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Theorem distrsr 10096
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 𝑢 𝑣 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-nr 10062 . . 3 R = ((P × P) / ~R )
2 addsrpr 10080 . . 3 (((𝑧P𝑤P) ∧ (𝑣P𝑢P)) → ([⟨𝑧, 𝑤⟩] ~R +R [⟨𝑣, 𝑢⟩] ~R ) = [⟨(𝑧 +P 𝑣), (𝑤 +P 𝑢)⟩] ~R )
3 mulsrpr 10081 . . 3 (((𝑥P𝑦P) ∧ ((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 10081 . . 3 (((𝑥P𝑦P) ∧ (𝑧P𝑤P)) → ([⟨𝑥, 𝑦⟩] ~R ·R [⟨𝑧, 𝑤⟩] ~R ) = [⟨((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)), ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧))⟩] ~R )
5 mulsrpr 10081 . . 3 (((𝑥P𝑦P) ∧ (𝑣P𝑢P)) → ([⟨𝑥, 𝑦⟩] ~R ·R [⟨𝑣, 𝑢⟩] ~R ) = [⟨((𝑥 ·P 𝑣) +P (𝑦 ·P 𝑢)), ((𝑥 ·P 𝑢) +P (𝑦 ·P 𝑣))⟩] ~R )
6 addsrpr 10080 . . 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 10024 . . . . 5 ((𝑧P𝑣P) → (𝑧 +P 𝑣) ∈ P)
8 addclpr 10024 . . . . 5 ((𝑤P𝑢P) → (𝑤 +P 𝑢) ∈ P)
97, 8anim12i 591 . . . 4 (((𝑧P𝑣P) ∧ (𝑤P𝑢P)) → ((𝑧 +P 𝑣) ∈ P ∧ (𝑤 +P 𝑢) ∈ P))
109an4s 904 . . 3 (((𝑧P𝑤P) ∧ (𝑣P𝑢P)) → ((𝑧 +P 𝑣) ∈ P ∧ (𝑤 +P 𝑢) ∈ P))
11 mulclpr 10026 . . . . . 6 ((𝑥P𝑧P) → (𝑥 ·P 𝑧) ∈ P)
12 mulclpr 10026 . . . . . 6 ((𝑦P𝑤P) → (𝑦 ·P 𝑤) ∈ P)
13 addclpr 10024 . . . . . 6 (((𝑥 ·P 𝑧) ∈ P ∧ (𝑦 ·P 𝑤) ∈ P) → ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P)
1411, 12, 13syl2an 495 . . . . 5 (((𝑥P𝑧P) ∧ (𝑦P𝑤P)) → ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P)
1514an4s 904 . . . 4 (((𝑥P𝑦P) ∧ (𝑧P𝑤P)) → ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P)
16 mulclpr 10026 . . . . . 6 ((𝑥P𝑤P) → (𝑥 ·P 𝑤) ∈ P)
17 mulclpr 10026 . . . . . 6 ((𝑦P𝑧P) → (𝑦 ·P 𝑧) ∈ P)
18 addclpr 10024 . . . . . 6 (((𝑥 ·P 𝑤) ∈ P ∧ (𝑦 ·P 𝑧) ∈ P) → ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) ∈ P)
1916, 17, 18syl2an 495 . . . . 5 (((𝑥P𝑤P) ∧ (𝑦P𝑧P)) → ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) ∈ P)
2019an42s 905 . . . 4 (((𝑥P𝑦P) ∧ (𝑧P𝑤P)) → ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) ∈ P)
2115, 20jca 555 . . 3 (((𝑥P𝑦P) ∧ (𝑧P𝑤P)) → (((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P ∧ ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) ∈ P))
22 mulclpr 10026 . . . . . 6 ((𝑥P𝑣P) → (𝑥 ·P 𝑣) ∈ P)
23 mulclpr 10026 . . . . . 6 ((𝑦P𝑢P) → (𝑦 ·P 𝑢) ∈ P)
24 addclpr 10024 . . . . . 6 (((𝑥 ·P 𝑣) ∈ P ∧ (𝑦 ·P 𝑢) ∈ P) → ((𝑥 ·P 𝑣) +P (𝑦 ·P 𝑢)) ∈ P)
2522, 23, 24syl2an 495 . . . . 5 (((𝑥P𝑣P) ∧ (𝑦P𝑢P)) → ((𝑥 ·P 𝑣) +P (𝑦 ·P 𝑢)) ∈ P)
2625an4s 904 . . . 4 (((𝑥P𝑦P) ∧ (𝑣P𝑢P)) → ((𝑥 ·P 𝑣) +P (𝑦 ·P 𝑢)) ∈ P)
27 mulclpr 10026 . . . . . 6 ((𝑥P𝑢P) → (𝑥 ·P 𝑢) ∈ P)
28 mulclpr 10026 . . . . . 6 ((𝑦P𝑣P) → (𝑦 ·P 𝑣) ∈ P)
29 addclpr 10024 . . . . . 6 (((𝑥 ·P 𝑢) ∈ P ∧ (𝑦 ·P 𝑣) ∈ P) → ((𝑥 ·P 𝑢) +P (𝑦 ·P 𝑣)) ∈ P)
3027, 28, 29syl2an 495 . . . . 5 (((𝑥P𝑢P) ∧ (𝑦P𝑣P)) → ((𝑥 ·P 𝑢) +P (𝑦 ·P 𝑣)) ∈ P)
3130an42s 905 . . . 4 (((𝑥P𝑦P) ∧ (𝑣P𝑢P)) → ((𝑥 ·P 𝑢) +P (𝑦 ·P 𝑣)) ∈ P)
3226, 31jca 555 . . 3 (((𝑥P𝑦P) ∧ (𝑣P𝑢P)) → (((𝑥 ·P 𝑣) +P (𝑦 ·P 𝑢)) ∈ P ∧ ((𝑥 ·P 𝑢) +P (𝑦 ·P 𝑣)) ∈ P))
33 distrpr 10034 . . . . 5 (𝑥 ·P (𝑧 +P 𝑣)) = ((𝑥 ·P 𝑧) +P (𝑥 ·P 𝑣))
