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Theorem distrnq0 6939
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 6905 . . . 4 Q0 = ((ω × N) / ~Q0 )
2 oveq1 5601 . . . . . . 7 ([⟨𝑧, 𝑤⟩] ~Q0 = 𝐵 → ([⟨𝑧, 𝑤⟩] ~Q0 +Q0 [⟨𝑣, 𝑢⟩] ~Q0 ) = (𝐵 +Q0 [⟨𝑣, 𝑢⟩] ~Q0 ))
32oveq2d 5610 . . . . . 6 ([⟨𝑧, 𝑤⟩] ~Q0 = 𝐵 → (𝐴 ·Q0 ([⟨𝑧, 𝑤⟩] ~Q0 +Q0 [⟨𝑣, 𝑢⟩] ~Q0 )) = (𝐴 ·Q0 (𝐵 +Q0 [⟨𝑣, 𝑢⟩] ~Q0 )))
4 oveq2 5602 . . . . . . 7 ([⟨𝑧, 𝑤⟩] ~Q0 = 𝐵 → (𝐴 ·Q0 [⟨𝑧, 𝑤⟩] ~Q0 ) = (𝐴 ·Q0 𝐵))
54oveq1d 5609 . . . . . 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 2099 . . . . 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 228 . . . 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 5602 . . . . . . 7 ([⟨𝑣, 𝑢⟩] ~Q0 = 𝐶 → (𝐵 +Q0 [⟨𝑣, 𝑢⟩] ~Q0 ) = (𝐵 +Q0 𝐶))
98oveq2d 5610 . . . . . 6 ([⟨𝑣, 𝑢⟩] ~Q0 = 𝐶 → (𝐴 ·Q0 (𝐵 +Q0 [⟨𝑣, 𝑢⟩] ~Q0 )) = (𝐴 ·Q0 (𝐵 +Q0 𝐶)))
10 oveq2 5602 . . . . . . 7 ([⟨𝑣, 𝑢⟩] ~Q0 = 𝐶 → (𝐴 ·Q0 [⟨𝑣, 𝑢⟩] ~Q0 ) = (𝐴 ·Q0 𝐶))
1110oveq2d 5610 . . . . . 6 ([⟨𝑣, 𝑢⟩] ~Q0 = 𝐶 → ((𝐴 ·Q0 𝐵) +Q0 (𝐴 ·Q0 [⟨𝑣, 𝑢⟩] ~Q0 )) = ((𝐴 ·Q0 𝐵) +Q0 (𝐴 ·Q0 𝐶)))
129, 11eqeq12d 2099 . . . . 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 228 . . . 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 5601 . . . . . . . 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 5601 . . . . . . . . 9 ([⟨𝑥, 𝑦⟩] ~Q0 = 𝐴 → ([⟨𝑥, 𝑦⟩] ~Q0 ·Q0 [⟨𝑧, 𝑤⟩] ~Q0 ) = (𝐴 ·Q0 [⟨𝑧, 𝑤⟩] ~Q0 ))
16 oveq1 5601 . . . . . . . . 9 ([⟨𝑥, 𝑦⟩] ~Q0 = 𝐴 → ([⟨𝑥, 𝑦⟩] ~Q0 ·Q0 [⟨𝑣, 𝑢⟩] ~Q0 ) = (𝐴 ·Q0 [⟨𝑣, 𝑢⟩] ~Q0 ))
1715, 16oveq12d 5612 . . . . . . . 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 2099 . . . . . . 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 228 . . . . . 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 552 . . . . . . . . . . . 12 (((𝑧 ∈ ω ∧ 𝑢N) ∧ (𝑤N𝑣 ∈ ω)) ↔ ((𝑧 ∈ ω ∧ 𝑤N) ∧ (𝑣 ∈ ω ∧ 𝑢N)))
2120anbi2i 445 . . . . . . . . . . 11 (((𝑥 ∈ ω ∧ 𝑦N) ∧ ((𝑧 ∈ ω ∧ 𝑢N) ∧ (𝑤N𝑣 ∈ ω))) ↔ ((𝑥 ∈ ω ∧ 𝑦N) ∧ ((𝑧 ∈ ω ∧ 𝑤N) ∧ (𝑣 ∈ ω ∧ 𝑢N))))
22 3anass 926 . . . . . . . . . . 11 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑢N) ∧ (𝑤N𝑣 ∈ ω)) ↔ ((𝑥 ∈ ω ∧ 𝑦N) ∧ ((𝑧 ∈ ω ∧ 𝑢N) ∧ (𝑤N𝑣 ∈ ω))))
23 3anass 926 . . . . . . . . . . 11 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑤N) ∧ (𝑣 ∈ ω ∧ 𝑢N)) ↔ ((𝑥 ∈ ω ∧ 𝑦N) ∧ ((𝑧 ∈ ω ∧ 𝑤N) ∧ (𝑣 ∈ ω ∧ 𝑢N))))
