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Theorem distrsrg 6801
Description: Multiplication of signed reals is distributive. (Contributed by Jim Kingdon, 4-Jan-2020.)
Assertion
Ref Expression
distrsrg ((𝐴R𝐵R𝐶R) → (𝐴 ·R (𝐵 +R 𝐶)) = ((𝐴 ·R 𝐵) +R (𝐴 ·R 𝐶)))

Proof of Theorem distrsrg
Dummy variables 𝑓 𝑔 𝑢 𝑣 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-nr 6769 . 2 R = ((P × P) / ~R )
2 addsrpr 6787 . 2 (((𝑧P𝑤P) ∧ (𝑣P𝑢P)) → ([⟨𝑧, 𝑤⟩] ~R +R [⟨𝑣, 𝑢⟩] ~R ) = [⟨(𝑧 +P 𝑣), (𝑤 +P 𝑢)⟩] ~R )
3 mulsrpr 6788 . 2 (((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 6788 . 2 (((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 6788 . 2 (((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 6787 . 2 (((((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 6592 . . . 4 ((𝑧P𝑣P) → (𝑧 +P 𝑣) ∈ P)
87ad2ant2r 478 . . 3 (((𝑧P𝑤P) ∧ (𝑣P𝑢P)) → (𝑧 +P 𝑣) ∈ P)
9 addclpr 6592 . . . 4 ((𝑤P𝑢P) → (𝑤 +P 𝑢) ∈ P)
109ad2ant2l 477 . . 3 (((𝑧P𝑤P) ∧ (𝑣P𝑢P)) → (𝑤 +P 𝑢) ∈ P)
118, 10jca 290 . 2 (((𝑧P𝑤P) ∧ (𝑣P𝑢P)) → ((𝑧 +P 𝑣) ∈ P ∧ (𝑤 +P 𝑢) ∈ P))
12 mulclpr 6627 . . . . 5 ((𝑥P𝑧P) → (𝑥 ·P 𝑧) ∈ P)
1312ad2ant2r 478 . . . 4 (((𝑥P𝑦P) ∧ (𝑧P𝑤P)) → (𝑥 ·P 𝑧) ∈ P)
14 mulclpr 6627 . . . . 5 ((𝑦P𝑤P) → (𝑦 ·P 𝑤) ∈ P)
1514ad2ant2l 477 . . . 4 (((𝑥P𝑦P) ∧ (𝑧P𝑤P)) → (𝑦 ·P 𝑤) ∈ P)
16 addclpr 6592 . . . 4 (((𝑥 ·P 𝑧) ∈ P ∧ (𝑦 ·P 𝑤) ∈ P) → ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P)
1713, 15, 16syl2anc 391 . . 3 (((𝑥P𝑦P) ∧ (𝑧P𝑤P)) → ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P)
18 mulclpr 6627 . . . . 5 ((𝑥P𝑤P) → (𝑥 ·P 𝑤) ∈ P)
1918ad2ant2rl 480 . . . 4 (((𝑥P𝑦P) ∧ (𝑧P𝑤P)) → (𝑥 ·P 𝑤) ∈ P)
20 mulclpr 6627 . . . . 5 ((𝑦P𝑧P) → (𝑦 ·P 𝑧) ∈ P)
2120ad2ant2lr 479 . . . 4 (((𝑥P𝑦P) ∧ (𝑧P𝑤P)) → (𝑦 ·P 𝑧) ∈ P)
22 addclpr 6592 . . . 4 (((𝑥 ·P 𝑤) ∈ P ∧ (𝑦 ·P 𝑧) ∈ P) → ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) ∈ P)
2319, 21, 22syl2anc 391 . . 3 (((𝑥P𝑦P) ∧ (𝑧P𝑤P)) → ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) ∈ P)
2417, 23jca 290 . 2 (((𝑥P𝑦P) ∧ (𝑧P𝑤P)) → (((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P ∧ ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) ∈ P))
25 mulclpr 6627 . . . . 5 ((𝑥P𝑣P) → (𝑥 ·P 𝑣) ∈ P)
2625ad2ant2r 478 . . . 4 (((𝑥P𝑦P) ∧ (𝑣P𝑢P)) → (𝑥 ·P 𝑣) ∈ P)
27 mulclpr 6627 . . . . 5 ((𝑦P𝑢P) → (𝑦 ·P 𝑢) ∈ P)
2827ad2ant2l 477 . . . 4 (((𝑥P𝑦P) ∧ (𝑣P𝑢P)) → (𝑦 ·P 𝑢) ∈ P)
29 addclpr 6592 . . . 4 (((𝑥 ·P 𝑣) ∈ P ∧ (𝑦 ·P 𝑢) ∈ P) → ((𝑥 ·P 𝑣) +P (𝑦 ·P 𝑢)) ∈ P)
3026, 28, 29syl2anc 391 . . 3 (((𝑥P𝑦P) ∧ (𝑣P𝑢P)) → ((𝑥 ·P 𝑣) +P (𝑦 ·P 𝑢)) ∈ P)
31 mulclpr 6627 . . . . 5 ((𝑥P𝑢P) → (𝑥 ·P 𝑢) ∈ P)
3231ad2ant2rl 480 . . . 4 (((𝑥P𝑦P) ∧ (𝑣P𝑢P)) → (𝑥 ·P 𝑢) ∈ P)
33 mulclpr 6627 . . . . 5 ((𝑦P𝑣P) → (𝑦 ·P 𝑣) ∈ P)
3433ad2ant2lr 479 . . . 4 (((𝑥P𝑦P) ∧ (𝑣P𝑢P)) → (𝑦 ·P 𝑣) ∈ P)
35 addclpr 6592 . . . 4 (((𝑥 ·P 𝑢) ∈ P ∧ (𝑦 ·P 𝑣) ∈ P) → ((𝑥 ·P 𝑢) +P (𝑦 ·P 𝑣)) ∈ P)
3632, 34, 35syl2anc 391 . . 3 (((𝑥P𝑦P) ∧ (𝑣P𝑢P)) → ((𝑥 ·P 𝑢) +P (𝑦 ·P 𝑣)) ∈ P)