34 distrpr 10034 . . . . 5 (𝑦 ·P (𝑤 +P 𝑢)) = ((𝑦 ·P 𝑤) +P (𝑦 ·P 𝑢))
3533, 34oveq12i 6817 . . . 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 6833 . . . . 5 (𝑥 ·P 𝑧) ∈ V
37 ovex 6833 . . . . 5 (𝑥 ·P 𝑣) ∈ V
38 ovex 6833 . . . . 5 (𝑦 ·P 𝑤) ∈ V
39 addcompr 10027 . . . . 5 (𝑓 +P 𝑔) = (𝑔 +P 𝑓)
40 addasspr 10028 . . . . 5 ((𝑓 +P 𝑔) +P ) = (𝑓 +P (𝑔 +P ))
41 ovex 6833 . . . . 5 (𝑦 ·P 𝑢) ∈ V
4236, 37, 38, 39, 40, 41caov4 7022 . . . 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)))
4335, 42eqtri 2774 . . 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 10034 . . . . 5 (𝑥 ·P (𝑤 +P 𝑢)) = ((𝑥 ·P 𝑤) +P (𝑥 ·P 𝑢))
45 distrpr 10034 . . . . 5 (𝑦 ·P (𝑧 +P 𝑣)) = ((𝑦 ·P 𝑧) +P (𝑦 ·P 𝑣))
4644, 45oveq12i 6817 . . . 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 6833 . . . . 5 (𝑥 ·P 𝑤) ∈ V
48 ovex 6833 . . . . 5 (𝑥 ·P 𝑢) ∈ V
49 ovex 6833 . . . . 5 (𝑦 ·P 𝑧) ∈ V
50 ovex 6833 . . . . 5 (𝑦 ·P 𝑣) ∈ V
5147, 48, 49, 39, 40, 50caov4 7022 . . . 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)))
5246, 51eqtri 2774 . . 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)))
531, 2, 3, 4, 5, 6, 10, 21, 32, 43, 52ecovdi 8014 . 2 ((𝐴R𝐵R𝐶R) → (𝐴 ·R (𝐵 +R 𝐶)) = ((𝐴 ·R 𝐵) +R (𝐴 ·R 𝐶)))
54 dmaddsr 10090 . . 3 dom +R = (R × R)
55 0nsr 10084 . . 3 ¬ ∅ ∈ R
56 dmmulsr 10091 . . 3 dom ·R = (R × R)
5754, 55, 56ndmovdistr 6980 . 2 (¬ (𝐴R𝐵R𝐶R) → (𝐴 ·R (𝐵 +R 𝐶)) = ((𝐴 ·R 𝐵) +R (𝐴 ·R 𝐶)))
5853, 57pm2.61i 176 1 (𝐴 ·R (𝐵 +R 𝐶)) = ((𝐴 ·R 𝐵) +R (𝐴 ·R 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wa 383  w3a 1072   = wceq 1624  wcel 2131  (class class class)co 6805  Pcnp 9865   +P cpp 9867   ·P cmp 9868   ~R cer 9870  Rcnr 9871   +R cplr 9875   ·R cmr 9876
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106  ax-inf2 8703
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-reu 3049  df-rmo 3050  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-pss 3723  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-tp 4318  df-op 4320  df-uni 4581  df-int 4620  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-tr 4897  df-id 5166  df-eprel 5171  df-po 5179  df-so 5180  df-fr 5217  df-we 5219  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-pred 5833  df-ord 5879  df-on 5880  df-lim 5881  df-suc 5882  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-ov 6808  df-oprab 6809  df-mpt2 6810  df-om 7223  df-1st 7325  df-2nd 7326  df-wrecs 7568  df-recs 7629  df-rdg 7667  df-1o 7721  df-oadd 7725  df-omul 7726  df-er 7903  df-ec 7905  df-qs 7909  df-ni 9878  df-pli 9879  df-mi 9880  df-lti 9881  df-plpq 9914  df-mpq 9915  df-ltpq 9916  df-enq 9917  df-nq 9918  df-erq 9919  df-plq 9920  df-mq 9921  df-1nq 9922  df-rq 9923  df-ltnq 9924  df-np 9987  df-plp 9989  df-mp 9990  df-ltp 9991  df-enr 10061  df-nr 10062  df-plr 10063  df-mr 10064
This theorem is referenced by:  pn0sr  10106  axmulass  10162  axdistr  10163
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