Theorem distrsrg 8074

Description: Multiplication of signed reals is distributive. (Contributed by Jim Kingdon, 4-Jan-2020.)

Assertion

Ref	Expression
distrsrg	⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R ∧ 𝐶 ∈ R) → (𝐴 ·R (𝐵 +R 𝐶)) = ((𝐴 ·R 𝐵) +R (𝐴 ·R 𝐶)))

Proof of Theorem distrsrg

Dummy variables 𝑓 𝑔 ℎ 𝑢 𝑣 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-nr 8042	. 2 ⊢ R = ((P × P) / ~R )
2		addsrpr 8060	. 2 ⊢ (((𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → ([⟨𝑧, 𝑤⟩] ~R +R [⟨𝑣, 𝑢⟩] ~R ) = [⟨(𝑧 +P 𝑣), (𝑤 +P 𝑢)⟩] ~R )
3		mulsrpr 8061	. 2 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ ((𝑧 +P 𝑣) ∈ P ∧ (𝑤 +P 𝑢) ∈ P)) → ([⟨𝑥, 𝑦⟩] ~R ·R [⟨(𝑧 +P 𝑣), (𝑤 +P 𝑢)⟩] ~R ) = [⟨((𝑥 ·P (𝑧 +P 𝑣)) +P (𝑦 ·P (𝑤 +P 𝑢))), ((𝑥 ·P (𝑤 +P 𝑢)) +P (𝑦 ·P (𝑧 +P 𝑣)))⟩] ~R )
4		mulsrpr 8061	. 2 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → ([⟨𝑥, 𝑦⟩] ~R ·R [⟨𝑧, 𝑤⟩] ~R ) = [⟨((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)), ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧))⟩] ~R )
5		mulsrpr 8061	. 2 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → ([⟨𝑥, 𝑦⟩] ~R ·R [⟨𝑣, 𝑢⟩] ~R ) = [⟨((𝑥 ·P 𝑣) +P (𝑦 ·P 𝑢)), ((𝑥 ·P 𝑢) +P (𝑦 ·P 𝑣))⟩] ~R )
6		addsrpr 8060	. 2 ⊢ (((((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P ∧ ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) ∈ P) ∧ (((𝑥 ·P 𝑣) +P (𝑦 ·P 𝑢)) ∈ P ∧ ((𝑥 ·P 𝑢) +P (𝑦 ·P 𝑣)) ∈ P)) → ([⟨((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)), ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧))⟩] ~R +R [⟨((𝑥 ·P 𝑣) +P (𝑦 ·P 𝑢)), ((𝑥 ·P 𝑢) +P (𝑦 ·P 𝑣))⟩] ~R ) = [⟨(((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) +P ((𝑥 ·P 𝑣) +P (𝑦 ·P 𝑢))), (((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) +P ((𝑥 ·P 𝑢) +P (𝑦 ·P 𝑣)))⟩] ~R )
7		addclpr 7852	. . . 4 ⊢ ((𝑧 ∈ P ∧ 𝑣 ∈ P) → (𝑧 +P 𝑣) ∈ P)
8	7	ad2ant2r 509	. . 3 ⊢ (((𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → (𝑧 +P 𝑣) ∈ P)
9		addclpr 7852	. . . 4 ⊢ ((𝑤 ∈ P ∧ 𝑢 ∈ P) → (𝑤 +P 𝑢) ∈ P)
10	9	ad2ant2l 508	. . 3 ⊢ (((𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → (𝑤 +P 𝑢) ∈ P)
11	8, 10	jca 306	. 2 ⊢ (((𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → ((𝑧 +P 𝑣) ∈ P ∧ (𝑤 +P 𝑢) ∈ P))
12		mulclpr 7887	. . . . 5 ⊢ ((𝑥 ∈ P ∧ 𝑧 ∈ P) → (𝑥 ·P 𝑧) ∈ P)
13	12	ad2ant2r 509	. . . 4 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → (𝑥 ·P 𝑧) ∈ P)
14		mulclpr 7887	. . . . 5 ⊢ ((𝑦 ∈ P ∧ 𝑤 ∈ P) → (𝑦 ·P 𝑤) ∈ P)
15	14	ad2ant2l 508	. . . 4 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → (𝑦 ·P 𝑤) ∈ P)
