Theorem nnadddir 41181

Description: Right-distributivity for natural numbers without ax-mulcom 11170. (Contributed by SN, 5-Feb-2024.)

Assertion

Ref	Expression
nnadddir	⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((𝐴 + 𝐵) · 𝐶) = ((𝐴 · 𝐶) + (𝐵 · 𝐶)))

Proof of Theorem nnadddir

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 7413	. . . . . 6 ⊢ (𝑥 = 1 → ((𝐴 + 𝐵) · 𝑥) = ((𝐴 + 𝐵) · 1))
2		oveq2 7413	. . . . . . 7 ⊢ (𝑥 = 1 → (𝐴 · 𝑥) = (𝐴 · 1))
3		oveq2 7413	. . . . . . 7 ⊢ (𝑥 = 1 → (𝐵 · 𝑥) = (𝐵 · 1))
4	2, 3	oveq12d 7423	. . . . . 6 ⊢ (𝑥 = 1 → ((𝐴 · 𝑥) + (𝐵 · 𝑥)) = ((𝐴 · 1) + (𝐵 · 1)))
5	1, 4	eqeq12d 2748	. . . . 5 ⊢ (𝑥 = 1 → (((𝐴 + 𝐵) · 𝑥) = ((𝐴 · 𝑥) + (𝐵 · 𝑥)) ↔ ((𝐴 + 𝐵) · 1) = ((𝐴 · 1) + (𝐵 · 1))))
6	5	imbi2d 340	. . . 4 ⊢ (𝑥 = 1 → (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝐴 + 𝐵) · 𝑥) = ((𝐴 · 𝑥) + (𝐵 · 𝑥))) ↔ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝐴 + 𝐵) · 1) = ((𝐴 · 1) + (𝐵 · 1)))))
7		oveq2 7413	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝐴 + 𝐵) · 𝑥) = ((𝐴 + 𝐵) · 𝑦))
8		oveq2 7413	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐴 · 𝑥) = (𝐴 · 𝑦))
9		oveq2 7413	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐵 · 𝑥) = (𝐵 · 𝑦))
10	8, 9	oveq12d 7423	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝐴 · 𝑥) + (𝐵 · 𝑥)) = ((𝐴 · 𝑦) + (𝐵 · 𝑦)))
11	7, 10	eqeq12d 2748	. . . . 5 ⊢ (𝑥 = 𝑦 → (((𝐴 + 𝐵) · 𝑥) = ((𝐴 · 𝑥) + (𝐵 · 𝑥)) ↔ ((𝐴 + 𝐵) · 𝑦) = ((𝐴 · 𝑦) + (𝐵 · 𝑦))))
12	11	imbi2d 340	. . . 4 ⊢ (𝑥 = 𝑦 → (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝐴 + 𝐵) · 𝑥) = ((𝐴 · 𝑥) + (𝐵 · 𝑥))) ↔ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝐴 + 𝐵) · 𝑦) = ((𝐴 · 𝑦) + (𝐵 · 𝑦)))))
13		oveq2 7413	. . . . . 6 ⊢ (𝑥 = (𝑦 + 1) → ((𝐴 + 𝐵) · 𝑥) = ((𝐴 + 𝐵) · (𝑦 + 1)))
14		oveq2 7413	. . . . . . 7 ⊢ (𝑥 = (𝑦 + 1) → (𝐴 · 𝑥) = (𝐴 · (𝑦 + 1)))
15		oveq2 7413	. . . . . . 7 ⊢ (𝑥 = (𝑦 + 1) → (𝐵 · 𝑥) = (𝐵 · (𝑦 + 1)))
16	14, 15	oveq12d 7423	. . . . . 6 ⊢ (𝑥 = (𝑦 + 1) → ((𝐴 · 𝑥) + (𝐵 · 𝑥)) = ((𝐴 · (𝑦 + 1)) + (𝐵 · (𝑦 + 1))))
17	13, 16	eqeq12d 2748	. . . . 5 ⊢ (𝑥 = (𝑦 + 1) → (((𝐴 + 𝐵) · 𝑥) = ((𝐴 · 𝑥) + (𝐵 · 𝑥)) ↔ ((𝐴 + 𝐵) · (𝑦 + 1)) = ((𝐴 · (𝑦 + 1)) + (𝐵 · (𝑦 + 1)))))
