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Theorem enq0tr 7235
Description: The equivalence relation for nonnegative fractions is transitive. Lemma for enq0er 7236. (Contributed by Jim Kingdon, 14-Nov-2019.)
Assertion
Ref Expression
enq0tr ((𝑓 ~Q0 𝑔𝑔 ~Q0 ) → 𝑓 ~Q0 )

Proof of Theorem enq0tr
Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑒 𝑠 𝑡 𝑢 𝑣 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2684 . . . . . . . . . 10 𝑓 ∈ V
2 vex 2684 . . . . . . . . . 10 𝑔 ∈ V
3 eleq1 2200 . . . . . . . . . . . 12 (𝑥 = 𝑓 → (𝑥 ∈ (ω × N) ↔ 𝑓 ∈ (ω × N)))
43anbi1d 460 . . . . . . . . . . 11 (𝑥 = 𝑓 → ((𝑥 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ↔ (𝑓 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N))))
5 eqeq1 2144 . . . . . . . . . . . . . 14 (𝑥 = 𝑓 → (𝑥 = ⟨𝑧, 𝑤⟩ ↔ 𝑓 = ⟨𝑧, 𝑤⟩))
65anbi1d 460 . . . . . . . . . . . . 13 (𝑥 = 𝑓 → ((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ↔ (𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩)))
76anbi1d 460 . . . . . . . . . . . 12 (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 1842 . . . . . . . . . . 11 (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 464 . . . . . . . . . 10 (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 2200 . . . . . . . . . . . 12 (𝑦 = 𝑔 → (𝑦 ∈ (ω × N) ↔ 𝑔 ∈ (ω × N)))
1110anbi2d 459 . . . . . . . . . . 11 (𝑦 = 𝑔 → ((𝑓 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ↔ (𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N))))
12 eqeq1 2144 . . . . . . . . . . . . . 14 (𝑦 = 𝑔 → (𝑦 = ⟨𝑣, 𝑢⟩ ↔ 𝑔 = ⟨𝑣, 𝑢⟩))
1312anbi2d 459 . . . . . . . . . . . . 13 (𝑦 = 𝑔 → ((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ↔ (𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩)))
1413anbi1d 460 . . . . . . . . . . . 12 (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 1842 . . . . . . . . . . 11 (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 464 . . . . . . . . . 10 (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 7225 . . . . . . . . . 10 ~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 4189 . . . . . . . . 9 (𝑓 ~Q0 𝑔 ↔ ((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N)) ∧ ∃𝑧𝑤𝑣𝑢((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣))))
19 vex 2684 . . . . . . . . . 10 ∈ V
20 eleq1 2200 . . . . . . . . . . . 12 (𝑥 = 𝑔 → (𝑥 ∈ (ω × N) ↔ 𝑔 ∈ (ω × N)))
2120anbi1d 460 . . . . . . . . . . 11 (𝑥 = 𝑔 → ((𝑥 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ↔ (𝑔 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N))))
22 eqeq1 2144 . . . . . . . . . . . . . 14 (𝑥 = 𝑔 → (𝑥 = ⟨𝑎, 𝑏⟩ ↔ 𝑔 = ⟨𝑎, 𝑏⟩))
2322anbi1d 460 . . . . . . . . . . . . 13 (𝑥 = 𝑔 → ((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑦 = ⟨𝑠, 𝑡⟩) ↔ (𝑔 = ⟨𝑎, 𝑏⟩ ∧ 𝑦 = ⟨𝑠, 𝑡⟩)))
2423anbi1d 460 . . . . . . . . . . . 12 (𝑥 = 𝑔 → (((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑦 = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)) ↔ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ 𝑦 = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠))))
25244exbidv 1842 . . . . . . . . . . 11 (𝑥 = 𝑔 → (∃𝑎𝑏𝑠𝑡((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑦 = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)) ↔ ∃𝑎𝑏𝑠𝑡((𝑔 = ⟨𝑎, 𝑏⟩ ∧ 𝑦 = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠))))
2621, 25anbi12d 464 . . . . . . . . . 10 (𝑥 = 𝑔 → (((𝑥 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ∧ ∃𝑎𝑏𝑠𝑡((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑦 = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠))) ↔ ((𝑔 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ∧ ∃𝑎𝑏𝑠𝑡((𝑔 = ⟨𝑎, 𝑏⟩ ∧ 𝑦 = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))))
27 eleq1 2200 . . . . . . . . . . . 12 (𝑦 = → (𝑦 ∈ (ω × N) ↔ ∈ (ω × N)))
2827anbi2d 459 . . . . . . . . . . 11 (𝑦 = → ((𝑔 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ↔ (𝑔 ∈ (ω × N) ∧ ∈ (ω × N))))
29 eqeq1 2144 . . . . . . . . . . . . . 14 (𝑦 = → (𝑦 = ⟨𝑠, 𝑡⟩ ↔ = ⟨𝑠, 𝑡⟩))