16		addclpr 7852	. . . 4 ⊢ (((𝑥 ·P 𝑧) ∈ P ∧ (𝑦 ·P 𝑤) ∈ P) → ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P)
17	13, 15, 16	syl2anc 411	. . 3 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P)
18		mulclpr 7887	. . . . 5 ⊢ ((𝑥 ∈ P ∧ 𝑤 ∈ P) → (𝑥 ·P 𝑤) ∈ P)
19	18	ad2ant2rl 511	. . . 4 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → (𝑥 ·P 𝑤) ∈ P)
20		mulclpr 7887	. . . . 5 ⊢ ((𝑦 ∈ P ∧ 𝑧 ∈ P) → (𝑦 ·P 𝑧) ∈ P)
21	20	ad2ant2lr 510	. . . 4 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → (𝑦 ·P 𝑧) ∈ P)
22		addclpr 7852	. . . 4 ⊢ (((𝑥 ·P 𝑤) ∈ P ∧ (𝑦 ·P 𝑧) ∈ P) → ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) ∈ P)
23	19, 21, 22	syl2anc 411	. . 3 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) ∈ P)
24	17, 23	jca 306	. 2 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → (((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P ∧ ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) ∈ P))
25		mulclpr 7887	. . . . 5 ⊢ ((𝑥 ∈ P ∧ 𝑣 ∈ P) → (𝑥 ·P 𝑣) ∈ P)
26	25	ad2ant2r 509	. . . 4 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → (𝑥 ·P 𝑣) ∈ P)
27		mulclpr 7887	. . . . 5 ⊢ ((𝑦 ∈ P ∧ 𝑢 ∈ P) → (𝑦 ·P 𝑢) ∈ P)
28	27	ad2ant2l 508	. . . 4 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → (𝑦 ·P 𝑢) ∈ P)
29		addclpr 7852	. . . 4 ⊢ (((𝑥 ·P 𝑣) ∈ P ∧ (𝑦 ·P 𝑢) ∈ P) → ((𝑥 ·P 𝑣) +P (𝑦 ·P 𝑢)) ∈ P)
30	26, 28, 29	syl2anc 411	. . 3 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → ((𝑥 ·P 𝑣) +P (𝑦 ·P 𝑢)) ∈ P)
31		mulclpr 7887	. . . . 5 ⊢ ((𝑥 ∈ P ∧ 𝑢 ∈ P) → (𝑥 ·P 𝑢) ∈ P)
32	31	ad2ant2rl 511	. . . 4 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → (𝑥 ·P 𝑢) ∈ P)
33		mulclpr 7887	. . . . 5 ⊢ ((𝑦 ∈ P ∧ 𝑣 ∈ P) → (𝑦 ·P 𝑣) ∈ P)
34	33	ad2ant2lr 510	. . . 4 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → (𝑦 ·P 𝑣) ∈ P)
35		addclpr 7852	. . . 4 ⊢ (((𝑥 ·P 𝑢) ∈ P ∧ (𝑦 ·P 𝑣) ∈ P) → ((𝑥 ·P 𝑢) +P (𝑦 ·P 𝑣)) ∈ P)
36	32, 34, 35	syl2anc 411	. . 3 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → ((𝑥 ·P 𝑢) +P (𝑦 ·P 𝑣)) ∈ P)
37	30, 36	jca 306	. 2 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → (((𝑥 ·P 𝑣) +P (𝑦 ·P 𝑢)) ∈ P ∧ ((𝑥 ·P 𝑢) +P (𝑦 ·P 𝑣)) ∈ P))
38		simp1l 1048	. . . . 5 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → 𝑥 ∈ P)
39		simp2l 1050	. . . . 5 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → 𝑧 ∈ P)
40		simp3l 1052	. . . . 5 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → 𝑣 ∈ P)
