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Theorem distrnq0 6557
 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 6523 . . . 4 Q0 = ((ω × N) / ~Q0 )
2 oveq1 5519 . . . . . . 7 ([⟨𝑧, 𝑤⟩] ~Q0 = 𝐵 → ([⟨𝑧, 𝑤⟩] ~Q0 +Q0 [⟨𝑣, 𝑢⟩] ~Q0 ) = (𝐵 +Q0 [⟨𝑣, 𝑢⟩] ~Q0 ))
32oveq2d 5528 . . . . . 6 ([⟨𝑧, 𝑤⟩] ~Q0 = 𝐵 → (𝐴 ·Q0 ([⟨𝑧, 𝑤⟩] ~Q0 +Q0 [⟨𝑣, 𝑢⟩] ~Q0 )) = (𝐴 ·Q0 (𝐵 +Q0 [⟨𝑣, 𝑢⟩] ~Q0 )))
4 oveq2 5520 . . . . . . 7 ([⟨𝑧, 𝑤⟩] ~Q0 = 𝐵 → (𝐴 ·Q0 [⟨𝑧, 𝑤⟩] ~Q0 ) = (𝐴 ·Q0 𝐵))
54oveq1d 5527 . . . . . 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 2054 . . . . 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 219 . . . 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 5520 . . . . . . 7 ([⟨𝑣, 𝑢⟩] ~Q0 = 𝐶 → (𝐵 +Q0 [⟨𝑣, 𝑢⟩] ~Q0 ) = (𝐵 +Q0 𝐶))
98oveq2d 5528 . . . . . 6 ([⟨𝑣, 𝑢⟩] ~Q0 = 𝐶 → (𝐴 ·Q0 (𝐵 +Q0 [⟨𝑣, 𝑢⟩] ~Q0 )) = (𝐴 ·Q0 (𝐵 +Q0 𝐶)))
10 oveq2 5520 . . . . . . 7 ([⟨𝑣, 𝑢⟩] ~Q0 = 𝐶 → (𝐴 ·Q0 [⟨𝑣, 𝑢⟩] ~Q0 ) = (𝐴 ·Q0 𝐶))
1110oveq2d 5528 . . . . . 6 ([⟨𝑣, 𝑢⟩] ~Q0 = 𝐶 → ((𝐴 ·Q0 𝐵) +Q0 (𝐴 ·Q0 [⟨𝑣, 𝑢⟩] ~Q0 )) = ((𝐴 ·Q0 𝐵) +Q0 (𝐴 ·Q0 𝐶)))
129, 11eqeq12d 2054 . . . . 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 219 . . . 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 5519 . . . . . . . 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 5519 . . . . . . . . 9 ([⟨𝑥, 𝑦⟩] ~Q0 = 𝐴 → ([⟨𝑥, 𝑦⟩] ~Q0 ·Q0 [⟨𝑧, 𝑤⟩] ~Q0 ) = (𝐴 ·Q0 [⟨𝑧, 𝑤⟩] ~Q0 ))
16 oveq1 5519 . . . . . . . . 9 ([⟨𝑥, 𝑦⟩] ~Q0 = 𝐴 → ([⟨𝑥, 𝑦⟩] ~Q0 ·Q0 [⟨𝑣, 𝑢⟩] ~Q0 ) = (𝐴 ·Q0 [⟨𝑣, 𝑢⟩] ~Q0 ))
1715, 16oveq12d 5530 . . . . . . . 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 2054 . . . . . . 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 219 . . . . . 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 521 . . . . . . . . . . . 12 (((𝑧 ∈ ω ∧ 𝑢N) ∧ (𝑤N𝑣 ∈ ω)) ↔ ((𝑧 ∈ ω ∧ 𝑤N) ∧ (𝑣 ∈ ω ∧ 𝑢N)))
2120anbi2i 430 . . . . . . . . . . 11 (((𝑥 ∈ ω ∧ 𝑦N) ∧ ((𝑧 ∈ ω ∧ 𝑢N) ∧ (𝑤N𝑣 ∈ ω))) ↔ ((𝑥 ∈ ω ∧ 𝑦N) ∧ ((𝑧 ∈ ω ∧ 𝑤N) ∧ (𝑣 ∈ ω ∧ 𝑢N))))
22 3anass 889 . . . . . . . . . . 11 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑢N) ∧ (𝑤N𝑣 ∈ ω)) ↔ ((𝑥 ∈ ω ∧ 𝑦N) ∧ ((𝑧 ∈ ω ∧ 𝑢N) ∧ (𝑤N𝑣 ∈ ω))))
23 3anass 889 . . . . . . . . . . 11 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑤N) ∧ (𝑣 ∈ ω ∧ 𝑢N)) ↔ ((𝑥 ∈ ω ∧ 𝑦N) ∧ ((𝑧 ∈ ω ∧ 𝑤N) ∧ (𝑣 ∈ ω ∧ 𝑢N))))
2421, 22, 233bitr4i 201 . . . . . . . . . 10 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑢N) ∧ (𝑤N𝑣 ∈ ω)) ↔ ((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑤N) ∧ (𝑣 ∈ ω ∧ 𝑢N)))
25 pinn 6407 . . . . . . . . . . . . . 14 (𝑦N𝑦 ∈ ω)
26 nnmcl 6060 . . . . . . . . . . . . . 14 ((𝑦 ∈ ω ∧ 𝑥 ∈ ω) → (𝑦 ·𝑜 𝑥) ∈ ω)
