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Theorem distrnq0 6614
Description: Multiplication of non-negative fractions is distributive. (Contributed by Jim Kingdon, 27-Nov-2019.)
Assertion
Ref Expression
distrnq0 ((𝐴Q0𝐵Q0𝐶Q0) → (𝐴 ·Q0 (𝐵 +Q0 𝐶)) = ((𝐴 ·Q0 𝐵) +Q0 (𝐴 ·Q0 𝐶)))

Proof of Theorem distrnq0
Dummy variables 𝑢 𝑣 𝑤 𝑥 𝑦 𝑧 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-nq0 6580 . . . 4 Q0 = ((ω × N) / ~Q0 )
2 oveq1 5546 . . . . . . 7 ([⟨𝑧, 𝑤⟩] ~Q0 = 𝐵 → ([⟨𝑧, 𝑤⟩] ~Q0 +Q0 [⟨𝑣, 𝑢⟩] ~Q0 ) = (𝐵 +Q0 [⟨𝑣, 𝑢⟩] ~Q0 ))
32oveq2d 5555 . . . . . 6 ([⟨𝑧, 𝑤⟩] ~Q0 = 𝐵 → (𝐴 ·Q0 ([⟨𝑧, 𝑤⟩] ~Q0 +Q0 [⟨𝑣, 𝑢⟩] ~Q0 )) = (𝐴 ·Q0 (𝐵 +Q0 [⟨𝑣, 𝑢⟩] ~Q0 )))
4 oveq2 5547 . . . . . . 7 ([⟨𝑧, 𝑤⟩] ~Q0 = 𝐵 → (𝐴 ·Q0 [⟨𝑧, 𝑤⟩] ~Q0 ) = (𝐴 ·Q0 𝐵))
54oveq1d 5554 . . . . . 6 ([⟨𝑧, 𝑤⟩] ~Q0 = 𝐵 → ((𝐴 ·Q0 [⟨𝑧, 𝑤⟩] ~Q0 ) +Q0 (𝐴 ·Q0 [⟨𝑣, 𝑢⟩] ~Q0 )) = ((𝐴 ·Q0 𝐵) +Q0 (𝐴 ·Q0 [⟨𝑣, 𝑢⟩] ~Q0 )))
63, 5eqeq12d 2070 . . . . 5 ([⟨𝑧, 𝑤⟩] ~Q0 = 𝐵 → ((𝐴 ·Q0 ([⟨𝑧, 𝑤⟩] ~Q0 +Q0 [⟨𝑣, 𝑢⟩] ~Q0 )) = ((𝐴 ·Q0 [⟨𝑧, 𝑤⟩] ~Q0 ) +Q0 (𝐴 ·Q0 [⟨𝑣, 𝑢⟩] ~Q0 )) ↔ (𝐴 ·Q0 (𝐵 +Q0 [⟨𝑣, 𝑢⟩] ~Q0 )) = ((𝐴 ·Q0 𝐵) +Q0 (𝐴 ·Q0 [⟨𝑣, 𝑢⟩] ~Q0 ))))
76imbi2d 223 . . . 4 ([⟨𝑧, 𝑤⟩] ~Q0 = 𝐵 → ((𝐴Q0 → (𝐴 ·Q0 ([⟨𝑧, 𝑤⟩] ~Q0 +Q0 [⟨𝑣, 𝑢⟩] ~Q0 )) = ((𝐴 ·Q0 [⟨𝑧, 𝑤⟩] ~Q0 ) +Q0 (𝐴 ·Q0 [⟨𝑣, 𝑢⟩] ~Q0 ))) ↔ (𝐴Q0 → (𝐴 ·Q0 (𝐵 +Q0 [⟨𝑣, 𝑢⟩] ~Q0 )) = ((𝐴 ·Q0 𝐵) +Q0 (𝐴 ·Q0 [⟨𝑣, 𝑢⟩] ~Q0 )))))
8 oveq2 5547 . . . . . . 7 ([⟨𝑣, 𝑢⟩] ~Q0 = 𝐶 → (𝐵 +Q0 [⟨𝑣, 𝑢⟩] ~Q0 ) = (𝐵 +Q0 𝐶))
98oveq2d 5555 . . . . . 6 ([⟨𝑣, 𝑢⟩] ~Q0 = 𝐶 → (𝐴 ·Q0 (𝐵 +Q0 [⟨𝑣, 𝑢⟩] ~Q0 )) = (𝐴 ·Q0 (𝐵 +Q0 𝐶)))
10 oveq2 5547 . . . . . . 7 ([⟨𝑣, 𝑢⟩] ~Q0 = 𝐶 → (𝐴 ·Q0 [⟨𝑣, 𝑢⟩] ~Q0 ) = (𝐴 ·Q0 𝐶))
1110oveq2d 5555 . . . . . 6 ([⟨𝑣, 𝑢⟩] ~Q0 = 𝐶 → ((𝐴 ·Q0 𝐵) +Q0 (𝐴 ·Q0 [⟨𝑣, 𝑢⟩] ~Q0 )) = ((𝐴 ·Q0 𝐵) +Q0 (𝐴 ·Q0 𝐶)))
129, 11eqeq12d 2070 . . . . 5 ([⟨𝑣, 𝑢⟩] ~Q0 = 𝐶 → ((𝐴 ·Q0 (𝐵 +Q0 [⟨𝑣, 𝑢⟩] ~Q0 )) = ((𝐴 ·Q0 𝐵) +Q0 (𝐴 ·Q0 [⟨𝑣, 𝑢⟩] ~Q0 )) ↔ (𝐴 ·Q0 (𝐵 +Q0 𝐶)) = ((𝐴 ·Q0 𝐵) +Q0 (𝐴 ·Q0 𝐶))))
1312imbi2d 223 . . . 4 ([⟨𝑣, 𝑢⟩] ~Q0 = 𝐶 → ((𝐴Q0 → (𝐴 ·Q0 (𝐵 +Q0 [⟨𝑣, 𝑢⟩] ~Q0 )) = ((𝐴 ·Q0 𝐵) +Q0 (𝐴 ·Q0 [⟨𝑣, 𝑢⟩] ~Q0 ))) ↔ (𝐴Q0 → (𝐴 ·Q0 (𝐵 +Q0 𝐶)) = ((𝐴 ·Q0 𝐵) +Q0 (𝐴 ·Q0 𝐶)))))
14 oveq1 5546 . . . . . . . 8 ([⟨𝑥, 𝑦⟩] ~Q0 = 𝐴 → ([⟨𝑥, 𝑦⟩] ~Q0 ·Q0 ([⟨𝑧, 𝑤⟩] ~Q0 +Q0 [⟨𝑣, 𝑢⟩] ~Q0 )) = (𝐴 ·Q0 ([⟨𝑧, 𝑤⟩] ~Q0 +Q0 [⟨𝑣, 𝑢⟩] ~Q0 )))
15 oveq1 5546 . . . . . . . . 9 ([⟨𝑥, 𝑦⟩] ~Q0 = 𝐴 → ([⟨𝑥, 𝑦⟩] ~Q0 ·Q0 [⟨𝑧, 𝑤⟩] ~Q0 ) = (𝐴 ·Q0 [⟨𝑧, 𝑤⟩] ~Q0 ))
16 oveq1 5546 . . . . . . . . 9 ([⟨𝑥, 𝑦⟩] ~Q0 = 𝐴 → ([⟨𝑥, 𝑦⟩] ~Q0 ·Q0 [⟨𝑣, 𝑢⟩] ~Q0 ) = (𝐴 ·Q0 [⟨𝑣, 𝑢⟩] ~Q0 ))
1715, 16oveq12d 5557 . . . . . . . 8 ([⟨𝑥, 𝑦⟩] ~Q0 = 𝐴 → (([⟨𝑥, 𝑦⟩] ~Q0 ·Q0 [⟨𝑧, 𝑤⟩] ~Q0 ) +Q0 ([⟨𝑥, 𝑦⟩] ~Q0 ·Q0 [⟨𝑣, 𝑢⟩] ~Q0 )) = ((𝐴 ·Q0 [⟨𝑧, 𝑤⟩] ~Q0 ) +Q0 (𝐴 ·Q0 [⟨𝑣, 𝑢⟩] ~Q0 )))
1814, 17eqeq12d 2070 . . . . . . 7 ([⟨𝑥, 𝑦⟩] ~Q0 = 𝐴 → (([⟨𝑥, 𝑦⟩] ~Q0 ·Q0 ([⟨𝑧, 𝑤⟩] ~Q0 +Q0 [⟨𝑣, 𝑢⟩] ~Q0 )) = (([⟨𝑥, 𝑦⟩] ~Q0 ·Q0 [⟨𝑧, 𝑤⟩] ~Q0 ) +Q0 ([⟨𝑥, 𝑦⟩] ~Q0 ·Q0 [⟨𝑣, 𝑢⟩] ~Q0 )) ↔ (𝐴 ·Q0 ([⟨𝑧, 𝑤⟩] ~Q0 +Q0 [⟨𝑣, 𝑢⟩] ~Q0 )) = ((𝐴 ·Q0 [⟨𝑧, 𝑤⟩] ~Q0 ) +Q0 (𝐴 ·Q0 [⟨𝑣, 𝑢⟩] ~Q0 ))))
