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Theorem distrnq0 6649

Description: Multiplication of non-negative fractions is distributive. (Contributed by Jim Kingdon, 27-Nov-2019.)

**Assertion**
Ref	Expression
distrnq0	⊢ ((𝐴 ∈ Q₀ ∧ 𝐵 ∈ Q₀ ∧ 𝐶 ∈ Q₀) → (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 ℎ are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-nq0 6615	. . . 4 ⊢ Q₀ = ((ω × N) / ~_Q0 )
2		oveq1 5539	. . . . . . 7 ⊢ ([⟨𝑧, 𝑤⟩] ~_Q0 = 𝐵 → ([⟨𝑧, 𝑤⟩] ~_Q0 +_Q0 [⟨𝑣, 𝑢⟩] ~_Q0 ) = (𝐵 +_Q0 [⟨𝑣, 𝑢⟩] ~_Q0 ))
3	2	oveq2d 5548	. . . . . 6 ⊢ ([⟨𝑧, 𝑤⟩] ~_Q0 = 𝐵 → (𝐴 ·_Q0 ([⟨𝑧, 𝑤⟩] ~_Q0 +_Q0 [⟨𝑣, 𝑢⟩] ~_Q0 )) = (𝐴 ·_Q0 (𝐵 +_Q0 [⟨𝑣, 𝑢⟩] ~_Q0 )))
4		oveq2 5540	. . . . . . 7 ⊢ ([⟨𝑧, 𝑤⟩] ~_Q0 = 𝐵 → (𝐴 ·_Q0 [⟨𝑧, 𝑤⟩] ~_Q0 ) = (𝐴 ·_Q0 𝐵))
5	4	oveq1d 5547	. . . . . 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 )))
6	3, 5	eqeq12d 2095	. . . . 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 ))))
7	6	imbi2d 228	. . . 4 ⊢ ([⟨𝑧, 𝑤⟩] ~_Q0 = 𝐵 → ((𝐴 ∈ Q₀ → (𝐴 ·_Q0 ([⟨𝑧, 𝑤⟩] ~_Q0 +_Q0 [⟨𝑣, 𝑢⟩] ~_Q0 )) = ((𝐴 ·_Q0 [⟨𝑧, 𝑤⟩] ~_Q0 ) +_Q0 (𝐴 ·_Q0 [⟨𝑣, 𝑢⟩] ~_Q0 ))) ↔ (𝐴 ∈ Q₀ → (𝐴 ·_Q0 (𝐵 +_Q0 [⟨𝑣, 𝑢⟩] ~_Q0 )) = ((𝐴 ·_Q0 𝐵) +_Q0 (𝐴 ·_Q0 [⟨𝑣, 𝑢⟩] ~_Q0 )))))
8		oveq2 5540	. . . . . . 7 ⊢ ([⟨𝑣, 𝑢⟩] ~_Q0 = 𝐶 → (𝐵 +_Q0 [⟨𝑣, 𝑢⟩] ~_Q0 ) = (𝐵 +_Q0 𝐶))
9	8	oveq2d 5548	. . . . . 6 ⊢ ([⟨𝑣, 𝑢⟩] ~_Q0 = 𝐶 → (𝐴 ·_Q0 (𝐵 +_Q0 [⟨𝑣, 𝑢⟩] ~_Q0 )) = (𝐴 ·_Q0 (𝐵 +_Q0 𝐶)))
10		oveq2 5540	. . . . . . 7 ⊢ ([⟨𝑣, 𝑢⟩] ~_Q0 = 𝐶 → (𝐴 ·_Q0 [⟨𝑣, 𝑢⟩] ~_Q0 ) = (𝐴 ·_Q0 𝐶))
11	10	oveq2d 5548	. . . . . 6 ⊢ ([⟨𝑣, 𝑢⟩] ~_Q0 = 𝐶 → ((𝐴 ·_Q0 𝐵) +_Q0 (𝐴 ·_Q0 [⟨𝑣, 𝑢⟩] ~_Q0 )) = ((𝐴 ·_Q0 𝐵) +_Q0 (𝐴 ·_Q0 𝐶)))
12	9, 11	eqeq12d 2095	. . . . 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))))
13	12	imbi2d 228	. . . 4 ⊢ ([⟨𝑣, 𝑢⟩] ~_Q0 = 𝐶 → ((𝐴 ∈ Q₀ → (𝐴 ·_Q0 (𝐵 +_Q0 [⟨𝑣, 𝑢⟩] ~_Q0 )) = ((𝐴 ·_Q0 𝐵) +_Q0 (𝐴 ·_Q0 [⟨𝑣, 𝑢⟩] ~_Q0 ))) ↔ (𝐴 ∈ Q₀ → (𝐴 ·_Q0 (𝐵 +_Q0 𝐶)) = ((𝐴 ·_Q0 𝐵) +_Q0 (𝐴 ·_Q0 𝐶)))))
14		oveq1 5539	. . . . . . . 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 5539	. . . . . . . . 9 ⊢ ([⟨𝑥, 𝑦⟩] ~_Q0 = 𝐴 → ([⟨𝑥, 𝑦⟩] ~_Q0 ·_Q0 [⟨𝑧, 𝑤⟩] ~_Q0 ) = (𝐴 ·_Q0 [⟨𝑧, 𝑤⟩] ~_Q0 ))
16		oveq1 5539	. . . . . . . . 9 ⊢ ([⟨𝑥, 𝑦⟩] ~_Q0 = 𝐴 → ([⟨𝑥, 𝑦⟩] ~_Q0 ·_Q0 [⟨𝑣, 𝑢⟩] ~_Q0 ) = (𝐴 ·_Q0 [⟨𝑣, 𝑢⟩] ~_Q0 ))
17	15, 16	oveq12d 5550	. . . . . . . 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 )))
18	14, 17	eqeq12d 2095	. . . . . . 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 ))))
19	18	imbi2d 228	. . . . . 6 ⊢ ([⟨𝑥, 𝑦⟩] ~_Q0 = 𝐴 → ((((𝑧 ∈ ω ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ N)) → ([⟨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 ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ N)) → (𝐴 ·_Q0 ([⟨𝑧, 𝑤⟩] ~_Q0 +_Q0 [⟨𝑣, 𝑢⟩] ~_Q0 )) = ((𝐴 ·_Q0 [⟨𝑧, 𝑤⟩] ~_Q0 ) +_Q0 (𝐴 ·_Q0 [⟨𝑣, 𝑢⟩] ~_Q0 )))))
20		an42 551	. . . . . . . . . . . 12 ⊢ (((𝑧 ∈ ω ∧ 𝑢 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ ω)) ↔ ((𝑧 ∈ ω ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ N)))
