Theorem distrsr 11002

Description: Multiplication of signed reals is distributive. (Contributed by NM, 2-Sep-1995.) (Revised by Mario Carneiro, 28-Apr-2015.) (New usage is discouraged.)

Assertion

Ref	Expression
distrsr	⊢ (𝐴 ·R (𝐵 +R 𝐶)) = ((𝐴 ·R 𝐵) +R (𝐴 ·R 𝐶))

Proof of Theorem distrsr

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 10967	. . 3 ⊢ R = ((P × P) / ~R )
2		addsrpr 10986	. . 3 ⊢ (((𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → ([⟨𝑧, 𝑤⟩] ~R +R [⟨𝑣, 𝑢⟩] ~R ) = [⟨(𝑧 +P 𝑣), (𝑤 +P 𝑢)⟩] ~R )
3		mulsrpr 10987	. . 3 ⊢ (((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 10987	. . 3 ⊢ (((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 10987	. . 3 ⊢ (((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 10986	. . 3 ⊢ (((((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 10929	. . . . 5 ⊢ ((𝑧 ∈ P ∧ 𝑣 ∈ P) → (𝑧 +P 𝑣) ∈ P)
8		addclpr 10929	. . . . 5 ⊢ ((𝑤 ∈ P ∧ 𝑢 ∈ P) → (𝑤 +P 𝑢) ∈ P)
9	7, 8	anim12i 613	. . . 4 ⊢ (((𝑧 ∈ P ∧ 𝑣 ∈ P) ∧ (𝑤 ∈ P ∧ 𝑢 ∈ P)) → ((𝑧 +P 𝑣) ∈ P ∧ (𝑤 +P 𝑢) ∈ P))
10	9	an4s 660	. . 3 ⊢ (((𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → ((𝑧 +P 𝑣) ∈ P ∧ (𝑤 +P 𝑢) ∈ P))
11		mulclpr 10931	. . . . . 6 ⊢ ((𝑥 ∈ P ∧ 𝑧 ∈ P) → (𝑥 ·P 𝑧) ∈ P)
12		mulclpr 10931	. . . . . 6 ⊢ ((𝑦 ∈ P ∧ 𝑤 ∈ P) → (𝑦 ·P 𝑤) ∈ P)
13		addclpr 10929	. . . . . 6 ⊢ (((𝑥 ·P 𝑧) ∈ P ∧ (𝑦 ·P 𝑤) ∈ P) → ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P)
14	11, 12, 13	syl2an 596	. . . . 5 ⊢ (((𝑥 ∈ P ∧ 𝑧 ∈ P) ∧ (𝑦 ∈ P ∧ 𝑤 ∈ P)) → ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P)
15	14	an4s 660	. . . 4 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P)
16		mulclpr 10931	. . . . . 6 ⊢ ((𝑥 ∈ P ∧ 𝑤 ∈ P) → (𝑥 ·P 𝑤) ∈ P)
17		mulclpr 10931	. . . . . 6 ⊢ ((𝑦 ∈ P ∧ 𝑧 ∈ P) → (𝑦 ·P 𝑧) ∈ P)
18		addclpr 10929	. . . . . 6 ⊢ (((𝑥 ·P 𝑤) ∈ P ∧ (𝑦 ·P 𝑧) ∈ P) → ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) ∈ P)
19	16, 17, 18	syl2an 596	. . . . 5 ⊢ (((𝑥 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑦 ∈ P ∧ 𝑧 ∈ P)) → ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) ∈ P)
20	19	an42s 661	. . . 4 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) ∈ P)
21	15, 20	jca 511	. . 3 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → (((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P ∧ ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) ∈ P))
22		mulclpr 10931	. . . . . 6 ⊢ ((𝑥 ∈ P ∧ 𝑣 ∈ P) → (𝑥 ·P 𝑣) ∈ P)
23		mulclpr 10931	. . . . . 6 ⊢ ((𝑦 ∈ P ∧ 𝑢 ∈ P) → (𝑦 ·P 𝑢) ∈ P)
24		addclpr 10929	. . . . . 6 ⊢ (((𝑥 ·P 𝑣) ∈ P ∧ (𝑦 ·P 𝑢) ∈ P) → ((𝑥 ·P 𝑣) +P (𝑦 ·P 𝑢)) ∈ P)
25	22, 23, 24	syl2an 596	. . . . 5 ⊢ (((𝑥 ∈ P ∧ 𝑣 ∈ P) ∧ (𝑦 ∈ P ∧ 𝑢 ∈ P)) → ((𝑥 ·P 𝑣) +P (𝑦 ·P 𝑢)) ∈ P)
26	25	an4s 660	. . . 4 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → ((𝑥 ·P 𝑣) +P (𝑦 ·P 𝑢)) ∈ P)
27		mulclpr 10931	. . . . . 6 ⊢ ((𝑥 ∈ P ∧ 𝑢 ∈ P) → (𝑥 ·P 𝑢) ∈ P)
28		mulclpr 10931	. . . . . 6 ⊢ ((𝑦 ∈ P ∧ 𝑣 ∈ P) → (𝑦 ·P 𝑣) ∈ P)