3029anbi2d 459 . . . . . . . . . . . . 13 (𝑦 = → ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ 𝑦 = ⟨𝑠, 𝑡⟩) ↔ (𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩)))
3130anbi1d 460 . . . . . . . . . . . 12 (𝑦 = → (((𝑔 = ⟨𝑎, 𝑏⟩ ∧ 𝑦 = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)) ↔ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠))))
32314exbidv 1842 . . . . . . . . . . 11 (𝑦 = → (∃𝑎𝑏𝑠𝑡((𝑔 = ⟨𝑎, 𝑏⟩ ∧ 𝑦 = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)) ↔ ∃𝑎𝑏𝑠𝑡((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠))))
3328, 32anbi12d 464 . . . . . . . . . 10 (𝑦 = → (((𝑔 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ∧ ∃𝑎𝑏𝑠𝑡((𝑔 = ⟨𝑎, 𝑏⟩ ∧ 𝑦 = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠))) ↔ ((𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ ∃𝑎𝑏𝑠𝑡((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))))
34 df-enq0 7225 . . . . . . . . . 10 ~Q0 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ∧ ∃𝑎𝑏𝑠𝑡((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑦 = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))}
352, 19, 26, 33, 34brab 4189 . . . . . . . . 9 (𝑔 ~Q0 ↔ ((𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ ∃𝑎𝑏𝑠𝑡((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠))))
3618, 35anbi12i 455 . . . . . . . 8 ((𝑓 ~Q0 𝑔𝑔 ~Q0 ) ↔ (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N)) ∧ ∃𝑧𝑤𝑣𝑢((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣))) ∧ ((𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ ∃𝑎𝑏𝑠𝑡((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))))
3736biimpi 119 . . . . . . 7 ((𝑓 ~Q0 𝑔𝑔 ~Q0 ) → (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N)) ∧ ∃𝑧𝑤𝑣𝑢((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣))) ∧ ((𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ ∃𝑎𝑏𝑠𝑡((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))))
38 an4 575 . . . . . . 7 ((((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N)) ∧ ∃𝑧𝑤𝑣𝑢((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣))) ∧ ((𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ ∃𝑎𝑏𝑠𝑡((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))) ↔ (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N)) ∧ (𝑔 ∈ (ω × N) ∧ ∈ (ω × N))) ∧ (∃𝑧𝑤𝑣𝑢((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ∃𝑎𝑏𝑠𝑡((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))))
3937, 38sylib 121 . . . . . 6 ((𝑓 ~Q0 𝑔𝑔 ~Q0 ) → (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N)) ∧ (𝑔 ∈ (ω × N) ∧ ∈ (ω × N))) ∧ (∃𝑧𝑤𝑣𝑢((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ∃𝑎𝑏𝑠𝑡((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))))
40 3anass 966 . . . . . . . 8 ((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ↔ (𝑓 ∈ (ω × N) ∧ (𝑔 ∈ (ω × N) ∧ ∈ (ω × N))))
41 anass 398 . . . . . . . . 9 (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N)) ∧ (𝑔 ∈ (ω × N) ∧ ∈ (ω × N))) ↔ (𝑓 ∈ (ω × N) ∧ (𝑔 ∈ (ω × N) ∧ (𝑔 ∈ (ω × N) ∧ ∈ (ω × N)))))
42 anass 398 . . . . . . . . . 10 (((𝑔 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N)) ∧ ∈ (ω × N)) ↔ (𝑔 ∈ (ω × N) ∧ (𝑔 ∈ (ω × N) ∧ ∈ (ω × N))))
4342anbi2i 452 . . . . . . . . 9 ((𝑓 ∈ (ω × N) ∧ ((𝑔 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N)) ∧ ∈ (ω × N))) ↔ (𝑓 ∈ (ω × N) ∧ (𝑔 ∈ (ω × N) ∧ (𝑔 ∈ (ω × N) ∧ ∈ (ω × N)))))
44 anidm 393 . . . . . . . . . . 11 ((𝑔 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N)) ↔ 𝑔 ∈ (ω × N))
4544anbi1i 453 . . . . . . . . . 10 (((𝑔 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N)) ∧ ∈ (ω × N)) ↔ (𝑔 ∈ (ω × N) ∧ ∈ (ω × N)))
4645anbi2i 452 . . . . . . . . 9 ((𝑓 ∈ (ω × N) ∧ ((𝑔 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N)) ∧ ∈ (ω × N))) ↔ (𝑓 ∈ (ω × N) ∧ (𝑔 ∈ (ω × N) ∧ ∈ (ω × N))))
4741, 43, 463bitr2i 207 . . . . . . . 8 (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N)) ∧ (𝑔 ∈ (ω × N) ∧ ∈ (ω × N))) ↔ (𝑓 ∈ (ω × N) ∧ (𝑔 ∈ (ω × N) ∧ ∈ (ω × N))))
