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Theorem addassnq0 6652

Description: Addition of non-negaative fractions is associative. (Contributed by Jim Kingdon, 29-Nov-2019.)

**Assertion**
Ref	Expression
addassnq0	⊢ ((𝐴 ∈ Q₀ ∧ 𝐵 ∈ Q₀ ∧ 𝐶 ∈ Q₀) → ((𝐴 +_Q0 𝐵) +_Q0 𝐶) = (𝐴 +_Q0 (𝐵 +_Q0 𝐶)))

Proof of Theorem addassnq0 Dummy variables 𝑥 𝑦 𝑧 𝑤 𝑣 𝑢 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		oveq2 5540	. . . . . . 7 ⊢ ([⟨𝑧, 𝑤⟩] ~_Q0 = 𝐵 → (𝐴 +_Q0 [⟨𝑧, 𝑤⟩] ~_Q0 ) = (𝐴 +_Q0 𝐵))
3	2	oveq1d 5547	. . . . . 6 ⊢ ([⟨𝑧, 𝑤⟩] ~_Q0 = 𝐵 → ((𝐴 +_Q0 [⟨𝑧, 𝑤⟩] ~_Q0 ) +_Q0 [⟨𝑣, 𝑢⟩] ~_Q0 ) = ((𝐴 +_Q0 𝐵) +_Q0 [⟨𝑣, 𝑢⟩] ~_Q0 ))
4		oveq1 5539	. . . . . . 7 ⊢ ([⟨𝑧, 𝑤⟩] ~_Q0 = 𝐵 → ([⟨𝑧, 𝑤⟩] ~_Q0 +_Q0 [⟨𝑣, 𝑢⟩] ~_Q0 ) = (𝐵 +_Q0 [⟨𝑣, 𝑢⟩] ~_Q0 ))
5	4	oveq2d 5548	. . . . . 6 ⊢ ([⟨𝑧, 𝑤⟩] ~_Q0 = 𝐵 → (𝐴 +_Q0 ([⟨𝑧, 𝑤⟩] ~_Q0 +_Q0 [⟨𝑣, 𝑢⟩] ~_Q0 )) = (𝐴 +_Q0 (𝐵 +_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 [⟨𝑣, 𝑢⟩] ~_Q0 )) ↔ ((𝐴 +_Q0 𝐵) +_Q0 [⟨𝑣, 𝑢⟩] ~_Q0 ) = (𝐴 +_Q0 (𝐵 +_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 [⟨𝑣, 𝑢⟩] ~_Q0 ))) ↔ (𝐴 ∈ Q₀ → ((𝐴 +_Q0 𝐵) +_Q0 [⟨𝑣, 𝑢⟩] ~_Q0 ) = (𝐴 +_Q0 (𝐵 +_Q0 [⟨𝑣, 𝑢⟩] ~_Q0 )))))
8		oveq2 5540	. . . . . 6 ⊢ ([⟨𝑣, 𝑢⟩] ~_Q0 = 𝐶 → ((𝐴 +_Q0 𝐵) +_Q0 [⟨𝑣, 𝑢⟩] ~_Q0 ) = ((𝐴 +_Q0 𝐵) +_Q0 𝐶))
9		oveq2 5540	. . . . . . 7 ⊢ ([⟨𝑣, 𝑢⟩] ~_Q0 = 𝐶 → (𝐵 +_Q0 [⟨𝑣, 𝑢⟩] ~_Q0 ) = (𝐵 +_Q0 𝐶))
10	9	oveq2d 5548	. . . . . 6 ⊢ ([⟨𝑣, 𝑢⟩] ~_Q0 = 𝐶 → (𝐴 +_Q0 (𝐵 +_Q0 [⟨𝑣, 𝑢⟩] ~_Q0 )) = (𝐴 +_Q0 (𝐵 +_Q0 𝐶)))
11	8, 10	eqeq12d 2095	. . . . 5 ⊢ ([⟨𝑣, 𝑢⟩] ~_Q0 = 𝐶 → (((𝐴 +_Q0 𝐵) +_Q0 [⟨𝑣, 𝑢⟩] ~_Q0 ) = (𝐴 +_Q0 (𝐵 +_Q0 [⟨𝑣, 𝑢⟩] ~_Q0 )) ↔ ((𝐴 +_Q0 𝐵) +_Q0 𝐶) = (𝐴 +_Q0 (𝐵 +_Q0 𝐶))))
12	11	imbi2d 228	. . . 4 ⊢ ([⟨𝑣, 𝑢⟩] ~_Q0 = 𝐶 → ((𝐴 ∈ Q₀ → ((𝐴 +_Q0 𝐵) +_Q0 [⟨𝑣, 𝑢⟩] ~_Q0 ) = (𝐴 +_Q0 (𝐵 +_Q0 [⟨𝑣, 𝑢⟩] ~_Q0 ))) ↔ (𝐴 ∈ Q₀ → ((𝐴 +_Q0 𝐵) +_Q0 𝐶) = (𝐴 +_Q0 (𝐵 +_Q0 𝐶)))))
