Theorem 1idsr 11138

Description: 1 is an identity element for multiplication. (Contributed by NM, 2-May-1996.) (New usage is discouraged.)

Assertion

Ref	Expression
1idsr	⊢ (𝐴 ∈ R → (𝐴 ·R 1R) = 𝐴)

Proof of Theorem 1idsr

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

Step	Hyp	Ref	Expression
1		df-nr 11096	. 2 ⊢ R = ((P × P) / ~R )
2		oveq1 7438	. . 3 ⊢ ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → ([⟨𝑥, 𝑦⟩] ~R ·R 1R) = (𝐴 ·R 1R))
3		id 22	. . 3 ⊢ ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → [⟨𝑥, 𝑦⟩] ~R = 𝐴)
4	2, 3	eqeq12d 2753	. 2 ⊢ ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → (([⟨𝑥, 𝑦⟩] ~R ·R 1R) = [⟨𝑥, 𝑦⟩] ~R ↔ (𝐴 ·R 1R) = 𝐴))
5		df-1r 11101	. . . 4 ⊢ 1R = [⟨(1P +P 1P), 1P⟩] ~R
6	5	oveq2i 7442	. . 3 ⊢ ([⟨𝑥, 𝑦⟩] ~R ·R 1R) = ([⟨𝑥, 𝑦⟩] ~R ·R [⟨(1P +P 1P), 1P⟩] ~R )
7		1pr 11055	. . . . . 6 ⊢ 1P ∈ P
8		addclpr 11058	. . . . . 6 ⊢ ((1P ∈ P ∧ 1P ∈ P) → (1P +P 1P) ∈ P)
9	7, 7, 8	mp2an 692	. . . . 5 ⊢ (1P +P 1P) ∈ P
10		mulsrpr 11116	. . . . 5 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ ((1P +P 1P) ∈ P ∧ 1P ∈ P)) → ([⟨𝑥, 𝑦⟩] ~R ·R [⟨(1P +P 1P), 1P⟩] ~R ) = [⟨((𝑥 ·P (1P +P 1P)) +P (𝑦 ·P 1P)), ((𝑥 ·P 1P) +P (𝑦 ·P (1P +P 1P)))⟩] ~R )
11	9, 7, 10	mpanr12 705	. . . 4 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → ([⟨𝑥, 𝑦⟩] ~R ·R [⟨(1P +P 1P), 1P⟩] ~R ) = [⟨((𝑥 ·P (1P +P 1P)) +P (𝑦 ·P 1P)), ((𝑥 ·P 1P) +P (𝑦 ·P (1P +P 1P)))⟩] ~R )
12		distrpr 11068	. . . . . . . 8 ⊢ (𝑥 ·P (1P +P 1P)) = ((𝑥 ·P 1P) +P (𝑥 ·P 1P))
13		1idpr 11069	. . . . . . . . 9 ⊢ (𝑥 ∈ P → (𝑥 ·P 1P) = 𝑥)
14	13	oveq1d 7446	. . . . . . . 8 ⊢ (𝑥 ∈ P → ((𝑥 ·P 1P) +P (𝑥 ·P 1P)) = (𝑥 +P (𝑥 ·P 1P)))
15	12, 14	eqtr2id 2790	. . . . . . 7 ⊢ (𝑥 ∈ P → (𝑥 +P (𝑥 ·P 1P)) = (𝑥 ·P (1P +P 1P)))
16		distrpr 11068	. . . . . . . 8 ⊢ (𝑦 ·P (1P +P 1P)) = ((𝑦 ·P 1P) +P (𝑦 ·P 1P))
17		1idpr 11069	. . . . . . . . 9 ⊢ (𝑦 ∈ P → (𝑦 ·P 1P) = 𝑦)
18	17	oveq1d 7446	. . . . . . . 8 ⊢ (𝑦 ∈ P → ((𝑦 ·P 1P) +P (𝑦 ·P 1P)) = (𝑦 +P (𝑦 ·P 1P)))
19	16, 18	eqtrid 2789	. . . . . . 7 ⊢ (𝑦 ∈ P → (𝑦 ·P (1P +P 1P)) = (𝑦 +P (𝑦 ·P 1P)))