4840, 47bitr4i 186 . . . . . . 7 ((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ↔ ((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N)) ∧ (𝑔 ∈ (ω × N) ∧ ∈ (ω × N))))
4948anbi1i 453 . . . . . 6 (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ (∃𝑧𝑤𝑣𝑢((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ∃𝑎𝑏𝑠𝑡((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))) ↔ (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N)) ∧ (𝑔 ∈ (ω × N) ∧ ∈ (ω × N))) ∧ (∃𝑧𝑤𝑣𝑢((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ∃𝑎𝑏𝑠𝑡((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))))
5039, 49sylibr 133 . . . . 5 ((𝑓 ~Q0 𝑔𝑔 ~Q0 ) → ((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ (∃𝑧𝑤𝑣𝑢((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ∃𝑎𝑏𝑠𝑡((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))))
51 ee8anv 1905 . . . . . 6 (∃𝑧𝑤𝑣𝑢𝑎𝑏𝑠𝑡(((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠))) ↔ (∃𝑧𝑤𝑣𝑢((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ∃𝑎𝑏𝑠𝑡((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠))))
5251anbi2i 452 . . . . 5 (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ ∃𝑧𝑤𝑣𝑢𝑎𝑏𝑠𝑡(((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))) ↔ ((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ (∃𝑧𝑤𝑣𝑢((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ∃𝑎𝑏𝑠𝑡((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))))
5350, 52sylibr 133 . . . 4 ((𝑓 ~Q0 𝑔𝑔 ~Q0 ) → ((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ ∃𝑧𝑤𝑣𝑢𝑎𝑏𝑠𝑡(((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))))
54 19.42vvvv 1885 . . . . . . 7 (∃𝑎𝑏𝑠𝑡((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))) ↔ ((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ ∃𝑎𝑏𝑠𝑡(((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))))
55542exbii 1585 . . . . . 6 (∃𝑣𝑢𝑎𝑏𝑠𝑡((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))) ↔ ∃𝑣𝑢((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ ∃𝑎𝑏𝑠𝑡(((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))))
56552exbii 1585 . . . . 5 (∃𝑧𝑤𝑣𝑢𝑎𝑏𝑠𝑡((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))) ↔ ∃𝑧𝑤𝑣𝑢((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ ∃𝑎𝑏𝑠𝑡(((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))))
57 19.42vvvv 1885 . . . . 5 (∃𝑧𝑤𝑣𝑢((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ ∃𝑎𝑏𝑠𝑡(((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))) ↔ ((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ ∃𝑧𝑤𝑣𝑢𝑎𝑏𝑠𝑡(((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))))
5856, 57bitri 183 . . . 4 (∃𝑧𝑤𝑣𝑢𝑎𝑏𝑠𝑡((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))) ↔ ((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ ∃𝑧𝑤𝑣𝑢𝑎𝑏𝑠𝑡(((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))))
5953, 58sylibr 133 . . 3 ((𝑓 ~Q0 𝑔𝑔 ~Q0 ) → ∃𝑧𝑤𝑣𝑢𝑎𝑏𝑠𝑡((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))))
60 3simpb 979 . . . . . . . . 9 ((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) → (𝑓 ∈ (ω × N) ∧ ∈ (ω × N)))
6160adantr 274 . . . . . . . 8 (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))) → (𝑓 ∈ (ω × N) ∧ ∈ (ω × N)))
62 simplll 522 . . . . . . . . . 10 ((((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠))) → 𝑓 = ⟨𝑧, 𝑤⟩)
63 simprlr 527 . . . . . . . . . 10 ((((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠))) → = ⟨𝑠, 𝑡⟩)
6462, 63jca 304 . . . . . . . . 9 ((((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠))) → (𝑓 = ⟨𝑧, 𝑤⟩ ∧ = ⟨𝑠, 𝑡⟩))
6564adantl 275 . . . . . . . 8 (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))) → (𝑓 = ⟨𝑧, 𝑤⟩ ∧ = ⟨𝑠, 𝑡⟩))
66 oveq1 5774 . . . . . . . . . . . . . . . 16 (𝑣 = ∅ → (𝑣 ·o 𝑡) = (∅ ·o 𝑡))
6763adantl 275 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))) → = ⟨𝑠, 𝑡⟩)
68 simpl3 986 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))) → ∈ (ω × N))
6967, 68eqeltrrd 2215 . . . . . . . . . . . . . . . . . . . . 21 (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))) → ⟨𝑠, 𝑡⟩ ∈ (ω × N))
