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Theorem addassnq0 6560

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 6523	. . . 4 ⊢ Q₀ = ((ω × N) / ~_Q0 )
2		oveq2 5520	. . . . . . 7 ⊢ ([⟨𝑧, 𝑤⟩] ~_Q0 = 𝐵 → (𝐴 +_Q0 [⟨𝑧, 𝑤⟩] ~_Q0 ) = (𝐴 +_Q0 𝐵))
3	2	oveq1d 5527	. . . . . 6 ⊢ ([⟨𝑧, 𝑤⟩] ~_Q0 = 𝐵 → ((𝐴 +_Q0 [⟨𝑧, 𝑤⟩] ~_Q0 ) +_Q0 [⟨𝑣, 𝑢⟩] ~_Q0 ) = ((𝐴 +_Q0 𝐵) +_Q0 [⟨𝑣, 𝑢⟩] ~_Q0 ))
4		oveq1 5519	. . . . . . 7 ⊢ ([⟨𝑧, 𝑤⟩] ~_Q0 = 𝐵 → ([⟨𝑧, 𝑤⟩] ~_Q0 +_Q0 [⟨𝑣, 𝑢⟩] ~_Q0 ) = (𝐵 +_Q0 [⟨𝑣, 𝑢⟩] ~_Q0 ))
5	4	oveq2d 5528	. . . . . 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 2054	. . . . 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 219	. . . 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 5520	. . . . . 6 ⊢ ([⟨𝑣, 𝑢⟩] ~_Q0 = 𝐶 → ((𝐴 +_Q0 𝐵) +_Q0 [⟨𝑣, 𝑢⟩] ~_Q0 ) = ((𝐴 +_Q0 𝐵) +_Q0 𝐶))
9		oveq2 5520	. . . . . . 7 ⊢ ([⟨𝑣, 𝑢⟩] ~_Q0 = 𝐶 → (𝐵 +_Q0 [⟨𝑣, 𝑢⟩] ~_Q0 ) = (𝐵 +_Q0 𝐶))
10	9	oveq2d 5528	. . . . . 6 ⊢ ([⟨𝑣, 𝑢⟩] ~_Q0 = 𝐶 → (𝐴 +_Q0 (𝐵 +_Q0 [⟨𝑣, 𝑢⟩] ~_Q0 )) = (𝐴 +_Q0 (𝐵 +_Q0 𝐶)))
11	8, 10	eqeq12d 2054	. . . . 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 219	. . . 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 5519	. . . . . . . . 9 ⊢ ([⟨𝑥, 𝑦⟩] ~_Q0 = 𝐴 → ([⟨𝑥, 𝑦⟩] ~_Q0 +_Q0 [⟨𝑧, 𝑤⟩] ~_Q0 ) = (𝐴 +_Q0 [⟨𝑧, 𝑤⟩] ~_Q0 ))
14	13	oveq1d 5527	. . . . . . . 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	. . . . . . . 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 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 [⟨𝑣, 𝑢⟩] ~_Q0 )) ↔ ((𝐴 +_Q0 [⟨𝑧, 𝑤⟩] ~_Q0 ) +_Q0 [⟨𝑣, 𝑢⟩] ~_Q0 ) = (𝐴 +_Q0 ([⟨𝑧, 𝑤⟩] ~_Q0 +_Q0 [⟨𝑣, 𝑢⟩] ~_Q0 ))))
17	16	imbi2d 219	. . . . . 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 928	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ N)) → 𝑥 ∈ ω)
19		simp2r 931	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ N)) → 𝑤 ∈ N)
20		pinn 6407	. . . . . . . . . . . . . 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 933	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ N)) → 𝑢 ∈ N)
23		pinn 6407	. . . . . . . . . . . . . 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 6060	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ ω ∧ 𝑢 ∈ ω) → (𝑤 ·_𝑜 𝑢) ∈ ω)
26	21, 24, 25	syl2anc 391	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ N)) → (𝑤 ·_𝑜 𝑢) ∈ ω)
27		nnmcl 6060	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ω ∧ (𝑤 ·_𝑜 𝑢) ∈ ω) → (𝑥 ·_𝑜 (𝑤 ·_𝑜 𝑢)) ∈ ω)
28	18, 26, 27	syl2anc 391	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ N)) → (𝑥 ·_𝑜 (𝑤 ·_𝑜 𝑢)) ∈ ω)
29		simp1r 929	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ N)) → 𝑦 ∈ N)
30		pinn 6407	. . . . . . . . . . . . 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 930	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ N)) → 𝑧 ∈ ω)
33		nnmcl 6060	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ ω ∧ 𝑢 ∈ ω) → (𝑧 ·_𝑜 𝑢) ∈ ω)
34	32, 24, 33	syl2anc 391	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ N)) → (𝑧 ·_𝑜 𝑢) ∈ ω)
35		nnmcl 6060	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ω ∧ (𝑧 ·_𝑜 𝑢) ∈ ω) → (𝑦 ·_𝑜 (𝑧 ·_𝑜 𝑢)) ∈ ω)
36	31, 34, 35	syl2anc 391	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ N)) → (𝑦 ·_𝑜 (𝑧 ·_𝑜 𝑢)) ∈ ω)
37		simp3l 932	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ N)) → 𝑣 ∈ ω)
38		nnmcl 6060	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ ω ∧ 𝑣 ∈ ω) → (𝑤 ·_𝑜 𝑣) ∈ ω)
39	21, 37, 38	syl2anc 391	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ N)) → (𝑤 ·_𝑜 𝑣) ∈ ω)
40		nnmcl 6060	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ω ∧ (𝑤 ·_𝑜 𝑣) ∈ ω) → (𝑦 ·_𝑜 (𝑤 ·_𝑜 𝑣)) ∈ ω)
41	31, 39, 40	syl2anc 391	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ N)) → (𝑦 ·_𝑜 (𝑤 ·_𝑜 𝑣)) ∈ ω)
42		nnaass 6064	. . . . . . . . . . 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 1135	. . . . . . . . . 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 6068	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∈ ω ∧ 𝑔 ∈ ω) → (𝑓 ·_𝑜 𝑔) = (𝑔 ·_𝑜 𝑓))