2725, 26sylan 267 . . . . . . . . . . . . 13 ((𝑦N𝑥 ∈ ω) → (𝑦 ·𝑜 𝑥) ∈ ω)
2827ancoms 255 . . . . . . . . . . . 12 ((𝑥 ∈ ω ∧ 𝑦N) → (𝑦 ·𝑜 𝑥) ∈ ω)
29 pinn 6407 . . . . . . . . . . . . 13 (𝑢N𝑢 ∈ ω)
30 nnmcl 6060 . . . . . . . . . . . . 13 ((𝑧 ∈ ω ∧ 𝑢 ∈ ω) → (𝑧 ·𝑜 𝑢) ∈ ω)
3129, 30sylan2 270 . . . . . . . . . . . 12 ((𝑧 ∈ ω ∧ 𝑢N) → (𝑧 ·𝑜 𝑢) ∈ ω)
32 pinn 6407 . . . . . . . . . . . . 13 (𝑤N𝑤 ∈ ω)
33 nnmcl 6060 . . . . . . . . . . . . 13 ((𝑤 ∈ ω ∧ 𝑣 ∈ ω) → (𝑤 ·𝑜 𝑣) ∈ ω)
3432, 33sylan 267 . . . . . . . . . . . 12 ((𝑤N𝑣 ∈ ω) → (𝑤 ·𝑜 𝑣) ∈ ω)
35 nndi 6065 . . . . . . . . . . . 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 1177 . . . . . . . . . . 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 929 . . . . . . . . . . . 12 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑢N) ∧ (𝑤N𝑣 ∈ ω)) → 𝑦N)
38 simp1l 928 . . . . . . . . . . . 12 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑢N) ∧ (𝑤N𝑣 ∈ ω)) → 𝑥 ∈ ω)
39313ad2ant2 926 . . . . . . . . . . . . 13 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑢N) ∧ (𝑤N𝑣 ∈ ω)) → (𝑧 ·𝑜 𝑢) ∈ ω)
40343ad2ant3 927 . . . . . . . . . . . . 13 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑢N) ∧ (𝑤N𝑣 ∈ ω)) → (𝑤 ·𝑜 𝑣) ∈ ω)
41 nnacl 6059 . . . . . . . . . . . . 13 (((𝑧 ·𝑜 𝑢) ∈ ω ∧ (𝑤 ·𝑜 𝑣) ∈ ω) → ((𝑧 ·𝑜 𝑢) +𝑜 (𝑤 ·𝑜 𝑣)) ∈ ω)
4239, 40, 41syl2anc 391 . . . . . . . . . . . 12 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑢N) ∧ (𝑤N𝑣 ∈ ω)) → ((𝑧 ·𝑜 𝑢) +𝑜 (𝑤 ·𝑜 𝑣)) ∈ ω)
43 nnmass 6066 . . . . . . . . . . . . 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 1168 . . . . . . . . . . . 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 1135 . . . . . . . . . . 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 6068 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → (𝑥 ·𝑜 𝑦) = (𝑦 ·𝑜 𝑥))
4725, 46sylan2 270 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ω ∧ 𝑦N) → (𝑥 ·𝑜 𝑦) = (𝑦 ·𝑜 𝑥))
4847oveq1d 5527 . . . . . . . . . . . . . . 15 ((𝑥 ∈ ω ∧ 𝑦N) → ((𝑥 ·𝑜 𝑦) ·𝑜 (𝑧 ·𝑜 𝑢)) = ((𝑦 ·𝑜 𝑥) ·𝑜 (𝑧 ·𝑜 𝑢)))
4948adantr 261 . . . . . . . . . . . . . 14 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑢N)) → ((𝑥 ·𝑜 𝑦) ·𝑜 (𝑧 ·𝑜 𝑢)) = ((𝑦 ·𝑜 𝑥) ·𝑜 (𝑧 ·𝑜 𝑢)))
50 simpll 481 . . . . . . . . . . . . . . 15 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑢N)) → 𝑥 ∈ ω)
5125ad2antlr 458 . . . . . . . . . . . . . . 15 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑢N)) → 𝑦 ∈ ω)
52 simprl 483 . . . . . . . . . . . . . . 15 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑢N)) → 𝑧 ∈ ω)
53 nnmcom 6068 . . . . . . . . . . . . . . . 16 ((𝑓 ∈ ω ∧ 𝑔 ∈ ω) → (𝑓 ·𝑜 𝑔) = (𝑔 ·𝑜 𝑓))