1918imbi2d 223 . . . . . 6 ([⟨𝑥, 𝑦⟩] ~Q0 = 𝐴 → ((((𝑧 ∈ ω ∧ 𝑤N) ∧ (𝑣 ∈ ω ∧ 𝑢N)) → ([⟨𝑥, 𝑦⟩] ~Q0 ·Q0 ([⟨𝑧, 𝑤⟩] ~Q0 +Q0 [⟨𝑣, 𝑢⟩] ~Q0 )) = (([⟨𝑥, 𝑦⟩] ~Q0 ·Q0 [⟨𝑧, 𝑤⟩] ~Q0 ) +Q0 ([⟨𝑥, 𝑦⟩] ~Q0 ·Q0 [⟨𝑣, 𝑢⟩] ~Q0 ))) ↔ (((𝑧 ∈ ω ∧ 𝑤N) ∧ (𝑣 ∈ ω ∧ 𝑢N)) → (𝐴 ·Q0 ([⟨𝑧, 𝑤⟩] ~Q0 +Q0 [⟨𝑣, 𝑢⟩] ~Q0 )) = ((𝐴 ·Q0 [⟨𝑧, 𝑤⟩] ~Q0 ) +Q0 (𝐴 ·Q0 [⟨𝑣, 𝑢⟩] ~Q0 )))))
20 an42 529 . . . . . . . . . . . 12 (((𝑧 ∈ ω ∧ 𝑢N) ∧ (𝑤N𝑣 ∈ ω)) ↔ ((𝑧 ∈ ω ∧ 𝑤N) ∧ (𝑣 ∈ ω ∧ 𝑢N)))
2120anbi2i 438 . . . . . . . . . . 11 (((𝑥 ∈ ω ∧ 𝑦N) ∧ ((𝑧 ∈ ω ∧ 𝑢N) ∧ (𝑤N𝑣 ∈ ω))) ↔ ((𝑥 ∈ ω ∧ 𝑦N) ∧ ((𝑧 ∈ ω ∧ 𝑤N) ∧ (𝑣 ∈ ω ∧ 𝑢N))))
22 3anass 900 . . . . . . . . . . 11 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑢N) ∧ (𝑤N𝑣 ∈ ω)) ↔ ((𝑥 ∈ ω ∧ 𝑦N) ∧ ((𝑧 ∈ ω ∧ 𝑢N) ∧ (𝑤N𝑣 ∈ ω))))
23 3anass 900 . . . . . . . . . . 11 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑤N) ∧ (𝑣 ∈ ω ∧ 𝑢N)) ↔ ((𝑥 ∈ ω ∧ 𝑦N) ∧ ((𝑧 ∈ ω ∧ 𝑤N) ∧ (𝑣 ∈ ω ∧ 𝑢N))))
2421, 22, 233bitr4i 205 . . . . . . . . . 10 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑢N) ∧ (𝑤N𝑣 ∈ ω)) ↔ ((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑤N) ∧ (𝑣 ∈ ω ∧ 𝑢N)))
25 pinn 6464 . . . . . . . . . . . . . 14 (𝑦N𝑦 ∈ ω)
26 nnmcl 6090 . . . . . . . . . . . . . 14 ((𝑦 ∈ ω ∧ 𝑥 ∈ ω) → (𝑦 ·𝑜 𝑥) ∈ ω)
2725, 26sylan 271 . . . . . . . . . . . . 13 ((𝑦N𝑥 ∈ ω) → (𝑦 ·𝑜 𝑥) ∈ ω)
2827ancoms 259 . . . . . . . . . . . 12 ((𝑥 ∈ ω ∧ 𝑦N) → (𝑦 ·𝑜 𝑥) ∈ ω)
29 pinn 6464 . . . . . . . . . . . . 13 (𝑢N𝑢 ∈ ω)
30 nnmcl 6090 . . . . . . . . . . . . 13 ((𝑧 ∈ ω ∧ 𝑢 ∈ ω) → (𝑧 ·𝑜 𝑢) ∈ ω)
3129, 30sylan2 274 . . . . . . . . . . . 12 ((𝑧 ∈ ω ∧ 𝑢N) → (𝑧 ·𝑜 𝑢) ∈ ω)
32 pinn 6464 . . . . . . . . . . . . 13 (𝑤N𝑤 ∈ ω)
33 nnmcl 6090 . . . . . . . . . . . . 13 ((𝑤 ∈ ω ∧ 𝑣 ∈ ω) → (𝑤 ·𝑜 𝑣) ∈ ω)
3432, 33sylan 271 . . . . . . . . . . . 12 ((𝑤N𝑣 ∈ ω) → (𝑤 ·𝑜 𝑣) ∈ ω)
35 nndi 6095 . . . . . . . . . . . 12 (((𝑦 ·𝑜 𝑥) ∈ ω ∧ (𝑧 ·𝑜 𝑢) ∈ ω ∧ (𝑤 ·𝑜 𝑣) ∈ ω) → ((𝑦 ·𝑜 𝑥) ·𝑜 ((𝑧 ·𝑜 𝑢) +𝑜 (𝑤 ·𝑜 𝑣))) = (((𝑦 ·𝑜 𝑥) ·𝑜 (𝑧 ·𝑜 𝑢)) +𝑜 ((𝑦 ·𝑜 𝑥) ·𝑜 (𝑤 ·𝑜 𝑣))))
3628, 31, 34, 35syl3an 1188 . . . . . . . . . . 11 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑢N) ∧ (𝑤N𝑣 ∈ ω)) → ((𝑦 ·𝑜 𝑥) ·𝑜 ((𝑧 ·𝑜 𝑢) +𝑜 (𝑤 ·𝑜 𝑣))) = (((𝑦 ·𝑜 𝑥) ·𝑜 (𝑧 ·𝑜 𝑢)) +𝑜 ((𝑦 ·𝑜 𝑥) ·𝑜 (𝑤 ·𝑜 𝑣))))
37 simp1r 940 . . . . . . . . . . . 12 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑢N) ∧ (𝑤N𝑣 ∈ ω)) → 𝑦N)
38 simp1l 939 . . . . . . . . . . . 12 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑢N) ∧ (𝑤N𝑣 ∈ ω)) → 𝑥 ∈ ω)
39313ad2ant2 937 . . . . . . . . . . . . 13 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑢N) ∧ (𝑤N𝑣 ∈ ω)) → (𝑧 ·𝑜 𝑢) ∈ ω)
40343ad2ant3 938 . . . . . . . . . . . . 13 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑢N) ∧ (𝑤N𝑣 ∈ ω)) → (𝑤 ·𝑜 𝑣) ∈ ω)
41 nnacl 6089 . . . . . . . . . . . . 13 (((𝑧 ·𝑜 𝑢) ∈ ω ∧ (𝑤 ·𝑜 𝑣) ∈ ω) → ((𝑧 ·𝑜 𝑢) +𝑜 (𝑤 ·𝑜 𝑣)) ∈ ω)
4239, 40, 41syl2anc 397 . . . . . . . . . . . 12 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑢N) ∧ (𝑤N𝑣 ∈ ω)) → ((𝑧 ·𝑜 𝑢) +𝑜 (𝑤 ·𝑜 𝑣)) ∈ ω)
43 nnmass 6096 . . . . . . . . . . . . 13 ((𝑦 ∈ ω ∧ 𝑥 ∈ ω ∧ ((𝑧 ·𝑜 𝑢) +𝑜 (𝑤 ·𝑜 𝑣)) ∈ ω) → ((𝑦 ·𝑜 𝑥) ·𝑜 ((𝑧 ·𝑜 𝑢) +𝑜 (𝑤 ·𝑜 𝑣))) = (𝑦 ·𝑜 (𝑥 ·𝑜 ((𝑧 ·𝑜 𝑢) +𝑜 (𝑤 ·𝑜 𝑣)))))