70 opelxp 4564 . . . . . . . . . . . . . . . . . . . . 21 (⟨𝑠, 𝑡⟩ ∈ (ω × N) ↔ (𝑠 ∈ ω ∧ 𝑡N))
7169, 70sylib 121 . . . . . . . . . . . . . . . . . . . 20 (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))) → (𝑠 ∈ ω ∧ 𝑡N))
7271simprd 113 . . . . . . . . . . . . . . . . . . 19 (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))) → 𝑡N)
73 pinn 7110 . . . . . . . . . . . . . . . . . . 19 (𝑡N𝑡 ∈ ω)
7472, 73syl 14 . . . . . . . . . . . . . . . . . 18 (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))) → 𝑡 ∈ ω)
75 nnm0r 6368 . . . . . . . . . . . . . . . . . 18 (𝑡 ∈ ω → (∅ ·o 𝑡) = ∅)
7674, 75syl 14 . . . . . . . . . . . . . . . . 17 (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))) → (∅ ·o 𝑡) = ∅)
7776eqeq2d 2149 . . . . . . . . . . . . . . . 16 (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))) → ((𝑣 ·o 𝑡) = (∅ ·o 𝑡) ↔ (𝑣 ·o 𝑡) = ∅))
7866, 77syl5ib 153 . . . . . . . . . . . . . . 15 (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))) → (𝑣 = ∅ → (𝑣 ·o 𝑡) = ∅))
79 simprr 521 . . . . . . . . . . . . . . . . . 18 ((((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠))) → (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠))
80 eqtr2 2156 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑔 = ⟨𝑣, 𝑢⟩ ∧ 𝑔 = ⟨𝑎, 𝑏⟩) → ⟨𝑣, 𝑢⟩ = ⟨𝑎, 𝑏⟩)
81 vex 2684 . . . . . . . . . . . . . . . . . . . . . . 23 𝑣 ∈ V
82 vex 2684 . . . . . . . . . . . . . . . . . . . . . . 23 𝑢 ∈ V
8381, 82opth 4154 . . . . . . . . . . . . . . . . . . . . . 22 (⟨𝑣, 𝑢⟩ = ⟨𝑎, 𝑏⟩ ↔ (𝑣 = 𝑎𝑢 = 𝑏))
8480, 83sylib 121 . . . . . . . . . . . . . . . . . . . . 21 ((𝑔 = ⟨𝑣, 𝑢⟩ ∧ 𝑔 = ⟨𝑎, 𝑏⟩) → (𝑣 = 𝑎𝑢 = 𝑏))
85 oveq1 5774 . . . . . . . . . . . . . . . . . . . . . 22 (𝑣 = 𝑎 → (𝑣 ·o 𝑡) = (𝑎 ·o 𝑡))
86 oveq1 5774 . . . . . . . . . . . . . . . . . . . . . 22 (𝑢 = 𝑏 → (𝑢 ·o 𝑠) = (𝑏 ·o 𝑠))
8785, 86eqeqan12d 2153 . . . . . . . . . . . . . . . . . . . . 21 ((𝑣 = 𝑎𝑢 = 𝑏) → ((𝑣 ·o 𝑡) = (𝑢 ·o 𝑠) ↔ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))
8884, 87syl 14 . . . . . . . . . . . . . . . . . . . 20 ((𝑔 = ⟨𝑣, 𝑢⟩ ∧ 𝑔 = ⟨𝑎, 𝑏⟩) → ((𝑣 ·o 𝑡) = (𝑢 ·o 𝑠) ↔ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))
8988ad2ant2lr 501 . . . . . . . . . . . . . . . . . . 19 (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩)) → ((𝑣 ·o 𝑡) = (𝑢 ·o 𝑠) ↔ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))
9089ad2ant2r 500 . . . . . . . . . . . . . . . . . 18 ((((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠))) → ((𝑣 ·o 𝑡) = (𝑢 ·o 𝑠) ↔ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))
9179, 90mpbird 166 . . . . . . . . . . . . . . . . 17 ((((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠))) → (𝑣 ·o 𝑡) = (𝑢 ·o 𝑠))
9291eqeq1d 2146 . . . . . . . . . . . . . . . 16 ((((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠))) → ((𝑣 ·o 𝑡) = ∅ ↔ (𝑢 ·o 𝑠) = ∅))
9392adantl 275 . . . . . . . . . . . . . . 15 (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))) → ((𝑣 ·o 𝑡) = ∅ ↔ (𝑢 ·o 𝑠) = ∅))
9478, 93sylibd 148 . . . . . . . . . . . . . 14 (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))) → (𝑣 = ∅ → (𝑢 ·o 𝑠) = ∅))
95 simpllr 523 . . . . . . . . . . . . . . . . . . . 20 ((((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠))) → 𝑔 = ⟨𝑣, 𝑢⟩)
9695adantl 275 . . . . . . . . . . . . . . . . . . 19 (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))) → 𝑔 = ⟨𝑣, 𝑢⟩)
97 simpl2 985 . . . . . . . . . . . . . . . . . . 19 (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))) → 𝑔 ∈ (ω × N))
9896, 97eqeltrrd 2215 . . . . . . . . . . . . . . . . . 18 (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))) → ⟨𝑣, 𝑢⟩ ∈ (ω × N))
99 opelxp 4564 . . . . . . . . . . . . . . . . . 18 (⟨𝑣, 𝑢⟩ ∈ (ω × N) ↔ (𝑣 ∈ ω ∧ 𝑢N))
10098, 99sylib 121 . . . . . . . . . . . . . . . . 17 (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))) → (𝑣 ∈ ω ∧ 𝑢N))
101100simprd 113 . . . . . . . . . . . . . . . 16 (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))) → 𝑢N)
