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Theorem nnnq0lem1 6429
Description: Decomposing non-negative fractions into natural numbers. Lemma for addnnnq0 6432 and mulnnnq0 6433. (Contributed by Jim Kingdon, 23-Nov-2019.)
Assertion
Ref Expression
nnnq0lem1 (((A ((𝜔 × N) / ~Q0 ) B ((𝜔 × N) / ~Q0 )) (((A = [⟨w, v⟩] ~Q0 B = [⟨u, 𝑡⟩] ~Q0 ) z = [𝐶] ~Q0 ) ((A = [⟨𝑠, f⟩] ~Q0 B = [⟨g, ⟩] ~Q0 ) 𝑞 = [𝐷] ~Q0 ))) → ((((w 𝜔 v N) (𝑠 𝜔 f N)) ((u 𝜔 𝑡 N) (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 6418 . . . . . 6 ~Q0 Er (𝜔 × N)
2 erdm 6052 . . . . . 6 ( ~Q0 Er (𝜔 × N) → dom ~Q0 = (𝜔 × N))
31, 2ax-mp 7 . . . . 5 dom ~Q0 = (𝜔 × N)
4 simpll 481 . . . . . 6 (((A ((𝜔 × N) / ~Q0 ) B ((𝜔 × N) / ~Q0 )) (((A = [⟨w, v⟩] ~Q0 B = [⟨u, 𝑡⟩] ~Q0 ) z = [𝐶] ~Q0 ) ((A = [⟨𝑠, f⟩] ~Q0 B = [⟨g, ⟩] ~Q0 ) 𝑞 = [𝐷] ~Q0 ))) → A ((𝜔 × N) / ~Q0 ))
5 simplll 485 . . . . . . . 8 ((((A = [⟨w, v⟩] ~Q0 B = [⟨u, 𝑡⟩] ~Q0 ) z = [𝐶] ~Q0 ) ((A = [⟨𝑠, f⟩] ~Q0 B = [⟨g, ⟩] ~Q0 ) 𝑞 = [𝐷] ~Q0 )) → A = [⟨w, v⟩] ~Q0 )
65eleq1d 2103 . . . . . . 7 ((((A = [⟨w, v⟩] ~Q0 B = [⟨u, 𝑡⟩] ~Q0 ) z = [𝐶] ~Q0 ) ((A = [⟨𝑠, f⟩] ~Q0 B = [⟨g, ⟩] ~Q0 ) 𝑞 = [𝐷] ~Q0 )) → (A ((𝜔 × N) / ~Q0 ) ↔ [⟨w, v⟩] ~Q0 ((𝜔 × N) / ~Q0 )))
76adantl 262 . . . . . 6 (((A ((𝜔 × N) / ~Q0 ) B ((𝜔 × N) / ~Q0 )) (((A = [⟨w, v⟩] ~Q0 B = [⟨u, 𝑡⟩] ~Q0 ) z = [𝐶] ~Q0 ) ((A = [⟨𝑠, f⟩] ~Q0 B = [⟨g, ⟩] ~Q0 ) 𝑞 = [𝐷] ~Q0 ))) → (A ((𝜔 × N) / ~Q0 ) ↔ [⟨w, v⟩] ~Q0 ((𝜔 × N) / ~Q0 )))
