Theorem nnadddir 12292

Description: Right-distributivity for natural numbers without ax-mulcom 11164. (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 7419	. . . . . 6 ⊢ (𝑥 = 1 → ((𝐴 + 𝐵) · 𝑥) = ((𝐴 + 𝐵) · 1))
2		oveq2 7419	. . . . . . 7 ⊢ (𝑥 = 1 → (𝐴 · 𝑥) = (𝐴 · 1))
3		oveq2 7419	. . . . . . 7 ⊢ (𝑥 = 1 → (𝐵 · 𝑥) = (𝐵 · 1))
4	2, 3	oveq12d 7429	. . . . . 6 ⊢ (𝑥 = 1 → ((𝐴 · 𝑥) + (𝐵 · 𝑥)) = ((𝐴 · 1) + (𝐵 · 1)))
5	1, 4	eqeq12d 2785	. . . . 5 ⊢ (𝑥 = 1 → (((𝐴 + 𝐵) · 𝑥) = ((𝐴 · 𝑥) + (𝐵 · 𝑥)) ↔ ((𝐴 + 𝐵) · 1) = ((𝐴 · 1) + (𝐵 · 1))))
6	5	imbi2d 343	. . . 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 7419	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝐴 + 𝐵) · 𝑥) = ((𝐴 + 𝐵) · 𝑦))
8		oveq2 7419	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐴 · 𝑥) = (𝐴 · 𝑦))
9		oveq2 7419	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐵 · 𝑥) = (𝐵 · 𝑦))
10	8, 9	oveq12d 7429	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝐴 · 𝑥) + (𝐵 · 𝑥)) = ((𝐴 · 𝑦) + (𝐵 · 𝑦)))
11	7, 10	eqeq12d 2785	. . . . 5 ⊢ (𝑥 = 𝑦 → (((𝐴 + 𝐵) · 𝑥) = ((𝐴 · 𝑥) + (𝐵 · 𝑥)) ↔ ((𝐴 + 𝐵) · 𝑦) = ((𝐴 · 𝑦) + (𝐵 · 𝑦))))
12	11	imbi2d 343	. . . 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 7419	. . . . . 6 ⊢ (𝑥 = (𝑦 + 1) → ((𝐴 + 𝐵) · 𝑥) = ((𝐴 + 𝐵) · (𝑦 + 1)))
14		oveq2 7419	. . . . . . 7 ⊢ (𝑥 = (𝑦 + 1) → (𝐴 · 𝑥) = (𝐴 · (𝑦 + 1)))
15		oveq2 7419	. . . . . . 7 ⊢ (𝑥 = (𝑦 + 1) → (𝐵 · 𝑥) = (𝐵 · (𝑦 + 1)))
16	14, 15	oveq12d 7429	. . . . . 6 ⊢ (𝑥 = (𝑦 + 1) → ((𝐴 · 𝑥) + (𝐵 · 𝑥)) = ((𝐴 · (𝑦 + 1)) + (𝐵 · (𝑦 + 1))))
17	13, 16	eqeq12d 2785	. . . . 5 ⊢ (𝑥 = (𝑦 + 1) → (((𝐴 + 𝐵) · 𝑥) = ((𝐴 · 𝑥) + (𝐵 · 𝑥)) ↔ ((𝐴 + 𝐵) · (𝑦 + 1)) = ((𝐴 · (𝑦 + 1)) + (𝐵 · (𝑦 + 1)))))
18	17	imbi2d 343	. . . 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 7419	. . . . . 6 ⊢ (𝑥 = 𝐶 → ((𝐴 + 𝐵) · 𝑥) = ((𝐴 + 𝐵) · 𝐶))
20		oveq2 7419	. . . . . . 7 ⊢ (𝑥 = 𝐶 → (𝐴 · 𝑥) = (𝐴 · 𝐶))
21		oveq2 7419	. . . . . . 7 ⊢ (𝑥 = 𝐶 → (𝐵 · 𝑥) = (𝐵 · 𝐶))
22	20, 21	oveq12d 7429	. . . . . 6 ⊢ (𝑥 = 𝐶 → ((𝐴 · 𝑥) + (𝐵 · 𝑥)) = ((𝐴 · 𝐶) + (𝐵 · 𝐶)))
23	19, 22	eqeq12d 2785	. . . . 5 ⊢ (𝑥 = 𝐶 → (((𝐴 + 𝐵) · 𝑥) = ((𝐴 · 𝑥) + (𝐵 · 𝑥)) ↔ ((𝐴 + 𝐵) · 𝐶) = ((𝐴 · 𝐶) + (𝐵 · 𝐶))))
24	23	imbi2d 343	. . . 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 12256	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 + 𝐵) ∈ ℕ)
26	25	nnred 12248	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 + 𝐵) ∈ ℝ)
27		ax-1rid 11170	. . . . . 6 ⊢ ((𝐴 + 𝐵) ∈ ℝ → ((𝐴 + 𝐵) · 1) = (𝐴 + 𝐵))
28	26, 27	syl 18	. . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝐴 + 𝐵) · 1) = (𝐴 + 𝐵))
29		nnre 12240	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℝ)
30		ax-1rid 11170	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (𝐴 · 1) = 𝐴)
31	29, 30	syl 18	. . . . . 6 ⊢ (𝐴 ∈ ℕ → (𝐴 · 1) = 𝐴)
32		nnre 12240	. . . . . . 7 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℝ)
33		ax-1rid 11170	. . . . . . 7 ⊢ (𝐵 ∈ ℝ → (𝐵 · 1) = 𝐵)
34	32, 33	syl 18	. . . . . 6 ⊢ (𝐵 ∈ ℕ → (𝐵 · 1) = 𝐵)
35	31, 34	oveqan12d 7430	. . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝐴 · 1) + (𝐵 · 1)) = (𝐴 + 𝐵))
36	28, 35	eqtr4d 2807	. . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝐴 + 𝐵) · 1) = ((𝐴 · 1) + (𝐵 · 1)))