102 pinn 7110 . . . . . . . . . . . . . . . 16 (𝑢N𝑢 ∈ ω)
103101, 102syl 14 . . . . . . . . . . . . . . 15 (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))) → 𝑢 ∈ ω)
10471simpld 111 . . . . . . . . . . . . . . 15 (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))) → 𝑠 ∈ ω)
105 nnm00 6418 . . . . . . . . . . . . . . 15 ((𝑢 ∈ ω ∧ 𝑠 ∈ ω) → ((𝑢 ·o 𝑠) = ∅ ↔ (𝑢 = ∅ ∨ 𝑠 = ∅)))
106103, 104, 105syl2anc 408 . . . . . . . . . . . . . 14 (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))) → ((𝑢 ·o 𝑠) = ∅ ↔ (𝑢 = ∅ ∨ 𝑠 = ∅)))
10794, 106sylibd 148 . . . . . . . . . . . . 13 (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))) → (𝑣 = ∅ → (𝑢 = ∅ ∨ 𝑠 = ∅)))
108 elni2 7115 . . . . . . . . . . . . . . . 16 (𝑢N ↔ (𝑢 ∈ ω ∧ ∅ ∈ 𝑢))
109108simprbi 273 . . . . . . . . . . . . . . 15 (𝑢N → ∅ ∈ 𝑢)
110101, 109syl 14 . . . . . . . . . . . . . 14 (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))) → ∅ ∈ 𝑢)
111 n0i 3363 . . . . . . . . . . . . . 14 (∅ ∈ 𝑢 → ¬ 𝑢 = ∅)
112 biorf 733 . . . . . . . . . . . . . 14 𝑢 = ∅ → (𝑠 = ∅ ↔ (𝑢 = ∅ ∨ 𝑠 = ∅)))
113110, 111, 1123syl 17 . . . . . . . . . . . . 13 (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))) → (𝑠 = ∅ ↔ (𝑢 = ∅ ∨ 𝑠 = ∅)))
114107, 113sylibrd 168 . . . . . . . . . . . 12 (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))) → (𝑣 = ∅ → 𝑠 = ∅))
11562adantl 275 . . . . . . . . . . . . . . . . . 18 (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))) → 𝑓 = ⟨𝑧, 𝑤⟩)
116 simpl1 984 . . . . . . . . . . . . . . . . . 18 (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))) → 𝑓 ∈ (ω × N))
117115, 116eqeltrrd 2215 . . . . . . . . . . . . . . . . 17 (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))) → ⟨𝑧, 𝑤⟩ ∈ (ω × N))
118 opelxp 4564 . . . . . . . . . . . . . . . . 17 (⟨𝑧, 𝑤⟩ ∈ (ω × N) ↔ (𝑧 ∈ ω ∧ 𝑤N))
119117, 118sylib 121 . . . . . . . . . . . . . . . 16 (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))) → (𝑧 ∈ ω ∧ 𝑤N))
120119simprd 113 . . . . . . . . . . . . . . 15 (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))) → 𝑤N)
121 pinn 7110 . . . . . . . . . . . . . . 15 (𝑤N𝑤 ∈ ω)
122120, 121syl 14 . . . . . . . . . . . . . 14 (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))) → 𝑤 ∈ ω)
123 nnm0 6364 . . . . . . . . . . . . . 14 (𝑤 ∈ ω → (𝑤 ·o ∅) = ∅)
124122, 123syl 14 . . . . . . . . . . . . 13 (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))) → (𝑤 ·o ∅) = ∅)
125 oveq2 5775 . . . . . . . . . . . . . 14 (𝑠 = ∅ → (𝑤 ·o 𝑠) = (𝑤 ·o ∅))
126125eqeq1d 2146 . . . . . . . . . . . . 13 (𝑠 = ∅ → ((𝑤 ·o 𝑠) = ∅ ↔ (𝑤 ·o ∅) = ∅))
127124, 126syl5ibrcom 156 . . . . . . . . . . . 12 (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))) → (𝑠 = ∅ → (𝑤 ·o 𝑠) = ∅))
128114, 127syld 45 . . . . . . . . . . 11 (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))) → (𝑣 = ∅ → (𝑤 ·o 𝑠) = ∅))
129 oveq2 5775 . . . . . . . . . . . . . . . 16 (𝑣 = ∅ → (𝑤 ·o 𝑣) = (𝑤 ·o ∅))
130124eqeq2d 2149 . . . . . . . . . . . . . . . 16 (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))) → ((𝑤 ·o 𝑣) = (𝑤 ·o ∅) ↔ (𝑤 ·o 𝑣) = ∅))
131129, 130syl5ib 153 . . . . . . . . . . . . . . 15 (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))) → (𝑣 = ∅ → (𝑤 ·o 𝑣) = ∅))
132 simprlr 527 . . . . . . . . . . . . . . . 16 (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))) → (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣))
133132eqeq1d 2146 . . . . . . . . . . . . . . 15 (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))) → ((𝑧 ·o 𝑢) = ∅ ↔ (𝑤 ·o 𝑣) = ∅))
134131, 133sylibrd 168 . . . . . . . . . . . . . 14 (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))) → (𝑣 = ∅ → (𝑧 ·o 𝑢) = ∅))
135119simpld 111 . . . . . . . . . . . . . . 15 (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))) → 𝑧 ∈ ω)
136 nnm00 6418 . . . . . . . . . . . . . . 15 ((𝑧 ∈ ω ∧ 𝑢 ∈ ω) → ((𝑧 ·o 𝑢) = ∅ ↔ (𝑧 = ∅ ∨ 𝑢 = ∅)))
