Theorem 1idsr 7943

Description: 1 is an identity element for multiplication. (Contributed by Jim Kingdon, 5-Jan-2020.)

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 7902	. 2 ⊢ R = ((P × P) / ~R )
2		oveq1 6001	. . 3 ⊢ ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → ([⟨𝑥, 𝑦⟩] ~R ·R 1R) = (𝐴 ·R 1R))
3		id 19	. . 3 ⊢ ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → [⟨𝑥, 𝑦⟩] ~R = 𝐴)
4	2, 3	eqeq12d 2244	. 2 ⊢ ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → (([⟨𝑥, 𝑦⟩] ~R ·R 1R) = [⟨𝑥, 𝑦⟩] ~R ↔ (𝐴 ·R 1R) = 𝐴))
5		df-1r 7907	. . . 4 ⊢ 1R = [⟨(1P +P 1P), 1P⟩] ~R
6	5	oveq2i 6005	. . 3 ⊢ ([⟨𝑥, 𝑦⟩] ~R ·R 1R) = ([⟨𝑥, 𝑦⟩] ~R ·R [⟨(1P +P 1P), 1P⟩] ~R )
7		1pr 7729	. . . . . 6 ⊢ 1P ∈ P
8		addclpr 7712	. . . . . 6 ⊢ ((1P ∈ P ∧ 1P ∈ P) → (1P +P 1P) ∈ P)
9	7, 7, 8	mp2an 426	. . . . 5 ⊢ (1P +P 1P) ∈ P
10		mulsrpr 7921	. . . . 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 439	. . . 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		distrprg 7763	. . . . . . . . 9 ⊢ ((𝑥 ∈ P ∧ 1P ∈ P ∧ 1P ∈ P) → (𝑥 ·P (1P +P 1P)) = ((𝑥 ·P 1P) +P (𝑥 ·P 1P)))
13	7, 7, 12	mp3an23 1363	. . . . . . . 8 ⊢ (𝑥 ∈ P → (𝑥 ·P (1P +P 1P)) = ((𝑥 ·P 1P) +P (𝑥 ·P 1P)))
14		1idpr 7767	. . . . . . . . 9 ⊢ (𝑥 ∈ P → (𝑥 ·P 1P) = 𝑥)
15	14	oveq1d 6009	. . . . . . . 8 ⊢ (𝑥 ∈ P → ((𝑥 ·P 1P) +P (𝑥 ·P 1P)) = (𝑥 +P (𝑥 ·P 1P)))
16	13, 15	eqtr2d 2263	. . . . . . 7 ⊢ (𝑥 ∈ P → (𝑥 +P (𝑥 ·P 1P)) = (𝑥 ·P (1P +P 1P)))
17		distrprg 7763	. . . . . . . . 9 ⊢ ((𝑦 ∈ P ∧ 1P ∈ P ∧ 1P ∈ P) → (𝑦 ·P (1P +P 1P)) = ((𝑦 ·P 1P) +P (𝑦 ·P 1P)))
18	7, 7, 17	mp3an23 1363	. . . . . . . 8 ⊢ (𝑦 ∈ P → (𝑦 ·P (1P +P 1P)) = ((𝑦 ·P 1P) +P (𝑦 ·P 1P)))
19		1idpr 7767	. . . . . . . . 9 ⊢ (𝑦 ∈ P → (𝑦 ·P 1P) = 𝑦)
20	19	oveq1d 6009	. . . . . . . 8 ⊢ (𝑦 ∈ P → ((𝑦 ·P 1P) +P (𝑦 ·P 1P)) = (𝑦 +P (𝑦 ·P 1P)))
21	18, 20	eqtrd 2262	. . . . . . 7 ⊢ (𝑦 ∈ P → (𝑦 ·P (1P +P 1P)) = (𝑦 +P (𝑦 ·P 1P)))
22	16, 21	oveqan12d 6013	. . . . . 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))))
23		simpl 109	. . . . . . 7 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → 𝑥 ∈ P)
24		mulclpr 7747	. . . . . . . 8 ⊢ ((𝑥 ∈ P ∧ 1P ∈ P) → (𝑥 ·P 1P) ∈ P)
25	23, 7, 24	sylancl 413	. . . . . . 7 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → (𝑥 ·P 1P) ∈ P)
26		mulclpr 7747	. . . . . . . . 9 ⊢ ((𝑦 ∈ P ∧ (1P +P 1P) ∈ P) → (𝑦 ·P (1P +P 1P)) ∈ P)
27	9, 26	mpan2 425	. . . . . . . 8 ⊢ (𝑦 ∈ P → (𝑦 ·P (1P +P 1P)) ∈ P)
28	27	adantl 277	. . . . . . 7 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → (𝑦 ·P (1P +P 1P)) ∈ P)
29		addassprg 7754	. . . . . . 7 ⊢ ((𝑥 ∈ P ∧ (𝑥 ·P 1P) ∈ P ∧ (𝑦 ·P (1P +P 1P)) ∈ P) → ((𝑥 +P (𝑥 ·P 1P)) +P (𝑦 ·P (1P +P 1P))) = (𝑥 +P ((𝑥 ·P 1P) +P (𝑦 ·P (1P +P 1P)))))
30	23, 25, 28, 29	syl3anc 1271	. . . . . 6 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → ((𝑥 +P (𝑥 ·P 1P)) +P (𝑦 ·P (1P +P 1P))) = (𝑥 +P ((𝑥 ·P 1P) +P (𝑦 ·P (1P +P 1P)))))