41		distrprg 7903	. . . . 5 ⊢ ((𝑥 ∈ P ∧ 𝑧 ∈ P ∧ 𝑣 ∈ P) → (𝑥 ·P (𝑧 +P 𝑣)) = ((𝑥 ·P 𝑧) +P (𝑥 ·P 𝑣)))
42	38, 39, 40, 41	syl3anc 1274	. . . 4 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → (𝑥 ·P (𝑧 +P 𝑣)) = ((𝑥 ·P 𝑧) +P (𝑥 ·P 𝑣)))
43		simp1r 1049	. . . . 5 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → 𝑦 ∈ P)
44		simp2r 1051	. . . . 5 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → 𝑤 ∈ P)
45		simp3r 1053	. . . . 5 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → 𝑢 ∈ P)
46		distrprg 7903	. . . . 5 ⊢ ((𝑦 ∈ P ∧ 𝑤 ∈ P ∧ 𝑢 ∈ P) → (𝑦 ·P (𝑤 +P 𝑢)) = ((𝑦 ·P 𝑤) +P (𝑦 ·P 𝑢)))
47	43, 44, 45, 46	syl3anc 1274	. . . 4 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → (𝑦 ·P (𝑤 +P 𝑢)) = ((𝑦 ·P 𝑤) +P (𝑦 ·P 𝑢)))
48	42, 47	oveq12d 6068	. . 3 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → ((𝑥 ·P (𝑧 +P 𝑣)) +P (𝑦 ·P (𝑤 +P 𝑢))) = (((𝑥 ·P 𝑧) +P (𝑥 ·P 𝑣)) +P ((𝑦 ·P 𝑤) +P (𝑦 ·P 𝑢))))
49	38, 39, 12	syl2anc 411	. . . 4 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → (𝑥 ·P 𝑧) ∈ P)
50	38, 40, 25	syl2anc 411	. . . 4 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → (𝑥 ·P 𝑣) ∈ P)
51	43, 44, 14	syl2anc 411	. . . 4 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → (𝑦 ·P 𝑤) ∈ P)
52		addcomprg 7893	. . . . 5 ⊢ ((𝑓 ∈ P ∧ 𝑔 ∈ P) → (𝑓 +P 𝑔) = (𝑔 +P 𝑓))
53	52	adantl 277	. . . 4 ⊢ ((((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) ∧ (𝑓 ∈ P ∧ 𝑔 ∈ P)) → (𝑓 +P 𝑔) = (𝑔 +P 𝑓))
54		addassprg 7894	. . . . 5 ⊢ ((𝑓 ∈ P ∧ 𝑔 ∈ P ∧ ℎ ∈ P) → ((𝑓 +P 𝑔) +P ℎ) = (𝑓 +P (𝑔 +P ℎ)))
55	54	adantl 277	. . . 4 ⊢ ((((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) ∧ (𝑓 ∈ P ∧ 𝑔 ∈ P ∧ ℎ ∈ P)) → ((𝑓 +P 𝑔) +P ℎ) = (𝑓 +P (𝑔 +P ℎ)))
56	43, 45, 27	syl2anc 411	. . . 4 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → (𝑦 ·P 𝑢) ∈ P)
57		addclpr 7852	. . . . 5 ⊢ ((𝑓 ∈ P ∧ 𝑔 ∈ P) → (𝑓 +P 𝑔) ∈ P)
58	57	adantl 277	. . . 4 ⊢ ((((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) ∧ (𝑓 ∈ P ∧ 𝑔 ∈ P)) → (𝑓 +P 𝑔) ∈ P)
59	49, 50, 51, 53, 55, 56, 58	caov4d 6239	. . 3 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → (((𝑥 ·P 𝑧) +P (𝑥 ·P 𝑣)) +P ((𝑦 ·P 𝑤) +P (𝑦 ·P 𝑢))) = (((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) +P ((𝑥 ·P 𝑣) +P (𝑦 ·P 𝑢))))
60	48, 59	eqtrd 2265	. 2 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → ((𝑥 ·P (𝑧 +P 𝑣)) +P (𝑦 ·P (𝑤 +P 𝑢))) = (((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) +P ((𝑥 ·P 𝑣) +P (𝑦 ·P 𝑢))))
61		distrprg 7903	. . . . 5 ⊢ ((𝑥 ∈ P ∧ 𝑤 ∈ P ∧ 𝑢 ∈ P) → (𝑥 ·P (𝑤 +P 𝑢)) = ((𝑥 ·P 𝑤) +P (𝑥 ·P 𝑢)))