137135, 103, 136syl2anc 408 . . . . . . . . . . . . . 14 (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))) → ((𝑧 ·o 𝑢) = ∅ ↔ (𝑧 = ∅ ∨ 𝑢 = ∅)))
138134, 137sylibd 148 . . . . . . . . . . . . 13 (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))) → (𝑣 = ∅ → (𝑧 = ∅ ∨ 𝑢 = ∅)))
139 biorf 733 . . . . . . . . . . . . . . 15 𝑢 = ∅ → (𝑧 = ∅ ↔ (𝑢 = ∅ ∨ 𝑧 = ∅)))
140 orcom 717 . . . . . . . . . . . . . . 15 ((𝑢 = ∅ ∨ 𝑧 = ∅) ↔ (𝑧 = ∅ ∨ 𝑢 = ∅))
141139, 140syl6bb 195 . . . . . . . . . . . . . 14 𝑢 = ∅ → (𝑧 = ∅ ↔ (𝑧 = ∅ ∨ 𝑢 = ∅)))
142110, 111, 1413syl 17 . . . . . . . . . . . . 13 (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))) → (𝑧 = ∅ ↔ (𝑧 = ∅ ∨ 𝑢 = ∅)))
143138, 142sylibrd 168 . . . . . . . . . . . 12 (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))) → (𝑣 = ∅ → 𝑧 = ∅))
144 oveq1 5774 . . . . . . . . . . . . . 14 (𝑧 = ∅ → (𝑧 ·o 𝑡) = (∅ ·o 𝑡))
145144eqeq1d 2146 . . . . . . . . . . . . 13 (𝑧 = ∅ → ((𝑧 ·o 𝑡) = ∅ ↔ (∅ ·o 𝑡) = ∅))
14676, 145syl5ibrcom 156 . . . . . . . . . . . 12 (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))) → (𝑧 = ∅ → (𝑧 ·o 𝑡) = ∅))
147143, 146syld 45 . . . . . . . . . . 11 (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))) → (𝑣 = ∅ → (𝑧 ·o 𝑡) = ∅))
148128, 147jcad 305 . . . . . . . . . 10 (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))) → (𝑣 = ∅ → ((𝑤 ·o 𝑠) = ∅ ∧ (𝑧 ·o 𝑡) = ∅)))
149 eqtr3 2157 . . . . . . . . . . 11 (((𝑤 ·o 𝑠) = ∅ ∧ (𝑧 ·o 𝑡) = ∅) → (𝑤 ·o 𝑠) = (𝑧 ·o 𝑡))
150149eqcomd 2143 . . . . . . . . . 10 (((𝑤 ·o 𝑠) = ∅ ∧ (𝑧 ·o 𝑡) = ∅) → (𝑧 ·o 𝑡) = (𝑤 ·o 𝑠))
151148, 150syl6 33 . . . . . . . . 9 (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))) → (𝑣 = ∅ → (𝑧 ·o 𝑡) = (𝑤 ·o 𝑠)))
152 simplr 519 . . . . . . . . . . . . . . . . 17 ((((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠))) → (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣))
15391, 152oveq12d 5785 . . . . . . . . . . . . . . . 16 ((((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠))) → ((𝑣 ·o 𝑡) ·o (𝑧 ·o 𝑢)) = ((𝑢 ·o 𝑠) ·o (𝑤 ·o 𝑣)))
154153adantl 275 . . . . . . . . . . . . . . 15 (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))) → ((𝑣 ·o 𝑡) ·o (𝑧 ·o 𝑢)) = ((𝑢 ·o 𝑠) ·o (𝑤 ·o 𝑣)))
155100simpld 111 . . . . . . . . . . . . . . . . 17 (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))) → 𝑣 ∈ ω)
156 nnmcl 6370 . . . . . . . . . . . . . . . . 17 ((𝑣 ∈ ω ∧ 𝑡 ∈ ω) → (𝑣 ·o 𝑡) ∈ ω)
157155, 74, 156syl2anc 408 . . . . . . . . . . . . . . . 16 (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))) → (𝑣 ·o 𝑡) ∈ ω)
158 nnmcom 6378 . . . . . . . . . . . . . . . . 17 ((𝑐 ∈ ω ∧ 𝑑 ∈ ω) → (𝑐 ·o 𝑑) = (𝑑 ·o 𝑐))
159158adantl 275 . . . . . . . . . . . . . . . 16 ((((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))) ∧ (𝑐 ∈ ω ∧ 𝑑 ∈ ω)) → (𝑐 ·o 𝑑) = (𝑑 ·o 𝑐))
160 nnmass 6376 . . . . . . . . . . . . . . . . 17 ((𝑐 ∈ ω ∧ 𝑑 ∈ ω ∧ 𝑒 ∈ ω) → ((𝑐 ·o 𝑑) ·o 𝑒) = (𝑐 ·o (𝑑 ·o 𝑒)))
161160adantl 275 . . . . . . . . . . . . . . . 16 ((((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))) ∧ (𝑐 ∈ ω ∧ 𝑑 ∈ ω ∧ 𝑒 ∈ ω)) → ((𝑐 ·o 𝑑) ·o 𝑒) = (𝑐 ·o (𝑑 ·o 𝑒)))
162157, 135, 103, 159, 161caov13d 5947 . . . . . . . . . . . . . . 15 (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))) → ((𝑣 ·o 𝑡) ·o (𝑧 ·o 𝑢)) = (𝑢 ·o (𝑧 ·o (𝑣 ·o 𝑡))))
163 nnmcl 6370 . . . . . . . . . . . . . . . . 17 ((𝑤 ∈ ω ∧ 𝑣 ∈ ω) → (𝑤 ·o 𝑣) ∈ ω)
164122, 155, 163syl2anc 408 . . . . . . . . . . . . . . . 16 (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))) → (𝑤 ·o 𝑣) ∈ ω)
165161, 103, 104, 164caovassd 5923 . . . . . . . . . . . . . . 15 (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))) → ((𝑢 ·o 𝑠) ·o (𝑤 ·o 𝑣)) = (𝑢 ·o (𝑠 ·o (𝑤 ·o 𝑣))))
166154, 162, 1653eqtr3d 2178 . . . . . . . . . . . . . 14 (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))) → (𝑢 ·o (𝑧 ·o (𝑣 ·o 𝑡))) = (𝑢 ·o (𝑠 ·o (𝑤 ·o 𝑣))))
167 nnmcl 6370 . . . . . . . . . . . . . . . 16 ((𝑧 ∈ ω ∧ (𝑣 ·o 𝑡) ∈ ω) → (𝑧 ·o (𝑣 ·o 𝑡)) ∈ ω)