5453adantl 262 . . . . . . . . . . . . . . 15 ((((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑢N)) ∧ (𝑓 ∈ ω ∧ 𝑔 ∈ ω)) → (𝑓 ·𝑜 𝑔) = (𝑔 ·𝑜 𝑓))
55 nnmass 6066 . . . . . . . . . . . . . . . 16 ((𝑓 ∈ ω ∧ 𝑔 ∈ ω ∧ ∈ ω) → ((𝑓 ·𝑜 𝑔) ·𝑜 ) = (𝑓 ·𝑜 (𝑔 ·𝑜 )))
5655adantl 262 . . . . . . . . . . . . . . 15 ((((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑢N)) ∧ (𝑓 ∈ ω ∧ 𝑔 ∈ ω ∧ ∈ ω)) → ((𝑓 ·𝑜 𝑔) ·𝑜 ) = (𝑓 ·𝑜 (𝑔 ·𝑜 )))
57 simprr 484 . . . . . . . . . . . . . . . 16 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑢N)) → 𝑢N)
5857, 29syl 14 . . . . . . . . . . . . . . 15 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑢N)) → 𝑢 ∈ ω)
59 nnmcl 6060 . . . . . . . . . . . . . . . 16 ((𝑓 ∈ ω ∧ 𝑔 ∈ ω) → (𝑓 ·𝑜 𝑔) ∈ ω)
6059adantl 262 . . . . . . . . . . . . . . 15 ((((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑢N)) ∧ (𝑓 ∈ ω ∧ 𝑔 ∈ ω)) → (𝑓 ·𝑜 𝑔) ∈ ω)
6150, 51, 52, 54, 56, 58, 60caov4d 5685 . . . . . . . . . . . . . 14 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑢N)) → ((𝑥 ·𝑜 𝑦) ·𝑜 (𝑧 ·𝑜 𝑢)) = ((𝑥 ·𝑜 𝑧) ·𝑜 (𝑦 ·𝑜 𝑢)))
6249, 61eqtr3d 2074 . . . . . . . . . . . . 13 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑢N)) → ((𝑦 ·𝑜 𝑥) ·𝑜 (𝑧 ·𝑜 𝑢)) = ((𝑥 ·𝑜 𝑧) ·𝑜 (𝑦 ·𝑜 𝑢)))
63623adant3 924 . . . . . . . . . . . 12 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑢N) ∧ (𝑤N𝑣 ∈ ω)) → ((𝑦 ·𝑜 𝑥) ·𝑜 (𝑧 ·𝑜 𝑢)) = ((𝑥 ·𝑜 𝑧) ·𝑜 (𝑦 ·𝑜 𝑢)))
6425ad2antlr 458 . . . . . . . . . . . . . 14 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑤N𝑣 ∈ ω)) → 𝑦 ∈ ω)
65 simpll 481 . . . . . . . . . . . . . 14 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑤N𝑣 ∈ ω)) → 𝑥 ∈ ω)
66 simprl 483 . . . . . . . . . . . . . . 15 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑤N𝑣 ∈ ω)) → 𝑤N)
6766, 32syl 14 . . . . . . . . . . . . . 14 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑤N𝑣 ∈ ω)) → 𝑤 ∈ ω)
6853adantl 262 . . . . . . . . . . . . . 14 ((((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑤N𝑣 ∈ ω)) ∧ (𝑓 ∈ ω ∧ 𝑔 ∈ ω)) → (𝑓 ·𝑜 𝑔) = (𝑔 ·𝑜 𝑓))
6955adantl 262 . . . . . . . . . . . . . 14 ((((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑤N𝑣 ∈ ω)) ∧ (𝑓 ∈ ω ∧ 𝑔 ∈ ω ∧ ∈ ω)) → ((𝑓 ·𝑜 𝑔) ·𝑜 ) = (𝑓 ·𝑜 (𝑔 ·𝑜 )))
70 simprr 484 . . . . . . . . . . . . . 14 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑤N𝑣 ∈ ω)) → 𝑣 ∈ ω)
7159adantl 262 . . . . . . . . . . . . . 14 ((((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑤N𝑣 ∈ ω)) ∧ (𝑓 ∈ ω ∧ 𝑔 ∈ ω)) → (𝑓 ·𝑜 𝑔) ∈ ω)
7264, 65, 67, 68, 69, 70, 71caov4d 5685 . . . . . . . . . . . . 13 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑤N𝑣 ∈ ω)) → ((𝑦 ·𝑜 𝑥) ·𝑜 (𝑤 ·𝑜 𝑣)) = ((𝑦 ·𝑜 𝑤) ·𝑜 (𝑥 ·𝑜 𝑣)))
