Theorem nnadddir 42305

Description: Right-distributivity for natural numbers without ax-mulcom 11219. (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 7439	. . . . . 6 ⊢ (𝑥 = 1 → ((𝐴 + 𝐵) · 𝑥) = ((𝐴 + 𝐵) · 1))
2		oveq2 7439	. . . . . . 7 ⊢ (𝑥 = 1 → (𝐴 · 𝑥) = (𝐴 · 1))
3		oveq2 7439	. . . . . . 7 ⊢ (𝑥 = 1 → (𝐵 · 𝑥) = (𝐵 · 1))
4	2, 3	oveq12d 7449	. . . . . 6 ⊢ (𝑥 = 1 → ((𝐴 · 𝑥) + (𝐵 · 𝑥)) = ((𝐴 · 1) + (𝐵 · 1)))
5	1, 4	eqeq12d 2753	. . . . 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 7439	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝐴 + 𝐵) · 𝑥) = ((𝐴 + 𝐵) · 𝑦))
8		oveq2 7439	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐴 · 𝑥) = (𝐴 · 𝑦))
9		oveq2 7439	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐵 · 𝑥) = (𝐵 · 𝑦))
10	8, 9	oveq12d 7449	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝐴 · 𝑥) + (𝐵 · 𝑥)) = ((𝐴 · 𝑦) + (𝐵 · 𝑦)))
11	7, 10	eqeq12d 2753	. . . . 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 7439	. . . . . 6 ⊢ (𝑥 = (𝑦 + 1) → ((𝐴 + 𝐵) · 𝑥) = ((𝐴 + 𝐵) · (𝑦 + 1)))
14		oveq2 7439	. . . . . . 7 ⊢ (𝑥 = (𝑦 + 1) → (𝐴 · 𝑥) = (𝐴 · (𝑦 + 1)))
15		oveq2 7439	. . . . . . 7 ⊢ (𝑥 = (𝑦 + 1) → (𝐵 · 𝑥) = (𝐵 · (𝑦 + 1)))
16	14, 15	oveq12d 7449	. . . . . 6 ⊢ (𝑥 = (𝑦 + 1) → ((𝐴 · 𝑥) + (𝐵 · 𝑥)) = ((𝐴 · (𝑦 + 1)) + (𝐵 · (𝑦 + 1))))
17	13, 16	eqeq12d 2753	. . . . 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 7439	. . . . . 6 ⊢ (𝑥 = 𝐶 → ((𝐴 + 𝐵) · 𝑥) = ((𝐴 + 𝐵) · 𝐶))
20		oveq2 7439	. . . . . . 7 ⊢ (𝑥 = 𝐶 → (𝐴 · 𝑥) = (𝐴 · 𝐶))
21		oveq2 7439	. . . . . . 7 ⊢ (𝑥 = 𝐶 → (𝐵 · 𝑥) = (𝐵 · 𝐶))
22	20, 21	oveq12d 7449	. . . . . 6 ⊢ (𝑥 = 𝐶 → ((𝐴 · 𝑥) + (𝐵 · 𝑥)) = ((𝐴 · 𝐶) + (𝐵 · 𝐶)))
23	19, 22	eqeq12d 2753	. . . . 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 12289	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 + 𝐵) ∈ ℕ)
26	25	nnred 12281	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 + 𝐵) ∈ ℝ)
27		ax-1rid 11225	. . . . . 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 12273	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℝ)
30		ax-1rid 11225	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (𝐴 · 1) = 𝐴)
31	29, 30	syl 17	. . . . . 6 ⊢ (𝐴 ∈ ℕ → (𝐴 · 1) = 𝐴)
32		nnre 12273	. . . . . . 7 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℝ)
33		ax-1rid 11225	. . . . . . 7 ⊢ (𝐵 ∈ ℝ → (𝐵 · 1) = 𝐵)
34	32, 33	syl 17	. . . . . 6 ⊢ (𝐵 ∈ ℕ → (𝐵 · 1) = 𝐵)
35	31, 34	oveqan12d 7450	. . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝐴 · 1) + (𝐵 · 1)) = (𝐴 + 𝐵))
36	28, 35	eqtr4d 2780	. . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝐴 + 𝐵) · 1) = ((𝐴 · 1) + (𝐵 · 1)))
37		simp2l 1200	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ ((𝐴 + 𝐵) · 𝑦) = ((𝐴 · 𝑦) + (𝐵 · 𝑦))) → 𝐴 ∈ ℕ)
38		simp2r 1201	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ ((𝐴 + 𝐵) · 𝑦) = ((𝐴 · 𝑦) + (𝐵 · 𝑦))) → 𝐵 ∈ ℕ)
39	37, 38	nnaddcld 12318	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ ((𝐴 + 𝐵) · 𝑦) = ((𝐴 · 𝑦) + (𝐵 · 𝑦))) → (𝐴 + 𝐵) ∈ ℕ)
40	39	nncnd 12282	. . . . . . . 8 ⊢ ((𝑦 ∈ ℕ ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ ((𝐴 + 𝐵) · 𝑦) = ((𝐴 · 𝑦) + (𝐵 · 𝑦))) → (𝐴 + 𝐵) ∈ ℂ)
41		simp1 1137	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ ((𝐴 + 𝐵) · 𝑦) = ((𝐴 · 𝑦) + (𝐵 · 𝑦))) → 𝑦 ∈ ℕ)
42	41	nncnd 12282	. . . . . . . 8 ⊢ ((𝑦 ∈ ℕ ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ ((𝐴 + 𝐵) · 𝑦) = ((𝐴 · 𝑦) + (𝐵 · 𝑦))) → 𝑦 ∈ ℂ)
43		1cnd 11256	. . . . . . . 8 ⊢ ((𝑦 ∈ ℕ ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ ((𝐴 + 𝐵) · 𝑦) = ((𝐴 · 𝑦) + (𝐵 · 𝑦))) → 1 ∈ ℂ)
44	40, 42, 43	adddid 11285	. . . . . . 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 12281	. . . . . . . . . . . 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 7447	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ ((𝐴 + 𝐵) · 𝑦) = ((𝐴 · 𝑦) + (𝐵 · 𝑦))) → ((𝐴 · 𝑦) + (𝐴 · 1)) = ((𝐴 · 𝑦) + 𝐴))