37		simp2l 1216	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ ((𝐴 + 𝐵) · 𝑦) = ((𝐴 · 𝑦) + (𝐵 · 𝑦))) → 𝐴 ∈ ℕ)
38		simp2r 1217	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ ((𝐴 + 𝐵) · 𝑦) = ((𝐴 · 𝑦) + (𝐵 · 𝑦))) → 𝐵 ∈ ℕ)
39	37, 38	nnaddcld 12288	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ ((𝐴 + 𝐵) · 𝑦) = ((𝐴 · 𝑦) + (𝐵 · 𝑦))) → (𝐴 + 𝐵) ∈ ℕ)
40	39	nncnd 12249	. . . . . . . 8 ⊢ ((𝑦 ∈ ℕ ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ ((𝐴 + 𝐵) · 𝑦) = ((𝐴 · 𝑦) + (𝐵 · 𝑦))) → (𝐴 + 𝐵) ∈ ℂ)
41		simp1 1152	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ ((𝐴 + 𝐵) · 𝑦) = ((𝐴 · 𝑦) + (𝐵 · 𝑦))) → 𝑦 ∈ ℕ)
42	41	nncnd 12249	. . . . . . . 8 ⊢ ((𝑦 ∈ ℕ ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ ((𝐴 + 𝐵) · 𝑦) = ((𝐴 · 𝑦) + (𝐵 · 𝑦))) → 𝑦 ∈ ℂ)
43		1cnd 11202	. . . . . . . 8 ⊢ ((𝑦 ∈ ℕ ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ ((𝐴 + 𝐵) · 𝑦) = ((𝐴 · 𝑦) + (𝐵 · 𝑦))) → 1 ∈ ℂ)
44	40, 42, 43	adddid 11233	. . . . . . 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 12248	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℕ ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ ((𝐴 + 𝐵) · 𝑦) = ((𝐴 · 𝑦) + (𝐵 · 𝑦))) → 𝐴 ∈ ℝ)
46	45, 30	syl 18	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ ((𝐴 + 𝐵) · 𝑦) = ((𝐴 · 𝑦) + (𝐵 · 𝑦))) → (𝐴 · 1) = 𝐴)
47	46	oveq2d 7427	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ ((𝐴 + 𝐵) · 𝑦) = ((𝐴 · 𝑦) + (𝐵 · 𝑦))) → ((𝐴 · 𝑦) + (𝐴 · 1)) = ((𝐴 · 𝑦) + 𝐴))
48	38	nnred 12248	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℕ ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ ((𝐴 + 𝐵) · 𝑦) = ((𝐴 · 𝑦) + (𝐵 · 𝑦))) → 𝐵 ∈ ℝ)
49	48, 33	syl 18	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ ((𝐴 + 𝐵) · 𝑦) = ((𝐴 · 𝑦) + (𝐵 · 𝑦))) → (𝐵 · 1) = 𝐵)
50	49	oveq2d 7427	. . . . . . . . . 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 7429	. . . . . . . . 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 12289	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℕ ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ ((𝐴 + 𝐵) · 𝑦) = ((𝐴 · 𝑦) + (𝐵 · 𝑦))) → (𝐴 · 𝑦) ∈ ℕ)
53	52	nncnd 12249	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ ((𝐴 + 𝐵) · 𝑦) = ((𝐴 · 𝑦) + (𝐵 · 𝑦))) → (𝐴 · 𝑦) ∈ ℂ)
54	37	nncnd 12249	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ ((𝐴 + 𝐵) · 𝑦) = ((𝐴 · 𝑦) + (𝐵 · 𝑦))) → 𝐴 ∈ ℂ)
55	38, 41	nnmulcld 12289	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ℕ ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ ((𝐴 + 𝐵) · 𝑦) = ((𝐴 · 𝑦) + (𝐵 · 𝑦))) → (𝐵 · 𝑦) ∈ ℕ)
56	55, 38	nnaddcld 12288	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℕ ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ ((𝐴 + 𝐵) · 𝑦) = ((𝐴 · 𝑦) + (𝐵 · 𝑦))) → ((𝐵 · 𝑦) + 𝐵) ∈ ℕ)
57	56	nncnd 12249	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ ((𝐴 + 𝐵) · 𝑦) = ((𝐴 · 𝑦) + (𝐵 · 𝑦))) → ((𝐵 · 𝑦) + 𝐵) ∈ ℂ)
58	53, 54, 57	addassd 11231	. . . . . . . . . 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 12249	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℕ ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ ((𝐴 + 𝐵) · 𝑦) = ((𝐴 · 𝑦) + (𝐵 · 𝑦))) → (𝐵 · 𝑦) ∈ ℂ)
