 𝑔 → (𝑦 ∈ (ω × N) ↔ 𝑔 ∈ (ω × N)))
1110anbi2d 457 . . . . . . . . 9 (𝑦 = 𝑔 → ((𝑓 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ↔ (𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N))))
12 eqeq1 2119 . . . . . . . . . . . 12 (𝑦 = 𝑔 → (𝑦 = ⟨𝑣, 𝑢⟩ ↔ 𝑔 = ⟨𝑣, 𝑢⟩))
1312anbi2d 457 . . . . . . . . . . 11 (𝑦 = 𝑔 → ((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ↔ (𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩)))
1413anbi1d 458 . . . . . . . . . 10 (𝑦 = 𝑔 → (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ↔ ((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣))))
15144exbidv 1822 . . . . . . . . 9 (𝑦 = 𝑔 → (∃𝑧𝑤𝑣𝑢((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ↔ ∃𝑧𝑤𝑣𝑢((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣))))
1611, 15anbi12d 462 . . . . . . . 8 (𝑦 = 𝑔 → (((𝑓 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ∧ ∃𝑧𝑤𝑣𝑢((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣))) ↔ ((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N)) ∧ ∃𝑧𝑤𝑣𝑢((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)))))
17 df-enq0 7174 . . . . . . . 8 ~Q0 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)))}
181, 2, 9, 16, 17brab 4152 . . . . . . 7 (𝑓 ~Q0 𝑔 ↔ ((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N)) ∧ ∃𝑧𝑤𝑣𝑢((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣))))
1918biimpi 119 . . . . . 6 (𝑓 ~Q0 𝑔 → ((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N)) ∧ ∃𝑧𝑤𝑣𝑢((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣))))
20 opeq12 3671 . . . . . . . . . . 11 ((𝑧 = 𝑎𝑤 = 𝑏) → ⟨𝑧, 𝑤⟩ = ⟨𝑎, 𝑏⟩)
2120eqeq2d 2124 . . . . . . . . . 10 ((𝑧 = 𝑎𝑤 = 𝑏) → (𝑓 = ⟨𝑧, 𝑤⟩ ↔ 𝑓 = ⟨𝑎, 𝑏⟩))
2221anbi1d 458 . . . . . . . . 9 ((𝑧 = 𝑎𝑤 = 𝑏) → ((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ↔ (𝑓 = ⟨𝑎, 𝑏⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩)))
23 simpl 108 . . . . . . . . . . 11 ((𝑧 = 𝑎𝑤 = 𝑏) → 𝑧 = 𝑎)
2423oveq1d 5741 . . . . . . . . . 10 ((𝑧 = 𝑎𝑤 = 𝑏) → (𝑧 ·o 𝑢) = (𝑎 ·o 𝑢))