13		oveq1 5539	. . . . . . . . 9 ⊢ ([⟨𝑥, 𝑦⟩] ~_Q0 = 𝐴 → ([⟨𝑥, 𝑦⟩] ~_Q0 +_Q0 [⟨𝑧, 𝑤⟩] ~_Q0 ) = (𝐴 +_Q0 [⟨𝑧, 𝑤⟩] ~_Q0 ))
14	13	oveq1d 5547	. . . . . . . 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	. . . . . . . 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 )))
16	14, 15	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 [⟨𝑣, 𝑢⟩] ~_Q0 )) ↔ ((𝐴 +_Q0 [⟨𝑧, 𝑤⟩] ~_Q0 ) +_Q0 [⟨𝑣, 𝑢⟩] ~_Q0 ) = (𝐴 +_Q0 ([⟨𝑧, 𝑤⟩] ~_Q0 +_Q0 [⟨𝑣, 𝑢⟩] ~_Q0 ))))
17	16	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 [⟨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 [⟨𝑣, 𝑢⟩] ~_Q0 )))))
18		simp1l 962	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ N)) → 𝑥 ∈ ω)
19		simp2r 965	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ N)) → 𝑤 ∈ N)
20		pinn 6499	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ N → 𝑤 ∈ ω)
21	19, 20	syl 14	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ N)) → 𝑤 ∈ ω)
22		simp3r 967	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ N)) → 𝑢 ∈ N)
23		pinn 6499	. . . . . . . . . . . . . 14 ⊢ (𝑢 ∈ N → 𝑢 ∈ ω)
24	22, 23	syl 14	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ N)) → 𝑢 ∈ ω)
25		nnmcl 6083	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ ω ∧ 𝑢 ∈ ω) → (𝑤 ·_𝑜 𝑢) ∈ ω)
26	21, 24, 25	syl2anc 403	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ N)) → (𝑤 ·_𝑜 𝑢) ∈ ω)
27		nnmcl 6083	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ω ∧ (𝑤 ·_𝑜 𝑢) ∈ ω) → (𝑥 ·_𝑜 (𝑤 ·_𝑜 𝑢)) ∈ ω)
28	18, 26, 27	syl2anc 403	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ N)) → (𝑥 ·_𝑜 (𝑤 ·_𝑜 𝑢)) ∈ ω)
29		simp1r 963	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ N)) → 𝑦 ∈ N)
30		pinn 6499	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ N → 𝑦 ∈ ω)
