Theorem 1idsr 4001

Description: 1 is an identity element for multiplication.

Assertion

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

Proof of Theorem 1idsr

Step	Hyp	Ref	Expression
1		df-nr 3961	. 2 ⊢ R = ((P × P) / ~R )
2		opreq1 3006	. . 3 ⊢ ([⟨x, y⟩] ~R = A → ([⟨x, y⟩] ~R ·R 1R) = (A ·R 1R))
3		id 9	. . 3 ⊢ ([⟨x, y⟩] ~R = A → [⟨x, y⟩] ~R = A)
4	2, 3	cleq12d 1115	. 2 ⊢ ([⟨x, y⟩] ~R = A → (([⟨x, y⟩] ~R ·R 1R) = [⟨x, y⟩] ~R ↔ (A ·R 1R) = A))
5		1pr 3911	. . . . . . 7 ⊢ 1P ∈ P
6		addclpr 3914	. . . . . . 7 ⊢ ((1P ∈ P ∧ 1P ∈ P) → (1P +P 1P) ∈ P)
7	5, 5, 6	mp2an 520	. . . . . 6 ⊢ (1P +P 1P) ∈ P
8	7, 5	pm3.2i 234	. . . . 5 ⊢ ((1P +P 1P) ∈ P ∧ 1P ∈ P)
9		mulsrpr 3979	. . . . 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 )
10	8, 9	mpan2 519	. . . 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 )
11		1idpr 3927	. . . . . . . . 9 ⊢ (x ∈ P → (x ·P 1P) = x)
12	11	opreq1d 3012	. . . . . . . 8 ⊢ (x ∈ P → ((x ·P 1P) +P (x ·P 1P)) = (x +P (x ·P 1P)))
13	5	elisseti 1355	. . . . . . . . 9 ⊢ 1P ∈ V
14	13, 13	distrpr 3926	. . . . . . . 8 ⊢ (x ·P (1P +P 1P)) = ((x ·P 1P) +P (x ·P 1P))
15	12, 14	syl5req 1137	. . . . . . 7 ⊢ (x ∈ P → (x +P (x ·P 1P)) = (x ·P (1P +P 1P)))
16		1idpr 3927	. . . . . . . . 9 ⊢ (y ∈ P → (y ·P 1P) = y)
17	16	opreq1d 3012	. . . . . . . 8 ⊢ (y ∈ P → ((y ·P 1P) +P (y ·P 1P)) = (y +P (y ·P 1P)))
18	13, 13	distrpr 3926	. . . . . . . 8 ⊢ (y ·P (1P +P 1P)) = ((y ·P 1P) +P (y ·P 1P))
19	17, 18	syl5eq 1136	. . . . . . 7 ⊢ (y ∈ P → (y ·P (1P +P 1P)) = (y +P (y ·P 1P)))
20	15, 19	opreqan12d 3015	. . . . . 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		oprex 3018	. . . . . . 7 ⊢ (x ·P 1P) ∈ V
22		oprex 3018	. . . . . . 7 ⊢ (y ·P (1P +P 1P)) ∈ V
23	21, 22	addasspr 3918	. . . . . 6 ⊢ ((x +P (x ·P 1P)) +P (y ·P (1P +P 1P))) = (x +P ((x ·P 1P) +P (y ·P (1P +P 1P))))
24		oprex 3018	. . . . . . 7 ⊢ (x ·P (1P +P 1P)) ∈ V
25		visset 1350	. . . . . . 7 ⊢ y ∈ V
26		oprex 3018	. . . . . . 7 ⊢ (y ·P 1P) ∈ V
27		visset 1350	. . . . . . . 8 ⊢ z ∈ V
28		visset 1350	. . . . . . . 8 ⊢ w ∈ V
29	27, 28	addcompr 3917	. . . . . . 7 ⊢ (z +P w) = (w +P z)
30		visset 1350	. . . . . . . 8 ⊢ v ∈ V
31	28, 30	addasspr 3918	. . . . . . 7 ⊢ ((z +P w) +P v) = (z +P (w +P v))