73723adant2 923 . . . . . . . . . . . 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 5530 . . . . . . . . . . 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 2080 . . . . . . . . . 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 125 . . . . . . . . 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 6415 . . . . . . . . . . . . . . . 16 ((𝑤N𝑢N) → (𝑤 ·N 𝑢) = (𝑤 ·𝑜 𝑢))
7978ancoms 255 . . . . . . . . . . . . . . 15 ((𝑢N𝑤N) → (𝑤 ·N 𝑢) = (𝑤 ·𝑜 𝑢))
8079ad2ant2lr 479 . . . . . . . . . . . . . 14 (((𝑧 ∈ ω ∧ 𝑢N) ∧ (𝑤N𝑣 ∈ ω)) → (𝑤 ·N 𝑢) = (𝑤 ·𝑜 𝑢))
81803adant1 922 . . . . . . . . . . . . 13 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑢N) ∧ (𝑤N𝑣 ∈ ω)) → (𝑤 ·N 𝑢) = (𝑤 ·𝑜 𝑢))
82663adant2 923 . . . . . . . . . . . . . . 15 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑢N) ∧ (𝑤N𝑣 ∈ ω)) → 𝑤N)
83573adant3 924 . . . . . . . . . . . . . . 15 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑢N) ∧ (𝑤N𝑣 ∈ ω)) → 𝑢N)
84 mulclpi 6426 . . . . . . . . . . . . . . 15 ((𝑤N𝑢N) → (𝑤 ·N 𝑢) ∈ N)
8582, 83, 84syl2anc 391 . . . . . . . . . . . . . 14 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑢N) ∧ (𝑤N𝑣 ∈ ω)) → (𝑤 ·N 𝑢) ∈ N)
86 pinn 6407 . . . . . . . . . . . . . 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 2115 . . . . . . . . . . . 12 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑢N) ∧ (𝑤N𝑣 ∈ ω)) → (𝑤 ·𝑜 𝑢) ∈ ω)
89 nnmass 6066 . . . . . . . . . . . 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 1135 . . . . . . . . . . 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 262 . . . . . . . . . . . 12 ((((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑢N) ∧ (𝑤N𝑣 ∈ ω)) ∧ (𝑓 ∈ ω ∧ 𝑔 ∈ ω)) → (𝑓 ·𝑜 𝑔) = (𝑔 ·𝑜 𝑓))
9355adantl 262 . . . . . . . . . . . 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 262 . . . . . . . . . . . 12 ((((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑢N) ∧ (𝑤N𝑣 ∈ ω)) ∧ (𝑓 ∈ ω ∧ 𝑔 ∈ ω)) → (𝑓 ·𝑜 𝑔) ∈ ω)
9677, 77, 91, 92, 93, 94, 95caov4d 5685 . . . . . . . . . . 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 2074 . . . . . . . . . 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 125 . . . . . . . . 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 3551 . . . . . . . . . 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 6142 . . . . . . . . 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 391 . . . . . . . 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 6547 . . . . . . . . . . . 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 5528 . . . . . . . . . . 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 262 . . . . . . . . . 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 273 . . . . . . . . . . . . 13 (((𝑧 ∈ ω ∧ 𝑢N) ∧ (𝑤N𝑣 ∈ ω)) → ((𝑧 ·𝑜 𝑢) +𝑜 (𝑤 ·𝑜 𝑣)) ∈ ω)
