Theorem nadddi 37188

Description: Natural multiplication distributes over natural addition. (Contributed by Scott Fenton, 27-Jul-2026.)

Assertion

Ref	Expression
nadddi	⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ·no (𝐵 +no 𝐶)) = ((𝐴 ·no 𝐵) +no (𝐴 ·no 𝐶)))

Proof of Theorem nadddi

Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑒 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq1 7458	. . 3 ⊢ (𝑎 = 𝑑 → (𝑎 ·no (𝑏 +no 𝑐)) = (𝑑 ·no (𝑏 +no 𝑐)))
2		oveq1 7458	. . . 4 ⊢ (𝑎 = 𝑑 → (𝑎 ·no 𝑏) = (𝑑 ·no 𝑏))
3		oveq1 7458	. . . 4 ⊢ (𝑎 = 𝑑 → (𝑎 ·no 𝑐) = (𝑑 ·no 𝑐))
4	2, 3	oveq12d 7469	. . 3 ⊢ (𝑎 = 𝑑 → ((𝑎 ·no 𝑏) +no (𝑎 ·no 𝑐)) = ((𝑑 ·no 𝑏) +no (𝑑 ·no 𝑐)))
5	1, 4	eqeq12d 2779	. 2 ⊢ (𝑎 = 𝑑 → ((𝑎 ·no (𝑏 +no 𝑐)) = ((𝑎 ·no 𝑏) +no (𝑎 ·no 𝑐)) ↔ (𝑑 ·no (𝑏 +no 𝑐)) = ((𝑑 ·no 𝑏) +no (𝑑 ·no 𝑐))))
6		oveq1 7458	. . . 4 ⊢ (𝑏 = 𝑒 → (𝑏 +no 𝑐) = (𝑒 +no 𝑐))
7	6	oveq2d 7467	. . 3 ⊢ (𝑏 = 𝑒 → (𝑑 ·no (𝑏 +no 𝑐)) = (𝑑 ·no (𝑒 +no 𝑐)))
8		oveq2 7459	. . . 4 ⊢ (𝑏 = 𝑒 → (𝑑 ·no 𝑏) = (𝑑 ·no 𝑒))
9	8	oveq1d 7466	. . 3 ⊢ (𝑏 = 𝑒 → ((𝑑 ·no 𝑏) +no (𝑑 ·no 𝑐)) = ((𝑑 ·no 𝑒) +no (𝑑 ·no 𝑐)))
10	7, 9	eqeq12d 2779	. 2 ⊢ (𝑏 = 𝑒 → ((𝑑 ·no (𝑏 +no 𝑐)) = ((𝑑 ·no 𝑏) +no (𝑑 ·no 𝑐)) ↔ (𝑑 ·no (𝑒 +no 𝑐)) = ((𝑑 ·no 𝑒) +no (𝑑 ·no 𝑐))))
11		oveq2 7459	. . . 4 ⊢ (𝑐 = 𝑓 → (𝑒 +no 𝑐) = (𝑒 +no 𝑓))
12	11	oveq2d 7467	. . 3 ⊢ (𝑐 = 𝑓 → (𝑑 ·no (𝑒 +no 𝑐)) = (𝑑 ·no (𝑒 +no 𝑓)))
13		oveq2 7459	. . . 4 ⊢ (𝑐 = 𝑓 → (𝑑 ·no 𝑐) = (𝑑 ·no 𝑓))
14	13	oveq2d 7467	. . 3 ⊢ (𝑐 = 𝑓 → ((𝑑 ·no 𝑒) +no (𝑑 ·no 𝑐)) = ((𝑑 ·no 𝑒) +no (𝑑 ·no 𝑓)))