4425, 43syl3an1 1179 . . . . . . . . . . . 12 ((𝑦N𝑥 ∈ ω ∧ ((𝑧 ·𝑜 𝑢) +𝑜 (𝑤 ·𝑜 𝑣)) ∈ ω) → ((𝑦 ·𝑜 𝑥) ·𝑜 ((𝑧 ·𝑜 𝑢) +𝑜 (𝑤 ·𝑜 𝑣))) = (𝑦 ·𝑜 (𝑥 ·𝑜 ((𝑧 ·𝑜 𝑢) +𝑜 (𝑤 ·𝑜 𝑣)))))
4537, 38, 42, 44syl3anc 1146 . . . . . . . . . . 11 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑢N) ∧ (𝑤N𝑣 ∈ ω)) → ((𝑦 ·𝑜 𝑥) ·𝑜 ((𝑧 ·𝑜 𝑢) +𝑜 (𝑤 ·𝑜 𝑣))) = (𝑦 ·𝑜 (𝑥 ·𝑜 ((𝑧 ·𝑜 𝑢) +𝑜 (𝑤 ·𝑜 𝑣)))))
46 nnmcom 6098 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → (𝑥 ·𝑜 𝑦) = (𝑦 ·𝑜 𝑥))
4725, 46sylan2 274 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ω ∧ 𝑦N) → (𝑥 ·𝑜 𝑦) = (𝑦 ·𝑜 𝑥))
4847oveq1d 5554 . . . . . . . . . . . . . . 15 ((𝑥 ∈ ω ∧ 𝑦N) → ((𝑥 ·𝑜 𝑦) ·𝑜 (𝑧 ·𝑜 𝑢)) = ((𝑦 ·𝑜 𝑥) ·𝑜 (𝑧 ·𝑜 𝑢)))
4948adantr 265 . . . . . . . . . . . . . 14 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑢N)) → ((𝑥 ·𝑜 𝑦) ·𝑜 (𝑧 ·𝑜 𝑢)) = ((𝑦 ·𝑜 𝑥) ·𝑜 (𝑧 ·𝑜 𝑢)))
50 simpll 489 . . . . . . . . . . . . . . 15 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑢N)) → 𝑥 ∈ ω)
5125ad2antlr 466 . . . . . . . . . . . . . . 15 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑢N)) → 𝑦 ∈ ω)
52 simprl 491 . . . . . . . . . . . . . . 15 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑢N)) → 𝑧 ∈ ω)
53 nnmcom 6098 . . . . . . . . . . . . . . . 16 ((𝑓 ∈ ω ∧ 𝑔 ∈ ω) → (𝑓 ·𝑜 𝑔) = (𝑔 ·𝑜 𝑓))
5453adantl 266 . . . . . . . . . . . . . . 15 ((((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑢N)) ∧ (𝑓 ∈ ω ∧ 𝑔 ∈ ω)) → (𝑓 ·𝑜 𝑔) = (𝑔 ·𝑜 𝑓))
55 nnmass 6096 . . . . . . . . . . . . . . . 16 ((𝑓 ∈ ω ∧ 𝑔 ∈ ω ∧ ∈ ω) → ((𝑓 ·𝑜 𝑔) ·𝑜 ) = (𝑓 ·𝑜 (𝑔 ·𝑜 )))
5655adantl 266 . . . . . . . . . . . . . . 15 ((((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑢N)) ∧ (𝑓 ∈ ω ∧ 𝑔 ∈ ω ∧ ∈ ω)) → ((𝑓 ·𝑜 𝑔) ·𝑜 ) = (𝑓 ·𝑜 (𝑔 ·𝑜 )))
57 simprr 492 . . . . . . . . . . . . . . . 16 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑢N)) → 𝑢N)
5857, 29syl 14 . . . . . . . . . . . . . . 15 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑢N)) → 𝑢 ∈ ω)
59 nnmcl 6090 . . . . . . . . . . . . . . . 16 ((𝑓 ∈ ω ∧ 𝑔 ∈ ω) → (𝑓 ·𝑜 𝑔) ∈ ω)
6059adantl 266 . . . . . . . . . . . . . . 15 ((((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑢N)) ∧ (𝑓 ∈ ω ∧ 𝑔 ∈ ω)) → (𝑓 ·𝑜 𝑔) ∈ ω)
6150, 51, 52, 54, 56, 58, 60caov4d 5712 . . . . . . . . . . . . . 14 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑢N)) → ((𝑥 ·𝑜 𝑦) ·𝑜 (𝑧 ·𝑜 𝑢)) = ((𝑥 ·𝑜 𝑧) ·𝑜 (𝑦 ·𝑜 𝑢)))
6249, 61eqtr3d 2090 . . . . . . . . . . . . 13 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑢N)) → ((𝑦 ·𝑜 𝑥) ·𝑜 (𝑧 ·𝑜 𝑢)) = ((𝑥 ·𝑜 𝑧) ·𝑜 (𝑦 ·𝑜 𝑢)))
63623adant3 935 . . . . . . . . . . . 12 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑢N) ∧ (𝑤N𝑣 ∈ ω)) → ((𝑦 ·𝑜 𝑥) ·𝑜 (𝑧 ·𝑜 𝑢)) = ((𝑥 ·𝑜 𝑧) ·𝑜 (𝑦 ·𝑜 𝑢)))
6425ad2antlr 466 . . . . . . . . . . . . . 14 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑤N𝑣 ∈ ω)) → 𝑦 ∈ ω)
65 simpll 489 . . . . . . . . . . . . . 14 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑤N𝑣 ∈ ω)) → 𝑥 ∈ ω)