25 simpr 109 . . . . . . . . . . 11 ((𝑧 = 𝑎𝑤 = 𝑏) → 𝑤 = 𝑏)
2625oveq1d 5741 . . . . . . . . . 10 ((𝑧 = 𝑎𝑤 = 𝑏) → (𝑤 ·o 𝑣) = (𝑏 ·o 𝑣))
2724, 26eqeq12d 2127 . . . . . . . . 9 ((𝑧 = 𝑎𝑤 = 𝑏) → ((𝑧 ·o 𝑢) = (𝑤 ·o 𝑣) ↔ (𝑎 ·o 𝑢) = (𝑏 ·o 𝑣)))
2822, 27anbi12d 462 . . . . . . . 8 ((𝑧 = 𝑎𝑤 = 𝑏) → (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ↔ ((𝑓 = ⟨𝑎, 𝑏⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑎 ·o 𝑢) = (𝑏 ·o 𝑣))))
29 opeq12 3671 . . . . . . . . . . 11 ((𝑣 = 𝑐𝑢 = 𝑑) → ⟨𝑣, 𝑢⟩ = ⟨𝑐, 𝑑⟩)
3029eqeq2d 2124 . . . . . . . . . 10 ((𝑣 = 𝑐𝑢 = 𝑑) → (𝑔 = ⟨𝑣, 𝑢⟩ ↔ 𝑔 = ⟨𝑐, 𝑑⟩))
3130anbi2d 457 . . . . . . . . 9 ((𝑣 = 𝑐𝑢 = 𝑑) → ((𝑓 = ⟨𝑎, 𝑏⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ↔ (𝑓 = ⟨𝑎, 𝑏⟩ ∧ 𝑔 = ⟨𝑐, 𝑑⟩)))
32 simpr 109 . . . . . . . . . . 11 ((𝑣 = 𝑐𝑢 = 𝑑) → 𝑢 = 𝑑)
3332oveq2d 5742 . . . . . . . . . 10 ((𝑣 = 𝑐𝑢 = 𝑑) → (𝑎 ·o 𝑢) = (𝑎 ·o 𝑑))
34 simpl 108 . . . . . . . . . . 11 ((𝑣 = 𝑐𝑢 = 𝑑) → 𝑣 = 𝑐)
3534oveq2d 5742 . . . . . . . . . 10 ((𝑣 = 𝑐𝑢 = 𝑑) → (𝑏 ·o 𝑣) = (𝑏 ·o 𝑐))
3633, 35eqeq12d 2127 . . . . . . . . 9 ((𝑣 = 𝑐𝑢 = 𝑑) → ((𝑎 ·o 𝑢) = (𝑏 ·o 𝑣) ↔ (𝑎 ·o 𝑑) = (𝑏 ·o 𝑐)))
3731, 36anbi12d 462 . . . . . . . 8 ((𝑣 = 𝑐𝑢 = 𝑑) → (((𝑓 = ⟨𝑎, 𝑏⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑎 ·o 𝑢) = (𝑏 ·o 𝑣)) ↔ ((𝑓 = ⟨𝑎, 𝑏⟩ ∧ 𝑔 = ⟨𝑐, 𝑑⟩) ∧ (𝑎 ·o 𝑑) = (𝑏 ·o 𝑐))))
3828, 37cbvex4v 1878 . . . . . . 7 (∃𝑧𝑤𝑣𝑢((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ↔ ∃𝑎𝑏𝑐𝑑((𝑓 = ⟨𝑎, 𝑏⟩ ∧ 𝑔 = ⟨𝑐, 𝑑⟩) ∧ (𝑎 ·o 𝑑) = (𝑏 ·o 𝑐)))
3938anbi2i 450 . . . . . 6 (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N)) ∧ ∃𝑧𝑤𝑣𝑢((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣))) ↔ ((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N)) ∧ ∃𝑎𝑏𝑐𝑑((𝑓 = ⟨𝑎, 𝑏⟩ ∧ 𝑔 = ⟨𝑐, 𝑑⟩) ∧ (𝑎 ·o 𝑑) = (𝑏 ·o 𝑐))))
4019, 39sylib 121 . . . . 5 (𝑓 ~Q0 𝑔 → ((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N)) ∧ ∃𝑎𝑏𝑐𝑑((𝑓 = ⟨𝑎, 𝑏⟩ ∧ 𝑔 = ⟨𝑐, 𝑑⟩) ∧ (𝑎 ·o 𝑑) = (𝑏 ·o 𝑐))))