31	29, 30	syl 14	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ N)) → 𝑦 ∈ ω)
32		simp2l 964	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ N)) → 𝑧 ∈ ω)
33		nnmcl 6083	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ ω ∧ 𝑢 ∈ ω) → (𝑧 ·_𝑜 𝑢) ∈ ω)
34	32, 24, 33	syl2anc 403	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ N)) → (𝑧 ·_𝑜 𝑢) ∈ ω)
35		nnmcl 6083	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ω ∧ (𝑧 ·_𝑜 𝑢) ∈ ω) → (𝑦 ·_𝑜 (𝑧 ·_𝑜 𝑢)) ∈ ω)
36	31, 34, 35	syl2anc 403	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ N)) → (𝑦 ·_𝑜 (𝑧 ·_𝑜 𝑢)) ∈ ω)
37		simp3l 966	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ N)) → 𝑣 ∈ ω)
38		nnmcl 6083	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ ω ∧ 𝑣 ∈ ω) → (𝑤 ·_𝑜 𝑣) ∈ ω)
39	21, 37, 38	syl2anc 403	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ N)) → (𝑤 ·_𝑜 𝑣) ∈ ω)
40		nnmcl 6083	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ω ∧ (𝑤 ·_𝑜 𝑣) ∈ ω) → (𝑦 ·_𝑜 (𝑤 ·_𝑜 𝑣)) ∈ ω)
41	31, 39, 40	syl2anc 403	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ N)) → (𝑦 ·_𝑜 (𝑤 ·_𝑜 𝑣)) ∈ ω)
42		nnaass 6087	. . . . . . . . . . 11 ⊢ (((𝑥 ·_𝑜 (𝑤 ·_𝑜 𝑢)) ∈ ω ∧ (𝑦 ·_𝑜 (𝑧 ·_𝑜 𝑢)) ∈ ω ∧ (𝑦 ·_𝑜 (𝑤 ·_𝑜 𝑣)) ∈ ω) → (((𝑥 ·_𝑜 (𝑤 ·_𝑜 𝑢)) +_𝑜 (𝑦 ·_𝑜 (𝑧 ·_𝑜 𝑢))) +_𝑜 (𝑦 ·_𝑜 (𝑤 ·_𝑜 𝑣))) = ((𝑥 ·_𝑜 (𝑤 ·_𝑜 𝑢)) +_𝑜 ((𝑦 ·_𝑜 (𝑧 ·_𝑜 𝑢)) +_𝑜 (𝑦 ·_𝑜 (𝑤 ·_𝑜 𝑣)))))
43	28, 36, 41, 42	syl3anc 1169	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ N)) → (((𝑥 ·_𝑜 (𝑤 ·_𝑜 𝑢)) +_𝑜 (𝑦 ·_𝑜 (𝑧 ·_𝑜 𝑢))) +_𝑜 (𝑦 ·_𝑜 (𝑤 ·_𝑜 𝑣))) = ((𝑥 ·_𝑜 (𝑤 ·_𝑜 𝑢)) +_𝑜 ((𝑦 ·_𝑜 (𝑧 ·_𝑜 𝑢)) +_𝑜 (𝑦 ·_𝑜 (𝑤 ·_𝑜 𝑣)))))
44		nnmcom 6091	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∈ ω ∧ 𝑔 ∈ ω) → (𝑓 ·_𝑜 𝑔) = (𝑔 ·_𝑜 𝑓))
45	44	adantl 271	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ N)) ∧ (𝑓 ∈ ω ∧ 𝑔 ∈ ω)) → (𝑓 ·_𝑜 𝑔) = (𝑔 ·_𝑜 𝑓))