84, 7mpbid 135 . . . . 5 (((A ((𝜔 × N) / ~Q0 ) B ((𝜔 × N) / ~Q0 )) (((A = [⟨w, v⟩] ~Q0 B = [⟨u, 𝑡⟩] ~Q0 ) z = [𝐶] ~Q0 ) ((A = [⟨𝑠, f⟩] ~Q0 B = [⟨g, ⟩] ~Q0 ) 𝑞 = [𝐷] ~Q0 ))) → [⟨w, v⟩] ~Q0 ((𝜔 × N) / ~Q0 ))
9 ecelqsdm 6112 . . . . 5 ((dom ~Q0 = (𝜔 × N) [⟨w, v⟩] ~Q0 ((𝜔 × N) / ~Q0 )) → ⟨w, v (𝜔 × N))
103, 8, 9sylancr 393 . . . 4 (((A ((𝜔 × N) / ~Q0 ) B ((𝜔 × N) / ~Q0 )) (((A = [⟨w, v⟩] ~Q0 B = [⟨u, 𝑡⟩] ~Q0 ) z = [𝐶] ~Q0 ) ((A = [⟨𝑠, f⟩] ~Q0 B = [⟨g, ⟩] ~Q0 ) 𝑞 = [𝐷] ~Q0 ))) → ⟨w, v (𝜔 × N))
11 opelxp 4317 . . . 4 (⟨w, v (𝜔 × N) ↔ (w 𝜔 v N))
1210, 11sylib 127 . . 3 (((A ((𝜔 × N) / ~Q0 ) B ((𝜔 × N) / ~Q0 )) (((A = [⟨w, v⟩] ~Q0 B = [⟨u, 𝑡⟩] ~Q0 ) z = [𝐶] ~Q0 ) ((A = [⟨𝑠, f⟩] ~Q0 B = [⟨g, ⟩] ~Q0 ) 𝑞 = [𝐷] ~Q0 ))) → (w 𝜔 v N))
13 simprll 489 . . . . . . . 8 ((((A = [⟨w, v⟩] ~Q0 B = [⟨u, 𝑡⟩] ~Q0 ) z = [𝐶] ~Q0 ) ((A = [⟨𝑠, f⟩] ~Q0 B = [⟨g, ⟩] ~Q0 ) 𝑞 = [𝐷] ~Q0 )) → A = [⟨𝑠, f⟩] ~Q0 )
1413eleq1d 2103 . . . . . . 7 ((((A = [⟨w, v⟩] ~Q0 B = [⟨u, 𝑡⟩] ~Q0 ) z = [𝐶] ~Q0 ) ((A = [⟨𝑠, f⟩] ~Q0 B = [⟨g, ⟩] ~Q0 ) 𝑞 = [𝐷] ~Q0 )) → (A ((𝜔 × N) / ~Q0 ) ↔ [⟨𝑠, f⟩] ~Q0 ((𝜔 × N) / ~Q0 )))
1514adantl 262 . . . . . 6 (((A ((𝜔 × N) / ~Q0 ) B ((𝜔 × N) / ~Q0 )) (((A = [⟨w, v⟩] ~Q0 B = [⟨u, 𝑡⟩] ~Q0 ) z = [𝐶] ~Q0 ) ((A = [⟨𝑠, f⟩] ~Q0 B = [⟨g, ⟩] ~Q0 ) 𝑞 = [𝐷] ~Q0 ))) → (A ((𝜔 × N) / ~Q0 ) ↔ [⟨𝑠, f⟩] ~Q0 ((𝜔 × N) / ~Q0 )))
164, 15mpbid 135 . . . . 5 (((A ((𝜔 × N) / ~Q0 ) B ((𝜔 × N) / ~Q0 )) (((A = [⟨w, v⟩] ~Q0 B = [⟨u, 𝑡⟩] ~Q0 ) z = [𝐶] ~Q0 ) ((A = [⟨𝑠, f⟩] ~Q0 B = [⟨g, ⟩] ~Q0 ) 𝑞 = [𝐷] ~Q0 ))) → [⟨𝑠, f⟩] ~Q0 ((𝜔 × N) / ~Q0 ))
17 ecelqsdm 6112 . . . . 5 ((dom ~Q0 = (𝜔 × N) [⟨𝑠, f⟩] ~Q0 ((𝜔 × N) / ~Q0 )) → ⟨𝑠, f (𝜔 × N))