31		mulclpr 7747	. . . . . . . 8 ⊢ ((𝑥 ∈ P ∧ (1P +P 1P) ∈ P) → (𝑥 ·P (1P +P 1P)) ∈ P)
32	23, 9, 31	sylancl 413	. . . . . . 7 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → (𝑥 ·P (1P +P 1P)) ∈ P)
33		simpr 110	. . . . . . 7 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → 𝑦 ∈ P)
34		mulclpr 7747	. . . . . . . 8 ⊢ ((𝑦 ∈ P ∧ 1P ∈ P) → (𝑦 ·P 1P) ∈ P)
35	33, 7, 34	sylancl 413	. . . . . . 7 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → (𝑦 ·P 1P) ∈ P)
36		addcomprg 7753	. . . . . . . 8 ⊢ ((𝑧 ∈ P ∧ 𝑤 ∈ P) → (𝑧 +P 𝑤) = (𝑤 +P 𝑧))
37	36	adantl 277	. . . . . . 7 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → (𝑧 +P 𝑤) = (𝑤 +P 𝑧))
38		addassprg 7754	. . . . . . . 8 ⊢ ((𝑧 ∈ P ∧ 𝑤 ∈ P ∧ 𝑣 ∈ P) → ((𝑧 +P 𝑤) +P 𝑣) = (𝑧 +P (𝑤 +P 𝑣)))
39	38	adantl 277	. . . . . . 7 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P ∧ 𝑣 ∈ P)) → ((𝑧 +P 𝑤) +P 𝑣) = (𝑧 +P (𝑤 +P 𝑣)))
40	32, 33, 35, 37, 39	caov12d 6178	. . . . . 6 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → ((𝑥 ·P (1P +P 1P)) +P (𝑦 +P (𝑦 ·P 1P))) = (𝑦 +P ((𝑥 ·P (1P +P 1P)) +P (𝑦 ·P 1P))))
41	22, 30, 40	3eqtr3d 2270	. . . . 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))))
42	9, 31	mpan2 425	. . . . . . . . 9 ⊢ (𝑥 ∈ P → (𝑥 ·P (1P +P 1P)) ∈ P)
43	7, 34	mpan2 425	. . . . . . . . 9 ⊢ (𝑦 ∈ P → (𝑦 ·P 1P) ∈ P)
44		addclpr 7712	. . . . . . . . 9 ⊢ (((𝑥 ·P (1P +P 1P)) ∈ P ∧ (𝑦 ·P 1P) ∈ P) → ((𝑥 ·P (1P +P 1P)) +P (𝑦 ·P 1P)) ∈ P)
45	42, 43, 44	syl2an 289	. . . . . . . 8 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → ((𝑥 ·P (1P +P 1P)) +P (𝑦 ·P 1P)) ∈ P)
46	7, 24	mpan2 425	. . . . . . . . 9 ⊢ (𝑥 ∈ P → (𝑥 ·P 1P) ∈ P)
47		addclpr 7712	. . . . . . . . 9 ⊢ (((𝑥 ·P 1P) ∈ P ∧ (𝑦 ·P (1P +P 1P)) ∈ P) → ((𝑥 ·P 1P) +P (𝑦 ·P (1P +P 1P))) ∈ P)
48	46, 27, 47	syl2an 289	. . . . . . . 8 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → ((𝑥 ·P 1P) +P (𝑦 ·P (1P +P 1P))) ∈ P)
49	45, 48	anim12i 338	. . . . . . 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))
50		enreceq 7911	. . . . . . 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)))))
51	49, 50	syldan 282	. . . . . 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)))))
52	51	anidms 397	. . . . 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)))))
53	41, 52	mpbird 167	. . . 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 )
54	11, 53	eqtr4d 2265	. . 3 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → ([⟨𝑥, 𝑦⟩] ~R ·R [⟨(1P +P 1P), 1P⟩] ~R ) = [⟨𝑥, 𝑦⟩] ~R )
55	6, 54	eqtrid 2274	. 2 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → ([⟨𝑥, 𝑦⟩] ~R ·R 1R) = [⟨𝑥, 𝑦⟩] ~R )
56	1, 4, 55	ecoptocl 6759	1 ⊢ (𝐴 ∈ R → (𝐴 ·R 1R) = 𝐴)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1002   = wceq 1395   ∈ wcel 2200  ⟨cop 3669  (class class class)co 5994  [cec 6668  Pcnp 7466  1_Pc1p 7467   +_P cpp 7468   ·_P cmp 7469   ~_R cer 7471  Rcnr 7472  1_Rc1r 7474   ·_R cmr 7477