29		addclpr 10929	. . . . . 6 ⊢ (((𝑥 ·P 𝑢) ∈ P ∧ (𝑦 ·P 𝑣) ∈ P) → ((𝑥 ·P 𝑢) +P (𝑦 ·P 𝑣)) ∈ P)
30	27, 28, 29	syl2an 596	. . . . 5 ⊢ (((𝑥 ∈ P ∧ 𝑢 ∈ P) ∧ (𝑦 ∈ P ∧ 𝑣 ∈ P)) → ((𝑥 ·P 𝑢) +P (𝑦 ·P 𝑣)) ∈ P)
31	30	an42s 661	. . . 4 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → ((𝑥 ·P 𝑢) +P (𝑦 ·P 𝑣)) ∈ P)
32	26, 31	jca 511	. . 3 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → (((𝑥 ·P 𝑣) +P (𝑦 ·P 𝑢)) ∈ P ∧ ((𝑥 ·P 𝑢) +P (𝑦 ·P 𝑣)) ∈ P))
33		distrpr 10939	. . . . 5 ⊢ (𝑥 ·P (𝑧 +P 𝑣)) = ((𝑥 ·P 𝑧) +P (𝑥 ·P 𝑣))
34		distrpr 10939	. . . . 5 ⊢ (𝑦 ·P (𝑤 +P 𝑢)) = ((𝑦 ·P 𝑤) +P (𝑦 ·P 𝑢))
35	33, 34	oveq12i 7370	. . . 4 ⊢ ((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)))
36		ovex 7391	. . . . 5 ⊢ (𝑥 ·P 𝑧) ∈ V
37		ovex 7391	. . . . 5 ⊢ (𝑥 ·P 𝑣) ∈ V
38		ovex 7391	. . . . 5 ⊢ (𝑦 ·P 𝑤) ∈ V
39		addcompr 10932	. . . . 5 ⊢ (𝑓 +P 𝑔) = (𝑔 +P 𝑓)
40		addasspr 10933	. . . . 5 ⊢ ((𝑓 +P 𝑔) +P ℎ) = (𝑓 +P (𝑔 +P ℎ))
41		ovex 7391	. . . . 5 ⊢ (𝑦 ·P 𝑢) ∈ V
42	36, 37, 38, 39, 40, 41	caov4 7589	. . . 4 ⊢ (((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)))
43	35, 42	eqtri 2759	. . 3 ⊢ ((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)))
44		distrpr 10939	. . . . 5 ⊢ (𝑥 ·P (𝑤 +P 𝑢)) = ((𝑥 ·P 𝑤) +P (𝑥 ·P 𝑢))
45		distrpr 10939	. . . . 5 ⊢ (𝑦 ·P (𝑧 +P 𝑣)) = ((𝑦 ·P 𝑧) +P (𝑦 ·P 𝑣))
46	44, 45	oveq12i 7370	. . . 4 ⊢ ((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)))
47		ovex 7391	. . . . 5 ⊢ (𝑥 ·P 𝑤) ∈ V
48		ovex 7391	. . . . 5 ⊢ (𝑥 ·P 𝑢) ∈ V
49		ovex 7391	. . . . 5 ⊢ (𝑦 ·P 𝑧) ∈ V
50		ovex 7391	. . . . 5 ⊢ (𝑦 ·P 𝑣) ∈ V
51	47, 48, 49, 39, 40, 50	caov4 7589	. . . 4 ⊢ (((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)))
52	46, 51	eqtri 2759	. . 3 ⊢ ((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)))
53	1, 2, 3, 4, 5, 6, 10, 21, 32, 43, 52	ecovdi 8762	. 2 ⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R ∧ 𝐶 ∈ R) → (𝐴 ·R (𝐵 +R 𝐶)) = ((𝐴 ·R 𝐵) +R (𝐴 ·R 𝐶)))
54		dmaddsr 10996	. . 3 ⊢ dom +R = (R × R)
55		0nsr 10990	. . 3 ⊢ ¬ ∅ ∈ R
56		dmmulsr 10997	. . 3 ⊢ dom ·R = (R × R)
57	54, 55, 56	ndmovdistr 7547	. 2 ⊢ (¬ (𝐴 ∈ R ∧ 𝐵 ∈ R ∧ 𝐶 ∈ R) → (𝐴 ·R (𝐵 +R 𝐶)) = ((𝐴 ·R 𝐵) +R (𝐴 ·R 𝐶)))
58	53, 57	pm2.61i 182	1 ⊢ (𝐴 ·R (𝐵 +R 𝐶)) = ((𝐴 ·R 𝐵) +R (𝐴 ·R 𝐶))

Syntax hints:   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  (class class class)co 7358  Pcnp 10770   +_P cpp 10772   ·_P cmp 10773   ~_R cer 10775  Rcnr 10776   +_R cplr 10780   ·_R cmr 10781

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-inf2 9550

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-oadd 8401  df-omul 8402  df-er 8635  df-ec 8637  df-qs 8641  df-ni 10783  df-pli 10784  df-mi 10785  df-lti 10786  df-plpq 10819  df-mpq 10820  df-ltpq 10821  df-enq 10822  df-nq 10823  df-erq 10824  df-plq 10825  df-mq 10826  df-1nq 10827  df-rq 10828  df-ltnq 10829  df-np 10892  df-plp 10894  df-mp 10895  df-ltp 10896  df-enr 10966  df-nr 10967  df-plr 10968  df-mr 10969

This theorem is referenced by:  pn0sr  11012  axmulass  11068  axdistr  11069