Theorem 1idsr 7709

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 7668	. 2 ⊢ R = ((P × P) / ~R )
2		oveq1 5849	. . 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 2180	. 2 ⊢ ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → (([⟨𝑥, 𝑦⟩] ~R ·R 1R) = [⟨𝑥, 𝑦⟩] ~R ↔ (𝐴 ·R 1R) = 𝐴))
5		df-1r 7673	. . . 4 ⊢ 1R = [⟨(1P +P 1P), 1P⟩] ~R
6	5	oveq2i 5853	. . 3 ⊢ ([⟨𝑥, 𝑦⟩] ~R ·R 1R) = ([⟨𝑥, 𝑦⟩] ~R ·R [⟨(1P +P 1P), 1P⟩] ~R )
7		1pr 7495	. . . . . 6 ⊢ 1P ∈ P
8		addclpr 7478	. . . . . 6 ⊢ ((1P ∈ P ∧ 1P ∈ P) → (1P +P 1P) ∈ P)
9	7, 7, 8	mp2an 423	. . . . 5 ⊢ (1P +P 1P) ∈ P
10		mulsrpr 7687	. . . . 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 436	. . . 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 7529	. . . . . . . . 9 ⊢ ((𝑥 ∈ P ∧ 1P ∈ P ∧ 1P ∈ P) → (𝑥 ·P (1P +P 1P)) = ((𝑥 ·P 1P) +P (𝑥 ·P 1P)))
13	7, 7, 12	mp3an23 1319	. . . . . . . 8 ⊢ (𝑥 ∈ P → (𝑥 ·P (1P +P 1P)) = ((𝑥 ·P 1P) +P (𝑥 ·P 1P)))
14		1idpr 7533	. . . . . . . . 9 ⊢ (𝑥 ∈ P → (𝑥 ·P 1P) = 𝑥)
15	14	oveq1d 5857	. . . . . . . 8 ⊢ (𝑥 ∈ P → ((𝑥 ·P 1P) +P (𝑥 ·P 1P)) = (𝑥 +P (𝑥 ·P 1P)))
16	13, 15	eqtr2d 2199	. . . . . . 7 ⊢ (𝑥 ∈ P → (𝑥 +P (𝑥 ·P 1P)) = (𝑥 ·P (1P +P 1P)))
17		distrprg 7529	. . . . . . . . 9 ⊢ ((𝑦 ∈ P ∧ 1P ∈ P ∧ 1P ∈ P) → (𝑦 ·P (1P +P 1P)) = ((𝑦 ·P 1P) +P (𝑦 ·P 1P)))
18	7, 7, 17	mp3an23 1319	. . . . . . . 8 ⊢ (𝑦 ∈ P → (𝑦 ·P (1P +P 1P)) = ((𝑦 ·P 1P) +P (𝑦 ·P 1P)))
19		1idpr 7533	. . . . . . . . 9 ⊢ (𝑦 ∈ P → (𝑦 ·P 1P) = 𝑦)
20	19	oveq1d 5857	. . . . . . . 8 ⊢ (𝑦 ∈ P → ((𝑦 ·P 1P) +P (𝑦 ·P 1P)) = (𝑦 +P (𝑦 ·P 1P)))
21	18, 20	eqtrd 2198	. . . . . . 7 ⊢ (𝑦 ∈ P → (𝑦 ·P (1P +P 1P)) = (𝑦 +P (𝑦 ·P 1P)))
22	16, 21	oveqan12d 5861	. . . . . 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 108	. . . . . . 7 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → 𝑥 ∈ P)
24		mulclpr 7513	. . . . . . . 8 ⊢ ((𝑥 ∈ P ∧ 1P ∈ P) → (𝑥 ·P 1P) ∈ P)
25	23, 7, 24	sylancl 410	. . . . . . 7 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → (𝑥 ·P 1P) ∈ P)
26		mulclpr 7513	. . . . . . . . 9 ⊢ ((𝑦 ∈ P ∧ (1P +P 1P) ∈ P) → (𝑦 ·P (1P +P 1P)) ∈ P)
27	9, 26	mpan2 422	. . . . . . . 8 ⊢ (𝑦 ∈ P → (𝑦 ·P (1P +P 1P)) ∈ P)
28	27	adantl 275	. . . . . . 7 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → (𝑦 ·P (1P +P 1P)) ∈ P)
29		addassprg 7520	. . . . . . 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 1228	. . . . . 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 7513	. . . . . . . 8 ⊢ ((𝑥 ∈ P ∧ (1P +P 1P) ∈ P) → (𝑥 ·P (1P +P 1P)) ∈ P)
32	23, 9, 31	sylancl 410	. . . . . . 7 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → (𝑥 ·P (1P +P 1P)) ∈ P)
33		simpr 109	. . . . . . 7 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → 𝑦 ∈ P)
34		mulclpr 7513	. . . . . . . 8 ⊢ ((𝑦 ∈ P ∧ 1P ∈ P) → (𝑦 ·P 1P) ∈ P)