21	20	anbi2i 444	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ ((𝑧 ∈ ω ∧ 𝑢 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ ω))) ↔ ((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ ((𝑧 ∈ ω ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ N))))
22		3anass 923	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑢 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ ω)) ↔ ((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ ((𝑧 ∈ ω ∧ 𝑢 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ ω))))
23		3anass 923	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ N)) ↔ ((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ ((𝑧 ∈ ω ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ N))))
24	21, 22, 23	3bitr4i 210	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑢 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ ω)) ↔ ((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ N)))
25		pinn 6499	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ N → 𝑦 ∈ ω)
26		nnmcl 6083	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ω ∧ 𝑥 ∈ ω) → (𝑦 ·_𝑜 𝑥) ∈ ω)
27	25, 26	sylan 277	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ N ∧ 𝑥 ∈ ω) → (𝑦 ·_𝑜 𝑥) ∈ ω)
28	27	ancoms 264	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ω ∧ 𝑦 ∈ N) → (𝑦 ·_𝑜 𝑥) ∈ ω)
29		pinn 6499	. . . . . . . . . . . . 13 ⊢ (𝑢 ∈ N → 𝑢 ∈ ω)
30		nnmcl 6083	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ ω ∧ 𝑢 ∈ ω) → (𝑧 ·_𝑜 𝑢) ∈ ω)
31	29, 30	sylan2 280	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ ω ∧ 𝑢 ∈ N) → (𝑧 ·_𝑜 𝑢) ∈ ω)
32		pinn 6499	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ N → 𝑤 ∈ ω)
33		nnmcl 6083	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ ω ∧ 𝑣 ∈ ω) → (𝑤 ·_𝑜 𝑣) ∈ ω)
34	32, 33	sylan 277	. . . . . . . . . . . 12 ⊢ ((𝑤 ∈ N ∧ 𝑣 ∈ ω) → (𝑤 ·_𝑜 𝑣) ∈ ω)
35		nndi 6088	. . . . . . . . . . . 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))))
36	28, 31, 34, 35	syl3an 1211	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑢 ∈ N) ∧ (𝑤 ∈ N ∧ 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))))
37		simp1r 963	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑢 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ ω)) → 𝑦 ∈ N)
38		simp1l 962	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑢 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ ω)) → 𝑥 ∈ ω)
39	31	3ad2ant2 960	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑢 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ ω)) → (𝑧 ·_𝑜 𝑢) ∈ ω)
40	34	3ad2ant3 961	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑢 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ ω)) → (𝑤 ·_𝑜 𝑣) ∈ ω)
41		nnacl 6082	. . . . . . . . . . . . 13 ⊢ (((𝑧 ·_𝑜 𝑢) ∈ ω ∧ (𝑤 ·_𝑜 𝑣) ∈ ω) → ((𝑧 ·_𝑜 𝑢) +_𝑜 (𝑤 ·_𝑜 𝑣)) ∈ ω)
42	39, 40, 41	syl2anc 403	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑢 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ ω)) → ((𝑧 ·_𝑜 𝑢) +_𝑜 (𝑤 ·_𝑜 𝑣)) ∈ ω)