41 19.42vv 1861 . . . . 5 (∃𝑎𝑏((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N)) ∧ ∃𝑐𝑑((𝑓 = ⟨𝑎, 𝑏⟩ ∧ 𝑔 = ⟨𝑐, 𝑑⟩) ∧ (𝑎 ·o 𝑑) = (𝑏 ·o 𝑐))) ↔ ((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N)) ∧ ∃𝑎𝑏𝑐𝑑((𝑓 = ⟨𝑎, 𝑏⟩ ∧ 𝑔 = ⟨𝑐, 𝑑⟩) ∧ (𝑎 ·o 𝑑) = (𝑏 ·o 𝑐))))
4240, 41sylibr 133 . . . 4 (𝑓 ~Q0 𝑔 → ∃𝑎𝑏((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N)) ∧ ∃𝑐𝑑((𝑓 = ⟨𝑎, 𝑏⟩ ∧ 𝑔 = ⟨𝑐, 𝑑⟩) ∧ (𝑎 ·o 𝑑) = (𝑏 ·o 𝑐))))
43 19.42vv 1861 . . . . 5 (∃𝑐𝑑((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N)) ∧ ((𝑓 = ⟨𝑎, 𝑏⟩ ∧ 𝑔 = ⟨𝑐, 𝑑⟩) ∧ (𝑎 ·o 𝑑) = (𝑏 ·o 𝑐))) ↔ ((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N)) ∧ ∃𝑐𝑑((𝑓 = ⟨𝑎, 𝑏⟩ ∧ 𝑔 = ⟨𝑐, 𝑑⟩) ∧ (𝑎 ·o 𝑑) = (𝑏 ·o 𝑐))))
44432exbii 1566 . . . 4 (∃𝑎𝑏𝑐𝑑((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N)) ∧ ((𝑓 = ⟨𝑎, 𝑏⟩ ∧ 𝑔 = ⟨𝑐, 𝑑⟩) ∧ (𝑎 ·o 𝑑) = (𝑏 ·o 𝑐))) ↔ ∃𝑎𝑏((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N)) ∧ ∃𝑐𝑑((𝑓 = ⟨𝑎, 𝑏⟩ ∧ 𝑔 = ⟨𝑐, 𝑑⟩) ∧ (𝑎 ·o 𝑑) = (𝑏 ·o 𝑐))))
4542, 44sylibr 133 . . 3 (𝑓 ~Q0 𝑔 → ∃𝑎𝑏𝑐𝑑((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N)) ∧ ((𝑓 = ⟨𝑎, 𝑏⟩ ∧ 𝑔 = ⟨𝑐, 𝑑⟩) ∧ (𝑎 ·o 𝑑) = (𝑏 ·o 𝑐))))
46 pm3.22 263 . . . . . . 7 ((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N)) → (𝑔 ∈ (ω × N) ∧ 𝑓 ∈ (ω × N)))
4746adantr 272 . . . . . 6 (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N)) ∧ ((𝑓 = ⟨𝑎, 𝑏⟩ ∧ 𝑔 = ⟨𝑐, 𝑑⟩) ∧ (𝑎 ·o 𝑑) = (𝑏 ·o 𝑐))) → (𝑔 ∈ (ω × N) ∧ 𝑓 ∈ (ω × N)))
48 pm3.22 263 . . . . . . 7 ((𝑓 = ⟨𝑎, 𝑏⟩ ∧ 𝑔 = ⟨𝑐, 𝑑⟩) → (𝑔 = ⟨𝑐, 𝑑⟩ ∧ 𝑓 = ⟨𝑎, 𝑏⟩))
4948ad2antrl 479 . . . . . 6 (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N)) ∧ ((𝑓 = ⟨𝑎, 𝑏⟩ ∧ 𝑔 = ⟨𝑐, 𝑑⟩) ∧ (𝑎 ·o 𝑑) = (𝑏 ·o 𝑐))) → (𝑔 = ⟨𝑐, 𝑑⟩ ∧ 𝑓 = ⟨𝑎, 𝑏⟩))
50 simprr 504 . . . . . . . 8 (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N)) ∧ ((𝑓 = ⟨𝑎, 𝑏⟩ ∧ 𝑔 = ⟨𝑐, 𝑑⟩) ∧ (𝑎 ·o 𝑑) = (𝑏 ·o 𝑐))) → (𝑎 ·o 𝑑) = (𝑏 ·o 𝑐))
51 eleq1 2175 . . . . . . . . . . . . . 14 (𝑓 = ⟨𝑎, 𝑏⟩ → (𝑓 ∈ (ω × N) ↔ ⟨𝑎, 𝑏⟩ ∈ (ω × N)))
52 opelxp 4527 . . . . . . . . . . . . . 14 (⟨𝑎, 𝑏⟩ ∈ (ω × N) ↔ (𝑎 ∈ ω ∧ 𝑏N))
5351, 52syl6bb 195 . . . . . . . . . . . . 13 (𝑓 = ⟨𝑎, 𝑏⟩ → (𝑓 ∈ (ω × N) ↔ (𝑎 ∈ ω ∧ 𝑏N)))
