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Theorem nnnq0lem1 6601
 Description: Decomposing non-negative fractions into natural numbers. Lemma for addnnnq0 6604 and mulnnnq0 6605. (Contributed by Jim Kingdon, 23-Nov-2019.)
Assertion
Ref Expression
nnnq0lem1 (((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) ∧ (((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [𝐶] ~Q0 ) ∧ ((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑔, ⟩] ~Q0 ) ∧ 𝑞 = [𝐷] ~Q0 ))) → ((((𝑤 ∈ ω ∧ 𝑣N) ∧ (𝑠 ∈ ω ∧ 𝑓N)) ∧ ((𝑢 ∈ ω ∧ 𝑡N) ∧ (𝑔 ∈ ω ∧ N))) ∧ ((𝑤 ·𝑜 𝑓) = (𝑣 ·𝑜 𝑠) ∧ (𝑢 ·𝑜 ) = (𝑡 ·𝑜 𝑔))))
Distinct variable groups:   𝑧,𝑤,𝑣,𝑢,𝑡,𝑠,𝑞,𝑓,𝑔,,𝐴   𝑧,𝐵,𝑤,𝑣,𝑢,𝑡,𝑠,𝑞,𝑓,𝑔,
Allowed substitution hints:   𝐶(𝑧,𝑤,𝑣,𝑢,𝑡,𝑓,𝑔,,𝑠,𝑞)   𝐷(𝑧,𝑤,𝑣,𝑢,𝑡,𝑓,𝑔,,𝑠,𝑞)

Proof of Theorem nnnq0lem1
StepHypRef Expression
1 enq0er 6590 . . . . . 6 ~Q0 Er (ω × N)
2 erdm 6146 . . . . . 6 ( ~Q0 Er (ω × N) → dom ~Q0 = (ω × N))
31, 2ax-mp 7 . . . . 5 dom ~Q0 = (ω × N)
4 simpll 489 . . . . . 6 (((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) ∧ (((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [𝐶] ~Q0 ) ∧ ((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑔, ⟩] ~Q0 ) ∧ 𝑞 = [𝐷] ~Q0 ))) → 𝐴 ∈ ((ω × N) / ~Q0 ))
5 simplll 493 . . . . . . . 8 ((((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [𝐶] ~Q0 ) ∧ ((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑔, ⟩] ~Q0 ) ∧ 𝑞 = [𝐷] ~Q0 )) → 𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 )
65eleq1d 2122 . . . . . . 7 ((((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [𝐶] ~Q0 ) ∧ ((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑔, ⟩] ~Q0 ) ∧ 𝑞 = [𝐷] ~Q0 )) → (𝐴 ∈ ((ω × N) / ~Q0 ) ↔ [⟨𝑤, 𝑣⟩] ~Q0 ∈ ((ω × N) / ~Q0 )))
76adantl 266 . . . . . 6 (((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) ∧ (((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [𝐶] ~Q0 ) ∧ ((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑔, ⟩] ~Q0 ) ∧ 𝑞 = [𝐷] ~Q0 ))) → (𝐴 ∈ ((ω × N) / ~Q0 ) ↔ [⟨𝑤, 𝑣⟩] ~Q0 ∈ ((ω × N) / ~Q0 )))
84, 7mpbid 139 . . . . 5 (((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) ∧ (((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [𝐶] ~Q0 ) ∧ ((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑔, ⟩] ~Q0 ) ∧ 𝑞 = [𝐷] ~Q0 ))) → [⟨𝑤, 𝑣⟩] ~Q0 ∈ ((ω × N) / ~Q0 ))
9 ecelqsdm 6206 . . . . 5 ((dom ~Q0 = (ω × N) ∧ [⟨𝑤, 𝑣⟩] ~Q0 ∈ ((ω × N) / ~Q0 )) → ⟨𝑤, 𝑣⟩ ∈ (ω × N))
103, 8, 9sylancr 399 . . . 4 (((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) ∧ (((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [𝐶] ~Q0 ) ∧ ((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑔, ⟩] ~Q0 ) ∧ 𝑞 = [𝐷] ~Q0 ))) → ⟨𝑤, 𝑣⟩ ∈ (ω × N))
11 opelxp 4401 . . . 4 (⟨𝑤, 𝑣⟩ ∈ (ω × N) ↔ (𝑤 ∈ ω ∧ 𝑣N))