106105an42s 523 . . . . . . . . . . . 12 (((𝑧 ∈ ω ∧ 𝑤N) ∧ (𝑣 ∈ ω ∧ 𝑢N)) → ((𝑧 ·𝑜 𝑢) +𝑜 (𝑤 ·𝑜 𝑣)) ∈ ω)
10784ad2ant2l 477 . . . . . . . . . . . . 13 (((𝑧 ∈ ω ∧ 𝑤N) ∧ (𝑣 ∈ ω ∧ 𝑢N)) → (𝑤 ·N 𝑢) ∈ N)
10878eleq1d 2106 . . . . . . . . . . . . . 14 ((𝑤N𝑢N) → ((𝑤 ·N 𝑢) ∈ N ↔ (𝑤 ·𝑜 𝑢) ∈ N))
109108ad2ant2l 477 . . . . . . . . . . . . 13 (((𝑧 ∈ ω ∧ 𝑤N) ∧ (𝑣 ∈ ω ∧ 𝑢N)) → ((𝑤 ·N 𝑢) ∈ N ↔ (𝑤 ·𝑜 𝑢) ∈ N))
110107, 109mpbid 135 . . . . . . . . . . . 12 (((𝑧 ∈ ω ∧ 𝑤N) ∧ (𝑣 ∈ ω ∧ 𝑢N)) → (𝑤 ·𝑜 𝑢) ∈ N)
111106, 110jca 290 . . . . . . . . . . 11 (((𝑧 ∈ ω ∧ 𝑤N) ∧ (𝑣 ∈ ω ∧ 𝑢N)) → (((𝑧 ·𝑜 𝑢) +𝑜 (𝑤 ·𝑜 𝑣)) ∈ ω ∧ (𝑤 ·𝑜 𝑢) ∈ N))
112 mulnnnq0 6548 . . . . . . . . . . . 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 6060 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ω ∧ ((𝑧 ·𝑜 𝑢) +𝑜 (𝑤 ·𝑜 𝑣)) ∈ ω) → (𝑥 ·𝑜 ((𝑧 ·𝑜 𝑢) +𝑜 (𝑤 ·𝑜 𝑣))) ∈ ω)
114 simpl 102 . . . . . . . . . . . . . . . . 17 ((𝑦N ∧ (𝑤 ·𝑜 𝑢) ∈ N) → 𝑦N)
115 mulpiord 6415 . . . . . . . . . . . . . . . . . 18 ((𝑦N ∧ (𝑤 ·𝑜 𝑢) ∈ N) → (𝑦 ·N (𝑤 ·𝑜 𝑢)) = (𝑦 ·𝑜 (𝑤 ·𝑜 𝑢)))
116 mulclpi 6426 . . . . . . . . . . . . . . . . . 18 ((𝑦N ∧ (𝑤 ·𝑜 𝑢) ∈ N) → (𝑦 ·N (𝑤 ·𝑜 𝑢)) ∈ N)
117115, 116eqeltrrd 2115 . . . . . . . . . . . . . . . . 17 ((𝑦N ∧ (𝑤 ·𝑜 𝑢) ∈ N) → (𝑦 ·𝑜 (𝑤 ·𝑜 𝑢)) ∈ N)
118114, 117jca 290 . . . . . . . . . . . . . . . 16 ((𝑦N ∧ (𝑤 ·𝑜 𝑢) ∈ N) → (𝑦N ∧ (𝑦 ·𝑜 (𝑤 ·𝑜 𝑢)) ∈ N))
119113, 118anim12i 321 . . . . . . . . . . . . . . 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 495 . . . . . . . . . . . . . . . 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 889 . . . . . . . . . . . . . . . 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 176 . . . . . . . . . . . . . . 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 127 . . . . . . . . . . . . . 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 522 . . . . . . . . . . . . 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 6543 . . . . . . . . . . . . 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 2075 . . . . . . . . . . 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 270 . . . . . . . . . 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 2072 . . . . . . . . 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 1100 . . . . . . . 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 6548 . . . . . . . . . . 11 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑤N)) → ([⟨𝑥, 𝑦⟩] ~Q0 ·Q0 [⟨𝑧, 𝑤⟩] ~Q0 ) = [⟨(𝑥 ·𝑜 𝑧), (𝑦 ·𝑜 𝑤)⟩] ~Q0 )
132 mulnnnq0 6548 . . . . . . . . . . 11 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑣 ∈ ω ∧ 𝑢N)) → ([⟨𝑥, 𝑦⟩] ~Q0 ·Q0 [⟨𝑣, 𝑢⟩] ~Q0 ) = [⟨(𝑥 ·𝑜 𝑣), (𝑦 ·𝑜 𝑢)⟩] ~Q0 )