2421, 22, 233bitr4i 210 . . . . . . . . . 10 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑢N) ∧ (𝑤N𝑣 ∈ ω)) ↔ ((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑤N) ∧ (𝑣 ∈ ω ∧ 𝑢N)))
25 pinn 6789 . . . . . . . . . . . . . 14 (𝑦N𝑦 ∈ ω)
26 nnmcl 6177 . . . . . . . . . . . . . 14 ((𝑦 ∈ ω ∧ 𝑥 ∈ ω) → (𝑦 ·𝑜 𝑥) ∈ ω)
2725, 26sylan 277 . . . . . . . . . . . . 13 ((𝑦N𝑥 ∈ ω) → (𝑦 ·𝑜 𝑥) ∈ ω)
2827ancoms 264 . . . . . . . . . . . 12 ((𝑥 ∈ ω ∧ 𝑦N) → (𝑦 ·𝑜 𝑥) ∈ ω)
29 pinn 6789 . . . . . . . . . . . . 13 (𝑢N𝑢 ∈ ω)
30 nnmcl 6177 . . . . . . . . . . . . 13 ((𝑧 ∈ ω ∧ 𝑢 ∈ ω) → (𝑧 ·𝑜 𝑢) ∈ ω)
3129, 30sylan2 280 . . . . . . . . . . . 12 ((𝑧 ∈ ω ∧ 𝑢N) → (𝑧 ·𝑜 𝑢) ∈ ω)
32 pinn 6789 . . . . . . . . . . . . 13 (𝑤N𝑤 ∈ ω)
33 nnmcl 6177 . . . . . . . . . . . . 13 ((𝑤 ∈ ω ∧ 𝑣 ∈ ω) → (𝑤 ·𝑜 𝑣) ∈ ω)
3432, 33sylan 277 . . . . . . . . . . . 12 ((𝑤N𝑣 ∈ ω) → (𝑤 ·𝑜 𝑣) ∈ ω)
35 nndi 6182 . . . . . . . . . . . 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 1214 . . . . . . . . . . 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 966 . . . . . . . . . . . 12 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑢N) ∧ (𝑤N𝑣 ∈ ω)) → 𝑦N)
38 simp1l 965 . . . . . . . . . . . 12 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑢N) ∧ (𝑤N𝑣 ∈ ω)) → 𝑥 ∈ ω)
39313ad2ant2 963 . . . . . . . . . . . . 13 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑢N) ∧ (𝑤N𝑣 ∈ ω)) → (𝑧 ·𝑜 𝑢) ∈ ω)
40343ad2ant3 964 . . . . . . . . . . . . 13 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑢N) ∧ (𝑤N𝑣 ∈ ω)) → (𝑤 ·𝑜 𝑣) ∈ ω)
41 nnacl 6176 . . . . . . . . . . . . 13 (((𝑧 ·𝑜 𝑢) ∈ ω ∧ (𝑤 ·𝑜 𝑣) ∈ ω) → ((𝑧 ·𝑜 𝑢) +𝑜 (𝑤 ·𝑜 𝑣)) ∈ ω)
4239, 40, 41syl2anc 403 . . . . . . . . . . . 12 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑢N) ∧ (𝑤N𝑣 ∈ ω)) → ((𝑧 ·𝑜 𝑢) +𝑜 (𝑤 ·𝑜 𝑣)) ∈ ω)
43 nnmass 6183 . . . . . . . . . . . . 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 1205 . . . . . . . . . . . 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 1172 . . . . . . . . . . 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 6185 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → (𝑥 ·𝑜 𝑦) = (𝑦 ·𝑜 𝑥))
4725, 46sylan2 280 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ω ∧ 𝑦N) → (𝑥 ·𝑜 𝑦) = (𝑦 ·𝑜 𝑥))
4847oveq1d 5609 . . . . . . . . . . . . . . 15 ((𝑥 ∈ ω ∧ 𝑦N) → ((𝑥 ·𝑜 𝑦) ·𝑜 (𝑧 ·𝑜 𝑢)) = ((𝑦 ·𝑜 𝑥) ·𝑜 (𝑧 ·𝑜 𝑢)))
4948adantr 270 . . . . . . . . . . . . . 14 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑢N)) → ((𝑥 ·𝑜 𝑦) ·𝑜 (𝑧 ·𝑜 𝑢)) = ((𝑦 ·𝑜 𝑥) ·𝑜 (𝑧 ·𝑜 𝑢)))