15	12, 14	eqeq12d 2779	. 2 ⊢ (𝑐 = 𝑓 → ((𝑑 ·no (𝑒 +no 𝑐)) = ((𝑑 ·no 𝑒) +no (𝑑 ·no 𝑐)) ↔ (𝑑 ·no (𝑒 +no 𝑓)) = ((𝑑 ·no 𝑒) +no (𝑑 ·no 𝑓))))
16		oveq1 7458	. . 3 ⊢ (𝑎 = 𝑑 → (𝑎 ·no (𝑒 +no 𝑓)) = (𝑑 ·no (𝑒 +no 𝑓)))
17		oveq1 7458	. . . 4 ⊢ (𝑎 = 𝑑 → (𝑎 ·no 𝑒) = (𝑑 ·no 𝑒))
18		oveq1 7458	. . . 4 ⊢ (𝑎 = 𝑑 → (𝑎 ·no 𝑓) = (𝑑 ·no 𝑓))
19	17, 18	oveq12d 7469	. . 3 ⊢ (𝑎 = 𝑑 → ((𝑎 ·no 𝑒) +no (𝑎 ·no 𝑓)) = ((𝑑 ·no 𝑒) +no (𝑑 ·no 𝑓)))
20	16, 19	eqeq12d 2779	. 2 ⊢ (𝑎 = 𝑑 → ((𝑎 ·no (𝑒 +no 𝑓)) = ((𝑎 ·no 𝑒) +no (𝑎 ·no 𝑓)) ↔ (𝑑 ·no (𝑒 +no 𝑓)) = ((𝑑 ·no 𝑒) +no (𝑑 ·no 𝑓))))
21		oveq1 7458	. . . 4 ⊢ (𝑏 = 𝑒 → (𝑏 +no 𝑓) = (𝑒 +no 𝑓))
22	21	oveq2d 7467	. . 3 ⊢ (𝑏 = 𝑒 → (𝑎 ·no (𝑏 +no 𝑓)) = (𝑎 ·no (𝑒 +no 𝑓)))
23		oveq2 7459	. . . 4 ⊢ (𝑏 = 𝑒 → (𝑎 ·no 𝑏) = (𝑎 ·no 𝑒))
24	23	oveq1d 7466	. . 3 ⊢ (𝑏 = 𝑒 → ((𝑎 ·no 𝑏) +no (𝑎 ·no 𝑓)) = ((𝑎 ·no 𝑒) +no (𝑎 ·no 𝑓)))
25	22, 24	eqeq12d 2779	. 2 ⊢ (𝑏 = 𝑒 → ((𝑎 ·no (𝑏 +no 𝑓)) = ((𝑎 ·no 𝑏) +no (𝑎 ·no 𝑓)) ↔ (𝑎 ·no (𝑒 +no 𝑓)) = ((𝑎 ·no 𝑒) +no (𝑎 ·no 𝑓))))
26	21	oveq2d 7467	. . 3 ⊢ (𝑏 = 𝑒 → (𝑑 ·no (𝑏 +no 𝑓)) = (𝑑 ·no (𝑒 +no 𝑓)))
27	8	oveq1d 7466	. . 3 ⊢ (𝑏 = 𝑒 → ((𝑑 ·no 𝑏) +no (𝑑 ·no 𝑓)) = ((𝑑 ·no 𝑒) +no (𝑑 ·no 𝑓)))
28	26, 27	eqeq12d 2779	. 2 ⊢ (𝑏 = 𝑒 → ((𝑑 ·no (𝑏 +no 𝑓)) = ((𝑑 ·no 𝑏) +no (𝑑 ·no 𝑓)) ↔ (𝑑 ·no (𝑒 +no 𝑓)) = ((𝑑 ·no 𝑒) +no (𝑑 ·no 𝑓))))
29	11	oveq2d 7467	. . 3 ⊢ (𝑐 = 𝑓 → (𝑎 ·no (𝑒 +no 𝑐)) = (𝑎 ·no (𝑒 +no 𝑓)))
30		oveq2 7459	. . . 4 ⊢ (𝑐 = 𝑓 → (𝑎 ·no 𝑐) = (𝑎 ·no 𝑓))