43		nnmass 6089	. . . . . . . . . . . . 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)))))
44	25, 43	syl3an1 1202	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ N ∧ 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)))))
45	37, 38, 42, 44	syl3anc 1169	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑢 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ 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 6091	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → (𝑥 ·_𝑜 𝑦) = (𝑦 ·_𝑜 𝑥))
47	25, 46	sylan2 280	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ω ∧ 𝑦 ∈ N) → (𝑥 ·_𝑜 𝑦) = (𝑦 ·_𝑜 𝑥))
48	47	oveq1d 5547	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ω ∧ 𝑦 ∈ N) → ((𝑥 ·_𝑜 𝑦) ·_𝑜 (𝑧 ·_𝑜 𝑢)) = ((𝑦 ·_𝑜 𝑥) ·_𝑜 (𝑧 ·_𝑜 𝑢)))
49	48	adantr 270	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑢 ∈ N)) → ((𝑥 ·_𝑜 𝑦) ·_𝑜 (𝑧 ·_𝑜 𝑢)) = ((𝑦 ·_𝑜 𝑥) ·_𝑜 (𝑧 ·_𝑜 𝑢)))
50		simpll 495	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑢 ∈ N)) → 𝑥 ∈ ω)
51	25	ad2antlr 472	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑢 ∈ N)) → 𝑦 ∈ ω)
52		simprl 497	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑢 ∈ N)) → 𝑧 ∈ ω)
53		nnmcom 6091	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 ∈ ω ∧ 𝑔 ∈ ω) → (𝑓 ·_𝑜 𝑔) = (𝑔 ·_𝑜 𝑓))
54	53	adantl 271	. . . . . . . . . . . . . . 15 ⊢ ((((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑢 ∈ N)) ∧ (𝑓 ∈ ω ∧ 𝑔 ∈ ω)) → (𝑓 ·_𝑜 𝑔) = (𝑔 ·_𝑜 𝑓))
55		nnmass 6089	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 ∈ ω ∧ 𝑔 ∈ ω ∧ ℎ ∈ ω) → ((𝑓 ·_𝑜 𝑔) ·_𝑜 ℎ) = (𝑓 ·_𝑜 (𝑔 ·_𝑜 ℎ)))
56	55	adantl 271	. . . . . . . . . . . . . . 15 ⊢ ((((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑢 ∈ N)) ∧ (𝑓 ∈ ω ∧ 𝑔 ∈ ω ∧ ℎ ∈ ω)) → ((𝑓 ·_𝑜 𝑔) ·_𝑜 ℎ) = (𝑓 ·_𝑜 (𝑔 ·_𝑜 ℎ)))
57		simprr 498	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑢 ∈ N)) → 𝑢 ∈ N)
58	57, 29	syl 14	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑢 ∈ N)) → 𝑢 ∈ ω)
59		nnmcl 6083	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 ∈ ω ∧ 𝑔 ∈ ω) → (𝑓 ·_𝑜 𝑔) ∈ ω)
60	59	adantl 271	. . . . . . . . . . . . . . 15 ⊢ ((((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑢 ∈ N)) ∧ (𝑓 ∈ ω ∧ 𝑔 ∈ ω)) → (𝑓 ·_𝑜 𝑔) ∈ ω)
61	50, 51, 52, 54, 56, 58, 60	caov4d 5705	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑢 ∈ N)) → ((𝑥 ·_𝑜 𝑦) ·_𝑜 (𝑧 ·_𝑜 𝑢)) = ((𝑥 ·_𝑜 𝑧) ·_𝑜 (𝑦 ·_𝑜 𝑢)))
62	49, 61	eqtr3d 2115	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑢 ∈ N)) → ((𝑦 ·_𝑜 𝑥) ·_𝑜 (𝑧 ·_𝑜 𝑢)) = ((𝑥 ·_𝑜 𝑧) ·_𝑜 (𝑦 ·_𝑜 𝑢)))
63	62	3adant3 958	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑢 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ ω)) → ((𝑦 ·_𝑜 𝑥) ·_𝑜 (𝑧 ·_𝑜 𝑢)) = ((𝑥 ·_𝑜 𝑧) ·_𝑜 (𝑦 ·_𝑜 𝑢)))
64	25	ad2antlr 472	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ ω)) → 𝑦 ∈ ω)
65		simpll 495	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ ω)) → 𝑥 ∈ ω)