18	17	imbi2d 340	. . . 4 ⊢ (𝑥 = (𝑦 + 1) → (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝐴 + 𝐵) · 𝑥) = ((𝐴 · 𝑥) + (𝐵 · 𝑥))) ↔ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝐴 + 𝐵) · (𝑦 + 1)) = ((𝐴 · (𝑦 + 1)) + (𝐵 · (𝑦 + 1))))))
19		oveq2 7413	. . . . . 6 ⊢ (𝑥 = 𝐶 → ((𝐴 + 𝐵) · 𝑥) = ((𝐴 + 𝐵) · 𝐶))
20		oveq2 7413	. . . . . . 7 ⊢ (𝑥 = 𝐶 → (𝐴 · 𝑥) = (𝐴 · 𝐶))
21		oveq2 7413	. . . . . . 7 ⊢ (𝑥 = 𝐶 → (𝐵 · 𝑥) = (𝐵 · 𝐶))
22	20, 21	oveq12d 7423	. . . . . 6 ⊢ (𝑥 = 𝐶 → ((𝐴 · 𝑥) + (𝐵 · 𝑥)) = ((𝐴 · 𝐶) + (𝐵 · 𝐶)))
23	19, 22	eqeq12d 2748	. . . . 5 ⊢ (𝑥 = 𝐶 → (((𝐴 + 𝐵) · 𝑥) = ((𝐴 · 𝑥) + (𝐵 · 𝑥)) ↔ ((𝐴 + 𝐵) · 𝐶) = ((𝐴 · 𝐶) + (𝐵 · 𝐶))))
24	23	imbi2d 340	. . . 4 ⊢ (𝑥 = 𝐶 → (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝐴 + 𝐵) · 𝑥) = ((𝐴 · 𝑥) + (𝐵 · 𝑥))) ↔ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝐴 + 𝐵) · 𝐶) = ((𝐴 · 𝐶) + (𝐵 · 𝐶)))))
25		nnaddcl 12231	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 + 𝐵) ∈ ℕ)
26	25	nnred 12223	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 + 𝐵) ∈ ℝ)
27		ax-1rid 11176	. . . . . 6 ⊢ ((𝐴 + 𝐵) ∈ ℝ → ((𝐴 + 𝐵) · 1) = (𝐴 + 𝐵))
28	26, 27	syl 17	. . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝐴 + 𝐵) · 1) = (𝐴 + 𝐵))
29		nnre 12215	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℝ)
30		ax-1rid 11176	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (𝐴 · 1) = 𝐴)
31	29, 30	syl 17	. . . . . 6 ⊢ (𝐴 ∈ ℕ → (𝐴 · 1) = 𝐴)
32		nnre 12215	. . . . . . 7 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℝ)
33		ax-1rid 11176	. . . . . . 7 ⊢ (𝐵 ∈ ℝ → (𝐵 · 1) = 𝐵)
34	32, 33	syl 17	. . . . . 6 ⊢ (𝐵 ∈ ℕ → (𝐵 · 1) = 𝐵)
35	31, 34	oveqan12d 7424	. . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝐴 · 1) + (𝐵 · 1)) = (𝐴 + 𝐵))
36	28, 35	eqtr4d 2775	. . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝐴 + 𝐵) · 1) = ((𝐴 · 1) + (𝐵 · 1)))
37		simp2l 1199	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ ((𝐴 + 𝐵) · 𝑦) = ((𝐴 · 𝑦) + (𝐵 · 𝑦))) → 𝐴 ∈ ℕ)
38		simp2r 1200	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ ((𝐴 + 𝐵) · 𝑦) = ((𝐴 · 𝑦) + (𝐵 · 𝑦))) → 𝐵 ∈ ℕ)