1210, 11sylib 131 . . 3 (((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) ∧ (((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [𝐶] ~Q0 ) ∧ ((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑔, ⟩] ~Q0 ) ∧ 𝑞 = [𝐷] ~Q0 ))) → (𝑤 ∈ ω ∧ 𝑣N))
13 simprll 497 . . . . . . . 8 ((((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [𝐶] ~Q0 ) ∧ ((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑔, ⟩] ~Q0 ) ∧ 𝑞 = [𝐷] ~Q0 )) → 𝐴 = [⟨𝑠, 𝑓⟩] ~Q0 )
1413eleq1d 2122 . . . . . . 7 ((((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [𝐶] ~Q0 ) ∧ ((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑔, ⟩] ~Q0 ) ∧ 𝑞 = [𝐷] ~Q0 )) → (𝐴 ∈ ((ω × N) / ~Q0 ) ↔ [⟨𝑠, 𝑓⟩] ~Q0 ∈ ((ω × N) / ~Q0 )))
1514adantl 266 . . . . . 6 (((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) ∧ (((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [𝐶] ~Q0 ) ∧ ((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑔, ⟩] ~Q0 ) ∧ 𝑞 = [𝐷] ~Q0 ))) → (𝐴 ∈ ((ω × N) / ~Q0 ) ↔ [⟨𝑠, 𝑓⟩] ~Q0 ∈ ((ω × N) / ~Q0 )))
164, 15mpbid 139 . . . . 5 (((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) ∧ (((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [𝐶] ~Q0 ) ∧ ((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑔, ⟩] ~Q0 ) ∧ 𝑞 = [𝐷] ~Q0 ))) → [⟨𝑠, 𝑓⟩] ~Q0 ∈ ((ω × N) / ~Q0 ))
17 ecelqsdm 6206 . . . . 5 ((dom ~Q0 = (ω × N) ∧ [⟨𝑠, 𝑓⟩] ~Q0 ∈ ((ω × N) / ~Q0 )) → ⟨𝑠, 𝑓⟩ ∈ (ω × N))
183, 16, 17sylancr 399 . . . 4 (((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) ∧ (((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [𝐶] ~Q0 ) ∧ ((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑔, ⟩] ~Q0 ) ∧ 𝑞 = [𝐷] ~Q0 ))) → ⟨𝑠, 𝑓⟩ ∈ (ω × N))
19 opelxp 4401 . . . 4 (⟨𝑠, 𝑓⟩ ∈ (ω × N) ↔ (𝑠 ∈ ω ∧ 𝑓N))
2018, 19sylib 131 . . 3 (((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) ∧ (((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [𝐶] ~Q0 ) ∧ ((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑔, ⟩] ~Q0 ) ∧ 𝑞 = [𝐷] ~Q0 ))) → (𝑠 ∈ ω ∧ 𝑓N))
2112, 20jca 294 . 2 (((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) ∧ (((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [𝐶] ~Q0 ) ∧ ((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑔, ⟩] ~Q0 ) ∧ 𝑞 = [𝐷] ~Q0 ))) → ((𝑤 ∈ ω ∧ 𝑣N) ∧ (𝑠 ∈ ω ∧ 𝑓N)))
22 simplr 490 . . . . . 6 (((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) ∧ (((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [𝐶] ~Q0 ) ∧ ((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑔, ⟩] ~Q0 ) ∧ 𝑞 = [𝐷] ~Q0 ))) → 𝐵 ∈ ((ω × N) / ~Q0 ))
23 simpllr 494 . . . . . . . 8 ((((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [𝐶] ~Q0 ) ∧ ((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑔, ⟩] ~Q0 ) ∧ 𝑞 = [𝐷] ~Q0 )) → 𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 )