31	30	oveq2d 7467	. . 3 ⊢ (𝑐 = 𝑓 → ((𝑎 ·no 𝑒) +no (𝑎 ·no 𝑐)) = ((𝑎 ·no 𝑒) +no (𝑎 ·no 𝑓)))
32	29, 31	eqeq12d 2779	. 2 ⊢ (𝑐 = 𝑓 → ((𝑎 ·no (𝑒 +no 𝑐)) = ((𝑎 ·no 𝑒) +no (𝑎 ·no 𝑐)) ↔ (𝑎 ·no (𝑒 +no 𝑓)) = ((𝑎 ·no 𝑒) +no (𝑎 ·no 𝑓))))
33		oveq1 7458	. . 3 ⊢ (𝑎 = 𝐴 → (𝑎 ·no (𝑏 +no 𝑐)) = (𝐴 ·no (𝑏 +no 𝑐)))
34		oveq1 7458	. . . 4 ⊢ (𝑎 = 𝐴 → (𝑎 ·no 𝑏) = (𝐴 ·no 𝑏))
35		oveq1 7458	. . . 4 ⊢ (𝑎 = 𝐴 → (𝑎 ·no 𝑐) = (𝐴 ·no 𝑐))
36	34, 35	oveq12d 7469	. . 3 ⊢ (𝑎 = 𝐴 → ((𝑎 ·no 𝑏) +no (𝑎 ·no 𝑐)) = ((𝐴 ·no 𝑏) +no (𝐴 ·no 𝑐)))
37	33, 36	eqeq12d 2779	. 2 ⊢ (𝑎 = 𝐴 → ((𝑎 ·no (𝑏 +no 𝑐)) = ((𝑎 ·no 𝑏) +no (𝑎 ·no 𝑐)) ↔ (𝐴 ·no (𝑏 +no 𝑐)) = ((𝐴 ·no 𝑏) +no (𝐴 ·no 𝑐))))
38		oveq1 7458	. . . 4 ⊢ (𝑏 = 𝐵 → (𝑏 +no 𝑐) = (𝐵 +no 𝑐))
39	38	oveq2d 7467	. . 3 ⊢ (𝑏 = 𝐵 → (𝐴 ·no (𝑏 +no 𝑐)) = (𝐴 ·no (𝐵 +no 𝑐)))
40		oveq2 7459	. . . 4 ⊢ (𝑏 = 𝐵 → (𝐴 ·no 𝑏) = (𝐴 ·no 𝐵))
41	40	oveq1d 7466	. . 3 ⊢ (𝑏 = 𝐵 → ((𝐴 ·no 𝑏) +no (𝐴 ·no 𝑐)) = ((𝐴 ·no 𝐵) +no (𝐴 ·no 𝑐)))
42	39, 41	eqeq12d 2779	. 2 ⊢ (𝑏 = 𝐵 → ((𝐴 ·no (𝑏 +no 𝑐)) = ((𝐴 ·no 𝑏) +no (𝐴 ·no 𝑐)) ↔ (𝐴 ·no (𝐵 +no 𝑐)) = ((𝐴 ·no 𝐵) +no (𝐴 ·no 𝑐))))
43		oveq2 7459	. . . 4 ⊢ (𝑐 = 𝐶 → (𝐵 +no 𝑐) = (𝐵 +no 𝐶))
44	43	oveq2d 7467	. . 3 ⊢ (𝑐 = 𝐶 → (𝐴 ·no (𝐵 +no 𝑐)) = (𝐴 ·no (𝐵 +no 𝐶)))
45		oveq2 7459	. . . 4 ⊢ (𝑐 = 𝐶 → (𝐴 ·no 𝑐) = (𝐴 ·no 𝐶))
46	45	oveq2d 7467	. . 3 ⊢ (𝑐 = 𝐶 → ((𝐴 ·no 𝐵) +no (𝐴 ·no 𝑐)) = ((𝐴 ·no 𝐵) +no (𝐴 ·no 𝐶)))
47	44, 46	eqeq12d 2779	. 2 ⊢ (𝑐 = 𝐶 → ((𝐴 ·no (𝐵 +no 𝑐)) = ((𝐴 ·no 𝐵) +no (𝐴 ·no 𝑐)) ↔ (𝐴 ·no (𝐵 +no 𝐶)) = ((𝐴 ·no 𝐵) +no (𝐴 ·no 𝐶))))
