Theorem nadddi 37097

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 7431	. . 3 ⊢ (𝑎 = 𝑑 → (𝑎 ·no (𝑏 +no 𝑐)) = (𝑑 ·no (𝑏 +no 𝑐)))
2		oveq1 7431	. . . 4 ⊢ (𝑎 = 𝑑 → (𝑎 ·no 𝑏) = (𝑑 ·no 𝑏))
3		oveq1 7431	. . . 4 ⊢ (𝑎 = 𝑑 → (𝑎 ·no 𝑐) = (𝑑 ·no 𝑐))
4	2, 3	oveq12d 7442	. . 3 ⊢ (𝑎 = 𝑑 → ((𝑎 ·no 𝑏) +no (𝑎 ·no 𝑐)) = ((𝑑 ·no 𝑏) +no (𝑑 ·no 𝑐)))
5	1, 4	eqeq12d 2776	. 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 7431	. . . 4 ⊢ (𝑏 = 𝑒 → (𝑏 +no 𝑐) = (𝑒 +no 𝑐))
7	6	oveq2d 7440	. . 3 ⊢ (𝑏 = 𝑒 → (𝑑 ·no (𝑏 +no 𝑐)) = (𝑑 ·no (𝑒 +no 𝑐)))
8		oveq2 7432	. . . 4 ⊢ (𝑏 = 𝑒 → (𝑑 ·no 𝑏) = (𝑑 ·no 𝑒))
9	8	oveq1d 7439	. . 3 ⊢ (𝑏 = 𝑒 → ((𝑑 ·no 𝑏) +no (𝑑 ·no 𝑐)) = ((𝑑 ·no 𝑒) +no (𝑑 ·no 𝑐)))
10	7, 9	eqeq12d 2776	. 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 7432	. . . 4 ⊢ (𝑐 = 𝑓 → (𝑒 +no 𝑐) = (𝑒 +no 𝑓))
12	11	oveq2d 7440	. . 3 ⊢ (𝑐 = 𝑓 → (𝑑 ·no (𝑒 +no 𝑐)) = (𝑑 ·no (𝑒 +no 𝑓)))
13		oveq2 7432	. . . 4 ⊢ (𝑐 = 𝑓 → (𝑑 ·no 𝑐) = (𝑑 ·no 𝑓))
14	13	oveq2d 7440	. . 3 ⊢ (𝑐 = 𝑓 → ((𝑑 ·no 𝑒) +no (𝑑 ·no 𝑐)) = ((𝑑 ·no 𝑒) +no (𝑑 ·no 𝑓)))
15	12, 14	eqeq12d 2776	. 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 7431	. . 3 ⊢ (𝑎 = 𝑑 → (𝑎 ·no (𝑒 +no 𝑓)) = (𝑑 ·no (𝑒 +no 𝑓)))
17		oveq1 7431	. . . 4 ⊢ (𝑎 = 𝑑 → (𝑎 ·no 𝑒) = (𝑑 ·no 𝑒))
18		oveq1 7431	. . . 4 ⊢ (𝑎 = 𝑑 → (𝑎 ·no 𝑓) = (𝑑 ·no 𝑓))
19	17, 18	oveq12d 7442	. . 3 ⊢ (𝑎 = 𝑑 → ((𝑎 ·no 𝑒) +no (𝑎 ·no 𝑓)) = ((𝑑 ·no 𝑒) +no (𝑑 ·no 𝑓)))
20	16, 19	eqeq12d 2776	. 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 7431	. . . 4 ⊢ (𝑏 = 𝑒 → (𝑏 +no 𝑓) = (𝑒 +no 𝑓))
22	21	oveq2d 7440	. . 3 ⊢ (𝑏 = 𝑒 → (𝑎 ·no (𝑏 +no 𝑓)) = (𝑎 ·no (𝑒 +no 𝑓)))
23		oveq2 7432	. . . 4 ⊢ (𝑏 = 𝑒 → (𝑎 ·no 𝑏) = (𝑎 ·no 𝑒))
24	23	oveq1d 7439	. . 3 ⊢ (𝑏 = 𝑒 → ((𝑎 ·no 𝑏) +no (𝑎 ·no 𝑓)) = ((𝑎 ·no 𝑒) +no (𝑎 ·no 𝑓)))
25	22, 24	eqeq12d 2776	. 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 7440	. . 3 ⊢ (𝑏 = 𝑒 → (𝑑 ·no (𝑏 +no 𝑓)) = (𝑑 ·no (𝑒 +no 𝑓)))
27	8	oveq1d 7439	. . 3 ⊢ (𝑏 = 𝑒 → ((𝑑 ·no 𝑏) +no (𝑑 ·no 𝑓)) = ((𝑑 ·no 𝑒) +no (𝑑 ·no 𝑓)))
28	26, 27	eqeq12d 2776	. 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 7440	. . 3 ⊢ (𝑐 = 𝑓 → (𝑎 ·no (𝑒 +no 𝑐)) = (𝑎 ·no (𝑒 +no 𝑓)))