39	37, 38	nnaddcld 12260	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ ((𝐴 + 𝐵) · 𝑦) = ((𝐴 · 𝑦) + (𝐵 · 𝑦))) → (𝐴 + 𝐵) ∈ ℕ)
40	39	nncnd 12224	. . . . . . . 8 ⊢ ((𝑦 ∈ ℕ ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ ((𝐴 + 𝐵) · 𝑦) = ((𝐴 · 𝑦) + (𝐵 · 𝑦))) → (𝐴 + 𝐵) ∈ ℂ)
41		simp1 1136	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ ((𝐴 + 𝐵) · 𝑦) = ((𝐴 · 𝑦) + (𝐵 · 𝑦))) → 𝑦 ∈ ℕ)
42	41	nncnd 12224	. . . . . . . 8 ⊢ ((𝑦 ∈ ℕ ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ ((𝐴 + 𝐵) · 𝑦) = ((𝐴 · 𝑦) + (𝐵 · 𝑦))) → 𝑦 ∈ ℂ)
43		1cnd 11205	. . . . . . . 8 ⊢ ((𝑦 ∈ ℕ ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ ((𝐴 + 𝐵) · 𝑦) = ((𝐴 · 𝑦) + (𝐵 · 𝑦))) → 1 ∈ ℂ)
44	40, 42, 43	adddid 11234	. . . . . . 7 ⊢ ((𝑦 ∈ ℕ ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ ((𝐴 + 𝐵) · 𝑦) = ((𝐴 · 𝑦) + (𝐵 · 𝑦))) → ((𝐴 + 𝐵) · (𝑦 + 1)) = (((𝐴 + 𝐵) · 𝑦) + ((𝐴 + 𝐵) · 1)))
45	37	nnred 12223	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℕ ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ ((𝐴 + 𝐵) · 𝑦) = ((𝐴 · 𝑦) + (𝐵 · 𝑦))) → 𝐴 ∈ ℝ)
46	45, 30	syl 17	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ ((𝐴 + 𝐵) · 𝑦) = ((𝐴 · 𝑦) + (𝐵 · 𝑦))) → (𝐴 · 1) = 𝐴)
47	46	oveq2d 7421	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ ((𝐴 + 𝐵) · 𝑦) = ((𝐴 · 𝑦) + (𝐵 · 𝑦))) → ((𝐴 · 𝑦) + (𝐴 · 1)) = ((𝐴 · 𝑦) + 𝐴))
48	38	nnred 12223	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℕ ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ ((𝐴 + 𝐵) · 𝑦) = ((𝐴 · 𝑦) + (𝐵 · 𝑦))) → 𝐵 ∈ ℝ)
49	48, 33	syl 17	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ ((𝐴 + 𝐵) · 𝑦) = ((𝐴 · 𝑦) + (𝐵 · 𝑦))) → (𝐵 · 1) = 𝐵)
50	49	oveq2d 7421	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ ((𝐴 + 𝐵) · 𝑦) = ((𝐴 · 𝑦) + (𝐵 · 𝑦))) → ((𝐵 · 𝑦) + (𝐵 · 1)) = ((𝐵 · 𝑦) + 𝐵))
51	47, 50	oveq12d 7423	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ ((𝐴 + 𝐵) · 𝑦) = ((𝐴 · 𝑦) + (𝐵 · 𝑦))) → (((𝐴 · 𝑦) + (𝐴 · 1)) + ((𝐵 · 𝑦) + (𝐵 · 1))) = (((𝐴 · 𝑦) + 𝐴) + ((𝐵 · 𝑦) + 𝐵)))
52	37, 41	nnmulcld 12261	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℕ ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ ((𝐴 + 𝐵) · 𝑦) = ((𝐴 · 𝑦) + (𝐵 · 𝑦))) → (𝐴 · 𝑦) ∈ ℕ)