3730, 36jca 290 . 2 (((𝑥P𝑦P) ∧ (𝑣P𝑢P)) → (((𝑥 ·P 𝑣) +P (𝑦 ·P 𝑢)) ∈ P ∧ ((𝑥 ·P 𝑢) +P (𝑦 ·P 𝑣)) ∈ P))
38 simp1l 928 . . . . 5 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑣P𝑢P)) → 𝑥P)
39 simp2l 930 . . . . 5 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑣P𝑢P)) → 𝑧P)
40 simp3l 932 . . . . 5 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑣P𝑢P)) → 𝑣P)
41 distrprg 6643 . . . . 5 ((𝑥P𝑧P𝑣P) → (𝑥 ·P (𝑧 +P 𝑣)) = ((𝑥 ·P 𝑧) +P (𝑥 ·P 𝑣)))
4238, 39, 40, 41syl3anc 1135 . . . 4 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑣P𝑢P)) → (𝑥 ·P (𝑧 +P 𝑣)) = ((𝑥 ·P 𝑧) +P (𝑥 ·P 𝑣)))
43 simp1r 929 . . . . 5 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑣P𝑢P)) → 𝑦P)
44 simp2r 931 . . . . 5 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑣P𝑢P)) → 𝑤P)
45 simp3r 933 . . . . 5 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑣P𝑢P)) → 𝑢P)
46 distrprg 6643 . . . . 5 ((𝑦P𝑤P𝑢P) → (𝑦 ·P (𝑤 +P 𝑢)) = ((𝑦 ·P 𝑤) +P (𝑦 ·P 𝑢)))
4743, 44, 45, 46syl3anc 1135 . . . 4 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑣P𝑢P)) → (𝑦 ·P (𝑤 +P 𝑢)) = ((𝑦 ·P 𝑤) +P (𝑦 ·P 𝑢)))
4842, 47oveq12d 5493 . . 3 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑣P𝑢P)) → ((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))))
4938, 39, 12syl2anc 391 . . . 4 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑣P𝑢P)) → (𝑥 ·P 𝑧) ∈ P)
5038, 40, 25syl2anc 391 . . . 4 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑣P𝑢P)) → (𝑥 ·P 𝑣) ∈ P)
5143, 44, 14syl2anc 391 . . . 4 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑣P𝑢P)) → (𝑦 ·P 𝑤) ∈ P)
52 addcomprg 6633 . . . . 5 ((𝑓P𝑔P) → (𝑓 +P 𝑔) = (𝑔 +P 𝑓))
5352adantl 262 . . . 4 ((((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑣P𝑢P)) ∧ (𝑓P𝑔P)) → (𝑓 +P 𝑔) = (𝑔 +P 𝑓))
54 addassprg 6634 . . . . 5 ((𝑓P𝑔PP) → ((𝑓 +P 𝑔) +P ) = (𝑓 +P (𝑔 +P )))
5554adantl 262 . . . 4 ((((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑣P𝑢P)) ∧ (𝑓P𝑔PP)) → ((𝑓 +P 𝑔) +P ) = (𝑓 +P (𝑔 +P )))
5643, 45, 27syl2anc 391 . . . 4 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑣P𝑢P)) → (𝑦 ·P 𝑢) ∈ P)
57 addclpr 6592 . . . . 5 ((𝑓P𝑔P) → (𝑓 +P 𝑔) ∈ P)
5857adantl 262 . . . 4 ((((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑣P𝑢P)) ∧ (𝑓P𝑔P)) → (𝑓 +P 𝑔) ∈ P)
5949, 50, 51, 53, 55, 56, 58caov4d 5648 . . 3 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑣P𝑢P)) → (((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))))
6048, 59eqtrd 2072 . 2 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑣P𝑢P)) → ((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))))
61 distrprg 6643 . . . . 5 ((𝑥P𝑤P𝑢P) → (𝑥 ·P (𝑤 +P 𝑢)) = ((𝑥 ·P 𝑤) +P (𝑥 ·P 𝑢)))
6238, 44, 45, 61syl3anc 1135 . . . 4 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑣P𝑢P)) → (𝑥 ·P (𝑤 +P 𝑢)) = ((𝑥 ·P 𝑤) +P (𝑥 ·P 𝑢)))