50 simpll 496 . . . . . . . . . . . . . . 15 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑢N)) → 𝑥 ∈ ω)
5125ad2antlr 473 . . . . . . . . . . . . . . 15 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑢N)) → 𝑦 ∈ ω)
52 simprl 498 . . . . . . . . . . . . . . 15 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑢N)) → 𝑧 ∈ ω)
53 nnmcom 6185 . . . . . . . . . . . . . . . 16 ((𝑓 ∈ ω ∧ 𝑔 ∈ ω) → (𝑓 ·𝑜 𝑔) = (𝑔 ·𝑜 𝑓))
5453adantl 271 . . . . . . . . . . . . . . 15 ((((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑢N)) ∧ (𝑓 ∈ ω ∧ 𝑔 ∈ ω)) → (𝑓 ·𝑜 𝑔) = (𝑔 ·𝑜 𝑓))
55 nnmass 6183 . . . . . . . . . . . . . . . 16 ((𝑓 ∈ ω ∧ 𝑔 ∈ ω ∧ ∈ ω) → ((𝑓 ·𝑜 𝑔) ·𝑜 ) = (𝑓 ·𝑜 (𝑔 ·𝑜 )))
5655adantl 271 . . . . . . . . . . . . . . 15 ((((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑢N)) ∧ (𝑓 ∈ ω ∧ 𝑔 ∈ ω ∧ ∈ ω)) → ((𝑓 ·𝑜 𝑔) ·𝑜 ) = (𝑓 ·𝑜 (𝑔 ·𝑜 )))
57 simprr 499 . . . . . . . . . . . . . . . 16 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑢N)) → 𝑢N)
5857, 29syl 14 . . . . . . . . . . . . . . 15 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑢N)) → 𝑢 ∈ ω)
59 nnmcl 6177 . . . . . . . . . . . . . . . 16 ((𝑓 ∈ ω ∧ 𝑔 ∈ ω) → (𝑓 ·𝑜 𝑔) ∈ ω)
6059adantl 271 . . . . . . . . . . . . . . 15 ((((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑢N)) ∧ (𝑓 ∈ ω ∧ 𝑔 ∈ ω)) → (𝑓 ·𝑜 𝑔) ∈ ω)
6150, 51, 52, 54, 56, 58, 60caov4d 5767 . . . . . . . . . . . . . 14 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑢N)) → ((𝑥 ·𝑜 𝑦) ·𝑜 (𝑧 ·𝑜 𝑢)) = ((𝑥 ·𝑜 𝑧) ·𝑜 (𝑦 ·𝑜 𝑢)))
6249, 61eqtr3d 2119 . . . . . . . . . . . . 13 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑢N)) → ((𝑦 ·𝑜 𝑥) ·𝑜 (𝑧 ·𝑜 𝑢)) = ((𝑥 ·𝑜 𝑧) ·𝑜 (𝑦 ·𝑜 𝑢)))
63623adant3 961 . . . . . . . . . . . 12 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑢N) ∧ (𝑤N𝑣 ∈ ω)) → ((𝑦 ·𝑜 𝑥) ·𝑜 (𝑧 ·𝑜 𝑢)) = ((𝑥 ·𝑜 𝑧) ·𝑜 (𝑦 ·𝑜 𝑢)))
6425ad2antlr 473 . . . . . . . . . . . . . 14 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑤N𝑣 ∈ ω)) → 𝑦 ∈ ω)
65 simpll 496 . . . . . . . . . . . . . 14 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑤N𝑣 ∈ ω)) → 𝑥 ∈ ω)
66 simprl 498 . . . . . . . . . . . . . . 15 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑤N𝑣 ∈ ω)) → 𝑤N)
6766, 32syl 14 . . . . . . . . . . . . . 14 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑤N𝑣 ∈ ω)) → 𝑤 ∈ ω)