53	52	nncnd 12224	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ ((𝐴 + 𝐵) · 𝑦) = ((𝐴 · 𝑦) + (𝐵 · 𝑦))) → (𝐴 · 𝑦) ∈ ℂ)
54	37	nncnd 12224	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ ((𝐴 + 𝐵) · 𝑦) = ((𝐴 · 𝑦) + (𝐵 · 𝑦))) → 𝐴 ∈ ℂ)
55	38, 41	nnmulcld 12261	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ℕ ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ ((𝐴 + 𝐵) · 𝑦) = ((𝐴 · 𝑦) + (𝐵 · 𝑦))) → (𝐵 · 𝑦) ∈ ℕ)
56	55, 38	nnaddcld 12260	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℕ ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ ((𝐴 + 𝐵) · 𝑦) = ((𝐴 · 𝑦) + (𝐵 · 𝑦))) → ((𝐵 · 𝑦) + 𝐵) ∈ ℕ)
57	56	nncnd 12224	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ ((𝐴 + 𝐵) · 𝑦) = ((𝐴 · 𝑦) + (𝐵 · 𝑦))) → ((𝐵 · 𝑦) + 𝐵) ∈ ℂ)
58	53, 54, 57	addassd 11232	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ ((𝐴 + 𝐵) · 𝑦) = ((𝐴 · 𝑦) + (𝐵 · 𝑦))) → (((𝐴 · 𝑦) + 𝐴) + ((𝐵 · 𝑦) + 𝐵)) = ((𝐴 · 𝑦) + (𝐴 + ((𝐵 · 𝑦) + 𝐵))))
59	55	nncnd 12224	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℕ ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ ((𝐴 + 𝐵) · 𝑦) = ((𝐴 · 𝑦) + (𝐵 · 𝑦))) → (𝐵 · 𝑦) ∈ ℂ)
60	38	nncnd 12224	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℕ ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ ((𝐴 + 𝐵) · 𝑦) = ((𝐴 · 𝑦) + (𝐵 · 𝑦))) → 𝐵 ∈ ℂ)
61	54, 59, 60	addassd 11232	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ ((𝐴 + 𝐵) · 𝑦) = ((𝐴 · 𝑦) + (𝐵 · 𝑦))) → ((𝐴 + (𝐵 · 𝑦)) + 𝐵) = (𝐴 + ((𝐵 · 𝑦) + 𝐵)))
62	61	oveq2d 7421	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ ((𝐴 + 𝐵) · 𝑦) = ((𝐴 · 𝑦) + (𝐵 · 𝑦))) → ((𝐴 · 𝑦) + ((𝐴 + (𝐵 · 𝑦)) + 𝐵)) = ((𝐴 · 𝑦) + (𝐴 + ((𝐵 · 𝑦) + 𝐵))))
63	59, 54, 60	addassd 11232	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℕ ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ ((𝐴 + 𝐵) · 𝑦) = ((𝐴 · 𝑦) + (𝐵 · 𝑦))) → (((𝐵 · 𝑦) + 𝐴) + 𝐵) = ((𝐵 · 𝑦) + (𝐴 + 𝐵)))
64	63	oveq2d 7421	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ ((𝐴 + 𝐵) · 𝑦) = ((𝐴 · 𝑦) + (𝐵 · 𝑦))) → ((𝐴 · 𝑦) + (((𝐵 · 𝑦) + 𝐴) + 𝐵)) = ((𝐴 · 𝑦) + ((𝐵 · 𝑦) + (𝐴 + 𝐵))))
65		nnaddcom 41179	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℕ ∧ (𝐵 · 𝑦) ∈ ℕ) → (𝐴 + (𝐵 · 𝑦)) = ((𝐵 · 𝑦) + 𝐴))