46		nndir 6092	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∈ ω ∧ 𝑔 ∈ ω ∧ ℎ ∈ ω) → ((𝑓 +_𝑜 𝑔) ·_𝑜 ℎ) = ((𝑓 ·_𝑜 ℎ) +_𝑜 (𝑔 ·_𝑜 ℎ)))
47	46	adantl 271	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ N)) ∧ (𝑓 ∈ ω ∧ 𝑔 ∈ ω ∧ ℎ ∈ ω)) → ((𝑓 +_𝑜 𝑔) ·_𝑜 ℎ) = ((𝑓 ·_𝑜 ℎ) +_𝑜 (𝑔 ·_𝑜 ℎ)))
48		nnmass 6089	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∈ ω ∧ 𝑔 ∈ ω ∧ ℎ ∈ ω) → ((𝑓 ·_𝑜 𝑔) ·_𝑜 ℎ) = (𝑓 ·_𝑜 (𝑔 ·_𝑜 ℎ)))
49	48	adantl 271	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ N)) ∧ (𝑓 ∈ ω ∧ 𝑔 ∈ ω ∧ ℎ ∈ ω)) → ((𝑓 ·_𝑜 𝑔) ·_𝑜 ℎ) = (𝑓 ·_𝑜 (𝑔 ·_𝑜 ℎ)))
50		nnmcl 6083	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∈ ω ∧ 𝑔 ∈ ω) → (𝑓 ·_𝑜 𝑔) ∈ ω)
51	50	adantl 271	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ N)) ∧ (𝑓 ∈ ω ∧ 𝑔 ∈ ω)) → (𝑓 ·_𝑜 𝑔) ∈ ω)
52	45, 47, 49, 51, 18, 31, 21, 32, 24	caovdilemd 5712	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ N)) → (((𝑥 ·_𝑜 𝑤) +_𝑜 (𝑦 ·_𝑜 𝑧)) ·_𝑜 𝑢) = ((𝑥 ·_𝑜 (𝑤 ·_𝑜 𝑢)) +_𝑜 (𝑦 ·_𝑜 (𝑧 ·_𝑜 𝑢))))
53		nnmass 6089	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ω ∧ 𝑤 ∈ ω ∧ 𝑣 ∈ ω) → ((𝑦 ·_𝑜 𝑤) ·_𝑜 𝑣) = (𝑦 ·_𝑜 (𝑤 ·_𝑜 𝑣)))
54	31, 21, 37, 53	syl3anc 1169	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ N)) → ((𝑦 ·_𝑜 𝑤) ·_𝑜 𝑣) = (𝑦 ·_𝑜 (𝑤 ·_𝑜 𝑣)))
55	52, 54	oveq12d 5550	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ N)) → ((((𝑥 ·_𝑜 𝑤) +_𝑜 (𝑦 ·_𝑜 𝑧)) ·_𝑜 𝑢) +_𝑜 ((𝑦 ·_𝑜 𝑤) ·_𝑜 𝑣)) = (((𝑥 ·_𝑜 (𝑤 ·_𝑜 𝑢)) +_𝑜 (𝑦 ·_𝑜 (𝑧 ·_𝑜 𝑢))) +_𝑜 (𝑦 ·_𝑜 (𝑤 ·_𝑜 𝑣))))
56		nndi 6088	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ω ∧ (𝑧 ·_𝑜 𝑢) ∈ ω ∧ (𝑤 ·_𝑜 𝑣) ∈ ω) → (𝑦 ·_𝑜 ((𝑧 ·_𝑜 𝑢) +_𝑜 (𝑤 ·_𝑜 𝑣))) = ((𝑦 ·_𝑜 (𝑧 ·_𝑜 𝑢)) +_𝑜 (𝑦 ·_𝑜 (𝑤 ·_𝑜 𝑣))))
57	31, 34, 39, 56	syl3anc 1169	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ N)) → (𝑦 ·_𝑜 ((𝑧 ·_𝑜 𝑢) +_𝑜 (𝑤 ·_𝑜 𝑣))) = ((𝑦 ·_𝑜 (𝑧 ·_𝑜 𝑢)) +_𝑜 (𝑦 ·_𝑜 (𝑤 ·_𝑜 𝑣))))