5453biimpcd 158 . . . . . . . . . . . 12 (𝑓 ∈ (ω × N) → (𝑓 = ⟨𝑎, 𝑏⟩ → (𝑎 ∈ ω ∧ 𝑏N)))
55 eleq1 2175 . . . . . . . . . . . . . 14 (𝑔 = ⟨𝑐, 𝑑⟩ → (𝑔 ∈ (ω × N) ↔ ⟨𝑐, 𝑑⟩ ∈ (ω × N)))
56 opelxp 4527 . . . . . . . . . . . . . 14 (⟨𝑐, 𝑑⟩ ∈ (ω × N) ↔ (𝑐 ∈ ω ∧ 𝑑N))
5755, 56syl6bb 195 . . . . . . . . . . . . 13 (𝑔 = ⟨𝑐, 𝑑⟩ → (𝑔 ∈ (ω × N) ↔ (𝑐 ∈ ω ∧ 𝑑N)))
5857biimpcd 158 . . . . . . . . . . . 12 (𝑔 ∈ (ω × N) → (𝑔 = ⟨𝑐, 𝑑⟩ → (𝑐 ∈ ω ∧ 𝑑N)))
5954, 58im2anan9 570 . . . . . . . . . . 11 ((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N)) → ((𝑓 = ⟨𝑎, 𝑏⟩ ∧ 𝑔 = ⟨𝑐, 𝑑⟩) → ((𝑎 ∈ ω ∧ 𝑏N) ∧ (𝑐 ∈ ω ∧ 𝑑N))))
6059imp 123 . . . . . . . . . 10 (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N)) ∧ (𝑓 = ⟨𝑎, 𝑏⟩ ∧ 𝑔 = ⟨𝑐, 𝑑⟩)) → ((𝑎 ∈ ω ∧ 𝑏N) ∧ (𝑐 ∈ ω ∧ 𝑑N)))
6160adantrr 468 . . . . . . . . 9 (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N)) ∧ ((𝑓 = ⟨𝑎, 𝑏⟩ ∧ 𝑔 = ⟨𝑐, 𝑑⟩) ∧ (𝑎 ·o 𝑑) = (𝑏 ·o 𝑐))) → ((𝑎 ∈ ω ∧ 𝑏N) ∧ (𝑐 ∈ ω ∧ 𝑑N)))
62 pinn 7059 . . . . . . . . . . . 12 (𝑑N𝑑 ∈ ω)
63 nnmcom 6337 . . . . . . . . . . . 12 ((𝑎 ∈ ω ∧ 𝑑 ∈ ω) → (𝑎 ·o 𝑑) = (𝑑 ·o 𝑎))
6462, 63sylan2 282 . . . . . . . . . . 11 ((𝑎 ∈ ω ∧ 𝑑N) → (𝑎 ·o 𝑑) = (𝑑 ·o 𝑎))
65 pinn 7059 . . . . . . . . . . . 12 (𝑏N𝑏 ∈ ω)
66 nnmcom 6337 . . . . . . . . . . . 12 ((𝑏 ∈ ω ∧ 𝑐 ∈ ω) → (𝑏 ·o 𝑐) = (𝑐 ·o 𝑏))
6765, 66sylan 279 . . . . . . . . . . 11 ((𝑏N𝑐 ∈ ω) → (𝑏 ·o 𝑐) = (𝑐 ·o 𝑏))
6864, 67eqeqan12d 2128 . . . . . . . . . 10 (((𝑎 ∈ ω ∧ 𝑑N) ∧ (𝑏N𝑐 ∈ ω)) → ((𝑎 ·o 𝑑) = (𝑏 ·o 𝑐) ↔ (𝑑 ·o 𝑎) = (𝑐 ·o 𝑏)))
6968an42s 561 . . . . . . . . 9 (((𝑎 ∈ ω ∧ 𝑏N) ∧ (𝑐 ∈ ω ∧ 𝑑N)) → ((𝑎 ·o 𝑑) = (𝑏 ·o 𝑐) ↔ (𝑑 ·o 𝑎) = (𝑐 ·o 𝑏)))
7061, 69syl 14 . . . . . . . 8 (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N)) ∧ ((𝑓 = ⟨𝑎, 𝑏⟩ ∧ 𝑔 = ⟨𝑐, 𝑑⟩) ∧ (𝑎 ·o 𝑑) = (𝑏 ·o 𝑐))) → ((𝑎 ·o 𝑑) = (𝑏 ·o 𝑐) ↔ (𝑑 ·o 𝑎) = (𝑐 ·o 𝑏)))
7150, 70mpbid 146 . . . . . . 7 (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N)) ∧ ((𝑓 = ⟨𝑎, 𝑏⟩ ∧ 𝑔 = ⟨𝑐, 𝑑⟩) ∧ (𝑎 ·o 𝑑) = (𝑏 ·o 𝑐))) → (𝑑 ·o 𝑎) = (𝑐 ·o 𝑏))