66	37, 55, 65	syl2anc 584	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ℕ ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ ((𝐴 + 𝐵) · 𝑦) = ((𝐴 · 𝑦) + (𝐵 · 𝑦))) → (𝐴 + (𝐵 · 𝑦)) = ((𝐵 · 𝑦) + 𝐴))
67	66	oveq1d 7420	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℕ ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ ((𝐴 + 𝐵) · 𝑦) = ((𝐴 · 𝑦) + (𝐵 · 𝑦))) → ((𝐴 + (𝐵 · 𝑦)) + 𝐵) = (((𝐵 · 𝑦) + 𝐴) + 𝐵))
68	67	oveq2d 7421	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ ((𝐴 + 𝐵) · 𝑦) = ((𝐴 · 𝑦) + (𝐵 · 𝑦))) → ((𝐴 · 𝑦) + ((𝐴 + (𝐵 · 𝑦)) + 𝐵)) = ((𝐴 · 𝑦) + (((𝐵 · 𝑦) + 𝐴) + 𝐵)))
69	53, 59, 40	addassd 11232	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ ((𝐴 + 𝐵) · 𝑦) = ((𝐴 · 𝑦) + (𝐵 · 𝑦))) → (((𝐴 · 𝑦) + (𝐵 · 𝑦)) + (𝐴 + 𝐵)) = ((𝐴 · 𝑦) + ((𝐵 · 𝑦) + (𝐴 + 𝐵))))
70	64, 68, 69	3eqtr4d 2782	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ ((𝐴 + 𝐵) · 𝑦) = ((𝐴 · 𝑦) + (𝐵 · 𝑦))) → ((𝐴 · 𝑦) + ((𝐴 + (𝐵 · 𝑦)) + 𝐵)) = (((𝐴 · 𝑦) + (𝐵 · 𝑦)) + (𝐴 + 𝐵)))
71	58, 62, 70	3eqtr2d 2778	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ ((𝐴 + 𝐵) · 𝑦) = ((𝐴 · 𝑦) + (𝐵 · 𝑦))) → (((𝐴 · 𝑦) + 𝐴) + ((𝐵 · 𝑦) + 𝐵)) = (((𝐴 · 𝑦) + (𝐵 · 𝑦)) + (𝐴 + 𝐵)))
72	51, 71	eqtrd 2772	. . . . . . . 8 ⊢ ((𝑦 ∈ ℕ ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ ((𝐴 + 𝐵) · 𝑦) = ((𝐴 · 𝑦) + (𝐵 · 𝑦))) → (((𝐴 · 𝑦) + (𝐴 · 1)) + ((𝐵 · 𝑦) + (𝐵 · 1))) = (((𝐴 · 𝑦) + (𝐵 · 𝑦)) + (𝐴 + 𝐵)))
73	54, 42, 43	adddid 11234	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ ((𝐴 + 𝐵) · 𝑦) = ((𝐴 · 𝑦) + (𝐵 · 𝑦))) → (𝐴 · (𝑦 + 1)) = ((𝐴 · 𝑦) + (𝐴 · 1)))
74	60, 42, 43	adddid 11234	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ ((𝐴 + 𝐵) · 𝑦) = ((𝐴 · 𝑦) + (𝐵 · 𝑦))) → (𝐵 · (𝑦 + 1)) = ((𝐵 · 𝑦) + (𝐵 · 1)))
75	73, 74	oveq12d 7423	. . . . . . . 8 ⊢ ((𝑦 ∈ ℕ ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ ((𝐴 + 𝐵) · 𝑦) = ((𝐴 · 𝑦) + (𝐵 · 𝑦))) → ((𝐴 · (𝑦 + 1)) + (𝐵 · (𝑦 + 1))) = (((𝐴 · 𝑦) + (𝐴 · 1)) + ((𝐵 · 𝑦) + (𝐵 · 1))))
76		simp3 1138	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ ((𝐴 + 𝐵) · 𝑦) = ((𝐴 · 𝑦) + (𝐵 · 𝑦))) → ((𝐴 + 𝐵) · 𝑦) = ((𝐴 · 𝑦) + (𝐵 · 𝑦)))
77	39	nnred 12223	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ ((𝐴 + 𝐵) · 𝑦) = ((𝐴 · 𝑦) + (𝐵 · 𝑦))) → (𝐴 + 𝐵) ∈ ℝ)