35	33, 7, 34	sylancl 410	. . . . . . 7 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → (𝑦 ·P 1P) ∈ P)
36		addcomprg 7519	. . . . . . . 8 ⊢ ((𝑧 ∈ P ∧ 𝑤 ∈ P) → (𝑧 +P 𝑤) = (𝑤 +P 𝑧))
37	36	adantl 275	. . . . . . 7 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → (𝑧 +P 𝑤) = (𝑤 +P 𝑧))
38		addassprg 7520	. . . . . . . 8 ⊢ ((𝑧 ∈ P ∧ 𝑤 ∈ P ∧ 𝑣 ∈ P) → ((𝑧 +P 𝑤) +P 𝑣) = (𝑧 +P (𝑤 +P 𝑣)))
39	38	adantl 275	. . . . . . 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 6023	. . . . . 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 2206	. . . . 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 422	. . . . . . . . 9 ⊢ (𝑥 ∈ P → (𝑥 ·P (1P +P 1P)) ∈ P)
43	7, 34	mpan2 422	. . . . . . . . 9 ⊢ (𝑦 ∈ P → (𝑦 ·P 1P) ∈ P)
44		addclpr 7478	. . . . . . . . 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 287	. . . . . . . 8 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → ((𝑥 ·P (1P +P 1P)) +P (𝑦 ·P 1P)) ∈ P)
46	7, 24	mpan2 422	. . . . . . . . 9 ⊢ (𝑥 ∈ P → (𝑥 ·P 1P) ∈ P)
47		addclpr 7478	. . . . . . . . 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 287	. . . . . . . 8 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → ((𝑥 ·P 1P) +P (𝑦 ·P (1P +P 1P))) ∈ P)
49	45, 48	anim12i 336	. . . . . . 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 7677	. . . . . . 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 280	. . . . . 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 395	. . . . 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 166	. . . 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 2201	. . 3 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → ([⟨𝑥, 𝑦⟩] ~R ·R [⟨(1P +P 1P), 1P⟩] ~R ) = [⟨𝑥, 𝑦⟩] ~R )
55	6, 54	syl5eq 2211	. 2 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → ([⟨𝑥, 𝑦⟩] ~R ·R 1R) = [⟨𝑥, 𝑦⟩] ~R )
56	1, 4, 55	ecoptocl 6588	1 ⊢ (𝐴 ∈ R → (𝐴 ·R 1R) = 𝐴)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∧ w3a 968   = wceq 1343   ∈ wcel 2136  ⟨cop 3579  (class class class)co 5842  [cec 6499  Pcnp 7232  1_Pc1p 7233   +_P cpp 7234   ·_P cmp 7235   ~_R cer 7237  Rcnr 7238  1_Rc1r 7240   ·_R cmr 7243

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-eprel 4267  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-irdg 6338  df-1o 6384  df-2o 6385  df-oadd 6388  df-omul 6389  df-er 6501  df-ec 6503  df-qs 6507  df-ni 7245  df-pli 7246  df-mi 7247  df-lti 7248  df-plpq 7285  df-mpq 7286  df-enq 7288  df-nqqs 7289  df-plqqs 7290  df-mqqs 7291  df-1nqqs 7292  df-rq 7293  df-ltnqqs 7294  df-enq0 7365  df-nq0 7366  df-0nq0 7367  df-plq0 7368  df-mq0 7369  df-inp 7407  df-i1p 7408  df-iplp 7409  df-imp 7410  df-enr 7667  df-nr 7668  df-mr 7670  df-1r 7673

This theorem is referenced by:  pn0sr  7712  axi2m1  7816  ax1rid  7818  axcnre  7822