7271eqcomd 2118 . . . . . 6 (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N)) ∧ ((𝑓 = ⟨𝑎, 𝑏⟩ ∧ 𝑔 = ⟨𝑐, 𝑑⟩) ∧ (𝑎 ·o 𝑑) = (𝑏 ·o 𝑐))) → (𝑐 ·o 𝑏) = (𝑑 ·o 𝑎))
7347, 49, 72jca32 306 . . . . 5 (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N)) ∧ ((𝑓 = ⟨𝑎, 𝑏⟩ ∧ 𝑔 = ⟨𝑐, 𝑑⟩) ∧ (𝑎 ·o 𝑑) = (𝑏 ·o 𝑐))) → ((𝑔 ∈ (ω × N) ∧ 𝑓 ∈ (ω × N)) ∧ ((𝑔 = ⟨𝑐, 𝑑⟩ ∧ 𝑓 = ⟨𝑎, 𝑏⟩) ∧ (𝑐 ·o 𝑏) = (𝑑 ·o 𝑎))))
74732eximi 1561 . . . 4 (∃𝑐𝑑((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N)) ∧ ((𝑓 = ⟨𝑎, 𝑏⟩ ∧ 𝑔 = ⟨𝑐, 𝑑⟩) ∧ (𝑎 ·o 𝑑) = (𝑏 ·o 𝑐))) → ∃𝑐𝑑((𝑔 ∈ (ω × N) ∧ 𝑓 ∈ (ω × N)) ∧ ((𝑔 = ⟨𝑐, 𝑑⟩ ∧ 𝑓 = ⟨𝑎, 𝑏⟩) ∧ (𝑐 ·o 𝑏) = (𝑑 ·o 𝑎))))
75742eximi 1561 . . 3 (∃𝑎𝑏𝑐𝑑((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N)) ∧ ((𝑓 = ⟨𝑎, 𝑏⟩ ∧ 𝑔 = ⟨𝑐, 𝑑⟩) ∧ (𝑎 ·o 𝑑) = (𝑏 ·o 𝑐))) → ∃𝑎𝑏𝑐𝑑((𝑔 ∈ (ω × N) ∧ 𝑓 ∈ (ω × N)) ∧ ((𝑔 = ⟨𝑐, 𝑑⟩ ∧ 𝑓 = ⟨𝑎, 𝑏⟩) ∧ (𝑐 ·o 𝑏) = (𝑑 ·o 𝑎))))
7645, 75syl 14 . 2 (𝑓 ~Q0 𝑔 → ∃𝑎𝑏𝑐𝑑((𝑔 ∈ (ω × N) ∧ 𝑓 ∈ (ω × N)) ∧ ((𝑔 = ⟨𝑐, 𝑑⟩ ∧ 𝑓 = ⟨𝑎, 𝑏⟩) ∧ (𝑐 ·o 𝑏) = (𝑑 ·o 𝑎))))
77 exrot4 1650 . . 3 (∃𝑎𝑏𝑐𝑑((𝑔 ∈ (ω × N) ∧ 𝑓 ∈ (ω × N)) ∧ ((𝑔 = ⟨𝑐, 𝑑⟩ ∧ 𝑓 = ⟨𝑎, 𝑏⟩) ∧ (𝑐 ·o 𝑏) = (𝑑 ·o 𝑎))) ↔ ∃𝑐𝑑𝑎𝑏((𝑔 ∈ (ω × N) ∧ 𝑓 ∈ (ω × N)) ∧ ((𝑔 = ⟨𝑐, 𝑑⟩ ∧ 𝑓 = ⟨𝑎, 𝑏⟩) ∧ (𝑐 ·o 𝑏) = (𝑑 ·o 𝑎))))
78 19.42vv 1861 . . . . 5 (∃𝑎𝑏((𝑔 ∈ (ω × N) ∧ 𝑓 ∈ (ω × N)) ∧ ((𝑔 = ⟨𝑐, 𝑑⟩ ∧ 𝑓 = ⟨𝑎, 𝑏⟩) ∧ (𝑐 ·o 𝑏) = (𝑑 ·o 𝑎))) ↔ ((𝑔 ∈ (ω × N) ∧ 𝑓 ∈ (ω × N)) ∧ ∃𝑎𝑏((𝑔 = ⟨𝑐, 𝑑⟩ ∧ 𝑓 = ⟨𝑎, 𝑏⟩) ∧ (𝑐 ·o 𝑏) = (𝑑 ·o 𝑎))))
79782exbii 1566 . . . 4 (∃𝑐𝑑𝑎𝑏((𝑔 ∈ (ω × N) ∧ 𝑓 ∈ (ω × N)) ∧ ((𝑔 = ⟨𝑐, 𝑑⟩ ∧ 𝑓 = ⟨𝑎, 𝑏⟩) ∧ (𝑐 ·o 𝑏) = (𝑑 ·o 𝑎))) ↔ ∃𝑐𝑑((𝑔 ∈ (ω × N) ∧ 𝑓 ∈ (ω × N)) ∧ ∃𝑎𝑏((𝑔 = ⟨𝑐, 𝑑⟩ ∧ 𝑓 = ⟨𝑎, 𝑏⟩) ∧ (𝑐 ·o 𝑏) = (𝑑 ·o 𝑎))))
80 19.42vv 1861 . . . . 5 (∃𝑐𝑑((𝑔 ∈ (ω × N) ∧ 𝑓 ∈ (ω × N)) ∧ ∃𝑎𝑏((𝑔 = ⟨𝑐, 𝑑⟩ ∧ 𝑓 = ⟨𝑎, 𝑏⟩) ∧ (𝑐 ·o 𝑏) = (𝑑 ·o 𝑎))) ↔ ((𝑔 ∈ (ω × N) ∧ 𝑓 ∈ (ω × N)) ∧ ∃𝑐𝑑𝑎𝑏((𝑔 = ⟨𝑐, 𝑑⟩ ∧ 𝑓 = ⟨𝑎, 𝑏⟩) ∧ (𝑐 ·o 𝑏) = (𝑑 ·o 𝑎))))
81 opeq12 3671 . . . . . . . . . 10 ((𝑧 = 𝑐𝑤 = 𝑑) → ⟨𝑧, 𝑤⟩ = ⟨𝑐, 𝑑⟩)
8281eqeq2d 2124 . . . . . . . . 9 ((𝑧 = 𝑐𝑤 = 𝑑) → (𝑔 = ⟨𝑧, 𝑤⟩ ↔ 𝑔 = ⟨𝑐, 𝑑⟩))