66 simprl 491 . . . . . . . . . . . . . . 15 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑤N𝑣 ∈ ω)) → 𝑤N)
6766, 32syl 14 . . . . . . . . . . . . . 14 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑤N𝑣 ∈ ω)) → 𝑤 ∈ ω)
6853adantl 266 . . . . . . . . . . . . . 14 ((((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑤N𝑣 ∈ ω)) ∧ (𝑓 ∈ ω ∧ 𝑔 ∈ ω)) → (𝑓 ·𝑜 𝑔) = (𝑔 ·𝑜 𝑓))
6955adantl 266 . . . . . . . . . . . . . 14 ((((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑤N𝑣 ∈ ω)) ∧ (𝑓 ∈ ω ∧ 𝑔 ∈ ω ∧ ∈ ω)) → ((𝑓 ·𝑜 𝑔) ·𝑜 ) = (𝑓 ·𝑜 (𝑔 ·𝑜 )))
70 simprr 492 . . . . . . . . . . . . . 14 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑤N𝑣 ∈ ω)) → 𝑣 ∈ ω)
7159adantl 266 . . . . . . . . . . . . . 14 ((((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑤N𝑣 ∈ ω)) ∧ (𝑓 ∈ ω ∧ 𝑔 ∈ ω)) → (𝑓 ·𝑜 𝑔) ∈ ω)
7264, 65, 67, 68, 69, 70, 71caov4d 5712 . . . . . . . . . . . . 13 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑤N𝑣 ∈ ω)) → ((𝑦 ·𝑜 𝑥) ·𝑜 (𝑤 ·𝑜 𝑣)) = ((𝑦 ·𝑜 𝑤) ·𝑜 (𝑥 ·𝑜 𝑣)))
73723adant2 934 . . . . . . . . . . . 12 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑢N) ∧ (𝑤N𝑣 ∈ ω)) → ((𝑦 ·𝑜 𝑥) ·𝑜 (𝑤 ·𝑜 𝑣)) = ((𝑦 ·𝑜 𝑤) ·𝑜 (𝑥 ·𝑜 𝑣)))
7463, 73oveq12d 5557 . . . . . . . . . . 11 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑢N) ∧ (𝑤N𝑣 ∈ ω)) → (((𝑦 ·𝑜 𝑥) ·𝑜 (𝑧 ·𝑜 𝑢)) +𝑜 ((𝑦 ·𝑜 𝑥) ·𝑜 (𝑤 ·𝑜 𝑣))) = (((𝑥 ·𝑜 𝑧) ·𝑜 (𝑦 ·𝑜 𝑢)) +𝑜 ((𝑦 ·𝑜 𝑤) ·𝑜 (𝑥 ·𝑜 𝑣))))
7536, 45, 743eqtr3d 2096 . . . . . . . . . 10 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑢N) ∧ (𝑤N𝑣 ∈ ω)) → (𝑦 ·𝑜 (𝑥 ·𝑜 ((𝑧 ·𝑜 𝑢) +𝑜 (𝑤 ·𝑜 𝑣)))) = (((𝑥 ·𝑜 𝑧) ·𝑜 (𝑦 ·𝑜 𝑢)) +𝑜 ((𝑦 ·𝑜 𝑤) ·𝑜 (𝑥 ·𝑜 𝑣))))
7624, 75sylbir 129 . . . . . . . . 9 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑤N) ∧ (𝑣 ∈ ω ∧ 𝑢N)) → (𝑦 ·𝑜 (𝑥 ·𝑜 ((𝑧 ·𝑜 𝑢) +𝑜 (𝑤 ·𝑜 𝑣)))) = (((𝑥 ·𝑜 𝑧) ·𝑜 (𝑦 ·𝑜 𝑢)) +𝑜 ((𝑦 ·𝑜 𝑤) ·𝑜 (𝑥 ·𝑜 𝑣))))
7737, 25syl 14 . . . . . . . . . . . 12 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑢N) ∧ (𝑤N𝑣 ∈ ω)) → 𝑦 ∈ ω)
78 mulpiord 6472 . . . . . . . . . . . . . . . 16 ((𝑤N𝑢N) → (𝑤 ·N 𝑢) = (𝑤 ·𝑜 𝑢))
7978ancoms 259 . . . . . . . . . . . . . . 15 ((𝑢N𝑤N) → (𝑤 ·N 𝑢) = (𝑤 ·𝑜 𝑢))
8079ad2ant2lr 487 . . . . . . . . . . . . . 14 (((𝑧 ∈ ω ∧ 𝑢N) ∧ (𝑤N𝑣 ∈ ω)) → (𝑤 ·N 𝑢) = (𝑤 ·𝑜 𝑢))
81803adant1 933 . . . . . . . . . . . . 13 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑢N) ∧ (𝑤N𝑣 ∈ ω)) → (𝑤 ·N 𝑢) = (𝑤 ·𝑜 𝑢))
82663adant2 934 . . . . . . . . . . . . . . 15 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑢N) ∧ (𝑤N𝑣 ∈ ω)) → 𝑤N)
83573adant3 935 . . . . . . . . . . . . . . 15 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑢N) ∧ (𝑤N𝑣 ∈ ω)) → 𝑢N)
84 mulclpi 6483 . . . . . . . . . . . . . . 15 ((𝑤N𝑢N) → (𝑤 ·N 𝑢) ∈ N)
8582, 83, 84syl2anc 397 . . . . . . . . . . . . . 14 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑢N) ∧ (𝑤N𝑣 ∈ ω)) → (𝑤 ·N 𝑢) ∈ N)
86 pinn 6464 . . . . . . . . . . . . . 14 ((𝑤 ·N 𝑢) ∈ N → (𝑤 ·N 𝑢) ∈ ω)
8785, 86syl 14 . . . . . . . . . . . . 13 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑢N) ∧ (𝑤N𝑣 ∈ ω)) → (𝑤 ·N 𝑢) ∈ ω)
