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Theorem enq0sym 6587
Description: The equivalence relation for non-negative fractions is symmetric. Lemma for enq0er 6590. (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 2577 . . . . . . . 8 𝑓 ∈ V
2 vex 2577 . . . . . . . 8 𝑔 ∈ V
3 eleq1 2116 . . . . . . . . . 10 (𝑥 = 𝑓 → (𝑥 ∈ (ω × N) ↔ 𝑓 ∈ (ω × N)))
43anbi1d 446 . . . . . . . . 9 (𝑥 = 𝑓 → ((𝑥 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ↔ (𝑓 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N))))
5 eqeq1 2062 . . . . . . . . . . . 12 (𝑥 = 𝑓 → (𝑥 = ⟨𝑧, 𝑤⟩ ↔ 𝑓 = ⟨𝑧, 𝑤⟩))
65anbi1d 446 . . . . . . . . . . 11 (𝑥 = 𝑓 → ((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ↔ (𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩)))
76anbi1d 446 . . . . . . . . . 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 1766 . . . . . . . . 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 450 . . . . . . . 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 2116 . . . . . . . . . 10 (𝑦 = 𝑔 → (𝑦 ∈ (ω × N) ↔ 𝑔 ∈ (ω × N)))
1110anbi2d 445 . . . . . . . . 9 (𝑦 = 𝑔 → ((𝑓 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ↔ (𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N))))
12 eqeq1 2062 . . . . . . . . . . . 12 (𝑦 = 𝑔 → (𝑦 = ⟨𝑣, 𝑢⟩ ↔ 𝑔 = ⟨𝑣, 𝑢⟩))
1312anbi2d 445 . . . . . . . . . . 11 (𝑦 = 𝑔 → ((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ↔ (𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩)))
1413anbi1d 446 . . . . . . . . . 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 1766 . . . . . . . . 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 450 . . . . . . . 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 6579 . . . . . . . 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 4036 . . . . . . 7 (𝑓 ~Q0 𝑔 ↔ ((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N)) ∧ ∃𝑧𝑤𝑣𝑢((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣))))
1918biimpi 117 . . . . . 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 3578 . . . . . . . . . . 11 ((𝑧 = 𝑎𝑤 = 𝑏) → ⟨𝑧, 𝑤⟩ = ⟨𝑎, 𝑏⟩)
2120eqeq2d 2067 . . . . . . . . . 10 ((𝑧 = 𝑎𝑤 = 𝑏) → (𝑓 = ⟨𝑧, 𝑤⟩ ↔ 𝑓 = ⟨𝑎, 𝑏⟩))
2221anbi1d 446 . . . . . . . . 9 ((𝑧 = 𝑎𝑤 = 𝑏) → ((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ↔ (𝑓 = ⟨𝑎, 𝑏⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩)))
23 simpl 106 . . . . . . . . . . 11 ((𝑧 = 𝑎𝑤 = 𝑏) → 𝑧 = 𝑎)
2423oveq1d 5554 . . . . . . . . . 10 ((𝑧 = 𝑎𝑤 = 𝑏) → (𝑧 ·𝑜 𝑢) = (𝑎 ·𝑜 𝑢))
25 simpr 107 . . . . . . . . . . 11 ((𝑧 = 𝑎𝑤 = 𝑏) → 𝑤 = 𝑏)
2625oveq1d 5554 . . . . . . . . . 10 ((𝑧 = 𝑎𝑤 = 𝑏) → (𝑤 ·𝑜 𝑣) = (𝑏 ·𝑜 𝑣))
2724, 26eqeq12d 2070 . . . . . . . . 9 ((𝑧 = 𝑎𝑤 = 𝑏) → ((𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣) ↔ (𝑎 ·𝑜 𝑢) = (𝑏 ·𝑜 𝑣)))
2822, 27anbi12d 450 . . . . . . . 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 3578 . . . . . . . . . . 11 ((𝑣 = 𝑐𝑢 = 𝑑) → ⟨𝑣, 𝑢⟩ = ⟨𝑐, 𝑑⟩)