6853adantl 271 . . . . . . . . . . . . . 14 ((((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑤N𝑣 ∈ ω)) ∧ (𝑓 ∈ ω ∧ 𝑔 ∈ ω)) → (𝑓 ·𝑜 𝑔) = (𝑔 ·𝑜 𝑓))
6955adantl 271 . . . . . . . . . . . . . 14 ((((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑤N𝑣 ∈ ω)) ∧ (𝑓 ∈ ω ∧ 𝑔 ∈ ω ∧ ∈ ω)) → ((𝑓 ·𝑜 𝑔) ·𝑜 ) = (𝑓 ·𝑜 (𝑔 ·𝑜 )))
70 simprr 499 . . . . . . . . . . . . . 14 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑤N𝑣 ∈ ω)) → 𝑣 ∈ ω)
7159adantl 271 . . . . . . . . . . . . . 14 ((((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑤N𝑣 ∈ ω)) ∧ (𝑓 ∈ ω ∧ 𝑔 ∈ ω)) → (𝑓 ·𝑜 𝑔) ∈ ω)
7264, 65, 67, 68, 69, 70, 71caov4d 5767 . . . . . . . . . . . . 13 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑤N𝑣 ∈ ω)) → ((𝑦 ·𝑜 𝑥) ·𝑜 (𝑤 ·𝑜 𝑣)) = ((𝑦 ·𝑜 𝑤) ·𝑜 (𝑥 ·𝑜 𝑣)))
73723adant2 960 . . . . . . . . . . . 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 5612 . . . . . . . . . . 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 2125 . . . . . . . . . 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 133 . . . . . . . . 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 6797 . . . . . . . . . . . . . . . 16 ((𝑤N𝑢N) → (𝑤 ·N 𝑢) = (𝑤 ·𝑜 𝑢))
7978ancoms 264 . . . . . . . . . . . . . . 15 ((𝑢N𝑤N) → (𝑤 ·N 𝑢) = (𝑤 ·𝑜 𝑢))
8079ad2ant2lr 494 . . . . . . . . . . . . . 14 (((𝑧 ∈ ω ∧ 𝑢N) ∧ (𝑤N𝑣 ∈ ω)) → (𝑤 ·N 𝑢) = (𝑤 ·𝑜 𝑢))
81803adant1 959 . . . . . . . . . . . . 13 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑢N) ∧ (𝑤N𝑣 ∈ ω)) → (𝑤 ·N 𝑢) = (𝑤 ·𝑜 𝑢))
82663adant2 960 . . . . . . . . . . . . . . 15 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑢N) ∧ (𝑤N𝑣 ∈ ω)) → 𝑤N)
83573adant3 961 . . . . . . . . . . . . . . 15 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑢N) ∧ (𝑤N𝑣 ∈ ω)) → 𝑢N)
84 mulclpi 6808 . . . . . . . . . . . . . . 15 ((𝑤N𝑢N) → (𝑤 ·N 𝑢) ∈ N)
8582, 83, 84syl2anc 403 . . . . . . . . . . . . . 14 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑢N) ∧ (𝑤N𝑣 ∈ ω)) → (𝑤 ·N 𝑢) ∈ N)
86 pinn 6789 . . . . . . . . . . . . . 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 2162 . . . . . . . . . . . 12 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑢N) ∧ (𝑤N𝑣 ∈ ω)) → (𝑤 ·𝑜 𝑢) ∈ ω)
89 nnmass 6183 . . . . . . . . . . . 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 1172 . . . . . . . . . . 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 271 . . . . . . . . . . . 12 ((((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑢N) ∧ (𝑤N𝑣 ∈ ω)) ∧ (𝑓 ∈ ω ∧ 𝑔 ∈ ω)) → (𝑓 ·𝑜 𝑔) = (𝑔 ·𝑜 𝑓))