8382anbi1d 458 . . . . . . . 8 ((𝑧 = 𝑐𝑤 = 𝑑) → ((𝑔 = ⟨𝑧, 𝑤⟩ ∧ 𝑓 = ⟨𝑣, 𝑢⟩) ↔ (𝑔 = ⟨𝑐, 𝑑⟩ ∧ 𝑓 = ⟨𝑣, 𝑢⟩)))
84 simpl 108 . . . . . . . . . 10 ((𝑧 = 𝑐𝑤 = 𝑑) → 𝑧 = 𝑐)
8584oveq1d 5741 . . . . . . . . 9 ((𝑧 = 𝑐𝑤 = 𝑑) → (𝑧 ·o 𝑢) = (𝑐 ·o 𝑢))
86 simpr 109 . . . . . . . . . 10 ((𝑧 = 𝑐𝑤 = 𝑑) → 𝑤 = 𝑑)
8786oveq1d 5741 . . . . . . . . 9 ((𝑧 = 𝑐𝑤 = 𝑑) → (𝑤 ·o 𝑣) = (𝑑 ·o 𝑣))
8885, 87eqeq12d 2127 . . . . . . . 8 ((𝑧 = 𝑐𝑤 = 𝑑) → ((𝑧 ·o 𝑢) = (𝑤 ·o 𝑣) ↔ (𝑐 ·o 𝑢) = (𝑑 ·o 𝑣)))
8983, 88anbi12d 462 . . . . . . 7 ((𝑧 = 𝑐𝑤 = 𝑑) → (((𝑔 = ⟨𝑧, 𝑤⟩ ∧ 𝑓 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ↔ ((𝑔 = ⟨𝑐, 𝑑⟩ ∧ 𝑓 = ⟨𝑣, 𝑢⟩) ∧ (𝑐 ·o 𝑢) = (𝑑 ·o 𝑣))))
90 opeq12 3671 . . . . . . . . . 10 ((𝑣 = 𝑎𝑢 = 𝑏) → ⟨𝑣, 𝑢⟩ = ⟨𝑎, 𝑏⟩)
9190eqeq2d 2124 . . . . . . . . 9 ((𝑣 = 𝑎𝑢 = 𝑏) → (𝑓 = ⟨𝑣, 𝑢⟩ ↔ 𝑓 = ⟨𝑎, 𝑏⟩))
9291anbi2d 457 . . . . . . . 8 ((𝑣 = 𝑎𝑢 = 𝑏) → ((𝑔 = ⟨𝑐, 𝑑⟩ ∧ 𝑓 = ⟨𝑣, 𝑢⟩) ↔ (𝑔 = ⟨𝑐, 𝑑⟩ ∧ 𝑓 = ⟨𝑎, 𝑏⟩)))
93 simpr 109 . . . . . . . . . 10 ((𝑣 = 𝑎𝑢 = 𝑏) → 𝑢 = 𝑏)
9493oveq2d 5742 . . . . . . . . 9 ((𝑣 = 𝑎𝑢 = 𝑏) → (𝑐 ·o 𝑢) = (𝑐 ·o 𝑏))
95 simpl 108 . . . . . . . . . 10 ((𝑣 = 𝑎𝑢 = 𝑏) → 𝑣 = 𝑎)
9695oveq2d 5742 . . . . . . . . 9 ((𝑣 = 𝑎𝑢 = 𝑏) → (𝑑 ·o 𝑣) = (𝑑 ·o 𝑎))
9794, 96eqeq12d 2127 . . . . . . . 8 ((𝑣 = 𝑎𝑢 = 𝑏) → ((𝑐 ·o 𝑢) = (𝑑 ·o 𝑣) ↔ (𝑐 ·o 𝑏) = (𝑑 ·o 𝑎)))
9892, 97anbi12d 462 . . . . . . 7 ((𝑣 = 𝑎𝑢 = 𝑏) → (((𝑔 = ⟨𝑐, 𝑑⟩ ∧ 𝑓 = ⟨𝑣, 𝑢⟩) ∧ (𝑐 ·o 𝑢) = (𝑑 ·o 𝑣)) ↔ ((𝑔 = ⟨𝑐, 𝑑⟩ ∧ 𝑓 = ⟨𝑎, 𝑏⟩) ∧ (𝑐 ·o 𝑏) = (𝑑 ·o 𝑎))))
9989, 98cbvex4v 1878 . . . . . 6 (∃𝑧𝑤𝑣𝑢((𝑔 = ⟨𝑧, 𝑤⟩ ∧ 𝑓 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ↔ ∃𝑐𝑑𝑎𝑏((𝑔 = ⟨𝑐, 𝑑⟩ ∧ 𝑓 = ⟨𝑎, 𝑏⟩) ∧ (𝑐 ·o 𝑏) = (𝑑 ·o 𝑎)))
100 eleq1 2175 . . . . . . . . . 10 (𝑥 = 𝑔 → (𝑥 ∈ (ω × N) ↔ 𝑔 ∈ (ω × N)))
101100anbi1d 458 . . . . . . . . 9 (𝑥 = 𝑔 → ((𝑥 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ↔ (𝑔 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N))))
102 eqeq1 2119 . . . . . . . . . . . 12 (𝑥 = 𝑔 → (𝑥 = ⟨𝑧, 𝑤⟩ ↔ 𝑔 = ⟨𝑧, 𝑤⟩))
103102anbi1d 458 . . . . . . . . . . 11 (𝑥 = 𝑔 → ((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ↔ (𝑔 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩)))
