Theorem mulasssr 10981

Description: Multiplication of signed reals is associative. (Contributed by NM, 2-Sep-1995.) (Revised by Mario Carneiro, 28-Apr-2015.) (New usage is discouraged.)

Assertion

Ref	Expression
mulasssr	⊢ ((𝐴 ·R 𝐵) ·R 𝐶) = (𝐴 ·R (𝐵 ·R 𝐶))

Proof of Theorem mulasssr

Dummy variables 𝑓 𝑔 ℎ 𝑢 𝑣 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-nr 10947	. . 3 ⊢ R = ((P × P) / ~R )
2		mulsrpr 10967	. . 3 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → ([⟨𝑥, 𝑦⟩] ~R ·R [⟨𝑧, 𝑤⟩] ~R ) = [⟨((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)), ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧))⟩] ~R )
3		mulsrpr 10967	. . 3 ⊢ (((𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → ([⟨𝑧, 𝑤⟩] ~R ·R [⟨𝑣, 𝑢⟩] ~R ) = [⟨((𝑧 ·P 𝑣) +P (𝑤 ·P 𝑢)), ((𝑧 ·P 𝑢) +P (𝑤 ·P 𝑣))⟩] ~R )
4		mulsrpr 10967	. . 3 ⊢ (((((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P ∧ ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → ([⟨((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)), ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧))⟩] ~R ·R [⟨𝑣, 𝑢⟩] ~R ) = [⟨((((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ·P 𝑣) +P (((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) ·P 𝑢)), ((((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ·P 𝑢) +P (((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) ·P 𝑣))⟩] ~R )
5		mulsrpr 10967	. . 3 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (((𝑧 ·P 𝑣) +P (𝑤 ·P 𝑢)) ∈ P ∧ ((𝑧 ·P 𝑢) +P (𝑤 ·P 𝑣)) ∈ P)) → ([⟨𝑥, 𝑦⟩] ~R ·R [⟨((𝑧 ·P 𝑣) +P (𝑤 ·P 𝑢)), ((𝑧 ·P 𝑢) +P (𝑤 ·P 𝑣))⟩] ~R ) = [⟨((𝑥 ·P ((𝑧 ·P 𝑣) +P (𝑤 ·P 𝑢))) +P (𝑦 ·P ((𝑧 ·P 𝑢) +P (𝑤 ·P 𝑣)))), ((𝑥 ·P ((𝑧 ·P 𝑢) +P (𝑤 ·P 𝑣))) +P (𝑦 ·P ((𝑧 ·P 𝑣) +P (𝑤 ·P 𝑢))))⟩] ~R )
6		mulclpr 10911	. . . . . 6 ⊢ ((𝑥 ∈ P ∧ 𝑧 ∈ P) → (𝑥 ·P 𝑧) ∈ P)
7		mulclpr 10911	. . . . . 6 ⊢ ((𝑦 ∈ P ∧ 𝑤 ∈ P) → (𝑦 ·P 𝑤) ∈ P)
8		addclpr 10909	. . . . . 6 ⊢ (((𝑥 ·P 𝑧) ∈ P ∧ (𝑦 ·P 𝑤) ∈ P) → ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P)
9	6, 7, 8	syl2an 596	. . . . 5 ⊢ (((𝑥 ∈ P ∧ 𝑧 ∈ P) ∧ (𝑦 ∈ P ∧ 𝑤 ∈ P)) → ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P)
10	9	an4s 660	. . . 4 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P)
11		mulclpr 10911	. . . . . 6 ⊢ ((𝑥 ∈ P ∧ 𝑤 ∈ P) → (𝑥 ·P 𝑤) ∈ P)
12		mulclpr 10911	. . . . . 6 ⊢ ((𝑦 ∈ P ∧ 𝑧 ∈ P) → (𝑦 ·P 𝑧) ∈ P)
13		addclpr 10909	. . . . . 6 ⊢ (((𝑥 ·P 𝑤) ∈ P ∧ (𝑦 ·P 𝑧) ∈ P) → ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) ∈ P)
14	11, 12, 13	syl2an 596	. . . . 5 ⊢ (((𝑥 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑦 ∈ P ∧ 𝑧 ∈ P)) → ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) ∈ P)
15	14	an42s 661	. . . 4 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) ∈ P)
16	10, 15	jca 511	. . 3 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → (((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P ∧ ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) ∈ P))
17		mulclpr 10911	. . . . . 6 ⊢ ((𝑧 ∈ P ∧ 𝑣 ∈ P) → (𝑧 ·P 𝑣) ∈ P)
18		mulclpr 10911	. . . . . 6 ⊢ ((𝑤 ∈ P ∧ 𝑢 ∈ P) → (𝑤 ·P 𝑢) ∈ P)
19		addclpr 10909	. . . . . 6 ⊢ (((𝑧 ·P 𝑣) ∈ P ∧ (𝑤 ·P 𝑢) ∈ P) → ((𝑧 ·P 𝑣) +P (𝑤 ·P 𝑢)) ∈ P)