8881, 87eqeltrrd 2131 . . . . . . . . . . . 12 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑢N) ∧ (𝑤N𝑣 ∈ ω)) → (𝑤 ·𝑜 𝑢) ∈ ω)
89 nnmass 6096 . . . . . . . . . . . 12 ((𝑦 ∈ ω ∧ 𝑦 ∈ ω ∧ (𝑤 ·𝑜 𝑢) ∈ ω) → ((𝑦 ·𝑜 𝑦) ·𝑜 (𝑤 ·𝑜 𝑢)) = (𝑦 ·𝑜 (𝑦 ·𝑜 (𝑤 ·𝑜 𝑢))))
9077, 77, 88, 89syl3anc 1146 . . . . . . . . . . 11 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑢N) ∧ (𝑤N𝑣 ∈ ω)) → ((𝑦 ·𝑜 𝑦) ·𝑜 (𝑤 ·𝑜 𝑢)) = (𝑦 ·𝑜 (𝑦 ·𝑜 (𝑤 ·𝑜 𝑢))))
9182, 32syl 14 . . . . . . . . . . . 12 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑢N) ∧ (𝑤N𝑣 ∈ ω)) → 𝑤 ∈ ω)
9253adantl 266 . . . . . . . . . . . 12 ((((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑢N) ∧ (𝑤N𝑣 ∈ ω)) ∧ (𝑓 ∈ ω ∧ 𝑔 ∈ ω)) → (𝑓 ·𝑜 𝑔) = (𝑔 ·𝑜 𝑓))
9355adantl 266 . . . . . . . . . . . 12 ((((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑢N) ∧ (𝑤N𝑣 ∈ ω)) ∧ (𝑓 ∈ ω ∧ 𝑔 ∈ ω ∧ ∈ ω)) → ((𝑓 ·𝑜 𝑔) ·𝑜 ) = (𝑓 ·𝑜 (𝑔 ·𝑜 )))
9483, 29syl 14 . . . . . . . . . . . 12 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑢N) ∧ (𝑤N𝑣 ∈ ω)) → 𝑢 ∈ ω)
9559adantl 266 . . . . . . . . . . . 12 ((((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑢N) ∧ (𝑤N𝑣 ∈ ω)) ∧ (𝑓 ∈ ω ∧ 𝑔 ∈ ω)) → (𝑓 ·𝑜 𝑔) ∈ ω)
9677, 77, 91, 92, 93, 94, 95caov4d 5712 . . . . . . . . . . 11 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑢N) ∧ (𝑤N𝑣 ∈ ω)) → ((𝑦 ·𝑜 𝑦) ·𝑜 (𝑤 ·𝑜 𝑢)) = ((𝑦 ·𝑜 𝑤) ·𝑜 (𝑦 ·𝑜 𝑢)))
9790, 96eqtr3d 2090 . . . . . . . . . 10 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑢N) ∧ (𝑤N𝑣 ∈ ω)) → (𝑦 ·𝑜 (𝑦 ·𝑜 (𝑤 ·𝑜 𝑢))) = ((𝑦 ·𝑜 𝑤) ·𝑜 (𝑦 ·𝑜 𝑢)))
9824, 97sylbir 129 . . . . . . . . 9 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑤N) ∧ (𝑣 ∈ ω ∧ 𝑢N)) → (𝑦 ·𝑜 (𝑦 ·𝑜 (𝑤 ·𝑜 𝑢))) = ((𝑦 ·𝑜 𝑤) ·𝑜 (𝑦 ·𝑜 𝑢)))
99 opeq12 3578 . . . . . . . . . 10 (((𝑦 ·𝑜 (𝑥 ·𝑜 ((𝑧 ·𝑜 𝑢) +𝑜 (𝑤 ·𝑜 𝑣)))) = (((𝑥 ·𝑜 𝑧) ·𝑜 (𝑦 ·𝑜 𝑢)) +𝑜 ((𝑦 ·𝑜 𝑤) ·𝑜 (𝑥 ·𝑜 𝑣))) ∧ (𝑦 ·𝑜 (𝑦 ·𝑜 (𝑤 ·𝑜 𝑢))) = ((𝑦 ·𝑜 𝑤) ·𝑜 (𝑦 ·𝑜 𝑢))) → ⟨(𝑦 ·𝑜 (𝑥 ·𝑜 ((𝑧 ·𝑜 𝑢) +𝑜 (𝑤 ·𝑜 𝑣)))), (𝑦 ·𝑜 (𝑦 ·𝑜 (𝑤 ·𝑜 𝑢)))⟩ = ⟨(((𝑥 ·𝑜 𝑧) ·𝑜 (𝑦 ·𝑜 𝑢)) +𝑜 ((𝑦 ·𝑜 𝑤) ·𝑜 (𝑥 ·𝑜 𝑣))), ((𝑦 ·𝑜 𝑤) ·𝑜 (𝑦 ·𝑜 𝑢))⟩)
10099eceq1d 6172 . . . . . . . . 9 (((𝑦 ·𝑜 (𝑥 ·𝑜 ((𝑧 ·𝑜 𝑢) +𝑜 (𝑤 ·𝑜 𝑣)))) = (((𝑥 ·𝑜 𝑧) ·𝑜 (𝑦 ·𝑜 𝑢)) +𝑜 ((𝑦 ·𝑜 𝑤) ·𝑜 (𝑥 ·𝑜 𝑣))) ∧ (𝑦 ·𝑜 (𝑦 ·𝑜 (𝑤 ·𝑜 𝑢))) = ((𝑦 ·𝑜 𝑤) ·𝑜 (𝑦 ·𝑜 𝑢))) → [⟨(𝑦 ·𝑜 (𝑥 ·𝑜 ((𝑧 ·𝑜 𝑢) +𝑜 (𝑤 ·𝑜 𝑣)))), (𝑦 ·𝑜 (𝑦 ·𝑜 (𝑤 ·𝑜 𝑢)))⟩] ~Q0 = [⟨(((𝑥 ·𝑜 𝑧) ·𝑜 (𝑦 ·𝑜 𝑢)) +𝑜 ((𝑦 ·𝑜 𝑤) ·𝑜 (𝑥 ·𝑜 𝑣))), ((𝑦 ·𝑜 𝑤) ·𝑜 (𝑦 ·𝑜 𝑢))⟩] ~Q0 )
10176, 98, 100syl2anc 397 . . . . . . . 8 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑤N) ∧ (𝑣 ∈ ω ∧ 𝑢N)) → [⟨(𝑦 ·𝑜 (𝑥 ·𝑜 ((𝑧 ·𝑜 𝑢) +𝑜 (𝑤 ·𝑜 𝑣)))), (𝑦 ·𝑜 (𝑦 ·𝑜 (𝑤 ·𝑜 𝑢)))⟩] ~Q0 = [⟨(((𝑥 ·𝑜 𝑧) ·𝑜 (𝑦 ·𝑜 𝑢)) +𝑜 ((𝑦 ·𝑜 𝑤) ·𝑜 (𝑥 ·𝑜 𝑣))), ((𝑦 ·𝑜 𝑤) ·𝑜 (𝑦 ·𝑜 𝑢))⟩] ~Q0 )
102 addnnnq0 6604 . . . . . . . . . . . 12 (((𝑧 ∈ ω ∧ 𝑤N) ∧ (𝑣 ∈ ω ∧ 𝑢N)) → ([⟨𝑧, 𝑤⟩] ~Q0 +Q0 [⟨𝑣, 𝑢⟩] ~Q0 ) = [⟨((𝑧 ·𝑜 𝑢) +𝑜 (𝑤 ·𝑜 𝑣)), (𝑤 ·𝑜 𝑢)⟩] ~Q0 )
103102oveq2d 5555 . . . . . . . . . . 11 (((𝑧 ∈ ω ∧ 𝑤N) ∧ (𝑣 ∈ ω ∧ 𝑢N)) → ([⟨𝑥, 𝑦⟩] ~Q0 ·Q0 ([⟨𝑧, 𝑤⟩] ~Q0 +Q0 [⟨𝑣, 𝑢⟩] ~Q0 )) = ([⟨𝑥, 𝑦⟩] ~Q0 ·Q0 [⟨((𝑧 ·𝑜 𝑢) +𝑜 (𝑤 ·𝑜 𝑣)), (𝑤 ·𝑜 𝑢)⟩] ~Q0 ))