20	15, 19	oveqan12d 7450	. . . . . 6 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → ((𝑥 +P (𝑥 ·P 1P)) +P (𝑦 ·P (1P +P 1P))) = ((𝑥 ·P (1P +P 1P)) +P (𝑦 +P (𝑦 ·P 1P))))
21		addasspr 11062	. . . . . 6 ⊢ ((𝑥 +P (𝑥 ·P 1P)) +P (𝑦 ·P (1P +P 1P))) = (𝑥 +P ((𝑥 ·P 1P) +P (𝑦 ·P (1P +P 1P))))
22		ovex 7464	. . . . . . 7 ⊢ (𝑥 ·P (1P +P 1P)) ∈ V
23		vex 3484	. . . . . . 7 ⊢ 𝑦 ∈ V
24		ovex 7464	. . . . . . 7 ⊢ (𝑦 ·P 1P) ∈ V
25		addcompr 11061	. . . . . . 7 ⊢ (𝑧 +P 𝑤) = (𝑤 +P 𝑧)
26		addasspr 11062	. . . . . . 7 ⊢ ((𝑧 +P 𝑤) +P 𝑣) = (𝑧 +P (𝑤 +P 𝑣))
27	22, 23, 24, 25, 26	caov12 7661	. . . . . 6 ⊢ ((𝑥 ·P (1P +P 1P)) +P (𝑦 +P (𝑦 ·P 1P))) = (𝑦 +P ((𝑥 ·P (1P +P 1P)) +P (𝑦 ·P 1P)))
28	20, 21, 27	3eqtr3g 2800	. . . . 5 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → (𝑥 +P ((𝑥 ·P 1P) +P (𝑦 ·P (1P +P 1P)))) = (𝑦 +P ((𝑥 ·P (1P +P 1P)) +P (𝑦 ·P 1P))))
29		mulclpr 11060	. . . . . . . . . 10 ⊢ ((𝑥 ∈ P ∧ (1P +P 1P) ∈ P) → (𝑥 ·P (1P +P 1P)) ∈ P)
30	9, 29	mpan2 691	. . . . . . . . 9 ⊢ (𝑥 ∈ P → (𝑥 ·P (1P +P 1P)) ∈ P)
31		mulclpr 11060	. . . . . . . . . 10 ⊢ ((𝑦 ∈ P ∧ 1P ∈ P) → (𝑦 ·P 1P) ∈ P)
32	7, 31	mpan2 691	. . . . . . . . 9 ⊢ (𝑦 ∈ P → (𝑦 ·P 1P) ∈ P)
33		addclpr 11058	. . . . . . . . 9 ⊢ (((𝑥 ·P (1P +P 1P)) ∈ P ∧ (𝑦 ·P 1P) ∈ P) → ((𝑥 ·P (1P +P 1P)) +P (𝑦 ·P 1P)) ∈ P)
34	30, 32, 33	syl2an 596	. . . . . . . 8 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → ((𝑥 ·P (1P +P 1P)) +P (𝑦 ·P 1P)) ∈ P)
35		mulclpr 11060	. . . . . . . . . 10 ⊢ ((𝑥 ∈ P ∧ 1P ∈ P) → (𝑥 ·P 1P) ∈ P)
36	7, 35	mpan2 691	. . . . . . . . 9 ⊢ (𝑥 ∈ P → (𝑥 ·P 1P) ∈ P)
37		mulclpr 11060	. . . . . . . . . 10 ⊢ ((𝑦 ∈ P ∧ (1P +P 1P) ∈ P) → (𝑦 ·P (1P +P 1P)) ∈ P)
38	9, 37	mpan2 691	. . . . . . . . 9 ⊢ (𝑦 ∈ P → (𝑦 ·P (1P +P 1P)) ∈ P)
39		addclpr 11058	. . . . . . . . 9 ⊢ (((𝑥 ·P 1P) ∈ P ∧ (𝑦 ·P (1P +P 1P)) ∈ P) → ((𝑥 ·P 1P) +P (𝑦 ·P (1P +P 1P))) ∈ P)
40	36, 38, 39	syl2an 596	. . . . . . . 8 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → ((𝑥 ·P 1P) +P (𝑦 ·P (1P +P 1P))) ∈ P)
41	34, 40	anim12i 613	. . . . . . 7 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑥 ∈ P ∧ 𝑦 ∈ P)) → (((𝑥 ·P (1P +P 1P)) +P (𝑦 ·P 1P)) ∈ P ∧ ((𝑥 ·P 1P) +P (𝑦 ·P (1P +P 1P))) ∈ P))