78	77, 27	syl 17	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ ((𝐴 + 𝐵) · 𝑦) = ((𝐴 · 𝑦) + (𝐵 · 𝑦))) → ((𝐴 + 𝐵) · 1) = (𝐴 + 𝐵))
79	76, 78	oveq12d 7423	. . . . . . . 8 ⊢ ((𝑦 ∈ ℕ ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ ((𝐴 + 𝐵) · 𝑦) = ((𝐴 · 𝑦) + (𝐵 · 𝑦))) → (((𝐴 + 𝐵) · 𝑦) + ((𝐴 + 𝐵) · 1)) = (((𝐴 · 𝑦) + (𝐵 · 𝑦)) + (𝐴 + 𝐵)))
80	72, 75, 79	3eqtr4d 2782	. . . . . . 7 ⊢ ((𝑦 ∈ ℕ ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ ((𝐴 + 𝐵) · 𝑦) = ((𝐴 · 𝑦) + (𝐵 · 𝑦))) → ((𝐴 · (𝑦 + 1)) + (𝐵 · (𝑦 + 1))) = (((𝐴 + 𝐵) · 𝑦) + ((𝐴 + 𝐵) · 1)))
81	44, 80	eqtr4d 2775	. . . . . 6 ⊢ ((𝑦 ∈ ℕ ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ ((𝐴 + 𝐵) · 𝑦) = ((𝐴 · 𝑦) + (𝐵 · 𝑦))) → ((𝐴 + 𝐵) · (𝑦 + 1)) = ((𝐴 · (𝑦 + 1)) + (𝐵 · (𝑦 + 1))))
82	81	3exp 1119	. . . . 5 ⊢ (𝑦 ∈ ℕ → ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (((𝐴 + 𝐵) · 𝑦) = ((𝐴 · 𝑦) + (𝐵 · 𝑦)) → ((𝐴 + 𝐵) · (𝑦 + 1)) = ((𝐴 · (𝑦 + 1)) + (𝐵 · (𝑦 + 1))))))
83	82	a2d 29	. . . 4 ⊢ (𝑦 ∈ ℕ → (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝐴 + 𝐵) · 𝑦) = ((𝐴 · 𝑦) + (𝐵 · 𝑦))) → ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝐴 + 𝐵) · (𝑦 + 1)) = ((𝐴 · (𝑦 + 1)) + (𝐵 · (𝑦 + 1))))))
84	6, 12, 18, 24, 36, 83	nnind 12226	. . 3 ⊢ (𝐶 ∈ ℕ → ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝐴 + 𝐵) · 𝐶) = ((𝐴 · 𝐶) + (𝐵 · 𝐶))))
85	84	com12 32	. 2 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐶 ∈ ℕ → ((𝐴 + 𝐵) · 𝐶) = ((𝐴 · 𝐶) + (𝐵 · 𝐶))))
86	85	3impia 1117	1 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((𝐴 + 𝐵) · 𝐶) = ((𝐴 · 𝐶) + (𝐵 · 𝐶)))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  (class class class)co 7405  ℝcr 11105  1c1 11107   + caddc 11109   · cmul 11111  ℕcn 12208

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7721  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-addass 11171  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rrecex 11178  ax-cnre 11179

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7408  df-om 7852  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-nn 12209

This theorem is referenced by:  nnmulcom  41183