58	57	oveq2d 5548	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ N)) → ((𝑥 ·_𝑜 (𝑤 ·_𝑜 𝑢)) +_𝑜 (𝑦 ·_𝑜 ((𝑧 ·_𝑜 𝑢) +_𝑜 (𝑤 ·_𝑜 𝑣)))) = ((𝑥 ·_𝑜 (𝑤 ·_𝑜 𝑢)) +_𝑜 ((𝑦 ·_𝑜 (𝑧 ·_𝑜 𝑢)) +_𝑜 (𝑦 ·_𝑜 (𝑤 ·_𝑜 𝑣)))))
59	43, 55, 58	3eqtr4d 2123	. . . . . . . . 9 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ N)) → ((((𝑥 ·_𝑜 𝑤) +_𝑜 (𝑦 ·_𝑜 𝑧)) ·_𝑜 𝑢) +_𝑜 ((𝑦 ·_𝑜 𝑤) ·_𝑜 𝑣)) = ((𝑥 ·_𝑜 (𝑤 ·_𝑜 𝑢)) +_𝑜 (𝑦 ·_𝑜 ((𝑧 ·_𝑜 𝑢) +_𝑜 (𝑤 ·_𝑜 𝑣)))))
60		nnmass 6089	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ω ∧ 𝑤 ∈ ω ∧ 𝑢 ∈ ω) → ((𝑦 ·_𝑜 𝑤) ·_𝑜 𝑢) = (𝑦 ·_𝑜 (𝑤 ·_𝑜 𝑢)))
61	31, 21, 24, 60	syl3anc 1169	. . . . . . . . 9 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ N)) → ((𝑦 ·_𝑜 𝑤) ·_𝑜 𝑢) = (𝑦 ·_𝑜 (𝑤 ·_𝑜 𝑢)))
62		opeq12 3572	. . . . . . . . . 10 ⊢ ((((((𝑥 ·_𝑜 𝑤) +_𝑜 (𝑦 ·_𝑜 𝑧)) ·_𝑜 𝑢) +_𝑜 ((𝑦 ·_𝑜 𝑤) ·_𝑜 𝑣)) = ((𝑥 ·_𝑜 (𝑤 ·_𝑜 𝑢)) +_𝑜 (𝑦 ·_𝑜 ((𝑧 ·_𝑜 𝑢) +_𝑜 (𝑤 ·_𝑜 𝑣)))) ∧ ((𝑦 ·_𝑜 𝑤) ·_𝑜 𝑢) = (𝑦 ·_𝑜 (𝑤 ·_𝑜 𝑢))) → ⟨((((𝑥 ·_𝑜 𝑤) +_𝑜 (𝑦 ·_𝑜 𝑧)) ·_𝑜 𝑢) +_𝑜 ((𝑦 ·_𝑜 𝑤) ·_𝑜 𝑣)), ((𝑦 ·_𝑜 𝑤) ·_𝑜 𝑢)⟩ = ⟨((𝑥 ·_𝑜 (𝑤 ·_𝑜 𝑢)) +_𝑜 (𝑦 ·_𝑜 ((𝑧 ·_𝑜 𝑢) +_𝑜 (𝑤 ·_𝑜 𝑣)))), (𝑦 ·_𝑜 (𝑤 ·_𝑜 𝑢))⟩)
63	62	eceq1d 6165	. . . . . . . . 9 ⊢ ((((((𝑥 ·_𝑜 𝑤) +_𝑜 (𝑦 ·_𝑜 𝑧)) ·_𝑜 𝑢) +_𝑜 ((𝑦 ·_𝑜 𝑤) ·_𝑜 𝑣)) = ((𝑥 ·_𝑜 (𝑤 ·_𝑜 𝑢)) +_𝑜 (𝑦 ·_𝑜 ((𝑧 ·_𝑜 𝑢) +_𝑜 (𝑤 ·_𝑜 𝑣)))) ∧ ((𝑦 ·_𝑜 𝑤) ·_𝑜 𝑢) = (𝑦 ·_𝑜 (𝑤 ·_𝑜 𝑢))) → [⟨((((𝑥 ·_𝑜 𝑤) +_𝑜 (𝑦 ·_𝑜 𝑧)) ·_𝑜 𝑢) +_𝑜 ((𝑦 ·_𝑜 𝑤) ·_𝑜 𝑣)), ((𝑦 ·_𝑜 𝑤) ·_𝑜 𝑢)⟩] ~_Q0 = [⟨((𝑥 ·_𝑜 (𝑤 ·_𝑜 𝑢)) +_𝑜 (𝑦 ·_𝑜 ((𝑧 ·_𝑜 𝑢) +_𝑜 (𝑤 ·_𝑜 𝑣)))), (𝑦 ·_𝑜 (𝑤 ·_𝑜 𝑢))⟩] ~_Q0 )
64	59, 61, 63	syl2anc 403	. . . . . . . 8 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ N)) → [⟨((((𝑥 ·_𝑜 𝑤) +_𝑜 (𝑦 ·_𝑜 𝑧)) ·_𝑜 𝑢) +_𝑜 ((𝑦 ·_𝑜 𝑤) ·_𝑜 𝑣)), ((𝑦 ·_𝑜 𝑤) ·_𝑜 𝑢)⟩] ~_Q0 = [⟨((𝑥 ·_𝑜 (𝑤 ·_𝑜 𝑢)) +_𝑜 (𝑦 ·_𝑜 ((𝑧 ·_𝑜 𝑢) +_𝑜 (𝑤 ·_𝑜 𝑣)))), (𝑦 ·_𝑜 (𝑤 ·_𝑜 𝑢))⟩] ~_Q0 )
