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Theorem distrsr 9600
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 9566 . . 3 R = ((P × P) / ~R )
2 addsrpr 9584 . . 3 (((𝑧P𝑤P) ∧ (𝑣P𝑢P)) → ([⟨𝑧, 𝑤⟩] ~R +R [⟨𝑣, 𝑢⟩] ~R ) = [⟨(𝑧 +P 𝑣), (𝑤 +P 𝑢)⟩] ~R )
3 mulsrpr 9585 . . 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 9585 . . 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 9585 . . 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 9584 . . 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 9528 . . . . 5 ((𝑧P𝑣P) → (𝑧 +P 𝑣) ∈ P)
8 addclpr 9528 . . . . 5 ((𝑤P𝑢P) → (𝑤 +P 𝑢) ∈ P)
97, 8anim12i 582 . . . 4 (((𝑧P𝑣P) ∧ (𝑤P𝑢P)) → ((𝑧 +P 𝑣) ∈ P ∧ (𝑤 +P 𝑢) ∈ P))
109an4s 855 . . 3 (((𝑧P𝑤P) ∧ (𝑣P𝑢P)) → ((𝑧 +P 𝑣) ∈ P ∧ (𝑤 +P 𝑢) ∈ P))
11 mulclpr 9530 . . . . . 6 ((𝑥P𝑧P) → (𝑥 ·P 𝑧) ∈ P)
12 mulclpr 9530 . . . . . 6 ((𝑦P𝑤P) → (𝑦 ·P 𝑤) ∈ P)
13 addclpr 9528 . . . . . 6 (((𝑥 ·P 𝑧) ∈ P ∧ (𝑦 ·P 𝑤) ∈ P) → ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P)
1411, 12, 13syl2an 487 . . . . 5 (((𝑥P𝑧P) ∧ (𝑦P𝑤P)) → ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P)
1514an4s 855 . . . 4 (((𝑥P𝑦P) ∧ (𝑧P𝑤P)) → ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P)
16 mulclpr 9530 . . . . . 6 ((𝑥P𝑤P) → (𝑥 ·P 𝑤) ∈ P)
17 mulclpr 9530 . . . . . 6 ((𝑦P𝑧P) → (𝑦 ·P 𝑧) ∈ P)
18 addclpr 9528 . . . . . 6 (((𝑥 ·P 𝑤) ∈ P ∧ (𝑦 ·P 𝑧) ∈ P) → ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) ∈ P)
1916, 17, 18syl2an 487 . . . . 5 (((𝑥P𝑤P) ∧ (𝑦P𝑧P)) → ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) ∈ P)
2019an42s 856 . . . 4 (((𝑥P𝑦P) ∧ (𝑧P𝑤P)) → ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) ∈ P)
2115, 20jca 547 . . 3 (((𝑥P𝑦P) ∧ (𝑧P𝑤P)) → (((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P ∧ ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) ∈ P))
22 mulclpr 9530 . . . . . 6 ((𝑥P𝑣P) → (𝑥 ·P 𝑣) ∈ P)
23 mulclpr 9530 . . . . . 6 ((𝑦P𝑢P) → (𝑦 ·P 𝑢) ∈ P)
24 addclpr 9528 . . . . . 6 (((𝑥 ·P 𝑣) ∈ P ∧ (𝑦 ·P 𝑢) ∈ P) → ((𝑥 ·P 𝑣) +P (𝑦 ·P 𝑢)) ∈ P)
2522, 23, 24syl2an 487 . . . . 5 (((𝑥P𝑣P) ∧ (𝑦P𝑢P)) → ((𝑥 ·P 𝑣) +P (𝑦 ·P 𝑢)) ∈ P)
2625an4s 855 . . . 4 (((𝑥P𝑦P) ∧ (𝑣P𝑢P)) → ((𝑥 ·P 𝑣) +P (𝑦 ·P 𝑢)) ∈ P)
27 mulclpr 9530 . . . . . 6 ((𝑥P𝑢P) → (𝑥 ·P 𝑢) ∈ P)
28 mulclpr 9530 . . . . . 6 ((𝑦P𝑣P) → (𝑦 ·P 𝑣) ∈ P)
29 addclpr 9528 . . . . . 6 (((𝑥 ·P 𝑢) ∈ P ∧ (𝑦 ·P 𝑣) ∈ P) → ((𝑥 ·P 𝑢) +P (𝑦 ·P 𝑣)) ∈ P)
3027, 28, 29syl2an 487 . . . . 5 (((𝑥P𝑢P) ∧ (𝑦P𝑣P)) → ((𝑥 ·P 𝑢) +P (𝑦 ·P 𝑣)) ∈ P)
3130an42s 856 . . . 4 (((𝑥P𝑦P) ∧ (𝑣P𝑢P)) → ((𝑥 ·P 𝑢) +P (𝑦 ·P 𝑣)) ∈ P)
3226, 31jca 547 . . 3 (((𝑥P𝑦P) ∧ (𝑣P𝑢P)) → (((𝑥 ·P 𝑣) +P (𝑦 ·P 𝑢)) ∈ P ∧ ((𝑥 ·P 𝑢) +P (𝑦 ·P 𝑣)) ∈ P))
33 distrpr 9538 . . . . 5 (𝑥 ·P (𝑧 +P 𝑣)) = ((𝑥 ·P 𝑧) +P (𝑥 ·P 𝑣))
34 distrpr 9538 . . . . 5 (𝑦 ·P (𝑤 +P 𝑢)) = ((𝑦 ·P 𝑤) +P (𝑦 ·P 𝑢))
3533, 34oveq12i 6375 . . . 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 6391 . . . . 5 (𝑥 ·P 𝑧) ∈ V
37 ovex 6391 . . . . 5 (𝑥 ·P 𝑣) ∈ V
38 ovex 6391 . . . . 5 (𝑦 ·P 𝑤) ∈ V
39 addcompr 9531 . . . . 5 (𝑓 +P 𝑔) = (𝑔 +P 𝑓)