183, 16, 17sylancr 393 . . . 4 (((A ((𝜔 × N) / ~Q0 ) B ((𝜔 × N) / ~Q0 )) (((A = [⟨w, v⟩] ~Q0 B = [⟨u, 𝑡⟩] ~Q0 ) z = [𝐶] ~Q0 ) ((A = [⟨𝑠, f⟩] ~Q0 B = [⟨g, ⟩] ~Q0 ) 𝑞 = [𝐷] ~Q0 ))) → ⟨𝑠, f (𝜔 × N))
19 opelxp 4317 . . . 4 (⟨𝑠, f (𝜔 × N) ↔ (𝑠 𝜔 f N))
2018, 19sylib 127 . . 3 (((A ((𝜔 × N) / ~Q0 ) B ((𝜔 × N) / ~Q0 )) (((A = [⟨w, v⟩] ~Q0 B = [⟨u, 𝑡⟩] ~Q0 ) z = [𝐶] ~Q0 ) ((A = [⟨𝑠, f⟩] ~Q0 B = [⟨g, ⟩] ~Q0 ) 𝑞 = [𝐷] ~Q0 ))) → (𝑠 𝜔 f N))
2112, 20jca 290 . 2 (((A ((𝜔 × N) / ~Q0 ) B ((𝜔 × N) / ~Q0 )) (((A = [⟨w, v⟩] ~Q0 B = [⟨u, 𝑡⟩] ~Q0 ) z = [𝐶] ~Q0 ) ((A = [⟨𝑠, f⟩] ~Q0 B = [⟨g, ⟩] ~Q0 ) 𝑞 = [𝐷] ~Q0 ))) → ((w 𝜔 v N) (𝑠 𝜔 f N)))
22 simplr 482 . . . . . 6 (((A ((𝜔 × N) / ~Q0 ) B ((𝜔 × N) / ~Q0 )) (((A = [⟨w, v⟩] ~Q0 B = [⟨u, 𝑡⟩] ~Q0 ) z = [𝐶] ~Q0 ) ((A = [⟨𝑠, f⟩] ~Q0 B = [⟨g, ⟩] ~Q0 ) 𝑞 = [𝐷] ~Q0 ))) → B ((𝜔 × N) / ~Q0 ))
23 simpllr 486 . . . . . . . 8 ((((A = [⟨w, v⟩] ~Q0 B = [⟨u, 𝑡⟩] ~Q0 ) z = [𝐶] ~Q0 ) ((A = [⟨𝑠, f⟩] ~Q0 B = [⟨g, ⟩] ~Q0 ) 𝑞 = [𝐷] ~Q0 )) → B = [⟨u, 𝑡⟩] ~Q0 )
2423eleq1d 2103 . . . . . . 7 ((((A = [⟨w, v⟩] ~Q0 B = [⟨u, 𝑡⟩] ~Q0 ) z = [𝐶] ~Q0 ) ((A = [⟨𝑠, f⟩] ~Q0 B = [⟨g, ⟩] ~Q0 ) 𝑞 = [𝐷] ~Q0 )) → (B ((𝜔 × N) / ~Q0 ) ↔ [⟨u, 𝑡⟩] ~Q0 ((𝜔 × N) / ~Q0 )))
2524adantl 262 . . . . . 6 (((A ((𝜔 × N) / ~Q0 ) B ((𝜔 × N) / ~Q0 )) (((A = [⟨w, v⟩] ~Q0 B = [⟨u, 𝑡⟩] ~Q0 ) z = [𝐶] ~Q0 ) ((A = [⟨𝑠, f⟩] ~Q0 B = [⟨g, ⟩] ~Q0 ) 𝑞 = [𝐷] ~Q0 ))) → (B ((𝜔 × N) / ~Q0 ) ↔ [⟨u, 𝑡⟩] ~Q0 ((𝜔 × N) / ~Q0 )))
