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Theorem distrsr 9768
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 9734 . . 3 R = ((P × P) / ~R )
2 addsrpr 9752 . . 3 (((𝑧P𝑤P) ∧ (𝑣P𝑢P)) → ([⟨𝑧, 𝑤⟩] ~R +R [⟨𝑣, 𝑢⟩] ~R ) = [⟨(𝑧 +P 𝑣), (𝑤 +P 𝑢)⟩] ~R )
3 mulsrpr 9753 . . 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 9753 . . 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 9753 . . 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 9752 . . 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 9696 . . . . 5 ((𝑧P𝑣P) → (𝑧 +P 𝑣) ∈ P)
8 addclpr 9696 . . . . 5 ((𝑤P𝑢P) → (𝑤 +P 𝑢) ∈ P)
97, 8anim12i 587 . . . 4 (((𝑧P𝑣P) ∧ (𝑤P𝑢P)) → ((𝑧 +P 𝑣) ∈ P ∧ (𝑤 +P 𝑢) ∈ P))
109an4s 864 . . 3 (((𝑧P𝑤P) ∧ (𝑣P𝑢P)) → ((𝑧 +P 𝑣) ∈ P ∧ (𝑤 +P 𝑢) ∈ P))
11 mulclpr 9698 . . . . . 6 ((𝑥P𝑧P) → (𝑥 ·P 𝑧) ∈ P)
12 mulclpr 9698 . . . . . 6 ((𝑦P𝑤P) → (𝑦 ·P 𝑤) ∈ P)
13 addclpr 9696 . . . . . 6 (((𝑥 ·P 𝑧) ∈ P ∧ (𝑦 ·P 𝑤) ∈ P) → ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P)
1411, 12, 13syl2an 492 . . . . 5 (((𝑥P𝑧P) ∧ (𝑦P𝑤P)) → ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P)
1514an4s 864 . . . 4 (((𝑥P𝑦P) ∧ (𝑧P𝑤P)) → ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P)
16 mulclpr 9698 . . . . . 6 ((𝑥P𝑤P) → (𝑥 ·P 𝑤) ∈ P)
17 mulclpr 9698 . . . . . 6 ((𝑦P𝑧P) → (𝑦 ·P 𝑧) ∈ P)
18 addclpr 9696 . . . . . 6 (((𝑥 ·P 𝑤) ∈ P ∧ (𝑦 ·P 𝑧) ∈ P) → ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) ∈ P)
1916, 17, 18syl2an 492 . . . . 5 (((𝑥P𝑤P) ∧ (𝑦P𝑧P)) → ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) ∈ P)
2019an42s 865 . . . 4 (((𝑥P𝑦P) ∧ (𝑧P𝑤P)) → ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) ∈ P)
2115, 20jca 552 . . 3 (((𝑥P𝑦P) ∧ (𝑧P𝑤P)) → (((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P ∧ ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) ∈ P))
22 mulclpr 9698 . . . . . 6 ((𝑥P𝑣P) → (𝑥 ·P 𝑣) ∈ P)
23 mulclpr 9698 . . . . . 6 ((𝑦P𝑢P) → (𝑦 ·P 𝑢) ∈ P)
24 addclpr 9696 . . . . . 6 (((𝑥 ·P 𝑣) ∈ P ∧ (𝑦 ·P 𝑢) ∈ P) → ((𝑥 ·P 𝑣) +P (𝑦 ·P 𝑢)) ∈ P)
2522, 23, 24syl2an 492 . . . . 5 (((𝑥P𝑣P) ∧ (𝑦P𝑢P)) → ((𝑥 ·P 𝑣) +P (𝑦 ·P 𝑢)) ∈ P)
2625an4s 864 . . . 4 (((𝑥P𝑦P) ∧ (𝑣P𝑢P)) → ((𝑥 ·P 𝑣) +P (𝑦 ·P 𝑢)) ∈ P)
27 mulclpr 9698 . . . . . 6 ((𝑥P𝑢P) → (𝑥 ·P 𝑢) ∈ P)
28 mulclpr 9698 . . . . . 6 ((𝑦P𝑣P) → (𝑦 ·P 𝑣) ∈ P)
29 addclpr 9696 . . . . . 6 (((𝑥 ·P 𝑢) ∈ P ∧ (𝑦 ·P 𝑣) ∈ P) → ((𝑥 ·P 𝑢) +P (𝑦 ·P 𝑣)) ∈ P)
3027, 28, 29syl2an 492 . . . . 5 (((𝑥P𝑢P) ∧ (𝑦P𝑣P)) → ((𝑥 ·P 𝑢) +P (𝑦 ·P 𝑣)) ∈ P)
3130an42s 865 . . . 4 (((𝑥P𝑦P) ∧ (𝑣P𝑢P)) → ((𝑥 ·P 𝑢) +P (𝑦 ·P 𝑣)) ∈ P)