66		simprl 497	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ ω)) → 𝑤 ∈ N)
67	66, 32	syl 14	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ ω)) → 𝑤 ∈ ω)
68	53	adantl 271	. . . . . . . . . . . . . 14 ⊢ ((((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ ω)) ∧ (𝑓 ∈ ω ∧ 𝑔 ∈ ω)) → (𝑓 ·_𝑜 𝑔) = (𝑔 ·_𝑜 𝑓))
69	55	adantl 271	. . . . . . . . . . . . . 14 ⊢ ((((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ ω)) ∧ (𝑓 ∈ ω ∧ 𝑔 ∈ ω ∧ ℎ ∈ ω)) → ((𝑓 ·_𝑜 𝑔) ·_𝑜 ℎ) = (𝑓 ·_𝑜 (𝑔 ·_𝑜 ℎ)))
70		simprr 498	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ ω)) → 𝑣 ∈ ω)
71	59	adantl 271	. . . . . . . . . . . . . 14 ⊢ ((((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ ω)) ∧ (𝑓 ∈ ω ∧ 𝑔 ∈ ω)) → (𝑓 ·_𝑜 𝑔) ∈ ω)
72	64, 65, 67, 68, 69, 70, 71	caov4d 5705	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ ω)) → ((𝑦 ·_𝑜 𝑥) ·_𝑜 (𝑤 ·_𝑜 𝑣)) = ((𝑦 ·_𝑜 𝑤) ·_𝑜 (𝑥 ·_𝑜 𝑣)))
73	72	3adant2 957	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑢 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ ω)) → ((𝑦 ·_𝑜 𝑥) ·_𝑜 (𝑤 ·_𝑜 𝑣)) = ((𝑦 ·_𝑜 𝑤) ·_𝑜 (𝑥 ·_𝑜 𝑣)))
74	63, 73	oveq12d 5550	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑢 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ 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))))
75	36, 45, 74	3eqtr3d 2121	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑢 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ 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))))
76	24, 75	sylbir 133	. . . . . . . . 9 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ N)) → (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))))
77	37, 25	syl 14	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑢 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ ω)) → 𝑦 ∈ ω)
78		mulpiord 6507	. . . . . . . . . . . . . . . 16 ⊢ ((𝑤 ∈ N ∧ 𝑢 ∈ N) → (𝑤 ·_N 𝑢) = (𝑤 ·_𝑜 𝑢))
79	78	ancoms 264	. . . . . . . . . . . . . . 15 ⊢ ((𝑢 ∈ N ∧ 𝑤 ∈ N) → (𝑤 ·_N 𝑢) = (𝑤 ·_𝑜 𝑢))
80	79	ad2ant2lr 493	. . . . . . . . . . . . . 14 ⊢ (((𝑧 ∈ ω ∧ 𝑢 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ ω)) → (𝑤 ·_N 𝑢) = (𝑤 ·_𝑜 𝑢))
81	80	3adant1 956	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑢 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ ω)) → (𝑤 ·_N 𝑢) = (𝑤 ·_𝑜 𝑢))
82	66	3adant2 957	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑢 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ ω)) → 𝑤 ∈ N)
83	57	3adant3 958	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑢 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ ω)) → 𝑢 ∈ N)
84		mulclpi 6518	. . . . . . . . . . . . . . 15 ⊢ ((𝑤 ∈ N ∧ 𝑢 ∈ N) → (𝑤 ·_N 𝑢) ∈ N)
85	82, 83, 84	syl2anc 403	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑢 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ ω)) → (𝑤 ·_N 𝑢) ∈ N)
86		pinn 6499	. . . . . . . . . . . . . 14 ⊢ ((𝑤 ·_N 𝑢) ∈ N → (𝑤 ·_N 𝑢) ∈ ω)