2622, 25mpbid 135 . . . . 5 (((A ((𝜔 × N) / ~Q0 ) B ((𝜔 × N) / ~Q0 )) (((A = [⟨w, v⟩] ~Q0 B = [⟨u, 𝑡⟩] ~Q0 ) z = [𝐶] ~Q0 ) ((A = [⟨𝑠, f⟩] ~Q0 B = [⟨g, ⟩] ~Q0 ) 𝑞 = [𝐷] ~Q0 ))) → [⟨u, 𝑡⟩] ~Q0 ((𝜔 × N) / ~Q0 ))
27 ecelqsdm 6112 . . . . 5 ((dom ~Q0 = (𝜔 × N) [⟨u, 𝑡⟩] ~Q0 ((𝜔 × N) / ~Q0 )) → ⟨u, 𝑡 (𝜔 × N))
283, 26, 27sylancr 393 . . . 4 (((A ((𝜔 × N) / ~Q0 ) B ((𝜔 × N) / ~Q0 )) (((A = [⟨w, v⟩] ~Q0 B = [⟨u, 𝑡⟩] ~Q0 ) z = [𝐶] ~Q0 ) ((A = [⟨𝑠, f⟩] ~Q0 B = [⟨g, ⟩] ~Q0 ) 𝑞 = [𝐷] ~Q0 ))) → ⟨u, 𝑡 (𝜔 × N))
29 opelxp 4317 . . . 4 (⟨u, 𝑡 (𝜔 × N) ↔ (u 𝜔 𝑡 N))
3028, 29sylib 127 . . 3 (((A ((𝜔 × N) / ~Q0 ) B ((𝜔 × N) / ~Q0 )) (((A = [⟨w, v⟩] ~Q0 B = [⟨u, 𝑡⟩] ~Q0 ) z = [𝐶] ~Q0 ) ((A = [⟨𝑠, f⟩] ~Q0 B = [⟨g, ⟩] ~Q0 ) 𝑞 = [𝐷] ~Q0 ))) → (u 𝜔 𝑡 N))
31 simprlr 490 . . . . . . . 8 ((((A = [⟨w, v⟩] ~Q0 B = [⟨u, 𝑡⟩] ~Q0 ) z = [𝐶] ~Q0 ) ((A = [⟨𝑠, f⟩] ~Q0 B = [⟨g, ⟩] ~Q0 ) 𝑞 = [𝐷] ~Q0 )) → B = [⟨g, ⟩] ~Q0 )
3231eleq1d 2103 . . . . . . 7 ((((A = [⟨w, v⟩] ~Q0 B = [⟨u, 𝑡⟩] ~Q0 ) z = [𝐶] ~Q0 ) ((A = [⟨𝑠, f⟩] ~Q0 B = [⟨g, ⟩] ~Q0 ) 𝑞 = [𝐷] ~Q0 )) → (B ((𝜔 × N) / ~Q0 ) ↔ [⟨g, ⟩] ~Q0 ((𝜔 × N) / ~Q0 )))
3332adantl 262 . . . . . 6 (((A ((𝜔 × N) / ~Q0 ) B ((𝜔 × N) / ~Q0 )) (((A = [⟨w, v⟩] ~Q0 B = [⟨u, 𝑡⟩] ~Q0 ) z = [𝐶] ~Q0 ) ((A = [⟨𝑠, f⟩] ~Q0 B = [⟨g, ⟩] ~Q0 ) 𝑞 = [𝐷] ~Q0 ))) → (B ((𝜔 × N) / ~Q0 ) ↔ [⟨g, ⟩] ~Q0 ((𝜔 × N) / ~Q0 )))
3422, 33mpbid 135 . . . . 5 (((A ((𝜔 × N) / ~Q0 ) B ((𝜔 × N) / ~Q0 )) (((A = [⟨w, v⟩] ~Q0 B = [⟨u, 𝑡⟩] ~Q0 ) z = [𝐶] ~Q0 ) ((A = [⟨𝑠, f⟩] ~Q0 B = [⟨g, ⟩] ~Q0 ) 𝑞 = [𝐷] ~Q0 ))) → [⟨g, ⟩] ~Q0 ((𝜔 × N) / ~Q0 ))
35 ecelqsdm 6112 . . . . 5 ((dom ~Q0 = (𝜔 × N) [⟨g, ⟩] ~Q0 ((𝜔 × N) / ~Q0 )) → ⟨g, (𝜔 × N))
