Theorem distrsrg 7892

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 7860	. 2 ⊢ R = ((P × P) / ~R )
2		addsrpr 7878	. 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 7879	. 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 7879	. 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 7879	. 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 7878	. 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 7670	. . . 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 7670	. . . 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 7705	. . . . 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 7705	. . . . 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 7670	. . . 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 7705	. . . . 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 7705	. . . . 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 7670	. . . 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 7705	. . . . 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 7705	. . . . 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 7670	. . . 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 7705	. . . . 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 7705	. . . . 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 7670	. . . 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 1024	. . . . 5 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → 𝑥 ∈ P)
39		simp2l 1026	. . . . 5 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → 𝑧 ∈ P)
40		simp3l 1028	. . . . 5 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → 𝑣 ∈ P)
41		distrprg 7721	. . . . 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 1250	. . . 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 1025	. . . . 5 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → 𝑦 ∈ P)
44		simp2r 1027	. . . . 5 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → 𝑤 ∈ P)
45		simp3r 1029	. . . . 5 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → 𝑢 ∈ P)
46		distrprg 7721	. . . . 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 1250	. . . 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 5975	. . 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 7711	. . . . 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 7712	. . . . 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 7670	. . . . 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 6144	. . 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 2239	. 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 7721	. . . . 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 1250	. . . 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 7721	. . . . 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 1250	. . . 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 5975	. . 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 6144	. . 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 2239	. 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 6747	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 981   = wceq 1373   ∈ wcel 2177  (class class class)co 5957  Pcnp 7424   +_P cpp 7426   ·_P cmp 7427   ~_R cer 7429  Rcnr 7430   +_R cplr 7434   ·_R cmr 7435

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4167  ax-sep 4170  ax-nul 4178  ax-pow 4226  ax-pr 4261  ax-un 4488  ax-setind 4593  ax-iinf 4644

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-int 3892  df-iun 3935  df-br 4052  df-opab 4114  df-mpt 4115  df-tr 4151  df-eprel 4344  df-id 4348  df-po 4351  df-iso 4352  df-iord 4421  df-on 4423  df-suc 4426  df-iom 4647  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-res 4695  df-ima 4696  df-iota 5241  df-fun 5282  df-fn 5283  df-f 5284  df-f1 5285  df-fo 5286  df-f1o 5287  df-fv 5288  df-ov 5960  df-oprab 5961  df-mpo 5962  df-1st 6239  df-2nd 6240  df-recs 6404  df-irdg 6469  df-1o 6515  df-2o 6516  df-oadd 6519  df-omul 6520  df-er 6633  df-ec 6635  df-qs 6639  df-ni 7437  df-pli 7438  df-mi 7439  df-lti 7440  df-plpq 7477  df-mpq 7478  df-enq 7480  df-nqqs 7481  df-plqqs 7482  df-mqqs 7483  df-1nqqs 7484  df-rq 7485  df-ltnqqs 7486  df-enq0 7557  df-nq0 7558  df-0nq0 7559  df-plq0 7560  df-mq0 7561  df-inp 7599  df-iplp 7601  df-imp 7602  df-enr 7859  df-nr 7860  df-plr 7861  df-mr 7862

This theorem is referenced by:  pn0sr  7904  axmulass  8006  axdistr  8007