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4521  ax-setind 4626  ax-iinf 4677

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-eprel 4377  df-id 4381  df-po 4384  df-iso 4385  df-iord 4454  df-on 4456  df-suc 4459  df-iom 4680  df-xp 4722  df-rel 4723  df-cnv 4724  df-co 4725  df-dm 4726  df-rn 4727  df-res 4728  df-ima 4729  df-iota 5274  df-fun 5316  df-fn 5317  df-f 5318  df-f1 5319  df-fo 5320  df-f1o 5321  df-fv 5322  df-ov 5997  df-oprab 5998  df-mpo 5999  df-1st 6276  df-2nd 6277  df-recs 6441  df-irdg 6506  df-1o 6552  df-2o 6553  df-oadd 6556  df-omul 6557  df-er 6670  df-ec 6672  df-qs 6676  df-ni 7479  df-pli 7480  df-mi 7481  df-lti 7482  df-plpq 7519  df-mpq 7520  df-enq 7522  df-nqqs 7523  df-plqqs 7524  df-mqqs 7525  df-1nqqs 7526  df-rq 7527  df-ltnqqs 7528  df-enq0 7599  df-nq0 7600  df-0nq0 7601  df-plq0 7602  df-mq0 7603  df-inp 7641  df-i1p 7642  df-iplp 7643  df-imp 7644  df-enr 7901  df-nr 7902  df-mr 7904  df-1r 7907

This theorem is referenced by:  pn0sr  7946  axi2m1  8050  ax1rid  8052  axcnre  8056