87	85, 86	syl 14	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑢 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ ω)) → (𝑤 ·_N 𝑢) ∈ ω)
88	81, 87	eqeltrrd 2156	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑢 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ ω)) → (𝑤 ·_𝑜 𝑢) ∈ ω)
89		nnmass 6089	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ω ∧ 𝑦 ∈ ω ∧ (𝑤 ·_𝑜 𝑢) ∈ ω) → ((𝑦 ·_𝑜 𝑦) ·_𝑜 (𝑤 ·_𝑜 𝑢)) = (𝑦 ·_𝑜 (𝑦 ·_𝑜 (𝑤 ·_𝑜 𝑢))))
90	77, 77, 88, 89	syl3anc 1169	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑢 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ ω)) → ((𝑦 ·_𝑜 𝑦) ·_𝑜 (𝑤 ·_𝑜 𝑢)) = (𝑦 ·_𝑜 (𝑦 ·_𝑜 (𝑤 ·_𝑜 𝑢))))
91	82, 32	syl 14	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑢 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ ω)) → 𝑤 ∈ ω)
92	53	adantl 271	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑢 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ ω)) ∧ (𝑓 ∈ ω ∧ 𝑔 ∈ ω)) → (𝑓 ·_𝑜 𝑔) = (𝑔 ·_𝑜 𝑓))
93	55	adantl 271	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑢 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ ω)) ∧ (𝑓 ∈ ω ∧ 𝑔 ∈ ω ∧ ℎ ∈ ω)) → ((𝑓 ·_𝑜 𝑔) ·_𝑜 ℎ) = (𝑓 ·_𝑜 (𝑔 ·_𝑜 ℎ)))
94	83, 29	syl 14	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑢 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ ω)) → 𝑢 ∈ ω)
95	59	adantl 271	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑢 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ ω)) ∧ (𝑓 ∈ ω ∧ 𝑔 ∈ ω)) → (𝑓 ·_𝑜 𝑔) ∈ ω)
96	77, 77, 91, 92, 93, 94, 95	caov4d 5705	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑢 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ ω)) → ((𝑦 ·_𝑜 𝑦) ·_𝑜 (𝑤 ·_𝑜 𝑢)) = ((𝑦 ·_𝑜 𝑤) ·_𝑜 (𝑦 ·_𝑜 𝑢)))
97	90, 96	eqtr3d 2115	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑢 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ ω)) → (𝑦 ·_𝑜 (𝑦 ·_𝑜 (𝑤 ·_𝑜 𝑢))) = ((𝑦 ·_𝑜 𝑤) ·_𝑜 (𝑦 ·_𝑜 𝑢)))
98	24, 97	sylbir 133	. . . . . . . . 9 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ N)) → (𝑦 ·_𝑜 (𝑦 ·_𝑜 (𝑤 ·_𝑜 𝑢))) = ((𝑦 ·_𝑜 𝑤) ·_𝑜 (𝑦 ·_𝑜 𝑢)))
99		opeq12 3572	. . . . . . . . . 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))⟩)
100	99	eceq1d 6165	. . . . . . . . 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 )
101	76, 98, 100	syl2anc 403	. . . . . . . 8 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ ω ∧ 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 (𝑦 ·_𝑜 𝑢)) +_𝑜 ((𝑦 ·_𝑜 𝑤) ·_𝑜 (𝑥 ·_𝑜 𝑣))), ((𝑦 ·_𝑜 𝑤) ·_𝑜 (𝑦 ·_𝑜 𝑢))⟩] ~_Q0 )
102		addnnnq0 6639	. . . . . . . . . . . 12 ⊢ (((𝑧 ∈ ω ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ N)) → ([⟨𝑧, 𝑤⟩] ~_Q0 +_Q0 [⟨𝑣, 𝑢⟩] ~_Q0 ) = [⟨((𝑧 ·_𝑜 𝑢) +_𝑜 (𝑤 ·_𝑜 𝑣)), (𝑤 ·_𝑜 𝑢)⟩] ~_Q0 )
103	102	oveq2d 5548	. . . . . . . . . . 11 ⊢ (((𝑧 ∈ ω ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ N)) → ([⟨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 ))
104	103	adantl 271	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ ((𝑧 ∈ ω ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ N))) → ([⟨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 ))