60	38	nncnd 12249	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℕ ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ ((𝐴 + 𝐵) · 𝑦) = ((𝐴 · 𝑦) + (𝐵 · 𝑦))) → 𝐵 ∈ ℂ)
61	54, 59, 60	addassd 11231	. . . . . . . . . . 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 7427	. . . . . . . . . 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 11231	. . . . . . . . . . . 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 7427	. . . . . . . . . . 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 12260	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℕ ∧ (𝐵 · 𝑦) ∈ ℕ) → (𝐴 + (𝐵 · 𝑦)) = ((𝐵 · 𝑦) + 𝐴))
66	37, 55, 65	syl2anc 595	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ℕ ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ ((𝐴 + 𝐵) · 𝑦) = ((𝐴 · 𝑦) + (𝐵 · 𝑦))) → (𝐴 + (𝐵 · 𝑦)) = ((𝐵 · 𝑦) + 𝐴))
67	66	oveq1d 7426	. . . . . . . . . . . 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 7427	. . . . . . . . . . 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 11231	. . . . . . . . . . 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 2814	. . . . . . . . . 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 2810	. . . . . . . . 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 2804	. . . . . . . 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 11233	. . . . . . . . 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 11233	. . . . . . . . 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 7429	. . . . . . . 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 1154	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ ((𝐴 + 𝐵) · 𝑦) = ((𝐴 · 𝑦) + (𝐵 · 𝑦))) → ((𝐴 + 𝐵) · 𝑦) = ((𝐴 · 𝑦) + (𝐵 · 𝑦)))
77	39	nnred 12248	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ ((𝐴 + 𝐵) · 𝑦) = ((𝐴 · 𝑦) + (𝐵 · 𝑦))) → (𝐴 + 𝐵) ∈ ℝ)
78	77, 27	syl 18	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ ((𝐴 + 𝐵) · 𝑦) = ((𝐴 · 𝑦) + (𝐵 · 𝑦))) → ((𝐴 + 𝐵) · 1) = (𝐴 + 𝐵))
79	76, 78	oveq12d 7429	. . . . . . . 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 2814	. . . . . . 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 2807	. . . . . 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 1135	. . . . 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 30	. . . 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 12251	. . 3 ⊢ (𝐶 ∈ ℕ → ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝐴 + 𝐵) · 𝐶) = ((𝐴 · 𝐶) + (𝐵 · 𝐶))))
85	84	com12 33	. 2 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐶 ∈ ℕ → ((𝐴 + 𝐵) · 𝐶) = ((𝐴 · 𝐶) + (𝐵 · 𝐶))))
86	85	3impia 1133	1 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((𝐴 + 𝐵) · 𝐶) = ((𝐴 · 𝐶) + (𝐵 · 𝐶)))

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149  (class class class)co 7411  ℝcr 11099  1c1 11101   + caddc 11103   · cmul 11105  ℕcn 12233

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pr 5405  ax-un 7733  ax-1cn 11158  ax-icn 11159  ax-addcl 11160  ax-addrcl 11161  ax-mulcl 11162  ax-mulrcl 11163  ax-addass 11165  ax-distr 11167  ax-i2m1 11168  ax-1ne0 11169  ax-1rid 11170  ax-rrecex 11172  ax-cnre 11173

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7414  df-om 7863  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-nn 12234

This theorem is referenced by:  nnmulcom  12294