48		simpl1 1210	. . . . 5 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) ∧ ((∀𝑑 ∈ 𝑎 ∀𝑒 ∈ 𝑏 ∀𝑓 ∈ 𝑐 (𝑑 ·no (𝑒 +no 𝑓)) = ((𝑑 ·no 𝑒) +no (𝑑 ·no 𝑓)) ∧ ∀𝑑 ∈ 𝑎 ∀𝑒 ∈ 𝑏 (𝑑 ·no (𝑒 +no 𝑐)) = ((𝑑 ·no 𝑒) +no (𝑑 ·no 𝑐)) ∧ ∀𝑑 ∈ 𝑎 ∀𝑓 ∈ 𝑐 (𝑑 ·no (𝑏 +no 𝑓)) = ((𝑑 ·no 𝑏) +no (𝑑 ·no 𝑓))) ∧ (∀𝑑 ∈ 𝑎 (𝑑 ·no (𝑏 +no 𝑐)) = ((𝑑 ·no 𝑏) +no (𝑑 ·no 𝑐)) ∧ ∀𝑒 ∈ 𝑏 ∀𝑓 ∈ 𝑐 (𝑎 ·no (𝑒 +no 𝑓)) = ((𝑎 ·no 𝑒) +no (𝑎 ·no 𝑓)) ∧ ∀𝑒 ∈ 𝑏 (𝑎 ·no (𝑒 +no 𝑐)) = ((𝑎 ·no 𝑒) +no (𝑎 ·no 𝑐))) ∧ ∀𝑓 ∈ 𝑐 (𝑎 ·no (𝑏 +no 𝑓)) = ((𝑎 ·no 𝑏) +no (𝑎 ·no 𝑓)))) → 𝑎 ∈ On)
49		simpl2 1211	. . . . 5 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) ∧ ((∀𝑑 ∈ 𝑎 ∀𝑒 ∈ 𝑏 ∀𝑓 ∈ 𝑐 (𝑑 ·no (𝑒 +no 𝑓)) = ((𝑑 ·no 𝑒) +no (𝑑 ·no 𝑓)) ∧ ∀𝑑 ∈ 𝑎 ∀𝑒 ∈ 𝑏 (𝑑 ·no (𝑒 +no 𝑐)) = ((𝑑 ·no 𝑒) +no (𝑑 ·no 𝑐)) ∧ ∀𝑑 ∈ 𝑎 ∀𝑓 ∈ 𝑐 (𝑑 ·no (𝑏 +no 𝑓)) = ((𝑑 ·no 𝑏) +no (𝑑 ·no 𝑓))) ∧ (∀𝑑 ∈ 𝑎 (𝑑 ·no (𝑏 +no 𝑐)) = ((𝑑 ·no 𝑏) +no (𝑑 ·no 𝑐)) ∧ ∀𝑒 ∈ 𝑏 ∀𝑓 ∈ 𝑐 (𝑎 ·no (𝑒 +no 𝑓)) = ((𝑎 ·no 𝑒) +no (𝑎 ·no 𝑓)) ∧ ∀𝑒 ∈ 𝑏 (𝑎 ·no (𝑒 +no 𝑐)) = ((𝑎 ·no 𝑒) +no (𝑎 ·no 𝑐))) ∧ ∀𝑓 ∈ 𝑐 (𝑎 ·no (𝑏 +no 𝑓)) = ((𝑎 ·no 𝑏) +no (𝑎 ·no 𝑓)))) → 𝑏 ∈ On)
50		simpl3 1212	. . . . 5 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) ∧ ((∀𝑑 ∈ 𝑎 ∀𝑒 ∈ 𝑏 ∀𝑓 ∈ 𝑐 (𝑑 ·no (𝑒 +no 𝑓)) = ((𝑑 ·no 𝑒) +no (𝑑 ·no 𝑓)) ∧ ∀𝑑 ∈ 𝑎 ∀𝑒 ∈ 𝑏 (𝑑 ·no (𝑒 +no 𝑐)) = ((𝑑 ·no 𝑒) +no (𝑑 ·no 𝑐)) ∧ ∀𝑑 ∈ 𝑎 ∀𝑓 ∈ 𝑐 (𝑑 ·no (𝑏 +no 𝑓)) = ((𝑑 ·no 𝑏) +no (𝑑 ·no 𝑓))) ∧ (∀𝑑 ∈ 𝑎 (𝑑 ·no (𝑏 +no 𝑐)) = ((𝑑 ·no 𝑏) +no (𝑑 ·no 𝑐)) ∧ ∀𝑒 ∈ 𝑏 ∀𝑓 ∈ 𝑐 (𝑎 ·no (𝑒 +no 𝑓)) = ((𝑎 ·no 𝑒) +no (𝑎 ·no 𝑓)) ∧ ∀𝑒 ∈ 𝑏 (𝑎 ·no (𝑒 +no 𝑐)) = ((𝑎 ·no 𝑒) +no (𝑎 ·no 𝑐))) ∧ ∀𝑓 ∈ 𝑐 (𝑎 ·no (𝑏 +no 𝑓)) = ((𝑎 ·no 𝑏) +no (𝑎 ·no 𝑓)))) → 𝑐 ∈ On)