62	38, 44, 45, 61	syl3anc 1274	. . . 4 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → (𝑥 ·P (𝑤 +P 𝑢)) = ((𝑥 ·P 𝑤) +P (𝑥 ·P 𝑢)))
63		distrprg 7903	. . . . 5 ⊢ ((𝑦 ∈ P ∧ 𝑧 ∈ P ∧ 𝑣 ∈ P) → (𝑦 ·P (𝑧 +P 𝑣)) = ((𝑦 ·P 𝑧) +P (𝑦 ·P 𝑣)))
64	43, 39, 40, 63	syl3anc 1274	. . . 4 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → (𝑦 ·P (𝑧 +P 𝑣)) = ((𝑦 ·P 𝑧) +P (𝑦 ·P 𝑣)))
65	62, 64	oveq12d 6068	. . 3 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → ((𝑥 ·P (𝑤 +P 𝑢)) +P (𝑦 ·P (𝑧 +P 𝑣))) = (((𝑥 ·P 𝑤) +P (𝑥 ·P 𝑢)) +P ((𝑦 ·P 𝑧) +P (𝑦 ·P 𝑣))))
66	38, 44, 18	syl2anc 411	. . . 4 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → (𝑥 ·P 𝑤) ∈ P)
67	38, 45, 31	syl2anc 411	. . . 4 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → (𝑥 ·P 𝑢) ∈ P)
68	43, 39, 20	syl2anc 411	. . . 4 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → (𝑦 ·P 𝑧) ∈ P)
69	43, 40, 33	syl2anc 411	. . . 4 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → (𝑦 ·P 𝑣) ∈ P)
70	66, 67, 68, 53, 55, 69, 58	caov4d 6239	. . 3 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → (((𝑥 ·P 𝑤) +P (𝑥 ·P 𝑢)) +P ((𝑦 ·P 𝑧) +P (𝑦 ·P 𝑣))) = (((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) +P ((𝑥 ·P 𝑢) +P (𝑦 ·P 𝑣))))
71	65, 70	eqtrd 2265	. 2 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → ((𝑥 ·P (𝑤 +P 𝑢)) +P (𝑦 ·P (𝑧 +P 𝑣))) = (((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) +P ((𝑥 ·P 𝑢) +P (𝑦 ·P 𝑣))))
72	1, 2, 3, 4, 5, 6, 11, 24, 37, 60, 71	ecovidi 6881	1 ⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R ∧ 𝐶 ∈ R) → (𝐴 ·R (𝐵 +R 𝐶)) = ((𝐴 ·R 𝐵) +R (𝐴 ·R 𝐶)))

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 1005   = wceq 1398   ∈ wcel 2203  (class class class)co 6050  Pcnp 7606   +_P cpp 7608   ·_P cmp 7609   ~_R cer 7611  Rcnr 7612   +_R cplr 7616   ·_R cmr 7617

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-eprel 4410  df-id 4414  df-po 4417  df-iso 4418  df-iord 4487  df-on 4489  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-irdg 6601  df-1o 6647  df-2o 6648  df-oadd 6651  df-omul 6652  df-er 6767  df-ec 6769  df-qs 6773  df-ni 7619  df-pli 7620  df-mi 7621  df-lti 7622  df-plpq 7659  df-mpq 7660  df-enq 7662  df-nqqs 7663  df-plqqs 7664  df-mqqs 7665  df-1nqqs 7666  df-rq 7667  df-ltnqqs 7668  df-enq0 7739  df-nq0 7740  df-0nq0 7741  df-plq0 7742  df-mq0 7743  df-inp 7781  df-iplp 7783  df-imp 7784  df-enr 8041  df-nr 8042  df-plr 8043  df-mr 8044

This theorem is referenced by:  pn0sr  8086  axmulass  8188  axdistr  8189