9355adantl 271 . . . . . . . . . . . 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 271 . . . . . . . . . . . 12 ((((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑢N) ∧ (𝑤N𝑣 ∈ ω)) ∧ (𝑓 ∈ ω ∧ 𝑔 ∈ ω)) → (𝑓 ·𝑜 𝑔) ∈ ω)
9677, 77, 91, 92, 93, 94, 95caov4d 5767 . . . . . . . . . . 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 2119 . . . . . . . . . 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 133 . . . . . . . . 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 3601 . . . . . . . . . 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 6261 . . . . . . . . 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 403 . . . . . . . 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 6929 . . . . . . . . . . . 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 5610 . . . . . . . . . . 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 271 . . . . . . . . . 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 283 . . . . . . . . . . . . 13 (((𝑧 ∈ ω ∧ 𝑢N) ∧ (𝑤N𝑣 ∈ ω)) → ((𝑧 ·𝑜 𝑢) +𝑜 (𝑤 ·𝑜 𝑣)) ∈ ω)
106105an42s 554 . . . . . . . . . . . 12 (((𝑧 ∈ ω ∧ 𝑤N) ∧ (𝑣 ∈ ω ∧ 𝑢N)) → ((𝑧 ·𝑜 𝑢) +𝑜 (𝑤 ·𝑜 𝑣)) ∈ ω)
10784ad2ant2l 492 . . . . . . . . . . . . 13 (((𝑧 ∈ ω ∧ 𝑤N) ∧ (𝑣 ∈ ω ∧ 𝑢N)) → (𝑤 ·N 𝑢) ∈ N)
10878eleq1d 2153 . . . . . . . . . . . . . 14 ((𝑤N𝑢N) → ((𝑤 ·N 𝑢) ∈ N ↔ (𝑤 ·𝑜 𝑢) ∈ N))
109108ad2ant2l 492 . . . . . . . . . . . . 13 (((𝑧 ∈ ω ∧ 𝑤N) ∧ (𝑣 ∈ ω ∧ 𝑢N)) → ((𝑤 ·N 𝑢) ∈ N ↔ (𝑤 ·𝑜 𝑢) ∈ N))
110107, 109mpbid 145 . . . . . . . . . . . 12 (((𝑧 ∈ ω ∧ 𝑤N) ∧ (𝑣 ∈ ω ∧ 𝑢N)) → (𝑤 ·𝑜 𝑢) ∈ N)
111106, 110jca 300 . . . . . . . . . . 11 (((𝑧 ∈ ω ∧ 𝑤N) ∧ (𝑣 ∈ ω ∧ 𝑢N)) → (((𝑧 ·𝑜 𝑢) +𝑜 (𝑤 ·𝑜 𝑣)) ∈ ω ∧ (𝑤 ·𝑜 𝑢) ∈ N))
112 mulnnnq0 6930 . . . . . . . . . . . 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 6177 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ω ∧ ((𝑧 ·𝑜 𝑢) +𝑜 (𝑤 ·𝑜 𝑣)) ∈ ω) → (𝑥 ·𝑜 ((𝑧 ·𝑜 𝑢) +𝑜 (𝑤 ·𝑜 𝑣))) ∈ ω)
114 simpl 107 . . . . . . . . . . . . . . . . 17 ((𝑦N ∧ (𝑤 ·𝑜 𝑢) ∈ N) → 𝑦N)
115 mulpiord 6797 . . . . . . . . . . . . . . . . . 18 ((𝑦N ∧ (𝑤 ·𝑜 𝑢) ∈ N) → (𝑦 ·N (𝑤 ·𝑜 𝑢)) = (𝑦 ·𝑜 (𝑤 ·𝑜 𝑢)))
116 mulclpi 6808 . . . . . . . . . . . . . . . . . 18 ((𝑦N ∧ (𝑤 ·𝑜 𝑢) ∈ N) → (𝑦 ·N (𝑤 ·𝑜 𝑢)) ∈ N)