105	31, 34, 41	syl2an 283	. . . . . . . . . . . . 13 ⊢ (((𝑧 ∈ ω ∧ 𝑢 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ ω)) → ((𝑧 ·_𝑜 𝑢) +_𝑜 (𝑤 ·_𝑜 𝑣)) ∈ ω)
106	105	an42s 553	. . . . . . . . . . . 12 ⊢ (((𝑧 ∈ ω ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ N)) → ((𝑧 ·_𝑜 𝑢) +_𝑜 (𝑤 ·_𝑜 𝑣)) ∈ ω)
107	84	ad2ant2l 491	. . . . . . . . . . . . 13 ⊢ (((𝑧 ∈ ω ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ N)) → (𝑤 ·_N 𝑢) ∈ N)
108	78	eleq1d 2147	. . . . . . . . . . . . . 14 ⊢ ((𝑤 ∈ N ∧ 𝑢 ∈ N) → ((𝑤 ·_N 𝑢) ∈ N ↔ (𝑤 ·_𝑜 𝑢) ∈ N))
109	108	ad2ant2l 491	. . . . . . . . . . . . 13 ⊢ (((𝑧 ∈ ω ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ N)) → ((𝑤 ·_N 𝑢) ∈ N ↔ (𝑤 ·_𝑜 𝑢) ∈ N))
110	107, 109	mpbid 145	. . . . . . . . . . . 12 ⊢ (((𝑧 ∈ ω ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ N)) → (𝑤 ·_𝑜 𝑢) ∈ N)
111	106, 110	jca 300	. . . . . . . . . . 11 ⊢ (((𝑧 ∈ ω ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ N)) → (((𝑧 ·_𝑜 𝑢) +_𝑜 (𝑤 ·_𝑜 𝑣)) ∈ ω ∧ (𝑤 ·_𝑜 𝑢) ∈ N))
112		mulnnnq0 6640	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (((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 6083	. . . . . . . . . . . . . . . 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 ⊢ ((𝑦 ∈ N ∧ (𝑤 ·_𝑜 𝑢) ∈ N) → 𝑦 ∈ N)
115		mulpiord 6507	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ N ∧ (𝑤 ·_𝑜 𝑢) ∈ N) → (𝑦 ·_N (𝑤 ·_𝑜 𝑢)) = (𝑦 ·_𝑜 (𝑤 ·_𝑜 𝑢)))
116		mulclpi 6518	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ N ∧ (𝑤 ·_𝑜 𝑢) ∈ N) → (𝑦 ·_N (𝑤 ·_𝑜 𝑢)) ∈ N)
117	115, 116	eqeltrrd 2156	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ N ∧ (𝑤 ·_𝑜 𝑢) ∈ N) → (𝑦 ·_𝑜 (𝑤 ·_𝑜 𝑢)) ∈ N)
118	114, 117	jca 300	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ N ∧ (𝑤 ·_𝑜 𝑢) ∈ N) → (𝑦 ∈ N ∧ (𝑦 ·_𝑜 (𝑤 ·_𝑜 𝑢)) ∈ N))
119	113, 118	anim12i 331	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ω ∧ ((𝑧 ·_𝑜 𝑢) +_𝑜 (𝑤 ·_𝑜 𝑣)) ∈ ω) ∧ (𝑦 ∈ N ∧ (𝑤 ·_𝑜 𝑢) ∈ N)) → ((𝑥 ·_𝑜 ((𝑧 ·_𝑜 𝑢) +_𝑜 (𝑤 ·_𝑜 𝑣))) ∈ ω ∧ (𝑦 ∈ N ∧ (𝑦 ·_𝑜 (𝑤 ·_𝑜 𝑢)) ∈ N)))
120		an12 525	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ·_𝑜 ((𝑧 ·_𝑜 𝑢) +_𝑜 (𝑤 ·_𝑜 𝑣))) ∈ ω ∧ (𝑦 ∈ N ∧ (𝑦 ·_𝑜 (𝑤 ·_𝑜 𝑢)) ∈ N)) ↔ (𝑦 ∈ N ∧ ((𝑥 ·_𝑜 ((𝑧 ·_𝑜 𝑢) +_𝑜 (𝑤 ·_𝑜 𝑣))) ∈ ω ∧ (𝑦 ·_𝑜 (𝑤 ·_𝑜 𝑢)) ∈ N)))
121		3anass 923	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ N ∧ (𝑥 ·_𝑜 ((𝑧 ·_𝑜 𝑢) +_𝑜 (𝑤 ·_𝑜 𝑣))) ∈ ω ∧ (𝑦 ·_𝑜 (𝑤 ·_𝑜 𝑢)) ∈ N) ↔ (𝑦 ∈ N ∧ ((𝑥 ·_𝑜 ((𝑧 ·_𝑜 𝑢) +_𝑜 (𝑤 ·_𝑜 𝑣))) ∈ ω ∧ (𝑦 ·_𝑜 (𝑤 ·_𝑜 𝑢)) ∈ N)))
122	120, 121	bitr4i 185	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ·_𝑜 ((𝑧 ·_𝑜 𝑢) +_𝑜 (𝑤 ·_𝑜 𝑣))) ∈ ω ∧ (𝑦 ∈ N ∧ (𝑦 ·_𝑜 (𝑤 ·_𝑜 𝑢)) ∈ N)) ↔ (𝑦 ∈ N ∧ (𝑥 ·_𝑜 ((𝑧 ·_𝑜 𝑢) +_𝑜 (𝑤 ·_𝑜 𝑣))) ∈ ω ∧ (𝑦 ·_𝑜 (𝑤 ·_𝑜 𝑢)) ∈ N))
123	119, 122	sylib 120	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ω ∧ ((𝑧 ·_𝑜 𝑢) +_𝑜 (𝑤 ·_𝑜 𝑣)) ∈ ω) ∧ (𝑦 ∈ N ∧ (𝑤 ·_𝑜 𝑢) ∈ N)) → (𝑦 ∈ N ∧ (𝑥 ·_𝑜 ((𝑧 ·_𝑜 𝑢) +_𝑜 (𝑤 ·_𝑜 𝑣))) ∈ ω ∧ (𝑦 ·_𝑜 (𝑤 ·_𝑜 𝑢)) ∈ N))