65		addnnnq0 6639	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑤 ∈ N)) → ([⟨𝑥, 𝑦⟩] ~_Q0 +_Q0 [⟨𝑧, 𝑤⟩] ~_Q0 ) = [⟨((𝑥 ·_𝑜 𝑤) +_𝑜 (𝑦 ·_𝑜 𝑧)), (𝑦 ·_𝑜 𝑤)⟩] ~_Q0 )
66	65	oveq1d 5547	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑤 ∈ N)) → (([⟨𝑥, 𝑦⟩] ~_Q0 +_Q0 [⟨𝑧, 𝑤⟩] ~_Q0 ) +_Q0 [⟨𝑣, 𝑢⟩] ~_Q0 ) = ([⟨((𝑥 ·_𝑜 𝑤) +_𝑜 (𝑦 ·_𝑜 𝑧)), (𝑦 ·_𝑜 𝑤)⟩] ~_Q0 +_Q0 [⟨𝑣, 𝑢⟩] ~_Q0 ))
67	66	adantr 270	. . . . . . . . . 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 ·_𝑜 𝑤) +_𝑜 (𝑦 ·_𝑜 𝑧)), (𝑦 ·_𝑜 𝑤)⟩] ~_Q0 +_Q0 [⟨𝑣, 𝑢⟩] ~_Q0 ))
68		addassnq0lemcl 6651	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑤 ∈ N)) → (((𝑥 ·_𝑜 𝑤) +_𝑜 (𝑦 ·_𝑜 𝑧)) ∈ ω ∧ (𝑦 ·_𝑜 𝑤) ∈ N))
69		addnnnq0 6639	. . . . . . . . . . 11 ⊢ (((((𝑥 ·_𝑜 𝑤) +_𝑜 (𝑦 ·_𝑜 𝑧)) ∈ ω ∧ (𝑦 ·_𝑜 𝑤) ∈ N) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ N)) → ([⟨((𝑥 ·_𝑜 𝑤) +_𝑜 (𝑦 ·_𝑜 𝑧)), (𝑦 ·_𝑜 𝑤)⟩] ~_Q0 +_Q0 [⟨𝑣, 𝑢⟩] ~_Q0 ) = [⟨((((𝑥 ·_𝑜 𝑤) +_𝑜 (𝑦 ·_𝑜 𝑧)) ·_𝑜 𝑢) +_𝑜 ((𝑦 ·_𝑜 𝑤) ·_𝑜 𝑣)), ((𝑦 ·_𝑜 𝑤) ·_𝑜 𝑢)⟩] ~_Q0 )
70	68, 69	sylan 277	. . . . . . . . . 10 ⊢ ((((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑤 ∈ N)) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ N)) → ([⟨((𝑥 ·_𝑜 𝑤) +_𝑜 (𝑦 ·_𝑜 𝑧)), (𝑦 ·_𝑜 𝑤)⟩] ~_Q0 +_Q0 [⟨𝑣, 𝑢⟩] ~_Q0 ) = [⟨((((𝑥 ·_𝑜 𝑤) +_𝑜 (𝑦 ·_𝑜 𝑧)) ·_𝑜 𝑢) +_𝑜 ((𝑦 ·_𝑜 𝑤) ·_𝑜 𝑣)), ((𝑦 ·_𝑜 𝑤) ·_𝑜 𝑢)⟩] ~_Q0 )
71	67, 70	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 ) = [⟨((((𝑥 ·_𝑜 𝑤) +_𝑜 (𝑦 ·_𝑜 𝑧)) ·_𝑜 𝑢) +_𝑜 ((𝑦 ·_𝑜 𝑤) ·_𝑜 𝑣)), ((𝑦 ·_𝑜 𝑤) ·_𝑜 𝑢)⟩] ~_Q0 )
72	71	3impa 1133	. . . . . . . 8 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ N)) → (([⟨𝑥, 𝑦⟩] ~_Q0 +_Q0 [⟨𝑧, 𝑤⟩] ~_Q0 ) +_Q0 [⟨𝑣, 𝑢⟩] ~_Q0 ) = [⟨((((𝑥 ·_𝑜 𝑤) +_𝑜 (𝑦 ·_𝑜 𝑧)) ·_𝑜 𝑢) +_𝑜 ((𝑦 ·_𝑜 𝑤) ·_𝑜 𝑣)), ((𝑦 ·_𝑜 𝑤) ·_𝑜 𝑢)⟩] ~_Q0 )
73		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 )
74	73	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 ))