117115, 116eqeltrrd 2162 . . . . . . . . . . . . . . . . 17 ((𝑦N ∧ (𝑤 ·𝑜 𝑢) ∈ N) → (𝑦 ·𝑜 (𝑤 ·𝑜 𝑢)) ∈ N)
118114, 117jca 300 . . . . . . . . . . . . . . . 16 ((𝑦N ∧ (𝑤 ·𝑜 𝑢) ∈ N) → (𝑦N ∧ (𝑦 ·𝑜 (𝑤 ·𝑜 𝑢)) ∈ N))
119113, 118anim12i 331 . . . . . . . . . . . . . . 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 526 . . . . . . . . . . . . . . . 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 926 . . . . . . . . . . . . . . . 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 185 . . . . . . . . . . . . . . 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 120 . . . . . . . . . . . . . 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 553 . . . . . . . . . . . . 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 6925 . . . . . . . . . . . . 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 2120 . . . . . . . . . . 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 280 . . . . . . . . . 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 2117 . . . . . . . . 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 1137 . . . . . . . 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 6930 . . . . . . . . . . 11 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑤N)) → ([⟨𝑥, 𝑦⟩] ~Q0 ·Q0 [⟨𝑧, 𝑤⟩] ~Q0 ) = [⟨(𝑥 ·𝑜 𝑧), (𝑦 ·𝑜 𝑤)⟩] ~Q0 )
132 mulnnnq0 6930 . . . . . . . . . . 11 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑣 ∈ ω ∧ 𝑢N)) → ([⟨𝑥, 𝑦⟩] ~Q0 ·Q0 [⟨𝑣, 𝑢⟩] ~Q0 ) = [⟨(𝑥 ·𝑜 𝑣), (𝑦 ·𝑜 𝑢)⟩] ~Q0 )
133131, 132oveqan12d 5613 . . . . . . . . . 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 6177 . . . . . . . . . . . . 13 ((𝑥 ∈ ω ∧ 𝑧 ∈ ω) → (𝑥 ·𝑜 𝑧) ∈ ω)
135 mulpiord 6797 . . . . . . . . . . . . . 14 ((𝑦N𝑤N) → (𝑦 ·N 𝑤) = (𝑦 ·𝑜 𝑤))
136 mulclpi 6808 . . . . . . . . . . . . . 14 ((𝑦N𝑤N) → (𝑦 ·N 𝑤) ∈ N)
137135, 136eqeltrrd 2162 . . . . . . . . . . . . 13 ((𝑦N𝑤N) → (𝑦 ·𝑜 𝑤) ∈ N)
138134, 137anim12i 331 . . . . . . . . . . . 12 (((𝑥 ∈ ω ∧ 𝑧 ∈ ω) ∧ (𝑦N𝑤N)) → ((𝑥 ·𝑜 𝑧) ∈ ω ∧ (𝑦 ·𝑜 𝑤) ∈ N))
139138an4s 553 . . . . . . . . . . 11 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑤N)) → ((𝑥 ·𝑜 𝑧) ∈ ω ∧ (𝑦 ·𝑜 𝑤) ∈ N))
140 nnmcl 6177 . . . . . . . . . . . . 13 ((𝑥 ∈ ω ∧ 𝑣 ∈ ω) → (𝑥 ·𝑜 𝑣) ∈ ω)
141 mulpiord 6797 . . . . . . . . . . . . . 14 ((𝑦N𝑢N) → (𝑦 ·N 𝑢) = (𝑦 ·𝑜 𝑢))
142 mulclpi 6808 . . . . . . . . . . . . . 14 ((𝑦N𝑢N) → (𝑦 ·N 𝑢) ∈ N)