45	44	adantl 262	. . . . . . . . . . . 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 6069	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∈ ω ∧ 𝑔 ∈ ω ∧ ℎ ∈ ω) → ((𝑓 +_𝑜 𝑔) ·_𝑜 ℎ) = ((𝑓 ·_𝑜 ℎ) +_𝑜 (𝑔 ·_𝑜 ℎ)))
47	46	adantl 262	. . . . . . . . . . . 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 6066	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∈ ω ∧ 𝑔 ∈ ω ∧ ℎ ∈ ω) → ((𝑓 ·_𝑜 𝑔) ·_𝑜 ℎ) = (𝑓 ·_𝑜 (𝑔 ·_𝑜 ℎ)))
49	48	adantl 262	. . . . . . . . . . . 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 6060	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∈ ω ∧ 𝑔 ∈ ω) → (𝑓 ·_𝑜 𝑔) ∈ ω)
51	50	adantl 262	. . . . . . . . . . . 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 5692	. . . . . . . . . . 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 6066	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ω ∧ 𝑤 ∈ ω ∧ 𝑣 ∈ ω) → ((𝑦 ·_𝑜 𝑤) ·_𝑜 𝑣) = (𝑦 ·_𝑜 (𝑤 ·_𝑜 𝑣)))
54	31, 21, 37, 53	syl3anc 1135	. . . . . . . . . . 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 5530	. . . . . . . . . 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 6065	. . . . . . . . . . . 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 1135	. . . . . . . . . . 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 5528	. . . . . . . . . 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 2082	. . . . . . . . 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 6066	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ω ∧ 𝑤 ∈ ω ∧ 𝑢 ∈ ω) → ((𝑦 ·_𝑜 𝑤) ·_𝑜 𝑢) = (𝑦 ·_𝑜 (𝑤 ·_𝑜 𝑢)))
61	31, 21, 24, 60	syl3anc 1135	. . . . . . . . 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 3551	. . . . . . . . . 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 6142	. . . . . . . . 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 391	. . . . . . . 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 6547	. . . . . . . . . . . 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 5527	. . . . . . . . . . 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 261	. . . . . . . . . 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 6559	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ ω ∧ 𝑤 ∈ N)) → (((𝑥 ·_𝑜 𝑤) +_𝑜 (𝑦 ·_𝑜 𝑧)) ∈ ω ∧ (𝑦 ·_𝑜 𝑤) ∈ N))
69		addnnnq0 6547	. . . . . . . . . . 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 267	. . . . . . . . . 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 2072	. . . . . . . . 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 1099	. . . . . . . 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 6547	. . . . . . . . . . . 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 5528	. . . . . . . . . . 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 262	. . . . . . . . . 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 6559	. . . . . . . . . . 11 ⊢ (((𝑧 ∈ ω ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ N)) → (((𝑧 ·_𝑜 𝑢) +_𝑜 (𝑤 ·_𝑜 𝑣)) ∈ ω ∧ (𝑤 ·_𝑜 𝑢) ∈ N))
77		addnnnq0 6547	. . . . . . . . . . 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 270	. . . . . . . . . 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 2072	. . . . . . . . 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 1100	. . . . . . . 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 2082	. . . . . . 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 1107	. . . . . 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 6193	. . . . 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 27	. . . 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 6194	. . 3 ⊢ ((𝐵 ∈ Q₀ ∧ 𝐶 ∈ Q₀) → (𝐴 ∈ Q₀ → ((𝐴 +_Q0 𝐵) +_Q0 𝐶) = (𝐴 +_Q0 (𝐵 +_Q0 𝐶))))
86	85	com12 27	. 2 ⊢ (𝐴 ∈ Q₀ → ((𝐵 ∈ Q₀ ∧ 𝐶 ∈ Q₀) → ((𝐴 +_Q0 𝐵) +_Q0 𝐶) = (𝐴 +_Q0 (𝐵 +_Q0 𝐶))))
87	86	3impib 1102	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 97 ∧ 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 ~_Q0 ceq0 6384 Q₀cnq0 6385 +_Q0 cplq0 6387

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

This theorem is referenced by: prarloclemcalc 6600

W3C validator