75	74	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 ))
76		addassnq0lemcl 6651	. . . . . . . . . . 11 ⊢ (((𝑧 ∈ ω ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ N)) → (((𝑧 ·_𝑜 𝑢) +_𝑜 (𝑤 ·_𝑜 𝑣)) ∈ ω ∧ (𝑤 ·_𝑜 𝑢) ∈ N))
77		addnnnq0 6639	. . . . . . . . . . 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 ) = [⟨((𝑥 ·_𝑜 (𝑤 ·_𝑜 𝑢)) +_𝑜 (𝑦 ·_𝑜 ((𝑧 ·_𝑜 𝑢) +_𝑜 (𝑤 ·_𝑜 𝑣)))), (𝑦 ·_𝑜 (𝑤 ·_𝑜 𝑢))⟩] ~_Q0 )
78	76, 77	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 ) = [⟨((𝑥 ·_𝑜 (𝑤 ·_𝑜 𝑢)) +_𝑜 (𝑦 ·_𝑜 ((𝑧 ·_𝑜 𝑢) +_𝑜 (𝑤 ·_𝑜 𝑣)))), (𝑦 ·_𝑜 (𝑤 ·_𝑜 𝑢))⟩] ~_Q0 )
79	75, 78	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 )) = [⟨((𝑥 ·_𝑜 (𝑤 ·_𝑜 𝑢)) +_𝑜 (𝑦 ·_𝑜 ((𝑧 ·_𝑜 𝑢) +_𝑜 (𝑤 ·_𝑜 𝑣)))), (𝑦 ·_𝑜 (𝑤 ·_𝑜 𝑢))⟩] ~_Q0 )
80	79	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 )) = [⟨((𝑥 ·_𝑜 (𝑤 ·_𝑜 𝑢)) +_𝑜 (𝑦 ·_𝑜 ((𝑧 ·_𝑜 𝑢) +_𝑜 (𝑤 ·_𝑜 𝑣)))), (𝑦 ·_𝑜 (𝑤 ·_𝑜 𝑢))⟩] ~_Q0 )
81	64, 72, 80	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 [⟨𝑣, 𝑢⟩] ~_Q0 )))
82	81	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 [⟨𝑣, 𝑢⟩] ~_Q0 ))))
83	1, 17, 82	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 [⟨𝑣, 𝑢⟩] ~_Q0 ))))
84	83	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 [⟨𝑣, 𝑢⟩] ~_Q0 ))))
85	1, 7, 12, 84	2ecoptocl 6217	. . 3 ⊢ ((𝐵 ∈ Q₀ ∧ 𝐶 ∈ Q₀) → (𝐴 ∈ Q₀ → ((𝐴 +_Q0 𝐵) +_Q0 𝐶) = (𝐴 +_Q0 (𝐵 +_Q0 𝐶))))
86	85	com12 30	. 2 ⊢ (𝐴 ∈ Q₀ → ((𝐵 ∈ Q₀ ∧ 𝐶 ∈ Q₀) → ((𝐴 +_Q0 𝐵) +_Q0 𝐶) = (𝐴 +_Q0 (𝐵 +_Q0 𝐶))))
87	86	3impib 1136	1 ⊢ ((𝐴 ∈ Q₀ ∧ 𝐵 ∈ Q₀ ∧ 𝐶 ∈ Q₀) → ((𝐴 +_Q0 𝐵) +_Q0 𝐶) = (𝐴 +_Q0 (𝐵 +_Q0 𝐶)))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 102 ∧ 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 ~_Q0 ceq0 6476 Q₀cnq0 6477 +_Q0 cplq0 6479

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

This theorem is referenced by: prarloclemcalc 6692

W3C validator