133131, 132oveqan12d 5531 . . . . . . . . . 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 6060 . . . . . . . . . . . . 13 ((𝑥 ∈ ω ∧ 𝑧 ∈ ω) → (𝑥 ·𝑜 𝑧) ∈ ω)
135 mulpiord 6415 . . . . . . . . . . . . . 14 ((𝑦N𝑤N) → (𝑦 ·N 𝑤) = (𝑦 ·𝑜 𝑤))
136 mulclpi 6426 . . . . . . . . . . . . . 14 ((𝑦N𝑤N) → (𝑦 ·N 𝑤) ∈ N)
137135, 136eqeltrrd 2115 . . . . . . . . . . . . 13 ((𝑦N𝑤N) → (𝑦 ·𝑜 𝑤) ∈ N)
138134, 137anim12i 321 . . . . . . . . . . . 12 (((𝑥 ∈ ω ∧ 𝑧 ∈ ω) ∧ (𝑦N𝑤N)) → ((𝑥 ·𝑜 𝑧) ∈ ω ∧ (𝑦 ·𝑜 𝑤) ∈ N))
139138an4s 522 . . . . . . . . . . 11 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑤N)) → ((𝑥 ·𝑜 𝑧) ∈ ω ∧ (𝑦 ·𝑜 𝑤) ∈ N))
140 nnmcl 6060 . . . . . . . . . . . . 13 ((𝑥 ∈ ω ∧ 𝑣 ∈ ω) → (𝑥 ·𝑜 𝑣) ∈ ω)
141 mulpiord 6415 . . . . . . . . . . . . . 14 ((𝑦N𝑢N) → (𝑦 ·N 𝑢) = (𝑦 ·𝑜 𝑢))
142 mulclpi 6426 . . . . . . . . . . . . . 14 ((𝑦N𝑢N) → (𝑦 ·N 𝑢) ∈ N)
143141, 142eqeltrrd 2115 . . . . . . . . . . . . 13 ((𝑦N𝑢N) → (𝑦 ·𝑜 𝑢) ∈ N)
144140, 143anim12i 321 . . . . . . . . . . . 12 (((𝑥 ∈ ω ∧ 𝑣 ∈ ω) ∧ (𝑦N𝑢N)) → ((𝑥 ·𝑜 𝑣) ∈ ω ∧ (𝑦 ·𝑜 𝑢) ∈ N))
145144an4s 522 . . . . . . . . . . 11 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑣 ∈ ω ∧ 𝑢N)) → ((𝑥 ·𝑜 𝑣) ∈ ω ∧ (𝑦 ·𝑜 𝑢) ∈ N))
146 addnnnq0 6547 . . . . . . . . . . 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 273 . . . . . . . . . 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 2072 . . . . . . . . 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 1190 . . . . . . . 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 2082 . . . . . . 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 1107 . . . . . 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 6193 . . . . 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 27 . . . 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 6194 . . 3 ((𝐵Q0𝐶Q0) → (𝐴Q0 → (𝐴 ·Q0 (𝐵 +Q0 𝐶)) = ((𝐴 ·Q0 𝐵) +Q0 (𝐴 ·Q0 𝐶))))
155154com12 27 . 2 (𝐴Q0 → ((𝐵Q0𝐶Q0) → (𝐴 ·Q0 (𝐵 +Q0 𝐶)) = ((𝐴 ·Q0 𝐵) +Q0 (𝐴 ·Q0 𝐶))))
1561553impib 1102 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 97   ↔ wb 98   ∧ w3a 885   = wceq 1243   ∈ wcel 1393  ⟨cop 3378  ωcom 4313  (class class class)co 5512   +𝑜 coa 5998   ·𝑜 comu 5999  [cec 6104  Ncnpi 6370   ·N cmi 6372   ~Q0 ceq0 6384  Q0cnq0 6385   +Q0 cplq0 6387   ·Q0 cmq0 6388 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311 This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-id 4030  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-mi 6404  df-enq0 6522  df-nq0 6523  df-plq0 6525  df-mq0 6526 This theorem is referenced by:  distnq0r  6561
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