51		simpr21 1279	. . . . 5 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) ∧ ((∀𝑑 ∈ 𝑎 ∀𝑒 ∈ 𝑏 ∀𝑓 ∈ 𝑐 (𝑑 ·no (𝑒 +no 𝑓)) = ((𝑑 ·no 𝑒) +no (𝑑 ·no 𝑓)) ∧ ∀𝑑 ∈ 𝑎 ∀𝑒 ∈ 𝑏 (𝑑 ·no (𝑒 +no 𝑐)) = ((𝑑 ·no 𝑒) +no (𝑑 ·no 𝑐)) ∧ ∀𝑑 ∈ 𝑎 ∀𝑓 ∈ 𝑐 (𝑑 ·no (𝑏 +no 𝑓)) = ((𝑑 ·no 𝑏) +no (𝑑 ·no 𝑓))) ∧ (∀𝑑 ∈ 𝑎 (𝑑 ·no (𝑏 +no 𝑐)) = ((𝑑 ·no 𝑏) +no (𝑑 ·no 𝑐)) ∧ ∀𝑒 ∈ 𝑏 ∀𝑓 ∈ 𝑐 (𝑎 ·no (𝑒 +no 𝑓)) = ((𝑎 ·no 𝑒) +no (𝑎 ·no 𝑓)) ∧ ∀𝑒 ∈ 𝑏 (𝑎 ·no (𝑒 +no 𝑐)) = ((𝑎 ·no 𝑒) +no (𝑎 ·no 𝑐))) ∧ ∀𝑓 ∈ 𝑐 (𝑎 ·no (𝑏 +no 𝑓)) = ((𝑎 ·no 𝑏) +no (𝑎 ·no 𝑓)))) → ∀𝑑 ∈ 𝑎 (𝑑 ·no (𝑏 +no 𝑐)) = ((𝑑 ·no 𝑏) +no (𝑑 ·no 𝑐)))
52		simpr23 1281	. . . . 5 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) ∧ ((∀𝑑 ∈ 𝑎 ∀𝑒 ∈ 𝑏 ∀𝑓 ∈ 𝑐 (𝑑 ·no (𝑒 +no 𝑓)) = ((𝑑 ·no 𝑒) +no (𝑑 ·no 𝑓)) ∧ ∀𝑑 ∈ 𝑎 ∀𝑒 ∈ 𝑏 (𝑑 ·no (𝑒 +no 𝑐)) = ((𝑑 ·no 𝑒) +no (𝑑 ·no 𝑐)) ∧ ∀𝑑 ∈ 𝑎 ∀𝑓 ∈ 𝑐 (𝑑 ·no (𝑏 +no 𝑓)) = ((𝑑 ·no 𝑏) +no (𝑑 ·no 𝑓))) ∧ (∀𝑑 ∈ 𝑎 (𝑑 ·no (𝑏 +no 𝑐)) = ((𝑑 ·no 𝑏) +no (𝑑 ·no 𝑐)) ∧ ∀𝑒 ∈ 𝑏 ∀𝑓 ∈ 𝑐 (𝑎 ·no (𝑒 +no 𝑓)) = ((𝑎 ·no 𝑒) +no (𝑎 ·no 𝑓)) ∧ ∀𝑒 ∈ 𝑏 (𝑎 ·no (𝑒 +no 𝑐)) = ((𝑎 ·no 𝑒) +no (𝑎 ·no 𝑐))) ∧ ∀𝑓 ∈ 𝑐 (𝑎 ·no (𝑏 +no 𝑓)) = ((𝑎 ·no 𝑏) +no (𝑎 ·no 𝑓)))) → ∀𝑒 ∈ 𝑏 (𝑎 ·no (𝑒 +no 𝑐)) = ((𝑎 ·no 𝑒) +no (𝑎 ·no 𝑐)))