3226, 31jca 552 . . 3 (((𝑥P𝑦P) ∧ (𝑣P𝑢P)) → (((𝑥 ·P 𝑣) +P (𝑦 ·P 𝑢)) ∈ P ∧ ((𝑥 ·P 𝑢) +P (𝑦 ·P 𝑣)) ∈ P))
33 distrpr 9706 . . . . 5 (𝑥 ·P (𝑧 +P 𝑣)) = ((𝑥 ·P 𝑧) +P (𝑥 ·P 𝑣))
34 distrpr 9706 . . . . 5 (𝑦 ·P (𝑤 +P 𝑢)) = ((𝑦 ·P 𝑤) +P (𝑦 ·P 𝑢))
3533, 34oveq12i 6539 . . . 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 6555 . . . . 5 (𝑥 ·P 𝑧) ∈ V
37 ovex 6555 . . . . 5 (𝑥 ·P 𝑣) ∈ V
38 ovex 6555 . . . . 5 (𝑦 ·P 𝑤) ∈ V
39 addcompr 9699 . . . . 5 (𝑓 +P 𝑔) = (𝑔 +P 𝑓)
40 addasspr 9700 . . . . 5 ((𝑓 +P 𝑔) +P ) = (𝑓 +P (𝑔 +P ))
41 ovex 6555 . . . . 5 (𝑦 ·P 𝑢) ∈ V
4236, 37, 38, 39, 40, 41caov4 6740 . . . 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 2631 . . 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 9706 . . . . 5 (𝑥 ·P (𝑤 +P 𝑢)) = ((𝑥 ·P 𝑤) +P (𝑥 ·P 𝑢))
45 distrpr 9706 . . . . 5 (𝑦 ·P (𝑧 +P 𝑣)) = ((𝑦 ·P 𝑧) +P (𝑦 ·P 𝑣))
4644, 45oveq12i 6539 . . . 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 6555 . . . . 5 (𝑥 ·P 𝑤) ∈ V
48 ovex 6555 . . . . 5 (𝑥 ·P 𝑢) ∈ V
49 ovex 6555 . . . . 5 (𝑦 ·P 𝑧) ∈ V
50 ovex 6555 . . . . 5 (𝑦 ·P 𝑣) ∈ V
5147, 48, 49, 39, 40, 50caov4 6740 . . . 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 2631 . . 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 7720 . 2 ((𝐴R𝐵R𝐶R) → (𝐴 ·R (𝐵 +R 𝐶)) = ((𝐴 ·R 𝐵) +R (𝐴 ·R 𝐶)))
54 dmaddsr 9762 . . 3 dom +R = (R × R)
55 0nsr 9756 . . 3 ¬ ∅ ∈ R
56 dmmulsr 9763 . . 3 dom ·R = (R × R)
5754, 55, 56ndmovdistr 6698 . 2 (¬ (𝐴R𝐵R𝐶R) → (𝐴 ·R (𝐵 +R 𝐶)) = ((𝐴 ·R 𝐵) +R (𝐴 ·R 𝐶)))
5853, 57pm2.61i 174 1 (𝐴 ·R (𝐵 +R 𝐶)) = ((𝐴 ·R 𝐵) +R (𝐴 ·R 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wa 382  w3a 1030   = wceq 1474  wcel 1976  (class class class)co 6527  Pcnp 9537   +P cpp 9539   ·P cmp 9540   ~R cer 9542  Rcnr 9543   +R cplr 9547   ·R cmr 9548
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-inf2 8398
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6935  df-1st 7036  df-2nd 7037  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-1o 7424  df-oadd 7428  df-omul 7429  df-er 7606  df-ec 7608  df-qs 7612  df-ni 9550  df-pli 9551  df-mi 9552  df-lti 9553  df-plpq 9586  df-mpq 9587  df-ltpq 9588  df-enq 9589  df-nq 9590  df-erq 9591  df-plq 9592  df-mq 9593  df-1nq 9594  df-rq 9595  df-ltnq 9596  df-np 9659  df-plp 9661  df-mp 9662  df-ltp 9663  df-enr 9733  df-nr 9734  df-plr 9735  df-mr 9736
This theorem is referenced by:  pn0sr  9778  axmulass  9834  axdistr  9835
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