124	123	an4s 552	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (((𝑧 ·_𝑜 𝑢) +_𝑜 (𝑤 ·_𝑜 𝑣)) ∈ ω ∧ (𝑤 ·_𝑜 𝑢) ∈ N)) → (𝑦 ∈ N ∧ (𝑥 ·_𝑜 ((𝑧 ·_𝑜 𝑢) +_𝑜 (𝑤 ·_𝑜 𝑣))) ∈ ω ∧ (𝑦 ·_𝑜 (𝑤 ·_𝑜 𝑢)) ∈ N))
125		mulcanenq0ec 6635	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ N ∧ (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 )
126	124, 125	syl 14	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (((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 )
127	112, 126	eqtr4d 2116	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (((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 )
128	111, 127	sylan2 280	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ ((𝑧 ∈ ω ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ ω ∧ 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 )
129	104, 128	eqtrd 2113	. . . . . . . . 9 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ ((𝑧 ∈ ω ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ N))) → ([⟨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 )
130	129	3impb 1134	. . . . . . . 8 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ N)) → ([⟨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 6640	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑤 ∈ N)) → ([⟨𝑥, 𝑦⟩] ~_Q0 ·_Q0 [⟨𝑧, 𝑤⟩] ~_Q0 ) = [⟨(𝑥 ·_𝑜 𝑧), (𝑦 ·_𝑜 𝑤)⟩] ~_Q0 )
132		mulnnnq0 6640	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ N)) → ([⟨𝑥, 𝑦⟩] ~_Q0 ·_Q0 [⟨𝑣, 𝑢⟩] ~_Q0 ) = [⟨(𝑥 ·_𝑜 𝑣), (𝑦 ·_𝑜 𝑢)⟩] ~_Q0 )
133	131, 132	oveqan12d 5551	. . . . . . . . . 10 ⊢ ((((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑤 ∈ N)) ∧ ((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ N))) → (([⟨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 6083	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ω ∧ 𝑧 ∈ ω) → (𝑥 ·_𝑜 𝑧) ∈ ω)
135		mulpiord 6507	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ N ∧ 𝑤 ∈ N) → (𝑦 ·_N 𝑤) = (𝑦 ·_𝑜 𝑤))
136		mulclpi 6518	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ N ∧ 𝑤 ∈ N) → (𝑦 ·_N 𝑤) ∈ N)
137	135, 136	eqeltrrd 2156	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ N ∧ 𝑤 ∈ N) → (𝑦 ·_𝑜 𝑤) ∈ N)
138	134, 137	anim12i 331	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ω ∧ 𝑧 ∈ ω) ∧ (𝑦 ∈ N ∧ 𝑤 ∈ N)) → ((𝑥 ·_𝑜 𝑧) ∈ ω ∧ (𝑦 ·_𝑜 𝑤) ∈ N))
139	138	an4s 552	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑤 ∈ N)) → ((𝑥 ·_𝑜 𝑧) ∈ ω ∧ (𝑦 ·_𝑜 𝑤) ∈ N))
140		nnmcl 6083	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ω ∧ 𝑣 ∈ ω) → (𝑥 ·_𝑜 𝑣) ∈ ω)
141		mulpiord 6507	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ N ∧ 𝑢 ∈ N) → (𝑦 ·_N 𝑢) = (𝑦 ·_𝑜 𝑢))
142		mulclpi 6518	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ N ∧ 𝑢 ∈ N) → (𝑦 ·_N 𝑢) ∈ N)
143	141, 142	eqeltrrd 2156	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ N ∧ 𝑢 ∈ N) → (𝑦 ·_𝑜 𝑢) ∈ N)
144	140, 143	anim12i 331	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ω ∧ 𝑣 ∈ ω) ∧ (𝑦 ∈ N ∧ 𝑢 ∈ N)) → ((𝑥 ·_𝑜 𝑣) ∈ ω ∧ (𝑦 ·_𝑜 𝑢) ∈ N))
