Theorem mulasssr 11182

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 11148	. . 3 ⊢ R = ((P × P) / ~R )
2		mulsrpr 11168	. . 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 11168	. . 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 11168	. . 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 11168	. . 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 11112	. . . . . 6 ⊢ ((𝑥 ∈ P ∧ 𝑧 ∈ P) → (𝑥 ·P 𝑧) ∈ P)
7		mulclpr 11112	. . . . . 6 ⊢ ((𝑦 ∈ P ∧ 𝑤 ∈ P) → (𝑦 ·P 𝑤) ∈ P)
8		addclpr 11110	. . . . . 6 ⊢ (((𝑥 ·P 𝑧) ∈ P ∧ (𝑦 ·P 𝑤) ∈ P) → ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P)
9	6, 7, 8	syl2an 608	. . . . 5 ⊢ (((𝑥 ∈ P ∧ 𝑧 ∈ P) ∧ (𝑦 ∈ P ∧ 𝑤 ∈ P)) → ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P)
10	9	an4s 673	. . . 4 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P)
11		mulclpr 11112	. . . . . 6 ⊢ ((𝑥 ∈ P ∧ 𝑤 ∈ P) → (𝑥 ·P 𝑤) ∈ P)
12		mulclpr 11112	. . . . . 6 ⊢ ((𝑦 ∈ P ∧ 𝑧 ∈ P) → (𝑦 ·P 𝑧) ∈ P)
13		addclpr 11110	. . . . . 6 ⊢ (((𝑥 ·P 𝑤) ∈ P ∧ (𝑦 ·P 𝑧) ∈ P) → ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) ∈ P)
14	11, 12, 13	syl2an 608	. . . . 5 ⊢ (((𝑥 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑦 ∈ P ∧ 𝑧 ∈ P)) → ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) ∈ P)
15	14	an42s 674	. . . 4 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) ∈ P)
16	10, 15	jca 521	. . 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 11112	. . . . . 6 ⊢ ((𝑧 ∈ P ∧ 𝑣 ∈ P) → (𝑧 ·P 𝑣) ∈ P)
18		mulclpr 11112	. . . . . 6 ⊢ ((𝑤 ∈ P ∧ 𝑢 ∈ P) → (𝑤 ·P 𝑢) ∈ P)
19		addclpr 11110	. . . . . 6 ⊢ (((𝑧 ·P 𝑣) ∈ P ∧ (𝑤 ·P 𝑢) ∈ P) → ((𝑧 ·P 𝑣) +P (𝑤 ·P 𝑢)) ∈ P)
20	17, 18, 19	syl2an 608	. . . . 5 ⊢ (((𝑧 ∈ P ∧ 𝑣 ∈ P) ∧ (𝑤 ∈ P ∧ 𝑢 ∈ P)) → ((𝑧 ·P 𝑣) +P (𝑤 ·P 𝑢)) ∈ P)
21	20	an4s 673	. . . 4 ⊢ (((𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → ((𝑧 ·P 𝑣) +P (𝑤 ·P 𝑢)) ∈ P)
22		mulclpr 11112	. . . . . 6 ⊢ ((𝑧 ∈ P ∧ 𝑢 ∈ P) → (𝑧 ·P 𝑢) ∈ P)
23		mulclpr 11112	. . . . . 6 ⊢ ((𝑤 ∈ P ∧ 𝑣 ∈ P) → (𝑤 ·P 𝑣) ∈ P)
24		addclpr 11110	. . . . . 6 ⊢ (((𝑧 ·P 𝑢) ∈ P ∧ (𝑤 ·P 𝑣) ∈ P) → ((𝑧 ·P 𝑢) +P (𝑤 ·P 𝑣)) ∈ P)
25	22, 23, 24	syl2an 608	. . . . 5 ⊢ (((𝑧 ∈ P ∧ 𝑢 ∈ P) ∧ (𝑤 ∈ P ∧ 𝑣 ∈ P)) → ((𝑧 ·P 𝑢) +P (𝑤 ·P 𝑣)) ∈ P)
26	25	an42s 674	. . . 4 ⊢ (((𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → ((𝑧 ·P 𝑢) +P (𝑤 ·P 𝑣)) ∈ P)
27	21, 26	jca 521	. . 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 3454	. . . 4 ⊢ 𝑥 ∈ V
29		vex 3454	. . . 4 ⊢ 𝑦 ∈ V
30		vex 3454	. . . 4 ⊢ 𝑧 ∈ V
31		mulcompr 11115	. . . 4 ⊢ (𝑓 ·P 𝑔) = (𝑔 ·P 𝑓)
32		distrpr 11120	. . . 4 ⊢ (𝑓 ·P (𝑔 +P ℎ)) = ((𝑓 ·P 𝑔) +P (𝑓 ·P ℎ))
33		vex 3454	. . . 4 ⊢ 𝑤 ∈ V
34		vex 3454	. . . 4 ⊢ 𝑣 ∈ V
35		mulasspr 11116	. . . 4 ⊢ ((𝑓 ·P 𝑔) ·P ℎ) = (𝑓 ·P (𝑔 ·P ℎ))
36		vex 3454	. . . 4 ⊢ 𝑢 ∈ V
37		addcompr 11113	. . . 4 ⊢ (𝑓 +P 𝑔) = (𝑔 +P 𝑓)
38		addasspr 11114	. . . 4 ⊢ ((𝑓 +P 𝑔) +P ℎ) = (𝑓 +P (𝑔 +P ℎ))
39	28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38	caovlem2 7661	. . 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 7661	. . 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 8852	. 2 ⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R ∧ 𝐶 ∈ R) → ((𝐴 ·R 𝐵) ·R 𝐶) = (𝐴 ·R (𝐵 ·R 𝐶)))
42		dmmulsr 11178	. . 3 ⊢ dom ·R = (R × R)
43		0nsr 11171	. . 3 ⊢ ¬ ∅ ∈ R
44	42, 43	ndmovass 7613	. 2 ⊢ (¬ (𝐴 ∈ R ∧ 𝐵 ∈ R ∧ 𝐶 ∈ R) → ((𝐴 ·R 𝐵) ·R 𝐶) = (𝐴 ·R (𝐵 ·R 𝐶)))
45	41, 44	pm2.61i 184	1 ⊢ ((𝐴 ·R 𝐵) ·R 𝐶) = (𝐴 ·R (𝐵 ·R 𝐶))

This proof depends on syntax axioms:   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  (class class class)co 7424  Pcnp 10951   +_P cpp 10953   ·_P cmp 10954   ~_R cer 10956  Rcnr 10957   ·_R cmr 10962

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755  ax-inf2 9649

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6307  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7882  df-1st 8005  df-2nd 8006  df-frecs 8306  df-wrecs 8337  df-recs 8386  df-rdg 8425  df-1o 8483  df-oadd 8487  df-omul 8488  df-er 8724  df-ec 8726  df-qs 8730  df-ni 10964  df-pli 10965  df-mi 10966  df-lti 10967  df-plpq 11000  df-mpq 11001  df-ltpq 11002  df-enq 11003  df-nq 11004  df-erq 11005  df-plq 11006  df-mq 11007  df-1nq 11008  df-rq 11009  df-ltnq 11010  df-np 11073  df-plp 11075  df-mp 11076  df-ltp 11077  df-enr 11147  df-nr 11148  df-mr 11150

This theorem is used by:  sqgt0sr  11198  recexsr  11199  axmulass  11249