104103anbi1d 458 . . . . . . . . . 10 (𝑥 = 𝑔 → (((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ↔ ((𝑔 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣))))
1051044exbidv 1822 . . . . . . . . 9 (𝑥 = 𝑔 → (∃𝑧𝑤𝑣𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ↔ ∃𝑧𝑤𝑣𝑢((𝑔 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣))))
106101, 105anbi12d 462 . . . . . . . 8 (𝑥 = 𝑔 → (((𝑥 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣))) ↔ ((𝑔 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ∧ ∃𝑧𝑤𝑣𝑢((𝑔 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)))))
107 eleq1 2175 . . . . . . . . . 10 (𝑦 = 𝑓 → (𝑦 ∈ (ω × N) ↔ 𝑓 ∈ (ω × N)))
108107anbi2d 457 . . . . . . . . 9 (𝑦 = 𝑓 → ((𝑔 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ↔ (𝑔 ∈ (ω × N) ∧ 𝑓 ∈ (ω × N))))
109 eqeq1 2119 . . . . . . . . . . . 12 (𝑦 = 𝑓 → (𝑦 = ⟨𝑣, 𝑢⟩ ↔ 𝑓 = ⟨𝑣, 𝑢⟩))
110109anbi2d 457 . . . . . . . . . . 11 (𝑦 = 𝑓 → ((𝑔 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ↔ (𝑔 = ⟨𝑧, 𝑤⟩ ∧ 𝑓 = ⟨𝑣, 𝑢⟩)))
111110anbi1d 458 . . . . . . . . . 10 (𝑦 = 𝑓 → (((𝑔 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ↔ ((𝑔 = ⟨𝑧, 𝑤⟩ ∧ 𝑓 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣))))
1121114exbidv 1822 . . . . . . . . 9 (𝑦 = 𝑓 → (∃𝑧𝑤𝑣𝑢((𝑔 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ↔ ∃𝑧𝑤𝑣𝑢((𝑔 = ⟨𝑧, 𝑤⟩ ∧ 𝑓 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣))))
113108, 112anbi12d 462 . . . . . . . 8 (𝑦 = 𝑓 → (((𝑔 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ∧ ∃𝑧𝑤𝑣𝑢((𝑔 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣))) ↔ ((𝑔 ∈ (ω × N) ∧ 𝑓 ∈ (ω × N)) ∧ ∃𝑧𝑤𝑣𝑢((𝑔 = ⟨𝑧, 𝑤⟩ ∧ 𝑓 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)))))
1142, 1, 106, 113, 17brab 4152 . . . . . . 7 (𝑔 ~Q0 𝑓 ↔ ((𝑔 ∈ (ω × N) ∧ 𝑓 ∈ (ω × N)) ∧ ∃𝑧𝑤𝑣𝑢((𝑔 = ⟨𝑧, 𝑤⟩ ∧ 𝑓 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣))))
115114biimpri 132 . . . . . 6 (((𝑔 ∈ (ω × N) ∧ 𝑓 ∈ (ω × N)) ∧ ∃𝑧𝑤𝑣𝑢((𝑔 = ⟨𝑧, 𝑤⟩ ∧ 𝑓 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣))) → 𝑔 ~Q0 𝑓)