104103adantl 266 . . . . . . . . . 10 (((𝑥 ∈ ω ∧ 𝑦N) ∧ ((𝑧 ∈ ω ∧ 𝑤N) ∧ (𝑣 ∈ ω ∧ 𝑢N))) → ([⟨𝑥, 𝑦⟩] ~Q0 ·Q0 ([⟨𝑧, 𝑤⟩] ~Q0 +Q0 [⟨𝑣, 𝑢⟩] ~Q0 )) = ([⟨𝑥, 𝑦⟩] ~Q0 ·Q0 [⟨((𝑧 ·𝑜 𝑢) +𝑜 (𝑤 ·𝑜 𝑣)), (𝑤 ·𝑜 𝑢)⟩] ~Q0 ))
10531, 34, 41syl2an 277 . . . . . . . . . . . . 13 (((𝑧 ∈ ω ∧ 𝑢N) ∧ (𝑤N𝑣 ∈ ω)) → ((𝑧 ·𝑜 𝑢) +𝑜 (𝑤 ·𝑜 𝑣)) ∈ ω)
106105an42s 531 . . . . . . . . . . . 12 (((𝑧 ∈ ω ∧ 𝑤N) ∧ (𝑣 ∈ ω ∧ 𝑢N)) → ((𝑧 ·𝑜 𝑢) +𝑜 (𝑤 ·𝑜 𝑣)) ∈ ω)
10784ad2ant2l 485 . . . . . . . . . . . . 13 (((𝑧 ∈ ω ∧ 𝑤N) ∧ (𝑣 ∈ ω ∧ 𝑢N)) → (𝑤 ·N 𝑢) ∈ N)
10878eleq1d 2122 . . . . . . . . . . . . . 14 ((𝑤N𝑢N) → ((𝑤 ·N 𝑢) ∈ N ↔ (𝑤 ·𝑜 𝑢) ∈ N))
109108ad2ant2l 485 . . . . . . . . . . . . 13 (((𝑧 ∈ ω ∧ 𝑤N) ∧ (𝑣 ∈ ω ∧ 𝑢N)) → ((𝑤 ·N 𝑢) ∈ N ↔ (𝑤 ·𝑜 𝑢) ∈ N))
110107, 109mpbid 139 . . . . . . . . . . . 12 (((𝑧 ∈ ω ∧ 𝑤N) ∧ (𝑣 ∈ ω ∧ 𝑢N)) → (𝑤 ·𝑜 𝑢) ∈ N)
111106, 110jca 294 . . . . . . . . . . 11 (((𝑧 ∈ ω ∧ 𝑤N) ∧ (𝑣 ∈ ω ∧ 𝑢N)) → (((𝑧 ·𝑜 𝑢) +𝑜 (𝑤 ·𝑜 𝑣)) ∈ ω ∧ (𝑤 ·𝑜 𝑢) ∈ N))
112 mulnnnq0 6605 . . . . . . . . . . . 12 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (((𝑧 ·𝑜 𝑢) +𝑜 (𝑤 ·𝑜 𝑣)) ∈ ω ∧ (𝑤 ·𝑜 𝑢) ∈ N)) → ([⟨𝑥, 𝑦⟩] ~Q0 ·Q0 [⟨((𝑧 ·𝑜 𝑢) +𝑜 (𝑤 ·𝑜 𝑣)), (𝑤 ·𝑜 𝑢)⟩] ~Q0 ) = [⟨(𝑥 ·𝑜 ((𝑧 ·𝑜 𝑢) +𝑜 (𝑤 ·𝑜 𝑣))), (𝑦 ·𝑜 (𝑤 ·𝑜 𝑢))⟩] ~Q0 )
113 nnmcl 6090 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ω ∧ ((𝑧 ·𝑜 𝑢) +𝑜 (𝑤 ·𝑜 𝑣)) ∈ ω) → (𝑥 ·𝑜 ((𝑧 ·𝑜 𝑢) +𝑜 (𝑤 ·𝑜 𝑣))) ∈ ω)
114 simpl 106 . . . . . . . . . . . . . . . . 17 ((𝑦N ∧ (𝑤 ·𝑜 𝑢) ∈ N) → 𝑦N)
115 mulpiord 6472 . . . . . . . . . . . . . . . . . 18 ((𝑦N ∧ (𝑤 ·𝑜 𝑢) ∈ N) → (𝑦 ·N (𝑤 ·𝑜 𝑢)) = (𝑦 ·𝑜 (𝑤 ·𝑜 𝑢)))
116 mulclpi 6483 . . . . . . . . . . . . . . . . . 18 ((𝑦N ∧ (𝑤 ·𝑜 𝑢) ∈ N) → (𝑦 ·N (𝑤 ·𝑜 𝑢)) ∈ N)
117115, 116eqeltrrd 2131 . . . . . . . . . . . . . . . . 17 ((𝑦N ∧ (𝑤 ·𝑜 𝑢) ∈ N) → (𝑦 ·𝑜 (𝑤 ·𝑜 𝑢)) ∈ N)
118114, 117jca 294 . . . . . . . . . . . . . . . 16 ((𝑦N ∧ (𝑤 ·𝑜 𝑢) ∈ N) → (𝑦N ∧ (𝑦 ·𝑜 (𝑤 ·𝑜 𝑢)) ∈ N))
119113, 118anim12i 325 . . . . . . . . . . . . . . 15 (((𝑥 ∈ ω ∧ ((𝑧 ·𝑜 𝑢) +𝑜 (𝑤 ·𝑜 𝑣)) ∈ ω) ∧ (𝑦N ∧ (𝑤 ·𝑜 𝑢) ∈ N)) → ((𝑥 ·𝑜 ((𝑧 ·𝑜 𝑢) +𝑜 (𝑤 ·𝑜 𝑣))) ∈ ω ∧ (𝑦N ∧ (𝑦 ·𝑜 (𝑤 ·𝑜 𝑢)) ∈ N)))
120 an12 503 . . . . . . . . . . . . . . . 16 (((𝑥 ·𝑜 ((𝑧 ·𝑜 𝑢) +𝑜 (𝑤 ·𝑜 𝑣))) ∈ ω ∧ (𝑦N ∧ (𝑦 ·𝑜 (𝑤 ·𝑜 𝑢)) ∈ N)) ↔ (𝑦N ∧ ((𝑥 ·𝑜 ((𝑧 ·𝑜 𝑢) +𝑜 (𝑤 ·𝑜 𝑣))) ∈ ω ∧ (𝑦 ·𝑜 (𝑤 ·𝑜 𝑢)) ∈ N)))
121 3anass 900 . . . . . . . . . . . . . . . 16 ((𝑦N ∧ (𝑥 ·𝑜 ((𝑧 ·𝑜 𝑢) +𝑜 (𝑤 ·𝑜 𝑣))) ∈ ω ∧ (𝑦 ·𝑜 (𝑤 ·𝑜 𝑢)) ∈ N) ↔ (𝑦N ∧ ((𝑥 ·𝑜 ((𝑧 ·𝑜 𝑢) +𝑜 (𝑤 ·𝑜 𝑣))) ∈ ω ∧ (𝑦 ·𝑜 (𝑤 ·𝑜 𝑢)) ∈ N)))
122120, 121bitr4i 180 . . . . . . . . . . . . . . 15 (((𝑥 ·𝑜 ((𝑧 ·𝑜 𝑢) +𝑜 (𝑤 ·𝑜 𝑣))) ∈ ω ∧ (𝑦N ∧ (𝑦 ·𝑜 (𝑤 ·𝑜 𝑢)) ∈ N)) ↔ (𝑦N ∧ (𝑥 ·𝑜 ((𝑧 ·𝑜 𝑢) +𝑜 (𝑤 ·𝑜 𝑣))) ∈ ω ∧ (𝑦 ·𝑜 (𝑤 ·𝑜 𝑢)) ∈ N))
123119, 122sylib 131 . . . . . . . . . . . . . 14 (((𝑥 ∈ ω ∧ ((𝑧 ·𝑜 𝑢) +𝑜 (𝑤 ·𝑜 𝑣)) ∈ ω) ∧ (𝑦N ∧ (𝑤 ·𝑜 𝑢) ∈ N)) → (𝑦N ∧ (𝑥 ·𝑜 ((𝑧 ·𝑜 𝑢) +𝑜 (𝑤 ·𝑜 𝑣))) ∈ ω ∧ (𝑦 ·𝑜 (𝑤 ·𝑜 𝑢)) ∈ N))
124123an4s 530 . . . . . . . . . . . . 13 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (((𝑧 ·𝑜 𝑢) +𝑜 (𝑤 ·𝑜 𝑣)) ∈ ω ∧ (𝑤 ·𝑜 𝑢) ∈ N)) → (𝑦N ∧ (𝑥 ·𝑜 ((𝑧 ·𝑜 𝑢) +𝑜 (𝑤 ·𝑜 𝑣))) ∈ ω ∧ (𝑦 ·𝑜 (𝑤 ·𝑜 𝑢)) ∈ N))