145	144	an4s 552	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ N)) → ((𝑥 ·_𝑜 𝑣) ∈ ω ∧ (𝑦 ·_𝑜 𝑢) ∈ N))
146		addnnnq0 6639	. . . . . . . . . . 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 )
147	139, 145, 146	syl2an 283	. . . . . . . . . 10 ⊢ ((((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑤 ∈ N)) ∧ ((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑣 ∈ ω ∧ 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 )
148	133, 147	eqtrd 2113	. . . . . . . . 9 ⊢ ((((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑤 ∈ N)) ∧ ((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ N))) → (([⟨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 )
149	148	3impdi 1224	. . . . . . . 8 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ N)) → (([⟨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 )
150	101, 130, 149	3eqtr4d 2123	. . . . . . 7 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ N)) → ([⟨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 )))
151	150	3expib 1141	. . . . . 6 ⊢ ((𝑥 ∈ ω ∧ 𝑦 ∈ N) → (((𝑧 ∈ ω ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ N)) → ([⟨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 ))))
152	1, 19, 151	ecoptocl 6216	. . . . 5 ⊢ (𝐴 ∈ Q₀ → (((𝑧 ∈ ω ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ N)) → (𝐴 ·_Q0 ([⟨𝑧, 𝑤⟩] ~_Q0 +_Q0 [⟨𝑣, 𝑢⟩] ~_Q0 )) = ((𝐴 ·_Q0 [⟨𝑧, 𝑤⟩] ~_Q0 ) +_Q0 (𝐴 ·_Q0 [⟨𝑣, 𝑢⟩] ~_Q0 ))))
153	152	com12 30	. . . 4 ⊢ (((𝑧 ∈ ω ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ N)) → (𝐴 ∈ Q₀ → (𝐴 ·_Q0 ([⟨𝑧, 𝑤⟩] ~_Q0 +_Q0 [⟨𝑣, 𝑢⟩] ~_Q0 )) = ((𝐴 ·_Q0 [⟨𝑧, 𝑤⟩] ~_Q0 ) +_Q0 (𝐴 ·_Q0 [⟨𝑣, 𝑢⟩] ~_Q0 ))))
154	1, 7, 13, 153	2ecoptocl 6217	. . 3 ⊢ ((𝐵 ∈ Q₀ ∧ 𝐶 ∈ Q₀) → (𝐴 ∈ Q₀ → (𝐴 ·_Q0 (𝐵 +_Q0 𝐶)) = ((𝐴 ·_Q0 𝐵) +_Q0 (𝐴 ·_Q0 𝐶))))
155	154	com12 30	. 2 ⊢ (𝐴 ∈ Q₀ → ((𝐵 ∈ Q₀ ∧ 𝐶 ∈ Q₀) → (𝐴 ·_Q0 (𝐵 +_Q0 𝐶)) = ((𝐴 ·_Q0 𝐵) +_Q0 (𝐴 ·_Q0 𝐶))))
156	155	3impib 1136	1 ⊢ ((𝐴 ∈ Q₀ ∧ 𝐵 ∈ Q₀ ∧ 𝐶 ∈ Q₀) → (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 919 = wceq 1284 ∈ wcel 1433 ⟨cop 3401 ωcom 4331 (class class class)co 5532 +_𝑜 coa 6021 ·_𝑜 comu 6022 [cec 6127 Ncnpi 6462 ·_N cmi 6464 ~_Q0 ceq0 6476 Q₀cnq0 6477 +_Q0 cplq0 6479 ·_Q0 cmq0 6480

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 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-coll 3893 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-iinf 4329

This theorem depends on definitions: df-bi 115 df-dc 776 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-tr 3876 df-id 4048 df-iord 4121 df-on 4123 df-suc 4126 df-iom 4332 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1st 5787 df-2nd 5788 df-recs 5943 df-irdg 5980 df-oadd 6028 df-omul 6029 df-er 6129 df-ec 6131 df-qs 6135 df-ni 6494 df-mi 6496 df-enq0 6614 df-nq0 6615 df-plq0 6617 df-mq0 6618

This theorem is referenced by: distnq0r 6653

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