53		simpr3 1215	. . . . 5 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) ∧ ((∀𝑑 ∈ 𝑎 ∀𝑒 ∈ 𝑏 ∀𝑓 ∈ 𝑐 (𝑑 ·no (𝑒 +no 𝑓)) = ((𝑑 ·no 𝑒) +no (𝑑 ·no 𝑓)) ∧ ∀𝑑 ∈ 𝑎 ∀𝑒 ∈ 𝑏 (𝑑 ·no (𝑒 +no 𝑐)) = ((𝑑 ·no 𝑒) +no (𝑑 ·no 𝑐)) ∧ ∀𝑑 ∈ 𝑎 ∀𝑓 ∈ 𝑐 (𝑑 ·no (𝑏 +no 𝑓)) = ((𝑑 ·no 𝑏) +no (𝑑 ·no 𝑓))) ∧ (∀𝑑 ∈ 𝑎 (𝑑 ·no (𝑏 +no 𝑐)) = ((𝑑 ·no 𝑏) +no (𝑑 ·no 𝑐)) ∧ ∀𝑒 ∈ 𝑏 ∀𝑓 ∈ 𝑐 (𝑎 ·no (𝑒 +no 𝑓)) = ((𝑎 ·no 𝑒) +no (𝑎 ·no 𝑓)) ∧ ∀𝑒 ∈ 𝑏 (𝑎 ·no (𝑒 +no 𝑐)) = ((𝑎 ·no 𝑒) +no (𝑎 ·no 𝑐))) ∧ ∀𝑓 ∈ 𝑐 (𝑎 ·no (𝑏 +no 𝑓)) = ((𝑎 ·no 𝑏) +no (𝑎 ·no 𝑓)))) → ∀𝑓 ∈ 𝑐 (𝑎 ·no (𝑏 +no 𝑓)) = ((𝑎 ·no 𝑏) +no (𝑎 ·no 𝑓)))
54		simpr12 1277	. . . . 5 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) ∧ ((∀𝑑 ∈ 𝑎 ∀𝑒 ∈ 𝑏 ∀𝑓 ∈ 𝑐 (𝑑 ·no (𝑒 +no 𝑓)) = ((𝑑 ·no 𝑒) +no (𝑑 ·no 𝑓)) ∧ ∀𝑑 ∈ 𝑎 ∀𝑒 ∈ 𝑏 (𝑑 ·no (𝑒 +no 𝑐)) = ((𝑑 ·no 𝑒) +no (𝑑 ·no 𝑐)) ∧ ∀𝑑 ∈ 𝑎 ∀𝑓 ∈ 𝑐 (𝑑 ·no (𝑏 +no 𝑓)) = ((𝑑 ·no 𝑏) +no (𝑑 ·no 𝑓))) ∧ (∀𝑑 ∈ 𝑎 (𝑑 ·no (𝑏 +no 𝑐)) = ((𝑑 ·no 𝑏) +no (𝑑 ·no 𝑐)) ∧ ∀𝑒 ∈ 𝑏 ∀𝑓 ∈ 𝑐 (𝑎 ·no (𝑒 +no 𝑓)) = ((𝑎 ·no 𝑒) +no (𝑎 ·no 𝑓)) ∧ ∀𝑒 ∈ 𝑏 (𝑎 ·no (𝑒 +no 𝑐)) = ((𝑎 ·no 𝑒) +no (𝑎 ·no 𝑐))) ∧ ∀𝑓 ∈ 𝑐 (𝑎 ·no (𝑏 +no 𝑓)) = ((𝑎 ·no 𝑏) +no (𝑎 ·no 𝑓)))) → ∀𝑑 ∈ 𝑎 ∀𝑒 ∈ 𝑏 (𝑑 ·no (𝑒 +no 𝑐)) = ((𝑑 ·no 𝑒) +no (𝑑 ·no 𝑐)))
55		simpr13 1278	. . . . 5 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) ∧ ((∀𝑑 ∈ 𝑎 ∀𝑒 ∈ 𝑏 ∀𝑓 ∈ 𝑐 (𝑑 ·no (𝑒 +no 𝑓)) = ((𝑑 ·no 𝑒) +no (𝑑 ·no 𝑓)) ∧ ∀𝑑 ∈ 𝑎 ∀𝑒 ∈ 𝑏 (𝑑 ·no (𝑒 +no 𝑐)) = ((𝑑 ·no 𝑒) +no (𝑑 ·no 𝑐)) ∧ ∀𝑑 ∈ 𝑎 ∀𝑓 ∈ 𝑐 (𝑑 ·no (𝑏 +no 𝑓)) = ((𝑑 ·no 𝑏) +no (𝑑 ·no 𝑓))) ∧ (∀𝑑 ∈ 𝑎 (𝑑 ·no (𝑏 +no 𝑐)) = ((𝑑 ·no 𝑏) +no (𝑑 ·no 𝑐)) ∧ ∀𝑒 ∈ 𝑏 ∀𝑓 ∈ 𝑐 (𝑎 ·no (𝑒 +no 𝑓)) = ((𝑎 ·no 𝑒) +no (𝑎 ·no 𝑓)) ∧ ∀𝑒 ∈ 𝑏 (𝑎 ·no (𝑒 +no 𝑐)) = ((𝑎 ·no 𝑒) +no (𝑎 ·no 𝑐))) ∧ ∀𝑓 ∈ 𝑐 (𝑎 ·no (𝑏 +no 𝑓)) = ((𝑎 ·no 𝑏) +no (𝑎 ·no 𝑓)))) → ∀𝑑 ∈ 𝑎 ∀𝑓 ∈ 𝑐 (𝑑 ·no (𝑏 +no 𝑓)) = ((𝑑 ·no 𝑏) +no (𝑑 ·no 𝑓)))
