Theorem addasssrg 7452

Description: Addition of signed reals is associative. (Contributed by Jim Kingdon, 3-Jan-2020.)

Assertion

Ref	Expression
addasssrg	⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R ∧ 𝐶 ∈ R) → ((𝐴 +R 𝐵) +R 𝐶) = (𝐴 +R (𝐵 +R 𝐶)))

Proof of Theorem addasssrg

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

Step	Hyp	Ref	Expression
1		df-nr 7423	. 2 ⊢ R = ((P × P) / ~R )
2		addsrpr 7441	. 2 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → ([⟨𝑥, 𝑦⟩] ~R +R [⟨𝑧, 𝑤⟩] ~R ) = [⟨(𝑥 +P 𝑧), (𝑦 +P 𝑤)⟩] ~R )
3		addsrpr 7441	. 2 ⊢ (((𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → ([⟨𝑧, 𝑤⟩] ~R +R [⟨𝑣, 𝑢⟩] ~R ) = [⟨(𝑧 +P 𝑣), (𝑤 +P 𝑢)⟩] ~R )
4		addsrpr 7441	. 2 ⊢ ((((𝑥 +P 𝑧) ∈ P ∧ (𝑦 +P 𝑤) ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → ([⟨(𝑥 +P 𝑧), (𝑦 +P 𝑤)⟩] ~R +R [⟨𝑣, 𝑢⟩] ~R ) = [⟨((𝑥 +P 𝑧) +P 𝑣), ((𝑦 +P 𝑤) +P 𝑢)⟩] ~R )
5		addsrpr 7441	. 2 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ ((𝑧 +P 𝑣) ∈ P ∧ (𝑤 +P 𝑢) ∈ P)) → ([⟨𝑥, 𝑦⟩] ~R +R [⟨(𝑧 +P 𝑣), (𝑤 +P 𝑢)⟩] ~R ) = [⟨(𝑥 +P (𝑧 +P 𝑣)), (𝑦 +P (𝑤 +P 𝑢))⟩] ~R )
6		addclpr 7246	. . . 4 ⊢ ((𝑥 ∈ P ∧ 𝑧 ∈ P) → (𝑥 +P 𝑧) ∈ P)
7		addclpr 7246	. . . 4 ⊢ ((𝑦 ∈ P ∧ 𝑤 ∈ P) → (𝑦 +P 𝑤) ∈ P)
8	6, 7	anim12i 334	. . 3 ⊢ (((𝑥 ∈ P ∧ 𝑧 ∈ P) ∧ (𝑦 ∈ P ∧ 𝑤 ∈ P)) → ((𝑥 +P 𝑧) ∈ P ∧ (𝑦 +P 𝑤) ∈ P))
9	8	an4s 558	. 2 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → ((𝑥 +P 𝑧) ∈ P ∧ (𝑦 +P 𝑤) ∈ P))
10		addclpr 7246	. . . 4 ⊢ ((𝑧 ∈ P ∧ 𝑣 ∈ P) → (𝑧 +P 𝑣) ∈ P)
11		addclpr 7246	. . . 4 ⊢ ((𝑤 ∈ P ∧ 𝑢 ∈ P) → (𝑤 +P 𝑢) ∈ P)
12	10, 11	anim12i 334	. . 3 ⊢ (((𝑧 ∈ P ∧ 𝑣 ∈ P) ∧ (𝑤 ∈ P ∧ 𝑢 ∈ P)) → ((𝑧 +P 𝑣) ∈ P ∧ (𝑤 +P 𝑢) ∈ P))
13	12	an4s 558	. 2 ⊢ (((𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → ((𝑧 +P 𝑣) ∈ P ∧ (𝑤 +P 𝑢) ∈ P))
14		addassprg 7288	. . . . 5 ⊢ ((𝑥 ∈ P ∧ 𝑧 ∈ P ∧ 𝑣 ∈ P) → ((𝑥 +P 𝑧) +P 𝑣) = (𝑥 +P (𝑧 +P 𝑣)))
15	14	3adant1r 1177	. . . 4 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ 𝑧 ∈ P ∧ 𝑣 ∈ P) → ((𝑥 +P 𝑧) +P 𝑣) = (𝑥 +P (𝑧 +P 𝑣)))
16	15	3adant2r 1179	. . 3 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ 𝑣 ∈ P) → ((𝑥 +P 𝑧) +P 𝑣) = (𝑥 +P (𝑧 +P 𝑣)))
17	16	3adant3r 1181	. 2 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → ((𝑥 +P 𝑧) +P 𝑣) = (𝑥 +P (𝑧 +P 𝑣)))
18		addassprg 7288	. . . . 5 ⊢ ((𝑦 ∈ P ∧ 𝑤 ∈ P ∧ 𝑢 ∈ P) → ((𝑦 +P 𝑤) +P 𝑢) = (𝑦 +P (𝑤 +P 𝑢)))
19	18	3adant1l 1176	. . . 4 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ 𝑤 ∈ P ∧ 𝑢 ∈ P) → ((𝑦 +P 𝑤) +P 𝑢) = (𝑦 +P (𝑤 +P 𝑢)))
20	19	3adant2l 1178	. . 3 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ 𝑢 ∈ P) → ((𝑦 +P 𝑤) +P 𝑢) = (𝑦 +P (𝑤 +P 𝑢)))
21	20	3adant3l 1180	. 2 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → ((𝑦 +P 𝑤) +P 𝑢) = (𝑦 +P (𝑤 +P 𝑢)))
22	1, 2, 3, 4, 5, 9, 13, 17, 21	ecoviass 6469	1 ⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R ∧ 𝐶 ∈ R) → ((𝐴 +R 𝐵) +R 𝐶) = (𝐴 +R (𝐵 +R 𝐶)))

Syntax hints:   → wi 4   ∧ wa 103   ∧ w3a 930   = wceq 1299   ∈ wcel 1448  (class class class)co 5706  Pcnp 7000   +_P cpp 7002   ~_R cer 7005  Rcnr 7006   +_R cplr 7010

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-coll 3983  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-iinf 4440

This theorem depends on definitions:  df-bi 116  df-dc 787  df-3or 931  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-ral 2380  df-rex 2381  df-reu 2382  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-tr 3967  df-eprel 4149  df-id 4153  df-po 4156  df-iso 4157  df-iord 4226  df-on 4228  df-suc 4231  df-iom 4443  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-ov 5709  df-oprab 5710  df-mpo 5711  df-1st 5969  df-2nd 5970  df-recs 6132  df-irdg 6197  df-1o 6243  df-2o 6244  df-oadd 6247  df-omul 6248  df-er 6359  df-ec 6361  df-qs 6365  df-ni 7013  df-pli 7014  df-mi 7015  df-lti 7016  df-plpq 7053  df-mpq 7054  df-enq 7056  df-nqqs 7057  df-plqqs 7058  df-mqqs 7059  df-1nqqs 7060  df-rq 7061  df-ltnqqs 7062  df-enq0 7133  df-nq0 7134  df-0nq0 7135  df-plq0 7136  df-mq0 7137  df-inp 7175  df-iplp 7177  df-enr 7422  df-nr 7423  df-plr 7424

This theorem is referenced by:  caucvgsrlemoffval  7491  caucvgsrlemoffcau  7493  caucvgsrlemoffres  7495  caucvgsr  7497  axaddass  7557  axmulass  7558  axdistr  7559