11699, 115sylan2br 284 . . . . 5 (((𝑔 ∈ (ω × N) ∧ 𝑓 ∈ (ω × N)) ∧ ∃𝑐𝑑𝑎𝑏((𝑔 = ⟨𝑐, 𝑑⟩ ∧ 𝑓 = ⟨𝑎, 𝑏⟩) ∧ (𝑐 ·o 𝑏) = (𝑑 ·o 𝑎))) → 𝑔 ~Q0 𝑓)
11780, 116sylbi 120 . . . 4 (∃𝑐𝑑((𝑔 ∈ (ω × N) ∧ 𝑓 ∈ (ω × N)) ∧ ∃𝑎𝑏((𝑔 = ⟨𝑐, 𝑑⟩ ∧ 𝑓 = ⟨𝑎, 𝑏⟩) ∧ (𝑐 ·o 𝑏) = (𝑑 ·o 𝑎))) → 𝑔 ~Q0 𝑓)
11879, 117sylbi 120 . . 3 (∃𝑐𝑑𝑎𝑏((𝑔 ∈ (ω × N) ∧ 𝑓 ∈ (ω × N)) ∧ ((𝑔 = ⟨𝑐, 𝑑⟩ ∧ 𝑓 = ⟨𝑎, 𝑏⟩) ∧ (𝑐 ·o 𝑏) = (𝑑 ·o 𝑎))) → 𝑔 ~Q0 𝑓)
11977, 118sylbi 120 . 2 (∃𝑎𝑏𝑐𝑑((𝑔 ∈ (ω × N) ∧ 𝑓 ∈ (ω × N)) ∧ ((𝑔 = ⟨𝑐, 𝑑⟩ ∧ 𝑓 = ⟨𝑎, 𝑏⟩) ∧ (𝑐 ·o 𝑏) = (𝑑 ·o 𝑎))) → 𝑔 ~Q0 𝑓)
12076, 119syl 14 1 (𝑓 ~Q0 𝑔𝑔 ~Q0 𝑓)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1312  wex 1449  wcel 1461  cop 3494   class class class wbr 3893  ωcom 4462   × cxp 4495  (class class class)co 5726   ·o comu 6263  Ncnpi 7022   ~Q0 ceq0 7036
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-13 1472  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-coll 4001  ax-sep 4004  ax-nul 4012  ax-pow 4056  ax-pr 4089  ax-un 4313  ax-setind 4410  ax-iinf 4460
This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-fal 1318  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ne 2281  df-ral 2393  df-rex 2394  df-reu 2395  df-rab 2397  df-v 2657  df-sbc 2877  df-csb 2970  df-dif 3037  df-un 3039  df-in 3041  df-ss 3048  df-nul 3328  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-int 3736  df-iun 3779  df-br 3894  df-opab 3948  df-mpt 3949  df-tr 3985  df-id 4173  df-iord 4246  df-on 4248  df-suc 4251  df-iom 4463  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-rn 4508  df-res 4509  df-ima 4510  df-iota 5044  df-fun 5081  df-fn 5082  df-f 5083  df-f1 5084  df-fo 5085  df-f1o 5086  df-fv 5087  df-ov 5729  df-oprab 5730  df-mpo 5731  df-1st 5990  df-2nd 5991  df-recs 6154  df-irdg 6219  df-oadd 6269  df-omul 6270  df-ni 7054  df-enq0 7174
This theorem is referenced by:  enq0er  7185
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