168135, 157, 167syl2anc 408 . . . . . . . . . . . . . . 15 (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))) → (𝑧 ·o (𝑣 ·o 𝑡)) ∈ ω)
169 nnmcl 6370 . . . . . . . . . . . . . . . 16 ((𝑠 ∈ ω ∧ (𝑤 ·o 𝑣) ∈ ω) → (𝑠 ·o (𝑤 ·o 𝑣)) ∈ ω)
170104, 164, 169syl2anc 408 . . . . . . . . . . . . . . 15 (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))) → (𝑠 ·o (𝑤 ·o 𝑣)) ∈ ω)
171 nnmcan 6408 . . . . . . . . . . . . . . 15 (((𝑢 ∈ ω ∧ (𝑧 ·o (𝑣 ·o 𝑡)) ∈ ω ∧ (𝑠 ·o (𝑤 ·o 𝑣)) ∈ ω) ∧ ∅ ∈ 𝑢) → ((𝑢 ·o (𝑧 ·o (𝑣 ·o 𝑡))) = (𝑢 ·o (𝑠 ·o (𝑤 ·o 𝑣))) ↔ (𝑧 ·o (𝑣 ·o 𝑡)) = (𝑠 ·o (𝑤 ·o 𝑣))))
172103, 168, 170, 110, 171syl31anc 1219 . . . . . . . . . . . . . 14 (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))) → ((𝑢 ·o (𝑧 ·o (𝑣 ·o 𝑡))) = (𝑢 ·o (𝑠 ·o (𝑤 ·o 𝑣))) ↔ (𝑧 ·o (𝑣 ·o 𝑡)) = (𝑠 ·o (𝑤 ·o 𝑣))))
173166, 172mpbid 146 . . . . . . . . . . . . 13 (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))) → (𝑧 ·o (𝑣 ·o 𝑡)) = (𝑠 ·o (𝑤 ·o 𝑣)))
174135, 155, 74, 159, 161caov12d 5945 . . . . . . . . . . . . 13 (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))) → (𝑧 ·o (𝑣 ·o 𝑡)) = (𝑣 ·o (𝑧 ·o 𝑡)))
175104, 122, 155, 159, 161caov13d 5947 . . . . . . . . . . . . 13 (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))) → (𝑠 ·o (𝑤 ·o 𝑣)) = (𝑣 ·o (𝑤 ·o 𝑠)))
176173, 174, 1753eqtr3d 2178 . . . . . . . . . . . 12 (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))) → (𝑣 ·o (𝑧 ·o 𝑡)) = (𝑣 ·o (𝑤 ·o 𝑠)))
177176adantr 274 . . . . . . . . . . 11 ((((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))) ∧ ∅ ∈ 𝑣) → (𝑣 ·o (𝑧 ·o 𝑡)) = (𝑣 ·o (𝑤 ·o 𝑠)))
178 nnmcl 6370 . . . . . . . . . . . . . 14 ((𝑧 ∈ ω ∧ 𝑡 ∈ ω) → (𝑧 ·o 𝑡) ∈ ω)
179135, 74, 178syl2anc 408 . . . . . . . . . . . . 13 (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))) → (𝑧 ·o 𝑡) ∈ ω)
180 nnmcl 6370 . . . . . . . . . . . . . 14 ((𝑤 ∈ ω ∧ 𝑠 ∈ ω) → (𝑤 ·o 𝑠) ∈ ω)
181122, 104, 180syl2anc 408 . . . . . . . . . . . . 13 (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))) → (𝑤 ·o 𝑠) ∈ ω)
182155, 179, 1813jca 1161 . . . . . . . . . . . 12 (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))) → (𝑣 ∈ ω ∧ (𝑧 ·o 𝑡) ∈ ω ∧ (𝑤 ·o 𝑠) ∈ ω))
183 nnmcan 6408 . . . . . . . . . . . 12 (((𝑣 ∈ ω ∧ (𝑧 ·o 𝑡) ∈ ω ∧ (𝑤 ·o 𝑠) ∈ ω) ∧ ∅ ∈ 𝑣) → ((𝑣 ·o (𝑧 ·o 𝑡)) = (𝑣 ·o (𝑤 ·o 𝑠)) ↔ (𝑧 ·o 𝑡) = (𝑤 ·o 𝑠)))
184182, 183sylan 281 . . . . . . . . . . 11 ((((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))) ∧ ∅ ∈ 𝑣) → ((𝑣 ·o (𝑧 ·o 𝑡)) = (𝑣 ·o (𝑤 ·o 𝑠)) ↔ (𝑧 ·o 𝑡) = (𝑤 ·o 𝑠)))
185177, 184mpbid 146 . . . . . . . . . 10 ((((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))) ∧ ∅ ∈ 𝑣) → (𝑧 ·o 𝑡) = (𝑤 ·o 𝑠))
186185ex 114 . . . . . . . . 9 (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))) → (∅ ∈ 𝑣 → (𝑧 ·o 𝑡) = (𝑤 ·o 𝑠)))
187 0elnn 4527 . . . . . . . . . 10 (𝑣 ∈ ω → (𝑣 = ∅ ∨ ∅ ∈ 𝑣))
188155, 187syl 14 . . . . . . . . 9 (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))) → (𝑣 = ∅ ∨ ∅ ∈ 𝑣))
189151, 186, 188mpjaod 707 . . . . . . . 8 (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))) → (𝑧 ·o 𝑡) = (𝑤 ·o 𝑠))
19061, 65, 189jca32 308 . . . . . . 7 (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))) → ((𝑓 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ ((𝑓 = ⟨𝑧, 𝑤⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑧 ·o 𝑡) = (𝑤 ·o 𝑠))))
1911902eximi 1580 . . . . . 6 (∃𝑠𝑡((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))) → ∃𝑠𝑡((𝑓 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ ((𝑓 = ⟨𝑧, 𝑤⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑧 ·o 𝑡) = (𝑤 ·o 𝑠))))
192191exlimivv 1868 . . . . 5 (∃𝑎𝑏𝑠𝑡((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))) → ∃𝑠𝑡((𝑓 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ ((𝑓 = ⟨𝑧, 𝑤⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑧 ·o 𝑡) = (𝑤 ·o 𝑠))))
193192exlimivv 1868 . . . 4 (∃𝑣𝑢𝑎𝑏𝑠𝑡((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))) → ∃𝑠𝑡((𝑓 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ ((𝑓 = ⟨𝑧, 𝑤⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑧 ·o 𝑡) = (𝑤 ·o 𝑠))))