63 distrprg 6643 . . . . 5 ((𝑦P𝑧P𝑣P) → (𝑦 ·P (𝑧 +P 𝑣)) = ((𝑦 ·P 𝑧) +P (𝑦 ·P 𝑣)))
6443, 39, 40, 63syl3anc 1135 . . . 4 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑣P𝑢P)) → (𝑦 ·P (𝑧 +P 𝑣)) = ((𝑦 ·P 𝑧) +P (𝑦 ·P 𝑣)))
6562, 64oveq12d 5493 . . 3 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑣P𝑢P)) → ((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))))
6638, 44, 18syl2anc 391 . . . 4 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑣P𝑢P)) → (𝑥 ·P 𝑤) ∈ P)
6738, 45, 31syl2anc 391 . . . 4 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑣P𝑢P)) → (𝑥 ·P 𝑢) ∈ P)
6843, 39, 20syl2anc 391 . . . 4 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑣P𝑢P)) → (𝑦 ·P 𝑧) ∈ P)
6943, 40, 33syl2anc 391 . . . 4 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑣P𝑢P)) → (𝑦 ·P 𝑣) ∈ P)
7066, 67, 68, 53, 55, 69, 58caov4d 5648 . . 3 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑣P𝑢P)) → (((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))))
7165, 70eqtrd 2072 . 2 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑣P𝑢P)) → ((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))))
721, 2, 3, 4, 5, 6, 11, 24, 37, 60, 71ecovidi 6181 1 ((𝐴R𝐵R𝐶R) → (𝐴 ·R (𝐵 +R 𝐶)) = ((𝐴 ·R 𝐵) +R (𝐴 ·R 𝐶)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  w3a 885   = wceq 1243  wcel 1393  (class class class)co 5475  Pcnp 6346   +P cpp 6348   ·P cmp 6349   ~R cer 6351  Rcnr 6352   +R cplr 6356   ·R cmr 6357
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3869  ax-sep 3872  ax-nul 3880  ax-pow 3924  ax-pr 3941  ax-un 4142  ax-setind 4232  ax-iinf 4274
This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2308  df-rex 2309  df-reu 2310  df-rab 2312  df-v 2556  df-sbc 2762  df-csb 2850  df-dif 2917  df-un 2919  df-in 2921  df-ss 2928  df-nul 3222  df-pw 3358  df-sn 3378  df-pr 3379  df-op 3381  df-uni 3578  df-int 3613  df-iun 3656  df-br 3762  df-opab 3816  df-mpt 3817  df-tr 3852  df-eprel 4023  df-id 4027  df-po 4030  df-iso 4031  df-iord 4075  df-on 4077  df-suc 4080  df-iom 4277  df-xp 4314  df-rel 4315  df-cnv 4316  df-co 4317  df-dm 4318  df-rn 4319  df-res 4320  df-ima 4321  df-iota 4830  df-fun 4867  df-fn 4868  df-f 4869  df-f1 4870  df-fo 4871  df-f1o 4872  df-fv 4873  df-ov 5478  df-oprab 5479  df-mpt2 5480  df-1st 5730  df-2nd 5731  df-recs 5883  df-irdg 5920  df-1o 5964  df-2o 5965  df-oadd 5968  df-omul 5969  df-er 6069  df-ec 6071  df-qs 6075  df-ni 6359  df-pli 6360  df-mi 6361  df-lti 6362  df-plpq 6399  df-mpq 6400  df-enq 6402  df-nqqs 6403  df-plqqs 6404  df-mqqs 6405  df-1nqqs 6406  df-rq 6407  df-ltnqqs 6408  df-enq0 6479  df-nq0 6480  df-0nq0 6481  df-plq0 6482  df-mq0 6483  df-inp 6521  df-iplp 6523  df-imp 6524  df-enr 6768  df-nr 6769  df-plr 6770  df-mr 6771
This theorem is referenced by:  pn0sr  6813  axmulass  6904  axdistr  6905
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