3029eqeq2d 2067 . . . . . . . . . 10 ((𝑣 = 𝑐𝑢 = 𝑑) → (𝑔 = ⟨𝑣, 𝑢⟩ ↔ 𝑔 = ⟨𝑐, 𝑑⟩))
3130anbi2d 445 . . . . . . . . 9 ((𝑣 = 𝑐𝑢 = 𝑑) → ((𝑓 = ⟨𝑎, 𝑏⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ↔ (𝑓 = ⟨𝑎, 𝑏⟩ ∧ 𝑔 = ⟨𝑐, 𝑑⟩)))
32 simpr 107 . . . . . . . . . . 11 ((𝑣 = 𝑐𝑢 = 𝑑) → 𝑢 = 𝑑)
3332oveq2d 5555 . . . . . . . . . 10 ((𝑣 = 𝑐𝑢 = 𝑑) → (𝑎 ·𝑜 𝑢) = (𝑎 ·𝑜 𝑑))
34 simpl 106 . . . . . . . . . . 11 ((𝑣 = 𝑐𝑢 = 𝑑) → 𝑣 = 𝑐)
3534oveq2d 5555 . . . . . . . . . 10 ((𝑣 = 𝑐𝑢 = 𝑑) → (𝑏 ·𝑜 𝑣) = (𝑏 ·𝑜 𝑐))
3633, 35eqeq12d 2070 . . . . . . . . 9 ((𝑣 = 𝑐𝑢 = 𝑑) → ((𝑎 ·𝑜 𝑢) = (𝑏 ·𝑜 𝑣) ↔ (𝑎 ·𝑜 𝑑) = (𝑏 ·𝑜 𝑐)))
3731, 36anbi12d 450 . . . . . . . 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 1821 . . . . . . 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 438 . . . . . 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 131 . . . . 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 1804 . . . . 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 141 . . . 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 1804 . . . . 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 1513 . . . 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 141 . . 3 (𝑓 ~Q0 𝑔 → ∃𝑎𝑏𝑐𝑑((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N)) ∧ ((𝑓 = ⟨𝑎, 𝑏⟩ ∧ 𝑔 = ⟨𝑐, 𝑑⟩) ∧ (𝑎 ·𝑜 𝑑) = (𝑏 ·𝑜 𝑐))))
46 pm3.22 256 . . . . . . 7 ((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N)) → (𝑔 ∈ (ω × N) ∧ 𝑓 ∈ (ω × N)))
4746adantr 265 . . . . . 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 pm3.22 256 . . . . . . 7 ((𝑓 = ⟨𝑎, 𝑏⟩ ∧ 𝑔 = ⟨𝑐, 𝑑⟩) → (𝑔 = ⟨𝑐, 𝑑⟩ ∧ 𝑓 = ⟨𝑎, 𝑏⟩))
4948ad2antrl 467 . . . . . 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 492 . . . . . . . 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 2116 . . . . . . . . . . . . . 14 (𝑓 = ⟨𝑎, 𝑏⟩ → (𝑓 ∈ (ω × N) ↔ ⟨𝑎, 𝑏⟩ ∈ (ω × N)))
52 opelxp 4401 . . . . . . . . . . . . . 14 (⟨𝑎, 𝑏⟩ ∈ (ω × N) ↔ (𝑎 ∈ ω ∧ 𝑏N))
5351, 52syl6bb 189 . . . . . . . . . . . . 13 (𝑓 = ⟨𝑎, 𝑏⟩ → (𝑓 ∈ (ω × N) ↔ (𝑎 ∈ ω ∧ 𝑏N)))
5453biimpcd 152 . . . . . . . . . . . 12 (𝑓 ∈ (ω × N) → (𝑓 = ⟨𝑎, 𝑏⟩ → (𝑎 ∈ ω ∧ 𝑏N)))
55 eleq1 2116 . . . . . . . . . . . . . 14 (𝑔 = ⟨𝑐, 𝑑⟩ → (𝑔 ∈ (ω × N) ↔ ⟨𝑐, 𝑑⟩ ∈ (ω × N)))
56 opelxp 4401 . . . . . . . . . . . . . 14 (⟨𝑐, 𝑑⟩ ∈ (ω × N) ↔ (𝑐 ∈ ω ∧ 𝑑N))
5755, 56syl6bb 189 . . . . . . . . . . . . 13 (𝑔 = ⟨𝑐, 𝑑⟩ → (𝑔 ∈ (ω × N) ↔ (𝑐 ∈ ω ∧ 𝑑N)))
5857biimpcd 152 . . . . . . . . . . . 12 (𝑔 ∈ (ω × N) → (𝑔 = ⟨𝑐, 𝑑⟩ → (𝑐 ∈ ω ∧ 𝑑N)))
5954, 58im2anan9 540 . . . . . . . . . . 11 ((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N)) → ((𝑓 = ⟨𝑎, 𝑏⟩ ∧ 𝑔 = ⟨𝑐, 𝑑⟩) → ((𝑎 ∈ ω ∧ 𝑏N) ∧ (𝑐 ∈ ω ∧ 𝑑N))))
6059imp 119 . . . . . . . . . 10 (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N)) ∧ (𝑓 = ⟨𝑎, 𝑏⟩ ∧ 𝑔 = ⟨𝑐, 𝑑⟩)) → ((𝑎 ∈ ω ∧ 𝑏N) ∧ (𝑐 ∈ ω ∧ 𝑑N)))