1941932eximi 1580 . . 3 (∃𝑧𝑤𝑣𝑢𝑎𝑏𝑠𝑡((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·o 𝑡) = (𝑏 ·o 𝑠)))) → ∃𝑧𝑤𝑠𝑡((𝑓 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ ((𝑓 = ⟨𝑧, 𝑤⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑧 ·o 𝑡) = (𝑤 ·o 𝑠))))
19559, 194syl 14 . 2 ((𝑓 ~Q0 𝑔𝑔 ~Q0 ) → ∃𝑧𝑤𝑠𝑡((𝑓 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ ((𝑓 = ⟨𝑧, 𝑤⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑧 ·o 𝑡) = (𝑤 ·o 𝑠))))
196 19.42vvvv 1885 . . 3 (∃𝑧𝑤𝑠𝑡((𝑓 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ ((𝑓 = ⟨𝑧, 𝑤⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑧 ·o 𝑡) = (𝑤 ·o 𝑠))) ↔ ((𝑓 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ ∃𝑧𝑤𝑠𝑡((𝑓 = ⟨𝑧, 𝑤⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑧 ·o 𝑡) = (𝑤 ·o 𝑠))))
1975anbi1d 460 . . . . . . 7 (𝑥 = 𝑓 → ((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑠, 𝑡⟩) ↔ (𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑠, 𝑡⟩)))
198197anbi1d 460 . . . . . 6 (𝑥 = 𝑓 → (((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑠, 𝑡⟩) ∧ (𝑧 ·o 𝑡) = (𝑤 ·o 𝑠)) ↔ ((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑠, 𝑡⟩) ∧ (𝑧 ·o 𝑡) = (𝑤 ·o 𝑠))))
1991984exbidv 1842 . . . . 5 (𝑥 = 𝑓 → (∃𝑧𝑤𝑠𝑡((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑠, 𝑡⟩) ∧ (𝑧 ·o 𝑡) = (𝑤 ·o 𝑠)) ↔ ∃𝑧𝑤𝑠𝑡((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑠, 𝑡⟩) ∧ (𝑧 ·o 𝑡) = (𝑤 ·o 𝑠))))
2004, 199anbi12d 464 . . . 4 (𝑥 = 𝑓 → (((𝑥 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ∧ ∃𝑧𝑤𝑠𝑡((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑠, 𝑡⟩) ∧ (𝑧 ·o 𝑡) = (𝑤 ·o 𝑠))) ↔ ((𝑓 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ∧ ∃𝑧𝑤𝑠𝑡((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑠, 𝑡⟩) ∧ (𝑧 ·o 𝑡) = (𝑤 ·o 𝑠)))))
20127anbi2d 459 . . . . 5 (𝑦 = → ((𝑓 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ↔ (𝑓 ∈ (ω × N) ∧ ∈ (ω × N))))
20229anbi2d 459 . . . . . . 7 (𝑦 = → ((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑠, 𝑡⟩) ↔ (𝑓 = ⟨𝑧, 𝑤⟩ ∧ = ⟨𝑠, 𝑡⟩)))
203202anbi1d 460 . . . . . 6 (𝑦 = → (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑠, 𝑡⟩) ∧ (𝑧 ·o 𝑡) = (𝑤 ·o 𝑠)) ↔ ((𝑓 = ⟨𝑧, 𝑤⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑧 ·o 𝑡) = (𝑤 ·o 𝑠))))
2042034exbidv 1842 . . . . 5 (𝑦 = → (∃𝑧𝑤𝑠𝑡((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑠, 𝑡⟩) ∧ (𝑧 ·o 𝑡) = (𝑤 ·o 𝑠)) ↔ ∃𝑧𝑤𝑠𝑡((𝑓 = ⟨𝑧, 𝑤⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑧 ·o 𝑡) = (𝑤 ·o 𝑠))))
205201, 204anbi12d 464 . . . 4 (𝑦 = → (((𝑓 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ∧ ∃𝑧𝑤𝑠𝑡((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑠, 𝑡⟩) ∧ (𝑧 ·o 𝑡) = (𝑤 ·o 𝑠))) ↔ ((𝑓 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ ∃𝑧𝑤𝑠𝑡((𝑓 = ⟨𝑧, 𝑤⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑧 ·o 𝑡) = (𝑤 ·o 𝑠)))))
206 df-enq0 7225 . . . 4 ~Q0 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ∧ ∃𝑧𝑤𝑠𝑡((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑠, 𝑡⟩) ∧ (𝑧 ·o 𝑡) = (𝑤 ·o 𝑠)))}
2071, 19, 200, 205, 206brab 4189 . . 3 (𝑓 ~Q0 ↔ ((𝑓 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ ∃𝑧𝑤𝑠𝑡((𝑓 = ⟨𝑧, 𝑤⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑧 ·o 𝑡) = (𝑤 ·o 𝑠))))
208196, 207bitr4i 186 . 2 (∃𝑧𝑤𝑠𝑡((𝑓 ∈ (ω × N) ∧ ∈ (ω × N)) ∧ ((𝑓 = ⟨𝑧, 𝑤⟩ ∧ = ⟨𝑠, 𝑡⟩) ∧ (𝑧 ·o 𝑡) = (𝑤 ·o 𝑠))) ↔ 𝑓 ~Q0 )
209195, 208sylib 121 1 ((𝑓 ~Q0 𝑔𝑔 ~Q0 ) → 𝑓 ~Q0 )
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104  wo 697  w3a 962   = wceq 1331  wex 1468  wcel 1480  c0 3358  cop 3525   class class class wbr 3924  ωcom 4499   × cxp 4532  (class class class)co 5767   ·o comu 6304  Ncnpi 7073   ~Q0 ceq0 7087
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-id 4210  df-iord 4283  df-on 4285  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-recs 6195  df-irdg 6260  df-oadd 6310  df-omul 6311  df-ni 7105  df-enq0 7225
This theorem is referenced by:  enq0er  7236
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