40 addasspr 9532 . . . . 5 ((𝑓 +P 𝑔) +P ) = (𝑓 +P (𝑔 +P ))
41 ovex 6391 . . . . 5 (𝑦 ·P 𝑢) ∈ V
4236, 37, 38, 39, 40, 41caov4 6575 . . . 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 2527 . . 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 9538 . . . . 5 (𝑥 ·P (𝑤 +P 𝑢)) = ((𝑥 ·P 𝑤) +P (𝑥 ·P 𝑢))
45 distrpr 9538 . . . . 5 (𝑦 ·P (𝑧 +P 𝑣)) = ((𝑦 ·P 𝑧) +P (𝑦 ·P 𝑣))
4644, 45oveq12i 6375 . . . 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 6391 . . . . 5 (𝑥 ·P 𝑤) ∈ V
48 ovex 6391 . . . . 5 (𝑥 ·P 𝑢) ∈ V
49 ovex 6391 . . . . 5 (𝑦 ·P 𝑧) ∈ V
50 ovex 6391 . . . . 5 (𝑦 ·P 𝑣) ∈ V
5147, 48, 49, 39, 40, 50caov4 6575 . . . 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 2527 . . 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 7554 . 2 ((𝐴R𝐵R𝐶R) → (𝐴 ·R (𝐵 +R 𝐶)) = ((𝐴 ·R 𝐵) +R (𝐴 ·R 𝐶)))
54 dmaddsr 9594 . . 3 dom +R = (R × R)
55 0nsr 9588 . . 3 ¬ ∅ ∈ R
56 dmmulsr 9595 . . 3 dom ·R = (R × R)
5754, 55, 56ndmovdistr 6533 . 2 (¬ (𝐴R𝐵R𝐶R) → (𝐴 ·R (𝐵 +R 𝐶)) = ((𝐴 ·R 𝐵) +R (𝐴 ·R 𝐶)))
5853, 57pm2.61i 171 1 (𝐴 ·R (𝐵 +R 𝐶)) = ((𝐴 ·R 𝐵) +R (𝐴 ·R 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wa 378  w3a 1021   = wceq 1468  wcel 1937  (class class class)co 6363  Pcnp 9369   +P cpp 9371   ·P cmp 9372   ~R cer 9374  Rcnr 9375   +R cplr 9379   ·R cmr 9380
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1698  ax-4 1711  ax-5 1789  ax-6 1836  ax-7 1883  ax-8 1939  ax-9 1946  ax-10 1965  ax-11 1970  ax-12 1983  ax-13 2137  ax-ext 2485  ax-sep 4558  ax-nul 4567  ax-pow 4619  ax-pr 4680  ax-un 6659  ax-inf2 8231
This theorem depends on definitions:  df-bi 192  df-or 379  df-an 380  df-3or 1022  df-3an 1023  df-tru 1471  df-ex 1693  df-nf 1697  df-sb 1829  df-eu 2357  df-mo 2358  df-clab 2492  df-cleq 2498  df-clel 2501  df-nfc 2635  df-ne 2677  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3068  df-sbc 3292  df-csb 3386  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3758  df-if 3909  df-pw 3980  df-sn 3996  df-pr 3998  df-tp 4000  df-op 4002  df-uni 4229  df-int 4265  df-iun 4309  df-br 4435  df-opab 4494  df-mpt 4495  df-tr 4531  df-eprel 4791  df-id 4795  df-po 4801  df-so 4802  df-fr 4839  df-we 4841  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-pred 5431  df-ord 5477  df-on 5478  df-lim 5479  df-suc 5480  df-iota 5597  df-fun 5635  df-fn 5636  df-f 5637  df-f1 5638  df-fo 5639  df-f1o 5640  df-fv 5641  df-ov 6366  df-oprab 6367  df-mpt2 6368  df-om 6770  df-1st 6870  df-2nd 6871  df-wrecs 7105  df-recs 7167  df-rdg 7205  df-1o 7259  df-oadd 7263  df-omul 7264  df-er 7440  df-ec 7442  df-qs 7446  df-ni 9382  df-pli 9383  df-mi 9384  df-lti 9385  df-plpq 9418  df-mpq 9419  df-ltpq 9420  df-enq 9421  df-nq 9422  df-erq 9423  df-plq 9424  df-mq 9425  df-1nq 9426  df-rq 9427  df-ltnq 9428  df-np 9491  df-plp 9493  df-mp 9494  df-ltp 9495  df-enr 9565  df-nr 9566  df-plr 9567  df-mr 9568
This theorem is referenced by:  pn0sr  9610  axmulass  9666  axdistr  9667
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