363, 34, 35sylancr 393 . . . 4 (((A ((𝜔 × N) / ~Q0 ) B ((𝜔 × N) / ~Q0 )) (((A = [⟨w, v⟩] ~Q0 B = [⟨u, 𝑡⟩] ~Q0 ) z = [𝐶] ~Q0 ) ((A = [⟨𝑠, f⟩] ~Q0 B = [⟨g, ⟩] ~Q0 ) 𝑞 = [𝐷] ~Q0 ))) → ⟨g, (𝜔 × N))
37 opelxp 4317 . . . 4 (⟨g, (𝜔 × N) ↔ (g 𝜔 N))
3836, 37sylib 127 . . 3 (((A ((𝜔 × N) / ~Q0 ) B ((𝜔 × N) / ~Q0 )) (((A = [⟨w, v⟩] ~Q0 B = [⟨u, 𝑡⟩] ~Q0 ) z = [𝐶] ~Q0 ) ((A = [⟨𝑠, f⟩] ~Q0 B = [⟨g, ⟩] ~Q0 ) 𝑞 = [𝐷] ~Q0 ))) → (g 𝜔 N))
3930, 38jca 290 . 2 (((A ((𝜔 × N) / ~Q0 ) B ((𝜔 × N) / ~Q0 )) (((A = [⟨w, v⟩] ~Q0 B = [⟨u, 𝑡⟩] ~Q0 ) z = [𝐶] ~Q0 ) ((A = [⟨𝑠, f⟩] ~Q0 B = [⟨g, ⟩] ~Q0 ) 𝑞 = [𝐷] ~Q0 ))) → ((u 𝜔 𝑡 N) (g 𝜔 N)))
405, 13eqtr3d 2071 . . . . . 6 ((((A = [⟨w, v⟩] ~Q0 B = [⟨u, 𝑡⟩] ~Q0 ) z = [𝐶] ~Q0 ) ((A = [⟨𝑠, f⟩] ~Q0 B = [⟨g, ⟩] ~Q0 ) 𝑞 = [𝐷] ~Q0 )) → [⟨w, v⟩] ~Q0 = [⟨𝑠, f⟩] ~Q0 )
4140adantl 262 . . . . 5 (((A ((𝜔 × N) / ~Q0 ) B ((𝜔 × N) / ~Q0 )) (((A = [⟨w, v⟩] ~Q0 B = [⟨u, 𝑡⟩] ~Q0 ) z = [𝐶] ~Q0 ) ((A = [⟨𝑠, f⟩] ~Q0 B = [⟨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⟩] ~Q0 B = [⟨u, 𝑡⟩] ~Q0 ) z = [𝐶] ~Q0 ) ((A = [⟨𝑠, f⟩] ~Q0 B = [⟨g, ⟩] ~Q0 ) 𝑞 = [𝐷] ~Q0 ))) → ~Q0 Er (𝜔 × N))
4342, 10erth 6086 . . . . 5 (((A ((𝜔 × N) / ~Q0 ) B ((𝜔 × N) / ~Q0 )) (((A = [⟨w, v⟩] ~Q0 B = [⟨u, 𝑡⟩] ~Q0 ) z = [𝐶] ~Q0 ) ((A = [⟨𝑠, f⟩] ~Q0 B = [⟨g, ⟩] ~Q0 ) 𝑞 = [𝐷] ~Q0 ))) → (⟨w, v⟩ ~Q0𝑠, f⟩ ↔ [⟨w, v⟩] ~Q0 = [⟨𝑠, f⟩] ~Q0 ))
4441, 43mpbird 156 . . . 4 (((A ((𝜔 × N) / ~Q0 ) B ((𝜔 × N) / ~Q0 )) (((A = [⟨w, v⟩] ~Q0 B = [⟨u, 𝑡⟩] ~Q0 ) z = [𝐶] ~Q0 ) ((A = [⟨𝑠, f⟩] ~Q0 B = [⟨g, ⟩] ~Q0 ) 𝑞 = [𝐷] ~Q0 ))) → ⟨w, v⟩ ~Q0𝑠, f⟩)