6160adantrr 456 . . . . . . . . 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 6464 . . . . . . . . . . . 12 (𝑑N𝑑 ∈ ω)
63 nnmcom 6098 . . . . . . . . . . . 12 ((𝑎 ∈ ω ∧ 𝑑 ∈ ω) → (𝑎 ·𝑜 𝑑) = (𝑑 ·𝑜 𝑎))
6462, 63sylan2 274 . . . . . . . . . . 11 ((𝑎 ∈ ω ∧ 𝑑N) → (𝑎 ·𝑜 𝑑) = (𝑑 ·𝑜 𝑎))
65 pinn 6464 . . . . . . . . . . . 12 (𝑏N𝑏 ∈ ω)
66 nnmcom 6098 . . . . . . . . . . . 12 ((𝑏 ∈ ω ∧ 𝑐 ∈ ω) → (𝑏 ·𝑜 𝑐) = (𝑐 ·𝑜 𝑏))
6765, 66sylan 271 . . . . . . . . . . 11 ((𝑏N𝑐 ∈ ω) → (𝑏 ·𝑜 𝑐) = (𝑐 ·𝑜 𝑏))
6864, 67eqeqan12d 2071 . . . . . . . . . 10 (((𝑎 ∈ ω ∧ 𝑑N) ∧ (𝑏N𝑐 ∈ ω)) → ((𝑎 ·𝑜 𝑑) = (𝑏 ·𝑜 𝑐) ↔ (𝑑 ·𝑜 𝑎) = (𝑐 ·𝑜 𝑏)))
6968an42s 531 . . . . . . . . 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 139 . . . . . . 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 2061 . . . . . 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 297 . . . . 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 1508 . . . 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 1508 . . 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 1597 . . 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 1804 . . . . 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 1513 . . . 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 1804 . . . . 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 3578 . . . . . . . . . 10 ((𝑧 = 𝑐𝑤 = 𝑑) → ⟨𝑧, 𝑤⟩ = ⟨𝑐, 𝑑⟩)
8281eqeq2d 2067 . . . . . . . . 9 ((𝑧 = 𝑐𝑤 = 𝑑) → (𝑔 = ⟨𝑧, 𝑤⟩ ↔ 𝑔 = ⟨𝑐, 𝑑⟩))
8382anbi1d 446 . . . . . . . 8 ((𝑧 = 𝑐𝑤 = 𝑑) → ((𝑔 = ⟨𝑧, 𝑤⟩ ∧ 𝑓 = ⟨𝑣, 𝑢⟩) ↔ (𝑔 = ⟨𝑐, 𝑑⟩ ∧ 𝑓 = ⟨𝑣, 𝑢⟩)))
84 simpl 106 . . . . . . . . . 10 ((𝑧 = 𝑐𝑤 = 𝑑) → 𝑧 = 𝑐)
8584oveq1d 5554 . . . . . . . . 9 ((𝑧 = 𝑐𝑤 = 𝑑) → (𝑧 ·𝑜 𝑢) = (𝑐 ·𝑜 𝑢))
86 simpr 107 . . . . . . . . . 10 ((𝑧 = 𝑐𝑤 = 𝑑) → 𝑤 = 𝑑)
8786oveq1d 5554 . . . . . . . . 9 ((𝑧 = 𝑐𝑤 = 𝑑) → (𝑤 ·𝑜 𝑣) = (𝑑 ·𝑜 𝑣))
8885, 87eqeq12d 2070 . . . . . . . 8 ((𝑧 = 𝑐𝑤 = 𝑑) → ((𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣) ↔ (𝑐 ·𝑜 𝑢) = (𝑑 ·𝑜 𝑣)))
8983, 88anbi12d 450 . . . . . . 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 3578 . . . . . . . . . 10 ((𝑣 = 𝑎𝑢 = 𝑏) → ⟨𝑣, 𝑢⟩ = ⟨𝑎, 𝑏⟩)
9190eqeq2d 2067 . . . . . . . . 9 ((𝑣 = 𝑎𝑢 = 𝑏) → (𝑓 = ⟨𝑣, 𝑢⟩ ↔ 𝑓 = ⟨𝑎, 𝑏⟩))
9291anbi2d 445 . . . . . . . 8 ((𝑣 = 𝑎𝑢 = 𝑏) → ((𝑔 = ⟨𝑐, 𝑑⟩ ∧ 𝑓 = ⟨𝑣, 𝑢⟩) ↔ (𝑔 = ⟨𝑐, 𝑑⟩ ∧ 𝑓 = ⟨𝑎, 𝑏⟩)))
93 simpr 107 . . . . . . . . . 10 ((𝑣 = 𝑎𝑢 = 𝑏) → 𝑢 = 𝑏)
9493oveq2d 5555 . . . . . . . . 9 ((𝑣 = 𝑎𝑢 = 𝑏) → (𝑐 ·𝑜 𝑢) = (𝑐 ·𝑜 𝑏))
95 simpl 106 . . . . . . . . . 10 ((𝑣 = 𝑎𝑢 = 𝑏) → 𝑣 = 𝑎)
9695oveq2d 5555 . . . . . . . . 9 ((𝑣 = 𝑎𝑢 = 𝑏) → (𝑑 ·𝑜 𝑣) = (𝑑 ·𝑜 𝑎))
9794, 96eqeq12d 2070 . . . . . . . 8 ((𝑣 = 𝑎𝑢 = 𝑏) → ((𝑐 ·𝑜 𝑢) = (𝑑 ·𝑜 𝑣) ↔ (𝑐 ·𝑜 𝑏) = (𝑑 ·𝑜 𝑎)))
9892, 97anbi12d 450 . . . . . . 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 1821 . . . . . 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 2116 . . . . . . . . . 10 (𝑥 = 𝑔 → (𝑥 ∈ (ω × N) ↔ 𝑔 ∈ (ω × N)))