125 mulcanenq0ec 6600 . . . . . . . . . . . . 13 ((𝑦N ∧ (𝑥 ·𝑜 ((𝑧 ·𝑜 𝑢) +𝑜 (𝑤 ·𝑜 𝑣))) ∈ ω ∧ (𝑦 ·𝑜 (𝑤 ·𝑜 𝑢)) ∈ N) → [⟨(𝑦 ·𝑜 (𝑥 ·𝑜 ((𝑧 ·𝑜 𝑢) +𝑜 (𝑤 ·𝑜 𝑣)))), (𝑦 ·𝑜 (𝑦 ·𝑜 (𝑤 ·𝑜 𝑢)))⟩] ~Q0 = [⟨(𝑥 ·𝑜 ((𝑧 ·𝑜 𝑢) +𝑜 (𝑤 ·𝑜 𝑣))), (𝑦 ·𝑜 (𝑤 ·𝑜 𝑢))⟩] ~Q0 )
126124, 125syl 14 . . . . . . . . . . . 12 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (((𝑧 ·𝑜 𝑢) +𝑜 (𝑤 ·𝑜 𝑣)) ∈ ω ∧ (𝑤 ·𝑜 𝑢) ∈ N)) → [⟨(𝑦 ·𝑜 (𝑥 ·𝑜 ((𝑧 ·𝑜 𝑢) +𝑜 (𝑤 ·𝑜 𝑣)))), (𝑦 ·𝑜 (𝑦 ·𝑜 (𝑤 ·𝑜 𝑢)))⟩] ~Q0 = [⟨(𝑥 ·𝑜 ((𝑧 ·𝑜 𝑢) +𝑜 (𝑤 ·𝑜 𝑣))), (𝑦 ·𝑜 (𝑤 ·𝑜 𝑢))⟩] ~Q0 )
127112, 126eqtr4d 2091 . . . . . . . . . . 11 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (((𝑧 ·𝑜 𝑢) +𝑜 (𝑤 ·𝑜 𝑣)) ∈ ω ∧ (𝑤 ·𝑜 𝑢) ∈ N)) → ([⟨𝑥, 𝑦⟩] ~Q0 ·Q0 [⟨((𝑧 ·𝑜 𝑢) +𝑜 (𝑤 ·𝑜 𝑣)), (𝑤 ·𝑜 𝑢)⟩] ~Q0 ) = [⟨(𝑦 ·𝑜 (𝑥 ·𝑜 ((𝑧 ·𝑜 𝑢) +𝑜 (𝑤 ·𝑜 𝑣)))), (𝑦 ·𝑜 (𝑦 ·𝑜 (𝑤 ·𝑜 𝑢)))⟩] ~Q0 )
128111, 127sylan2 274 . . . . . . . . . 10 (((𝑥 ∈ ω ∧ 𝑦N) ∧ ((𝑧 ∈ ω ∧ 𝑤N) ∧ (𝑣 ∈ ω ∧ 𝑢N))) → ([⟨𝑥, 𝑦⟩] ~Q0 ·Q0 [⟨((𝑧 ·𝑜 𝑢) +𝑜 (𝑤 ·𝑜 𝑣)), (𝑤 ·𝑜 𝑢)⟩] ~Q0 ) = [⟨(𝑦 ·𝑜 (𝑥 ·𝑜 ((𝑧 ·𝑜 𝑢) +𝑜 (𝑤 ·𝑜 𝑣)))), (𝑦 ·𝑜 (𝑦 ·𝑜 (𝑤 ·𝑜 𝑢)))⟩] ~Q0 )
129104, 128eqtrd 2088 . . . . . . . . 9 (((𝑥 ∈ ω ∧ 𝑦N) ∧ ((𝑧 ∈ ω ∧ 𝑤N) ∧ (𝑣 ∈ ω ∧ 𝑢N))) → ([⟨𝑥, 𝑦⟩] ~Q0 ·Q0 ([⟨𝑧, 𝑤⟩] ~Q0 +Q0 [⟨𝑣, 𝑢⟩] ~Q0 )) = [⟨(𝑦 ·𝑜 (𝑥 ·𝑜 ((𝑧 ·𝑜 𝑢) +𝑜 (𝑤 ·𝑜 𝑣)))), (𝑦 ·𝑜 (𝑦 ·𝑜 (𝑤 ·𝑜 𝑢)))⟩] ~Q0 )
1301293impb 1111 . . . . . . . 8 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑤N) ∧ (𝑣 ∈ ω ∧ 𝑢N)) → ([⟨𝑥, 𝑦⟩] ~Q0 ·Q0 ([⟨𝑧, 𝑤⟩] ~Q0 +Q0 [⟨𝑣, 𝑢⟩] ~Q0 )) = [⟨(𝑦 ·𝑜 (𝑥 ·𝑜 ((𝑧 ·𝑜 𝑢) +𝑜 (𝑤 ·𝑜 𝑣)))), (𝑦 ·𝑜 (𝑦 ·𝑜 (𝑤 ·𝑜 𝑢)))⟩] ~Q0 )
131 mulnnnq0 6605 . . . . . . . . . . 11 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑤N)) → ([⟨𝑥, 𝑦⟩] ~Q0 ·Q0 [⟨𝑧, 𝑤⟩] ~Q0 ) = [⟨(𝑥 ·𝑜 𝑧), (𝑦 ·𝑜 𝑤)⟩] ~Q0 )
132 mulnnnq0 6605 . . . . . . . . . . 11 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑣 ∈ ω ∧ 𝑢N)) → ([⟨𝑥, 𝑦⟩] ~Q0 ·Q0 [⟨𝑣, 𝑢⟩] ~Q0 ) = [⟨(𝑥 ·𝑜 𝑣), (𝑦 ·𝑜 𝑢)⟩] ~Q0 )
133131, 132oveqan12d 5558 . . . . . . . . . 10 ((((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑤N)) ∧ ((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑣 ∈ ω ∧ 𝑢N))) → (([⟨𝑥, 𝑦⟩] ~Q0 ·Q0 [⟨𝑧, 𝑤⟩] ~Q0 ) +Q0 ([⟨𝑥, 𝑦⟩] ~Q0 ·Q0 [⟨𝑣, 𝑢⟩] ~Q0 )) = ([⟨(𝑥 ·𝑜 𝑧), (𝑦 ·𝑜 𝑤)⟩] ~Q0 +Q0 [⟨(𝑥 ·𝑜 𝑣), (𝑦 ·𝑜 𝑢)⟩] ~Q0 ))
134 nnmcl 6090 . . . . . . . . . . . . 13 ((𝑥 ∈ ω ∧ 𝑧 ∈ ω) → (𝑥 ·𝑜 𝑧) ∈ ω)
135 mulpiord 6472 . . . . . . . . . . . . . 14 ((𝑦N𝑤N) → (𝑦 ·N 𝑤) = (𝑦 ·𝑜 𝑤))
136 mulclpi 6483 . . . . . . . . . . . . . 14 ((𝑦N𝑤N) → (𝑦 ·N 𝑤) ∈ N)
137135, 136eqeltrrd 2131 . . . . . . . . . . . . 13 ((𝑦N𝑤N) → (𝑦 ·𝑜 𝑤) ∈ N)
138134, 137anim12i 325 . . . . . . . . . . . 12 (((𝑥 ∈ ω ∧ 𝑧 ∈ ω) ∧ (𝑦N𝑤N)) → ((𝑥 ·𝑜 𝑧) ∈ ω ∧ (𝑦 ·𝑜 𝑤) ∈ N))
139138an4s 530 . . . . . . . . . . 11 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑤N)) → ((𝑥 ·𝑜 𝑧) ∈ ω ∧ (𝑦 ·𝑜 𝑤) ∈ N))
140 nnmcl 6090 . . . . . . . . . . . . 13 ((𝑥 ∈ ω ∧ 𝑣 ∈ ω) → (𝑥 ·𝑜 𝑣) ∈ ω)
141 mulpiord 6472 . . . . . . . . . . . . . 14 ((𝑦N𝑢N) → (𝑦 ·N 𝑢) = (𝑦 ·𝑜 𝑢))
142 mulclpi 6483 . . . . . . . . . . . . . 14 ((𝑦N𝑢N) → (𝑦 ·N 𝑢) ∈ N)
143141, 142eqeltrrd 2131 . . . . . . . . . . . . 13 ((𝑦N𝑢N) → (𝑦 ·𝑜 𝑢) ∈ N)