30		oveq2 7432	. . . 4 ⊢ (𝑐 = 𝑓 → (𝑎 ·no 𝑐) = (𝑎 ·no 𝑓))
31	30	oveq2d 7440	. . 3 ⊢ (𝑐 = 𝑓 → ((𝑎 ·no 𝑒) +no (𝑎 ·no 𝑐)) = ((𝑎 ·no 𝑒) +no (𝑎 ·no 𝑓)))
32	29, 31	eqeq12d 2776	. 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 7431	. . 3 ⊢ (𝑎 = 𝐴 → (𝑎 ·no (𝑏 +no 𝑐)) = (𝐴 ·no (𝑏 +no 𝑐)))
34		oveq1 7431	. . . 4 ⊢ (𝑎 = 𝐴 → (𝑎 ·no 𝑏) = (𝐴 ·no 𝑏))
35		oveq1 7431	. . . 4 ⊢ (𝑎 = 𝐴 → (𝑎 ·no 𝑐) = (𝐴 ·no 𝑐))
36	34, 35	oveq12d 7442	. . 3 ⊢ (𝑎 = 𝐴 → ((𝑎 ·no 𝑏) +no (𝑎 ·no 𝑐)) = ((𝐴 ·no 𝑏) +no (𝐴 ·no 𝑐)))
37	33, 36	eqeq12d 2776	. 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 7431	. . . 4 ⊢ (𝑏 = 𝐵 → (𝑏 +no 𝑐) = (𝐵 +no 𝑐))
39	38	oveq2d 7440	. . 3 ⊢ (𝑏 = 𝐵 → (𝐴 ·no (𝑏 +no 𝑐)) = (𝐴 ·no (𝐵 +no 𝑐)))
40		oveq2 7432	. . . 4 ⊢ (𝑏 = 𝐵 → (𝐴 ·no 𝑏) = (𝐴 ·no 𝐵))
41	40	oveq1d 7439	. . 3 ⊢ (𝑏 = 𝐵 → ((𝐴 ·no 𝑏) +no (𝐴 ·no 𝑐)) = ((𝐴 ·no 𝐵) +no (𝐴 ·no 𝑐)))
42	39, 41	eqeq12d 2776	. 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 7432	. . . 4 ⊢ (𝑐 = 𝐶 → (𝐵 +no 𝑐) = (𝐵 +no 𝐶))
44	43	oveq2d 7440	. . 3 ⊢ (𝑐 = 𝐶 → (𝐴 ·no (𝐵 +no 𝑐)) = (𝐴 ·no (𝐵 +no 𝐶)))
45		oveq2 7432	. . . 4 ⊢ (𝑐 = 𝐶 → (𝐴 ·no 𝑐) = (𝐴 ·no 𝐶))
46	45	oveq2d 7440	. . 3 ⊢ (𝑐 = 𝐶 → ((𝐴 ·no 𝐵) +no (𝐴 ·no 𝑐)) = ((𝐴 ·no 𝐵) +no (𝐴 ·no 𝐶)))
47	44, 46	eqeq12d 2776	. 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 37096	. . . 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 37094	. . . 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 3926	. . 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 8687	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 1569   ∈ wcel 2144  ∀wral 3076  Oncon0 6365  (class class class)co 7424   +no cnadd 8682   ·_no cnmul 37060

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-ot 4858  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6307  df-ord 6368  df-on 6369  df-suc 6371  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7427  df-oprab 7428  df-mpo 7429  df-1st 8005  df-2nd 8006  df-frecs 8307  df-nadd 8683  df-nmul 37061

This theorem is used by:  nadddid  37098