45 enq0breq 6419 . . . . 5 (((w 𝜔 v N) (𝑠 𝜔 f N)) → (⟨w, v⟩ ~Q0𝑠, f⟩ ↔ (w ·𝑜 f) = (v ·𝑜 𝑠)))
4612, 20, 45syl2anc 391 . . . 4 (((A ((𝜔 × N) / ~Q0 ) B ((𝜔 × N) / ~Q0 )) (((A = [⟨w, v⟩] ~Q0 B = [⟨u, 𝑡⟩] ~Q0 ) z = [𝐶] ~Q0 ) ((A = [⟨𝑠, f⟩] ~Q0 B = [⟨g, ⟩] ~Q0 ) 𝑞 = [𝐷] ~Q0 ))) → (⟨w, v⟩ ~Q0𝑠, f⟩ ↔ (w ·𝑜 f) = (v ·𝑜 𝑠)))
4744, 46mpbid 135 . . 3 (((A ((𝜔 × N) / ~Q0 ) B ((𝜔 × N) / ~Q0 )) (((A = [⟨w, v⟩] ~Q0 B = [⟨u, 𝑡⟩] ~Q0 ) z = [𝐶] ~Q0 ) ((A = [⟨𝑠, f⟩] ~Q0 B = [⟨g, ⟩] ~Q0 ) 𝑞 = [𝐷] ~Q0 ))) → (w ·𝑜 f) = (v ·𝑜 𝑠))
4823, 31eqtr3d 2071 . . . . . 6 ((((A = [⟨w, v⟩] ~Q0 B = [⟨u, 𝑡⟩] ~Q0 ) z = [𝐶] ~Q0 ) ((A = [⟨𝑠, f⟩] ~Q0 B = [⟨g, ⟩] ~Q0 ) 𝑞 = [𝐷] ~Q0 )) → [⟨u, 𝑡⟩] ~Q0 = [⟨g, ⟩] ~Q0 )
4948adantl 262 . . . . 5 (((A ((𝜔 × N) / ~Q0 ) B ((𝜔 × N) / ~Q0 )) (((A = [⟨w, v⟩] ~Q0 B = [⟨u, 𝑡⟩] ~Q0 ) z = [𝐶] ~Q0 ) ((A = [⟨𝑠, f⟩] ~Q0 B = [⟨g, ⟩] ~Q0 ) 𝑞 = [𝐷] ~Q0 ))) → [⟨u, 𝑡⟩] ~Q0 = [⟨g, ⟩] ~Q0 )
5042, 28erth 6086 . . . . 5 (((A ((𝜔 × N) / ~Q0 ) B ((𝜔 × N) / ~Q0 )) (((A = [⟨w, v⟩] ~Q0 B = [⟨u, 𝑡⟩] ~Q0 ) z = [𝐶] ~Q0 ) ((A = [⟨𝑠, f⟩] ~Q0 B = [⟨g, ⟩] ~Q0 ) 𝑞 = [𝐷] ~Q0 ))) → (⟨u, 𝑡⟩ ~Q0g, ⟩ ↔ [⟨u, 𝑡⟩] ~Q0 = [⟨g, ⟩] ~Q0 ))
5149, 50mpbird 156 . . . 4 (((A ((𝜔 × N) / ~Q0 ) B ((𝜔 × N) / ~Q0 )) (((A = [⟨w, v⟩] ~Q0 B = [⟨u, 𝑡⟩] ~Q0 ) z = [𝐶] ~Q0 ) ((A = [⟨𝑠, f⟩] ~Q0 B = [⟨g, ⟩] ~Q0 ) 𝑞 = [𝐷] ~Q0 ))) → ⟨u, 𝑡⟩ ~Q0g, ⟩)
52 enq0breq 6419 . . . . 5 (((u 𝜔 𝑡 N) (g 𝜔 N)) → (⟨u, 𝑡⟩ ~Q0g, ⟩ ↔ (u ·𝑜 ) = (𝑡 ·𝑜 g)))