56	48, 49, 50, 51, 52, 53, 54, 55	nadddilem4 37187	. . . 4 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) ∧ ((∀𝑑 ∈ 𝑎 ∀𝑒 ∈ 𝑏 ∀𝑓 ∈ 𝑐 (𝑑 ·no (𝑒 +no 𝑓)) = ((𝑑 ·no 𝑒) +no (𝑑 ·no 𝑓)) ∧ ∀𝑑 ∈ 𝑎 ∀𝑒 ∈ 𝑏 (𝑑 ·no (𝑒 +no 𝑐)) = ((𝑑 ·no 𝑒) +no (𝑑 ·no 𝑐)) ∧ ∀𝑑 ∈ 𝑎 ∀𝑓 ∈ 𝑐 (𝑑 ·no (𝑏 +no 𝑓)) = ((𝑑 ·no 𝑏) +no (𝑑 ·no 𝑓))) ∧ (∀𝑑 ∈ 𝑎 (𝑑 ·no (𝑏 +no 𝑐)) = ((𝑑 ·no 𝑏) +no (𝑑 ·no 𝑐)) ∧ ∀𝑒 ∈ 𝑏 ∀𝑓 ∈ 𝑐 (𝑎 ·no (𝑒 +no 𝑓)) = ((𝑎 ·no 𝑒) +no (𝑎 ·no 𝑓)) ∧ ∀𝑒 ∈ 𝑏 (𝑎 ·no (𝑒 +no 𝑐)) = ((𝑎 ·no 𝑒) +no (𝑎 ·no 𝑐))) ∧ ∀𝑓 ∈ 𝑐 (𝑎 ·no (𝑏 +no 𝑓)) = ((𝑎 ·no 𝑏) +no (𝑎 ·no 𝑓)))) → (𝑎 ·no (𝑏 +no 𝑐)) ⊆ ((𝑎 ·no 𝑏) +no (𝑎 ·no 𝑐)))
57	48, 49, 50, 51, 52, 53, 54, 55	nadddilem2 37185	. . . 4 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) ∧ ((∀𝑑 ∈ 𝑎 ∀𝑒 ∈ 𝑏 ∀𝑓 ∈ 𝑐 (𝑑 ·no (𝑒 +no 𝑓)) = ((𝑑 ·no 𝑒) +no (𝑑 ·no 𝑓)) ∧ ∀𝑑 ∈ 𝑎 ∀𝑒 ∈ 𝑏 (𝑑 ·no (𝑒 +no 𝑐)) = ((𝑑 ·no 𝑒) +no (𝑑 ·no 𝑐)) ∧ ∀𝑑 ∈ 𝑎 ∀𝑓 ∈ 𝑐 (𝑑 ·no (𝑏 +no 𝑓)) = ((𝑑 ·no 𝑏) +no (𝑑 ·no 𝑓))) ∧ (∀𝑑 ∈ 𝑎 (𝑑 ·no (𝑏 +no 𝑐)) = ((𝑑 ·no 𝑏) +no (𝑑 ·no 𝑐)) ∧ ∀𝑒 ∈ 𝑏 ∀𝑓 ∈ 𝑐 (𝑎 ·no (𝑒 +no 𝑓)) = ((𝑎 ·no 𝑒) +no (𝑎 ·no 𝑓)) ∧ ∀𝑒 ∈ 𝑏 (𝑎 ·no (𝑒 +no 𝑐)) = ((𝑎 ·no 𝑒) +no (𝑎 ·no 𝑐))) ∧ ∀𝑓 ∈ 𝑐 (𝑎 ·no (𝑏 +no 𝑓)) = ((𝑎 ·no 𝑏) +no (𝑎 ·no 𝑓)))) → ((𝑎 ·no 𝑏) +no (𝑎 ·no 𝑐)) ⊆ (𝑎 ·no (𝑏 +no 𝑐)))