2423eleq1d 2122 . . . . . . 7 ((((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [𝐶] ~Q0 ) ∧ ((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑔, ⟩] ~Q0 ) ∧ 𝑞 = [𝐷] ~Q0 )) → (𝐵 ∈ ((ω × N) / ~Q0 ) ↔ [⟨𝑢, 𝑡⟩] ~Q0 ∈ ((ω × N) / ~Q0 )))
2524adantl 266 . . . . . 6 (((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) ∧ (((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [𝐶] ~Q0 ) ∧ ((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑔, ⟩] ~Q0 ) ∧ 𝑞 = [𝐷] ~Q0 ))) → (𝐵 ∈ ((ω × N) / ~Q0 ) ↔ [⟨𝑢, 𝑡⟩] ~Q0 ∈ ((ω × N) / ~Q0 )))
2622, 25mpbid 139 . . . . 5 (((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) ∧ (((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [𝐶] ~Q0 ) ∧ ((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑔, ⟩] ~Q0 ) ∧ 𝑞 = [𝐷] ~Q0 ))) → [⟨𝑢, 𝑡⟩] ~Q0 ∈ ((ω × N) / ~Q0 ))
27 ecelqsdm 6206 . . . . 5 ((dom ~Q0 = (ω × N) ∧ [⟨𝑢, 𝑡⟩] ~Q0 ∈ ((ω × N) / ~Q0 )) → ⟨𝑢, 𝑡⟩ ∈ (ω × N))
283, 26, 27sylancr 399 . . . 4 (((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) ∧ (((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [𝐶] ~Q0 ) ∧ ((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑔, ⟩] ~Q0 ) ∧ 𝑞 = [𝐷] ~Q0 ))) → ⟨𝑢, 𝑡⟩ ∈ (ω × N))
29 opelxp 4401 . . . 4 (⟨𝑢, 𝑡⟩ ∈ (ω × N) ↔ (𝑢 ∈ ω ∧ 𝑡N))
3028, 29sylib 131 . . 3 (((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) ∧ (((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [𝐶] ~Q0 ) ∧ ((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑔, ⟩] ~Q0 ) ∧ 𝑞 = [𝐷] ~Q0 ))) → (𝑢 ∈ ω ∧ 𝑡N))
31 simprlr 498 . . . . . . . 8 ((((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [𝐶] ~Q0 ) ∧ ((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑔, ⟩] ~Q0 ) ∧ 𝑞 = [𝐷] ~Q0 )) → 𝐵 = [⟨𝑔, ⟩] ~Q0 )
3231eleq1d 2122 . . . . . . 7 ((((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [𝐶] ~Q0 ) ∧ ((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑔, ⟩] ~Q0 ) ∧ 𝑞 = [𝐷] ~Q0 )) → (𝐵 ∈ ((ω × N) / ~Q0 ) ↔ [⟨𝑔, ⟩] ~Q0 ∈ ((ω × N) / ~Q0 )))
3332adantl 266 . . . . . 6 (((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) ∧ (((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [𝐶] ~Q0 ) ∧ ((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑔, ⟩] ~Q0 ) ∧ 𝑞 = [𝐷] ~Q0 ))) → (𝐵 ∈ ((ω × N) / ~Q0 ) ↔ [⟨𝑔, ⟩] ~Q0 ∈ ((ω × N) / ~Q0 )))
3422, 33mpbid 139 . . . . 5 (((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) ∧ (((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [𝐶] ~Q0 ) ∧ ((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑔, ⟩] ~Q0 ) ∧ 𝑞 = [𝐷] ~Q0 ))) → [⟨𝑔, ⟩] ~Q0 ∈ ((ω × N) / ~Q0 ))
35 ecelqsdm 6206 . . . . 5 ((dom ~Q0 = (ω × N) ∧ [⟨𝑔, ⟩] ~Q0 ∈ ((ω × N) / ~Q0 )) → ⟨𝑔, ⟩ ∈ (ω × N))
363, 34, 35sylancr 399 . . . 4 (((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) ∧ (((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [𝐶] ~Q0 ) ∧ ((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑔, ⟩] ~Q0 ) ∧ 𝑞 = [𝐷] ~Q0 ))) → ⟨𝑔, ⟩ ∈ (ω × N))