5330, 38, 52syl2anc 391 . . . 4 (((A ((𝜔 × N) / ~Q0 ) B ((𝜔 × N) / ~Q0 )) (((A = [⟨w, v⟩] ~Q0 B = [⟨u, 𝑡⟩] ~Q0 ) z = [𝐶] ~Q0 ) ((A = [⟨𝑠, f⟩] ~Q0 B = [⟨g, ⟩] ~Q0 ) 𝑞 = [𝐷] ~Q0 ))) → (⟨u, 𝑡⟩ ~Q0g, ⟩ ↔ (u ·𝑜 ) = (𝑡 ·𝑜 g)))
5451, 53mpbid 135 . . 3 (((A ((𝜔 × N) / ~Q0 ) B ((𝜔 × N) / ~Q0 )) (((A = [⟨w, v⟩] ~Q0 B = [⟨u, 𝑡⟩] ~Q0 ) z = [𝐶] ~Q0 ) ((A = [⟨𝑠, f⟩] ~Q0 B = [⟨g, ⟩] ~Q0 ) 𝑞 = [𝐷] ~Q0 ))) → (u ·𝑜 ) = (𝑡 ·𝑜 g))
5547, 54jca 290 . 2 (((A ((𝜔 × N) / ~Q0 ) B ((𝜔 × N) / ~Q0 )) (((A = [⟨w, v⟩] ~Q0 B = [⟨u, 𝑡⟩] ~Q0 ) z = [𝐶] ~Q0 ) ((A = [⟨𝑠, f⟩] ~Q0 B = [⟨g, ⟩] ~Q0 ) 𝑞 = [𝐷] ~Q0 ))) → ((w ·𝑜 f) = (v ·𝑜 𝑠) (u ·𝑜 ) = (𝑡 ·𝑜 g)))
5621, 39, 55jca31 292 1 (((A ((𝜔 × N) / ~Q0 ) B ((𝜔 × N) / ~Q0 )) (((A = [⟨w, v⟩] ~Q0 B = [⟨u, 𝑡⟩] ~Q0 ) z = [𝐶] ~Q0 ) ((A = [⟨𝑠, f⟩] ~Q0 B = [⟨g, ⟩] ~Q0 ) 𝑞 = [𝐷] ~Q0 ))) → ((((w 𝜔 v N) (𝑠 𝜔 f N)) ((u 𝜔 𝑡 N) (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 97  wb 98   = wceq 1242   wcel 1390  cop 3370   class class class wbr 3755  𝜔com 4256   × cxp 4286  dom cdm 4288  (class class class)co 5455   ·𝑜 comu 5938   Er wer 6039  [cec 6040   / cqs 6041  Ncnpi 6256   ~Q0 ceq0 6270
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254
This theorem depends on definitions:  df-bi 110  df-dc 742  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-id 4021  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-oadd 5944  df-omul 5945  df-er 6042  df-ec 6044  df-qs 6048  df-ni 6288  df-enq0 6407
This theorem is referenced by:  addnq0mo  6430  mulnq0mo  6431
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