58	56, 57	eqssd 3930	. . 3 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) ∧ ((∀𝑑 ∈ 𝑎 ∀𝑒 ∈ 𝑏 ∀𝑓 ∈ 𝑐 (𝑑 ·no (𝑒 +no 𝑓)) = ((𝑑 ·no 𝑒) +no (𝑑 ·no 𝑓)) ∧ ∀𝑑 ∈ 𝑎 ∀𝑒 ∈ 𝑏 (𝑑 ·no (𝑒 +no 𝑐)) = ((𝑑 ·no 𝑒) +no (𝑑 ·no 𝑐)) ∧ ∀𝑑 ∈ 𝑎 ∀𝑓 ∈ 𝑐 (𝑑 ·no (𝑏 +no 𝑓)) = ((𝑑 ·no 𝑏) +no (𝑑 ·no 𝑓))) ∧ (∀𝑑 ∈ 𝑎 (𝑑 ·no (𝑏 +no 𝑐)) = ((𝑑 ·no 𝑏) +no (𝑑 ·no 𝑐)) ∧ ∀𝑒 ∈ 𝑏 ∀𝑓 ∈ 𝑐 (𝑎 ·no (𝑒 +no 𝑓)) = ((𝑎 ·no 𝑒) +no (𝑎 ·no 𝑓)) ∧ ∀𝑒 ∈ 𝑏 (𝑎 ·no (𝑒 +no 𝑐)) = ((𝑎 ·no 𝑒) +no (𝑎 ·no 𝑐))) ∧ ∀𝑓 ∈ 𝑐 (𝑎 ·no (𝑏 +no 𝑓)) = ((𝑎 ·no 𝑏) +no (𝑎 ·no 𝑓)))) → (𝑎 ·no (𝑏 +no 𝑐)) = ((𝑎 ·no 𝑏) +no (𝑎 ·no 𝑐)))
59	58	ex 418	. 2 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → (((∀𝑑 ∈ 𝑎 ∀𝑒 ∈ 𝑏 ∀𝑓 ∈ 𝑐 (𝑑 ·no (𝑒 +no 𝑓)) = ((𝑑 ·no 𝑒) +no (𝑑 ·no 𝑓)) ∧ ∀𝑑 ∈ 𝑎 ∀𝑒 ∈ 𝑏 (𝑑 ·no (𝑒 +no 𝑐)) = ((𝑑 ·no 𝑒) +no (𝑑 ·no 𝑐)) ∧ ∀𝑑 ∈ 𝑎 ∀𝑓 ∈ 𝑐 (𝑑 ·no (𝑏 +no 𝑓)) = ((𝑑 ·no 𝑏) +no (𝑑 ·no 𝑓))) ∧ (∀𝑑 ∈ 𝑎 (𝑑 ·no (𝑏 +no 𝑐)) = ((𝑑 ·no 𝑏) +no (𝑑 ·no 𝑐)) ∧ ∀𝑒 ∈ 𝑏 ∀𝑓 ∈ 𝑐 (𝑎 ·no (𝑒 +no 𝑓)) = ((𝑎 ·no 𝑒) +no (𝑎 ·no 𝑓)) ∧ ∀𝑒 ∈ 𝑏 (𝑎 ·no (𝑒 +no 𝑐)) = ((𝑎 ·no 𝑒) +no (𝑎 ·no 𝑐))) ∧ ∀𝑓 ∈ 𝑐 (𝑎 ·no (𝑏 +no 𝑓)) = ((𝑎 ·no 𝑏) +no (𝑎 ·no 𝑓))) → (𝑎 ·no (𝑏 +no 𝑐)) = ((𝑎 ·no 𝑏) +no (𝑎 ·no 𝑐))))
60	5, 10, 15, 20, 25, 28, 32, 37, 42, 47, 59	on3ind 8717	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ·no (𝐵 +no 𝐶)) = ((𝐴 ·no 𝐵) +no (𝐴 ·no 𝐶)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ∀wral 3080  Oncon0 6388  (class class class)co 7451   +no cnadd 8712   ·_no cnmul 37151

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-ot 4864  df-uni 4873  df-int 4913  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456  df-1st 8035  df-2nd 8036  df-frecs 8337  df-nadd 8713  df-nmul 37152

This theorem is used by:  nadddid  37189