42		enreceq 11106	. . . . . . 7 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (((𝑥 ·P (1P +P 1P)) +P (𝑦 ·P 1P)) ∈ P ∧ ((𝑥 ·P 1P) +P (𝑦 ·P (1P +P 1P))) ∈ P)) → ([⟨𝑥, 𝑦⟩] ~R = [⟨((𝑥 ·P (1P +P 1P)) +P (𝑦 ·P 1P)), ((𝑥 ·P 1P) +P (𝑦 ·P (1P +P 1P)))⟩] ~R ↔ (𝑥 +P ((𝑥 ·P 1P) +P (𝑦 ·P (1P +P 1P)))) = (𝑦 +P ((𝑥 ·P (1P +P 1P)) +P (𝑦 ·P 1P)))))
43	41, 42	syldan 591	. . . . . 6 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑥 ∈ P ∧ 𝑦 ∈ P)) → ([⟨𝑥, 𝑦⟩] ~R = [⟨((𝑥 ·P (1P +P 1P)) +P (𝑦 ·P 1P)), ((𝑥 ·P 1P) +P (𝑦 ·P (1P +P 1P)))⟩] ~R ↔ (𝑥 +P ((𝑥 ·P 1P) +P (𝑦 ·P (1P +P 1P)))) = (𝑦 +P ((𝑥 ·P (1P +P 1P)) +P (𝑦 ·P 1P)))))
44	43	anidms 566	. . . . 5 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → ([⟨𝑥, 𝑦⟩] ~R = [⟨((𝑥 ·P (1P +P 1P)) +P (𝑦 ·P 1P)), ((𝑥 ·P 1P) +P (𝑦 ·P (1P +P 1P)))⟩] ~R ↔ (𝑥 +P ((𝑥 ·P 1P) +P (𝑦 ·P (1P +P 1P)))) = (𝑦 +P ((𝑥 ·P (1P +P 1P)) +P (𝑦 ·P 1P)))))
45	28, 44	mpbird 257	. . . 4 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → [⟨𝑥, 𝑦⟩] ~R = [⟨((𝑥 ·P (1P +P 1P)) +P (𝑦 ·P 1P)), ((𝑥 ·P 1P) +P (𝑦 ·P (1P +P 1P)))⟩] ~R )
46	11, 45	eqtr4d 2780	. . 3 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → ([⟨𝑥, 𝑦⟩] ~R ·R [⟨(1P +P 1P), 1P⟩] ~R ) = [⟨𝑥, 𝑦⟩] ~R )
47	6, 46	eqtrid 2789	. 2 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → ([⟨𝑥, 𝑦⟩] ~R ·R 1R) = [⟨𝑥, 𝑦⟩] ~R )
48	1, 4, 47	ecoptocl 8847	1 ⊢ (𝐴 ∈ R → (𝐴 ·R 1R) = 𝐴)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ⟨cop 4632  (class class class)co 7431  [cec 8743  Pcnp 10899  1_Pc1p 10900   +_P cpp 10901   ·_P cmp 10902   ~_R cer 10904  Rcnr 10905  1_Rc1r 10907   ·_R cmr 10910

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-oadd 8510  df-omul 8511  df-er 8745  df-ec 8747  df-qs 8751  df-ni 10912  df-pli 10913  df-mi 10914  df-lti 10915  df-plpq 10948  df-mpq 10949  df-ltpq 10950  df-enq 10951  df-nq 10952  df-erq 10953  df-plq 10954  df-mq 10955  df-1nq 10956  df-rq 10957  df-ltnq 10958  df-np 11021  df-1p 11022  df-plp 11023  df-mp 11024  df-ltp 11025  df-enr 11095  df-nr 11096  df-mr 11098  df-1r 11101

This theorem is referenced by:  pn0sr  11141  sqgt0sr  11146  axi2m1  11199  ax1rid  11201  axcnre  11204