48	38	nnred 12281	. . . . . . . . . . . 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 7447	. . . . . . . . . 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 7449	. . . . . . . . 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 12319	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℕ ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ ((𝐴 + 𝐵) · 𝑦) = ((𝐴 · 𝑦) + (𝐵 · 𝑦))) → (𝐴 · 𝑦) ∈ ℕ)
53	52	nncnd 12282	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ ((𝐴 + 𝐵) · 𝑦) = ((𝐴 · 𝑦) + (𝐵 · 𝑦))) → (𝐴 · 𝑦) ∈ ℂ)
54	37	nncnd 12282	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ ((𝐴 + 𝐵) · 𝑦) = ((𝐴 · 𝑦) + (𝐵 · 𝑦))) → 𝐴 ∈ ℂ)
55	38, 41	nnmulcld 12319	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ℕ ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ ((𝐴 + 𝐵) · 𝑦) = ((𝐴 · 𝑦) + (𝐵 · 𝑦))) → (𝐵 · 𝑦) ∈ ℕ)
56	55, 38	nnaddcld 12318	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℕ ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ ((𝐴 + 𝐵) · 𝑦) = ((𝐴 · 𝑦) + (𝐵 · 𝑦))) → ((𝐵 · 𝑦) + 𝐵) ∈ ℕ)
57	56	nncnd 12282	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ ((𝐴 + 𝐵) · 𝑦) = ((𝐴 · 𝑦) + (𝐵 · 𝑦))) → ((𝐵 · 𝑦) + 𝐵) ∈ ℂ)
58	53, 54, 57	addassd 11283	. . . . . . . . . 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 12282	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℕ ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ ((𝐴 + 𝐵) · 𝑦) = ((𝐴 · 𝑦) + (𝐵 · 𝑦))) → (𝐵 · 𝑦) ∈ ℂ)
60	38	nncnd 12282	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℕ ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ ((𝐴 + 𝐵) · 𝑦) = ((𝐴 · 𝑦) + (𝐵 · 𝑦))) → 𝐵 ∈ ℂ)
61	54, 59, 60	addassd 11283	. . . . . . . . . . 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 7447	. . . . . . . . . 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 11283	. . . . . . . . . . . 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 7447	. . . . . . . . . . 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 42303	. . . . . . . . . . . . . 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 7446	. . . . . . . . . . . 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 7447	. . . . . . . . . . 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 11283	. . . . . . . . . . 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 2787	. . . . . . . . . 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 2783	. . . . . . . . 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 2777	. . . . . . . 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 11285	. . . . . . . . 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 11285	. . . . . . . . 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 7449	. . . . . . . 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 1139	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ ((𝐴 + 𝐵) · 𝑦) = ((𝐴 · 𝑦) + (𝐵 · 𝑦))) → ((𝐴 + 𝐵) · 𝑦) = ((𝐴 · 𝑦) + (𝐵 · 𝑦)))
77	39	nnred 12281	. . . . . . . . . 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 7449	. . . . . . . 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 2787	. . . . . . 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 2780	. . . . . 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 1120	. . . . 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 12284	. . 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 1118	1 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((𝐴 + 𝐵) · 𝐶) = ((𝐴 · 𝐶) + (𝐵 · 𝐶)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108  (class class class)co 7431  ℝcr 11154  1c1 11156   + caddc 11158   · cmul 11160  ℕcn 12266

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-addass 11220  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rrecex 11227  ax-cnre 11228

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-nn 12267

This theorem is referenced by:  nnmulcom  42307