144140, 143anim12i 325 . . . . . . . . . . . 12 (((𝑥 ∈ ω ∧ 𝑣 ∈ ω) ∧ (𝑦N𝑢N)) → ((𝑥 ·𝑜 𝑣) ∈ ω ∧ (𝑦 ·𝑜 𝑢) ∈ N))
145144an4s 530 . . . . . . . . . . 11 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑣 ∈ ω ∧ 𝑢N)) → ((𝑥 ·𝑜 𝑣) ∈ ω ∧ (𝑦 ·𝑜 𝑢) ∈ N))
146 addnnnq0 6604 . . . . . . . . . . 11 ((((𝑥 ·𝑜 𝑧) ∈ ω ∧ (𝑦 ·𝑜 𝑤) ∈ N) ∧ ((𝑥 ·𝑜 𝑣) ∈ ω ∧ (𝑦 ·𝑜 𝑢) ∈ N)) → ([⟨(𝑥 ·𝑜 𝑧), (𝑦 ·𝑜 𝑤)⟩] ~Q0 +Q0 [⟨(𝑥 ·𝑜 𝑣), (𝑦 ·𝑜 𝑢)⟩] ~Q0 ) = [⟨(((𝑥 ·𝑜 𝑧) ·𝑜 (𝑦 ·𝑜 𝑢)) +𝑜 ((𝑦 ·𝑜 𝑤) ·𝑜 (𝑥 ·𝑜 𝑣))), ((𝑦 ·𝑜 𝑤) ·𝑜 (𝑦 ·𝑜 𝑢))⟩] ~Q0 )
147139, 145, 146syl2an 277 . . . . . . . . . 10 ((((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑤N)) ∧ ((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑣 ∈ ω ∧ 𝑢N))) → ([⟨(𝑥 ·𝑜 𝑧), (𝑦 ·𝑜 𝑤)⟩] ~Q0 +Q0 [⟨(𝑥 ·𝑜 𝑣), (𝑦 ·𝑜 𝑢)⟩] ~Q0 ) = [⟨(((𝑥 ·𝑜 𝑧) ·𝑜 (𝑦 ·𝑜 𝑢)) +𝑜 ((𝑦 ·𝑜 𝑤) ·𝑜 (𝑥 ·𝑜 𝑣))), ((𝑦 ·𝑜 𝑤) ·𝑜 (𝑦 ·𝑜 𝑢))⟩] ~Q0 )
148133, 147eqtrd 2088 . . . . . . . . 9 ((((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑤N)) ∧ ((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑣 ∈ ω ∧ 𝑢N))) → (([⟨𝑥, 𝑦⟩] ~Q0 ·Q0 [⟨𝑧, 𝑤⟩] ~Q0 ) +Q0 ([⟨𝑥, 𝑦⟩] ~Q0 ·Q0 [⟨𝑣, 𝑢⟩] ~Q0 )) = [⟨(((𝑥 ·𝑜 𝑧) ·𝑜 (𝑦 ·𝑜 𝑢)) +𝑜 ((𝑦 ·𝑜 𝑤) ·𝑜 (𝑥 ·𝑜 𝑣))), ((𝑦 ·𝑜 𝑤) ·𝑜 (𝑦 ·𝑜 𝑢))⟩] ~Q0 )
1491483impdi 1201 . . . . . . . 8 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑤N) ∧ (𝑣 ∈ ω ∧ 𝑢N)) → (([⟨𝑥, 𝑦⟩] ~Q0 ·Q0 [⟨𝑧, 𝑤⟩] ~Q0 ) +Q0 ([⟨𝑥, 𝑦⟩] ~Q0 ·Q0 [⟨𝑣, 𝑢⟩] ~Q0 )) = [⟨(((𝑥 ·𝑜 𝑧) ·𝑜 (𝑦 ·𝑜 𝑢)) +𝑜 ((𝑦 ·𝑜 𝑤) ·𝑜 (𝑥 ·𝑜 𝑣))), ((𝑦 ·𝑜 𝑤) ·𝑜 (𝑦 ·𝑜 𝑢))⟩] ~Q0 )
150101, 130, 1493eqtr4d 2098 . . . . . . 7 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑤N) ∧ (𝑣 ∈ ω ∧ 𝑢N)) → ([⟨𝑥, 𝑦⟩] ~Q0 ·Q0 ([⟨𝑧, 𝑤⟩] ~Q0 +Q0 [⟨𝑣, 𝑢⟩] ~Q0 )) = (([⟨𝑥, 𝑦⟩] ~Q0 ·Q0 [⟨𝑧, 𝑤⟩] ~Q0 ) +Q0 ([⟨𝑥, 𝑦⟩] ~Q0 ·Q0 [⟨𝑣, 𝑢⟩] ~Q0 )))
1511503expib 1118 . . . . . 6 ((𝑥 ∈ ω ∧ 𝑦N) → (((𝑧 ∈ ω ∧ 𝑤N) ∧ (𝑣 ∈ ω ∧ 𝑢N)) → ([⟨𝑥, 𝑦⟩] ~Q0 ·Q0 ([⟨𝑧, 𝑤⟩] ~Q0 +Q0 [⟨𝑣, 𝑢⟩] ~Q0 )) = (([⟨𝑥, 𝑦⟩] ~Q0 ·Q0 [⟨𝑧, 𝑤⟩] ~Q0 ) +Q0 ([⟨𝑥, 𝑦⟩] ~Q0 ·Q0 [⟨𝑣, 𝑢⟩] ~Q0 ))))
1521, 19, 151ecoptocl 6223 . . . . 5 (𝐴Q0 → (((𝑧 ∈ ω ∧ 𝑤N) ∧ (𝑣 ∈ ω ∧ 𝑢N)) → (𝐴 ·Q0 ([⟨𝑧, 𝑤⟩] ~Q0 +Q0 [⟨𝑣, 𝑢⟩] ~Q0 )) = ((𝐴 ·Q0 [⟨𝑧, 𝑤⟩] ~Q0 ) +Q0 (𝐴 ·Q0 [⟨𝑣, 𝑢⟩] ~Q0 ))))
153152com12 30 . . . 4 (((𝑧 ∈ ω ∧ 𝑤N) ∧ (𝑣 ∈ ω ∧ 𝑢N)) → (𝐴Q0 → (𝐴 ·Q0 ([⟨𝑧, 𝑤⟩] ~Q0 +Q0 [⟨𝑣, 𝑢⟩] ~Q0 )) = ((𝐴 ·Q0 [⟨𝑧, 𝑤⟩] ~Q0 ) +Q0 (𝐴 ·Q0 [⟨𝑣, 𝑢⟩] ~Q0 ))))
1541, 7, 13, 1532ecoptocl 6224 . . 3 ((𝐵Q0𝐶Q0) → (𝐴Q0 → (𝐴 ·Q0 (𝐵 +Q0 𝐶)) = ((𝐴 ·Q0 𝐵) +Q0 (𝐴 ·Q0 𝐶))))
155154com12 30 . 2 (𝐴Q0 → ((𝐵Q0𝐶Q0) → (𝐴 ·Q0 (𝐵 +Q0 𝐶)) = ((𝐴 ·Q0 𝐵) +Q0 (𝐴 ·Q0 𝐶))))
1561553impib 1113 1 ((𝐴Q0𝐵Q0𝐶Q0) → (𝐴 ·Q0 (𝐵 +Q0 𝐶)) = ((𝐴 ·Q0 𝐵) +Q0 (𝐴 ·Q0 𝐶)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  w3a 896   = wceq 1259  wcel 1409  cop 3405  ωcom 4340  (class class class)co 5539   +𝑜 coa 6028   ·𝑜 comu 6029  [cec 6134  Ncnpi 6427   ·N cmi 6429   ~Q0 ceq0 6441  Q0cnq0 6442   +Q0 cplq0 6444   ·Q0 cmq0 6445
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-mi 6461  df-enq0 6579  df-nq0 6580  df-plq0 6582  df-mq0 6583
This theorem is referenced by:  distnq0r  6618
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