143141, 142eqeltrrd 2162 . . . . . . . . . . . . 13 ((𝑦N𝑢N) → (𝑦 ·𝑜 𝑢) ∈ N)
144140, 143anim12i 331 . . . . . . . . . . . 12 (((𝑥 ∈ ω ∧ 𝑣 ∈ ω) ∧ (𝑦N𝑢N)) → ((𝑥 ·𝑜 𝑣) ∈ ω ∧ (𝑦 ·𝑜 𝑢) ∈ N))
145144an4s 553 . . . . . . . . . . 11 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑣 ∈ ω ∧ 𝑢N)) → ((𝑥 ·𝑜 𝑣) ∈ ω ∧ (𝑦 ·𝑜 𝑢) ∈ N))
146 addnnnq0 6929 . . . . . . . . . . 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 283 . . . . . . . . . 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 2117 . . . . . . . . 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 1227 . . . . . . . 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 2127 . . . . . . 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 1144 . . . . . 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 6312 . . . . 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 6313 . . 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 1139 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 102  wb 103  w3a 922   = wceq 1287  wcel 1436  cop 3428  ωcom 4371  (class class class)co 5594   +𝑜 coa 6113   ·𝑜 comu 6114  [cec 6223  Ncnpi 6752   ·N cmi 6754   ~Q0 ceq0 6766  Q0cnq0 6767   +Q0 cplq0 6769   ·Q0 cmq0 6770
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-coll 3922  ax-sep 3925  ax-nul 3933  ax-pow 3977  ax-pr 4003  ax-un 4227  ax-setind 4319  ax-iinf 4369
This theorem depends on definitions:  df-bi 115  df-dc 779  df-3or 923  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-ral 2360  df-rex 2361  df-reu 2362  df-rab 2364  df-v 2616  df-sbc 2829  df-csb 2922  df-dif 2988  df-un 2990  df-in 2992  df-ss 2999  df-nul 3273  df-pw 3411  df-sn 3431  df-pr 3432  df-op 3434  df-uni 3631  df-int 3666  df-iun 3709  df-br 3815  df-opab 3869  df-mpt 3870  df-tr 3905  df-id 4087  df-iord 4160  df-on 4162  df-suc 4165  df-iom 4372  df-xp 4410  df-rel 4411  df-cnv 4412  df-co 4413  df-dm 4414  df-rn 4415  df-res 4416  df-ima 4417  df-iota 4937  df-fun 4974  df-fn 4975  df-f 4976  df-f1 4977  df-fo 4978  df-f1o 4979  df-fv 4980  df-ov 5597  df-oprab 5598  df-mpt2 5599  df-1st 5849  df-2nd 5850  df-recs 6005  df-irdg 6070  df-oadd 6120  df-omul 6121  df-er 6225  df-ec 6227  df-qs 6231  df-ni 6784  df-mi 6786  df-enq0 6904  df-nq0 6905  df-plq0 6907  df-mq0 6908
This theorem is referenced by:  distnq0r  6943
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