20	17, 18, 19	syl2an 596	. . . . 5 ⊢ (((𝑧 ∈ P ∧ 𝑣 ∈ P) ∧ (𝑤 ∈ P ∧ 𝑢 ∈ P)) → ((𝑧 ·P 𝑣) +P (𝑤 ·P 𝑢)) ∈ P)
21	20	an4s 660	. . . 4 ⊢ (((𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → ((𝑧 ·P 𝑣) +P (𝑤 ·P 𝑢)) ∈ P)
22		mulclpr 10911	. . . . . 6 ⊢ ((𝑧 ∈ P ∧ 𝑢 ∈ P) → (𝑧 ·P 𝑢) ∈ P)
23		mulclpr 10911	. . . . . 6 ⊢ ((𝑤 ∈ P ∧ 𝑣 ∈ P) → (𝑤 ·P 𝑣) ∈ P)
24		addclpr 10909	. . . . . 6 ⊢ (((𝑧 ·P 𝑢) ∈ P ∧ (𝑤 ·P 𝑣) ∈ P) → ((𝑧 ·P 𝑢) +P (𝑤 ·P 𝑣)) ∈ P)
25	22, 23, 24	syl2an 596	. . . . 5 ⊢ (((𝑧 ∈ P ∧ 𝑢 ∈ P) ∧ (𝑤 ∈ P ∧ 𝑣 ∈ P)) → ((𝑧 ·P 𝑢) +P (𝑤 ·P 𝑣)) ∈ P)
26	25	an42s 661	. . . 4 ⊢ (((𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → ((𝑧 ·P 𝑢) +P (𝑤 ·P 𝑣)) ∈ P)
27	21, 26	jca 511	. . 3 ⊢ (((𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → (((𝑧 ·P 𝑣) +P (𝑤 ·P 𝑢)) ∈ P ∧ ((𝑧 ·P 𝑢) +P (𝑤 ·P 𝑣)) ∈ P))
28		vex 3440	. . . 4 ⊢ 𝑥 ∈ V
29		vex 3440	. . . 4 ⊢ 𝑦 ∈ V
30		vex 3440	. . . 4 ⊢ 𝑧 ∈ V
31		mulcompr 10914	. . . 4 ⊢ (𝑓 ·P 𝑔) = (𝑔 ·P 𝑓)
32		distrpr 10919	. . . 4 ⊢ (𝑓 ·P (𝑔 +P ℎ)) = ((𝑓 ·P 𝑔) +P (𝑓 ·P ℎ))
33		vex 3440	. . . 4 ⊢ 𝑤 ∈ V
34		vex 3440	. . . 4 ⊢ 𝑣 ∈ V
35		mulasspr 10915	. . . 4 ⊢ ((𝑓 ·P 𝑔) ·P ℎ) = (𝑓 ·P (𝑔 ·P ℎ))
36		vex 3440	. . . 4 ⊢ 𝑢 ∈ V
37		addcompr 10912	. . . 4 ⊢ (𝑓 +P 𝑔) = (𝑔 +P 𝑓)
38		addasspr 10913	. . . 4 ⊢ ((𝑓 +P 𝑔) +P ℎ) = (𝑓 +P (𝑔 +P ℎ))
39	28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38	caovlem2 7582	. . 3 ⊢ ((((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ·P 𝑣) +P (((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) ·P 𝑢)) = ((𝑥 ·P ((𝑧 ·P 𝑣) +P (𝑤 ·P 𝑢))) +P (𝑦 ·P ((𝑧 ·P 𝑢) +P (𝑤 ·P 𝑣))))
40	28, 29, 30, 31, 32, 33, 36, 35, 34, 37, 38	caovlem2 7582	. . 3 ⊢ ((((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ·P 𝑢) +P (((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) ·P 𝑣)) = ((𝑥 ·P ((𝑧 ·P 𝑢) +P (𝑤 ·P 𝑣))) +P (𝑦 ·P ((𝑧 ·P 𝑣) +P (𝑤 ·P 𝑢))))
41	1, 2, 3, 4, 5, 16, 27, 39, 40	ecovass 8748	. 2 ⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R ∧ 𝐶 ∈ R) → ((𝐴 ·R 𝐵) ·R 𝐶) = (𝐴 ·R (𝐵 ·R 𝐶)))
42		dmmulsr 10977	. . 3 ⊢ dom ·R = (R × R)
43		0nsr 10970	. . 3 ⊢ ¬ ∅ ∈ R
44	42, 43	ndmovass 7534	. 2 ⊢ (¬ (𝐴 ∈ R ∧ 𝐵 ∈ R ∧ 𝐶 ∈ R) → ((𝐴 ·R 𝐵) ·R 𝐶) = (𝐴 ·R (𝐵 ·R 𝐶)))
45	41, 44	pm2.61i 182	1 ⊢ ((𝐴 ·R 𝐵) ·R 𝐶) = (𝐴 ·R (𝐵 ·R 𝐶))

Syntax hints:   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111  (class class class)co 7346  Pcnp 10750   +_P cpp 10752   ·_P cmp 10753   ~_R cer 10755  Rcnr 10756   ·_R cmr 10761

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668  ax-inf2 9531

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-int 4896  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-1st 7921  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-1o 8385  df-oadd 8389  df-omul 8390  df-er 8622  df-ec 8624  df-qs 8628  df-ni 10763  df-pli 10764  df-mi 10765  df-lti 10766  df-plpq 10799  df-mpq 10800  df-ltpq 10801  df-enq 10802  df-nq 10803  df-erq 10804  df-plq 10805  df-mq 10806  df-1nq 10807  df-rq 10808  df-ltnq 10809  df-np 10872  df-plp 10874  df-mp 10875  df-ltp 10876  df-enr 10946  df-nr 10947  df-mr 10949

This theorem is referenced by:  sqgt0sr  10997  recexsr  10998  axmulass  11048