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Theorem distrsr 9950
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 9916 . . 3 R = ((P × P) / ~R )
2 addsrpr 9934 . . 3 (((𝑧P𝑤P) ∧ (𝑣P𝑢P)) → ([⟨𝑧, 𝑤⟩] ~R +R [⟨𝑣, 𝑢⟩] ~R ) = [⟨(𝑧 +P 𝑣), (𝑤 +P 𝑢)⟩] ~R )
3 mulsrpr 9935 . . 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 9935 . . 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 9935 . . 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 9934 . . 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 9878 . . . . 5 ((𝑧P𝑣P) → (𝑧 +P 𝑣) ∈ P)
8 addclpr 9878 . . . . 5 ((𝑤P𝑢P) → (𝑤 +P 𝑢) ∈ P)
97, 8anim12i 589 . . . 4 (((𝑧P𝑣P) ∧ (𝑤P𝑢P)) → ((𝑧 +P 𝑣) ∈ P ∧ (𝑤 +P 𝑢) ∈ P))
109an4s 886 . . 3 (((𝑧P𝑤P) ∧ (𝑣P𝑢P)) → ((𝑧 +P 𝑣) ∈ P ∧ (𝑤 +P 𝑢) ∈ P))
11 mulclpr 9880 . . . . . 6 ((𝑥P𝑧P) → (𝑥 ·P 𝑧) ∈ P)
12 mulclpr 9880 . . . . . 6 ((𝑦P𝑤P) → (𝑦 ·P 𝑤) ∈ P)
13 addclpr 9878 . . . . . 6 (((𝑥 ·P 𝑧) ∈ P ∧ (𝑦 ·P 𝑤) ∈ P) → ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P)
1411, 12, 13syl2an 493 . . . . 5 (((𝑥P𝑧P) ∧ (𝑦P𝑤P)) → ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P)
1514an4s 886 . . . 4 (((𝑥P𝑦P) ∧ (𝑧P𝑤P)) → ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P)
16 mulclpr 9880 . . . . . 6 ((𝑥P𝑤P) → (𝑥 ·P 𝑤) ∈ P)
17 mulclpr 9880 . . . . . 6 ((𝑦P𝑧P) → (𝑦 ·P 𝑧) ∈ P)
18 addclpr 9878 . . . . . 6 (((𝑥 ·P 𝑤) ∈ P ∧ (𝑦 ·P 𝑧) ∈ P) → ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) ∈ P)
1916, 17, 18syl2an 493 . . . . 5 (((𝑥P𝑤P) ∧ (𝑦P𝑧P)) → ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) ∈ P)
2019an42s 887 . . . 4 (((𝑥P𝑦P) ∧ (𝑧P𝑤P)) → ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) ∈ P)
2115, 20jca 553 . . 3 (((𝑥P𝑦P) ∧ (𝑧P𝑤P)) → (((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P ∧ ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) ∈ P))
22 mulclpr 9880 . . . . . 6 ((𝑥P𝑣P) → (𝑥 ·P 𝑣) ∈ P)
23 mulclpr 9880 . . . . . 6 ((𝑦P𝑢P) → (𝑦 ·P 𝑢) ∈ P)
24 addclpr 9878 . . . . . 6 (((𝑥 ·P 𝑣) ∈ P ∧ (𝑦 ·P 𝑢) ∈ P) → ((𝑥 ·P 𝑣) +P (𝑦 ·P 𝑢)) ∈ P)
2522, 23, 24syl2an 493 . . . . 5 (((𝑥P𝑣P) ∧ (𝑦P𝑢P)) → ((𝑥 ·P 𝑣) +P (𝑦 ·P 𝑢)) ∈ P)
2625an4s 886 . . . 4 (((𝑥P𝑦P) ∧ (𝑣P𝑢P)) → ((𝑥 ·P 𝑣) +P (𝑦 ·P 𝑢)) ∈ P)
27 mulclpr 9880 . . . . . 6 ((𝑥P𝑢P) → (𝑥 ·P 𝑢) ∈ P)
28 mulclpr 9880 . . . . . 6 ((𝑦P𝑣P) → (𝑦 ·P 𝑣) ∈ P)
29 addclpr 9878 . . . . . 6 (((𝑥 ·P 𝑢) ∈ P ∧ (𝑦 ·P 𝑣) ∈ P) → ((𝑥 ·P 𝑢) +P (𝑦 ·P 𝑣)) ∈ P)