37 opelxp 4401 . . . 4 (⟨𝑔, ⟩ ∈ (ω × N) ↔ (𝑔 ∈ ω ∧ N))
3836, 37sylib 131 . . 3 (((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) ∧ (((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [𝐶] ~Q0 ) ∧ ((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑔, ⟩] ~Q0 ) ∧ 𝑞 = [𝐷] ~Q0 ))) → (𝑔 ∈ ω ∧ N))
3930, 38jca 294 . 2 (((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) ∧ (((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [𝐶] ~Q0 ) ∧ ((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑔, ⟩] ~Q0 ) ∧ 𝑞 = [𝐷] ~Q0 ))) → ((𝑢 ∈ ω ∧ 𝑡N) ∧ (𝑔 ∈ ω ∧ N)))
405, 13eqtr3d 2090 . . . . . 6 ((((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [𝐶] ~Q0 ) ∧ ((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑔, ⟩] ~Q0 ) ∧ 𝑞 = [𝐷] ~Q0 )) → [⟨𝑤, 𝑣⟩] ~Q0 = [⟨𝑠, 𝑓⟩] ~Q0 )
4140adantl 266 . . . . 5 (((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) ∧ (((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [𝐶] ~Q0 ) ∧ ((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑔, ⟩] ~Q0 ) ∧ 𝑞 = [𝐷] ~Q0 ))) → [⟨𝑤, 𝑣⟩] ~Q0 = [⟨𝑠, 𝑓⟩] ~Q0 )
421a1i 9 . . . . . 6 (((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) ∧ (((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [𝐶] ~Q0 ) ∧ ((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑔, ⟩] ~Q0 ) ∧ 𝑞 = [𝐷] ~Q0 ))) → ~Q0 Er (ω × N))
4342, 10erth 6180 . . . . 5 (((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) ∧ (((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [𝐶] ~Q0 ) ∧ ((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑔, ⟩] ~Q0 ) ∧ 𝑞 = [𝐷] ~Q0 ))) → (⟨𝑤, 𝑣⟩ ~Q0𝑠, 𝑓⟩ ↔ [⟨𝑤, 𝑣⟩] ~Q0 = [⟨𝑠, 𝑓⟩] ~Q0 ))
4441, 43mpbird 160 . . . 4 (((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) ∧ (((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [𝐶] ~Q0 ) ∧ ((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑔, ⟩] ~Q0 ) ∧ 𝑞 = [𝐷] ~Q0 ))) → ⟨𝑤, 𝑣⟩ ~Q0𝑠, 𝑓⟩)
45 enq0breq 6591 . . . . 5 (((𝑤 ∈ ω ∧ 𝑣N) ∧ (𝑠 ∈ ω ∧ 𝑓N)) → (⟨𝑤, 𝑣⟩ ~Q0𝑠, 𝑓⟩ ↔ (𝑤 ·𝑜 𝑓) = (𝑣 ·𝑜 𝑠)))
4612, 20, 45syl2anc 397 . . . 4 (((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) ∧ (((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [𝐶] ~Q0 ) ∧ ((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑔, ⟩] ~Q0 ) ∧ 𝑞 = [𝐷] ~Q0 ))) → (⟨𝑤, 𝑣⟩ ~Q0𝑠, 𝑓⟩ ↔ (𝑤 ·𝑜 𝑓) = (𝑣 ·𝑜 𝑠)))
4744, 46mpbid 139 . . 3 (((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) ∧ (((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [𝐶] ~Q0 ) ∧ ((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑔, ⟩] ~Q0 ) ∧ 𝑞 = [𝐷] ~Q0 ))) → (𝑤 ·𝑜 𝑓) = (𝑣 ·𝑜 𝑠))
4823, 31eqtr3d 2090 . . . . . 6 ((((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [𝐶] ~Q0 ) ∧ ((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑔, ⟩] ~Q0 ) ∧ 𝑞 = [𝐷] ~Q0 )) → [⟨𝑢, 𝑡⟩] ~Q0 = [⟨𝑔, ⟩] ~Q0 )