101100anbi1d 446 . . . . . . . . 9 (𝑥 = 𝑔 → ((𝑥 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ↔ (𝑔 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N))))
102 eqeq1 2062 . . . . . . . . . . . 12 (𝑥 = 𝑔 → (𝑥 = ⟨𝑧, 𝑤⟩ ↔ 𝑔 = ⟨𝑧, 𝑤⟩))
103102anbi1d 446 . . . . . . . . . . 11 (𝑥 = 𝑔 → ((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ↔ (𝑔 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩)))
104103anbi1d 446 . . . . . . . . . 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 1766 . . . . . . . . 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 450 . . . . . . . 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 2116 . . . . . . . . . 10 (𝑦 = 𝑓 → (𝑦 ∈ (ω × N) ↔ 𝑓 ∈ (ω × N)))
108107anbi2d 445 . . . . . . . . 9 (𝑦 = 𝑓 → ((𝑔 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ↔ (𝑔 ∈ (ω × N) ∧ 𝑓 ∈ (ω × N))))
109 eqeq1 2062 . . . . . . . . . . . 12 (𝑦 = 𝑓 → (𝑦 = ⟨𝑣, 𝑢⟩ ↔ 𝑓 = ⟨𝑣, 𝑢⟩))
110109anbi2d 445 . . . . . . . . . . 11 (𝑦 = 𝑓 → ((𝑔 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ↔ (𝑔 = ⟨𝑧, 𝑤⟩ ∧ 𝑓 = ⟨𝑣, 𝑢⟩)))
111110anbi1d 446 . . . . . . . . . 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 1766 . . . . . . . . 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 450 . . . . . . . 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 4036 . . . . . . 7 (𝑔 ~Q0 𝑓 ↔ ((𝑔 ∈ (ω × N) ∧ 𝑓 ∈ (ω × N)) ∧ ∃𝑧𝑤𝑣𝑢((𝑔 = ⟨𝑧, 𝑤⟩ ∧ 𝑓 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣))))
115114biimpri 128 . . . . . 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 276 . . . . 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 118 . . . 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 118 . . 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 118 . 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 101  wb 102   = wceq 1259  wex 1397  wcel 1409  cop 3405   class class class wbr 3791  ωcom 4340   × cxp 4370  (class class class)co 5539   ·𝑜 comu 6029  Ncnpi 6427   ~Q0 ceq0 6441
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3899  ax-sep 3902  ax-nul 3910  ax-pow 3954  ax-pr 3971  ax-un 4197  ax-setind 4289  ax-iinf 4338
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2787  df-csb 2880  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-nul 3252  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-int 3643  df-iun 3686  df-br 3792  df-opab 3846  df-mpt 3847  df-tr 3882  df-id 4057  df-iord 4130  df-on 4132  df-suc 4135  df-iom 4341  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385  df-iota 4894  df-fun 4931  df-fn 4932  df-f 4933  df-f1 4934  df-fo 4935  df-f1o 4936  df-fv 4937  df-ov 5542  df-oprab 5543  df-mpt2 5544  df-1st 5794  df-2nd 5795  df-recs 5950  df-irdg 5987  df-oadd 6035  df-omul 6036  df-ni 6459  df-enq0 6579
This theorem is referenced by:  enq0er  6590
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