3027, 28, 29syl2an 493 . . . . 5 (((𝑥P𝑢P) ∧ (𝑦P𝑣P)) → ((𝑥 ·P 𝑢) +P (𝑦 ·P 𝑣)) ∈ P)
3130an42s 887 . . . 4 (((𝑥P𝑦P) ∧ (𝑣P𝑢P)) → ((𝑥 ·P 𝑢) +P (𝑦 ·P 𝑣)) ∈ P)
3226, 31jca 553 . . 3 (((𝑥P𝑦P) ∧ (𝑣P𝑢P)) → (((𝑥 ·P 𝑣) +P (𝑦 ·P 𝑢)) ∈ P ∧ ((𝑥 ·P 𝑢) +P (𝑦 ·P 𝑣)) ∈ P))
33 distrpr 9888 . . . . 5 (𝑥 ·P (𝑧 +P 𝑣)) = ((𝑥 ·P 𝑧) +P (𝑥 ·P 𝑣))
34 distrpr 9888 . . . . 5 (𝑦 ·P (𝑤 +P 𝑢)) = ((𝑦 ·P 𝑤) +P (𝑦 ·P 𝑢))
3533, 34oveq12i 6702 . . . 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 6718 . . . . 5 (𝑥 ·P 𝑧) ∈ V
37 ovex 6718 . . . . 5 (𝑥 ·P 𝑣) ∈ V
38 ovex 6718 . . . . 5 (𝑦 ·P 𝑤) ∈ V
39 addcompr 9881 . . . . 5 (𝑓 +P 𝑔) = (𝑔 +P 𝑓)
40 addasspr 9882 . . . . 5 ((𝑓 +P 𝑔) +P ) = (𝑓 +P (𝑔 +P ))
41 ovex 6718 . . . . 5 (𝑦 ·P 𝑢) ∈ V
4236, 37, 38, 39, 40, 41caov4 6907 . . . 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 2673 . . 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 9888 . . . . 5 (𝑥 ·P (𝑤 +P 𝑢)) = ((𝑥 ·P 𝑤) +P (𝑥 ·P 𝑢))
45 distrpr 9888 . . . . 5 (𝑦 ·P (𝑧 +P 𝑣)) = ((𝑦 ·P 𝑧) +P (𝑦 ·P 𝑣))
4644, 45oveq12i 6702 . . . 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 6718 . . . . 5 (𝑥 ·P 𝑤) ∈ V
48 ovex 6718 . . . . 5 (𝑥 ·P 𝑢) ∈ V
49 ovex 6718 . . . . 5 (𝑦 ·P 𝑧) ∈ V
50 ovex 6718 . . . . 5 (𝑦 ·P 𝑣) ∈ V
5147, 48, 49, 39, 40, 50caov4 6907 . . . 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 2673 . . 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 7898 . 2 ((𝐴R𝐵R𝐶R) → (𝐴 ·R (𝐵 +R 𝐶)) = ((𝐴 ·R 𝐵) +R (𝐴 ·R 𝐶)))
54 dmaddsr 9944 . . 3 dom +R = (R × R)
55 0nsr 9938 . . 3 ¬ ∅ ∈ R
56 dmmulsr 9945 . . 3 dom ·R = (R × R)
5754, 55, 56ndmovdistr 6865 . 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 1054   = wceq 1523  wcel 2030  (class class class)co 6690  Pcnp 9719   +P cpp 9721   ·P cmp 9722   ~R cer 9724  Rcnr 9725   +R cplr 9729   ·R cmr 9730
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-omul 7610  df-er 7787  df-ec 7789  df-qs 7793  df-ni 9732  df-pli 9733  df-mi 9734  df-lti 9735  df-plpq 9768  df-mpq 9769  df-ltpq 9770  df-enq 9771  df-nq 9772  df-erq 9773  df-plq 9774  df-mq 9775  df-1nq 9776  df-rq 9777  df-ltnq 9778  df-np 9841  df-plp 9843  df-mp 9844  df-ltp 9845  df-enr 9915  df-nr 9916  df-plr 9917  df-mr 9918
This theorem is referenced by:  pn0sr  9960  axmulass  10016  axdistr  10017
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