4948adantl 266 . . . . 5 (((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) ∧ (((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [𝐶] ~Q0 ) ∧ ((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑔, ⟩] ~Q0 ) ∧ 𝑞 = [𝐷] ~Q0 ))) → [⟨𝑢, 𝑡⟩] ~Q0 = [⟨𝑔, ⟩] ~Q0 )
5042, 28erth 6180 . . . . 5 (((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) ∧ (((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [𝐶] ~Q0 ) ∧ ((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑔, ⟩] ~Q0 ) ∧ 𝑞 = [𝐷] ~Q0 ))) → (⟨𝑢, 𝑡⟩ ~Q0𝑔, ⟩ ↔ [⟨𝑢, 𝑡⟩] ~Q0 = [⟨𝑔, ⟩] ~Q0 ))
5149, 50mpbird 160 . . . 4 (((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) ∧ (((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [𝐶] ~Q0 ) ∧ ((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑔, ⟩] ~Q0 ) ∧ 𝑞 = [𝐷] ~Q0 ))) → ⟨𝑢, 𝑡⟩ ~Q0𝑔, ⟩)
52 enq0breq 6591 . . . . 5 (((𝑢 ∈ ω ∧ 𝑡N) ∧ (𝑔 ∈ ω ∧ N)) → (⟨𝑢, 𝑡⟩ ~Q0𝑔, ⟩ ↔ (𝑢 ·𝑜 ) = (𝑡 ·𝑜 𝑔)))
5330, 38, 52syl2anc 397 . . . 4 (((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) ∧ (((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [𝐶] ~Q0 ) ∧ ((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑔, ⟩] ~Q0 ) ∧ 𝑞 = [𝐷] ~Q0 ))) → (⟨𝑢, 𝑡⟩ ~Q0𝑔, ⟩ ↔ (𝑢 ·𝑜 ) = (𝑡 ·𝑜 𝑔)))
5451, 53mpbid 139 . . 3 (((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) ∧ (((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [𝐶] ~Q0 ) ∧ ((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑔, ⟩] ~Q0 ) ∧ 𝑞 = [𝐷] ~Q0 ))) → (𝑢 ·𝑜 ) = (𝑡 ·𝑜 𝑔))
5547, 54jca 294 . 2 (((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) ∧ (((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [𝐶] ~Q0 ) ∧ ((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑔, ⟩] ~Q0 ) ∧ 𝑞 = [𝐷] ~Q0 ))) → ((𝑤 ·𝑜 𝑓) = (𝑣 ·𝑜 𝑠) ∧ (𝑢 ·𝑜 ) = (𝑡 ·𝑜 𝑔)))
5621, 39, 55jca31 296 1 (((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) ∧ (((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [𝐶] ~Q0 ) ∧ ((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0𝐵 = [⟨𝑔, ⟩] ~Q0 ) ∧ 𝑞 = [𝐷] ~Q0 ))) → ((((𝑤 ∈ ω ∧ 𝑣N) ∧ (𝑠 ∈ ω ∧ 𝑓N)) ∧ ((𝑢 ∈ ω ∧ 𝑡N) ∧ (𝑔 ∈ ω ∧ N))) ∧ ((𝑤 ·𝑜 𝑓) = (𝑣 ·𝑜 𝑠) ∧ (𝑢 ·𝑜 ) = (𝑡 ·𝑜 𝑔))))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 101   ↔ wb 102   = wceq 1259   ∈ wcel 1409  ⟨cop 3405   class class class wbr 3791  ωcom 4340   × cxp 4370  dom cdm 4372  (class class class)co 5539   ·𝑜 comu 6029   Er wer 6133  [cec 6134   / cqs 6135  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-dc 754  df-3or 897  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-er 6136  df-ec 6138  df-qs 6142  df-ni 6459  df-enq0 6579 This theorem is referenced by:  addnq0mo  6602  mulnq0mo  6603
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