32	24, 25, 26, 29, 31	caopr12 3075	. . . . . 6 ⊢ ((x ·P (1P +P 1P)) +P (y +P (y ·P 1P))) = (y +P ((x ·P (1P +P 1P)) +P (y ·P 1P)))
33	20, 23, 32	3eqtr3g 1146	. . . . 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))))
34		enreceq 3971	. . . . . . . 8 ⊢ (((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)))))
35		addclpr 3914	. . . . . . . . . 10 ⊢ (((x ·P (1P +P 1P)) ∈ P ∧ (y ·P 1P) ∈ P) → ((x ·P (1P +P 1P)) +P (y ·P 1P)) ∈ P)
36		mulclpr 3916	. . . . . . . . . . 11 ⊢ ((x ∈ P ∧ (1P +P 1P) ∈ P) → (x ·P (1P +P 1P)) ∈ P)
37	7, 36	mpan2 519	. . . . . . . . . 10 ⊢ (x ∈ P → (x ·P (1P +P 1P)) ∈ P)
38		mulclpr 3916	. . . . . . . . . . 11 ⊢ ((y ∈ P ∧ 1P ∈ P) → (y ·P 1P) ∈ P)
39	5, 38	mpan2 519	. . . . . . . . . 10 ⊢ (y ∈ P → (y ·P 1P) ∈ P)
40	35, 37, 39	syl2an 349	. . . . . . . . 9 ⊢ ((x ∈ P ∧ y ∈ P) → ((x ·P (1P +P 1P)) +P (y ·P 1P)) ∈ P)
41		addclpr 3914	. . . . . . . . . 10 ⊢ (((x ·P 1P) ∈ P ∧ (y ·P (1P +P 1P)) ∈ P) → ((x ·P 1P) +P (y ·P (1P +P 1P))) ∈ P)
42		mulclpr 3916	. . . . . . . . . . 11 ⊢ ((x ∈ P ∧ 1P ∈ P) → (x ·P 1P) ∈ P)
43	5, 42	mpan2 519	. . . . . . . . . 10 ⊢ (x ∈ P → (x ·P 1P) ∈ P)
44		mulclpr 3916	. . . . . . . . . . 11 ⊢ ((y ∈ P ∧ (1P +P 1P) ∈ P) → (y ·P (1P +P 1P)) ∈ P)
45	7, 44	mpan2 519	. . . . . . . . . 10 ⊢ (y ∈ P → (y ·P (1P +P 1P)) ∈ P)
46	41, 43, 45	syl2an 349	. . . . . . . . 9 ⊢ ((x ∈ P ∧ y ∈ P) → ((x ·P 1P) +P (y ·P (1P +P 1P))) ∈ P)
47	40, 46	anim12i 268	. . . . . . . 8 ⊢ (((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))
48	34, 47	sylan2 346	. . . . . . 7 ⊢ (((x ∈ P ∧ y ∈ P) ∧ ((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)))))
49	48	anabss5 384	. . . . . 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)))))
50	49	anidms 332	. . . . 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)))))
51	33, 50	mpbird 171	. . . 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 )
52	10, 51	eqtr4d 1131	. . 3 ⊢ ((x ∈ P ∧ y ∈ P) → ([⟨x, y⟩] ~R ·R [⟨(1P +P 1P), 1P⟩] ~R ) = [⟨x, y⟩] ~R )
53		df-1r 3966	. . . 4 ⊢ 1R = [⟨(1P +P 1P), 1P⟩] ~R
54	53	opreq2i 3010	. . 3 ⊢ ([⟨x, y⟩] ~R ·R 1R) = ([⟨x, y⟩] ~R ·R [⟨(1P +P 1P), 1P⟩] ~R )
55	52, 54	syl5eq 1136	. 2 ⊢ ((x ∈ P ∧ y ∈ P) → ([⟨x, y⟩] ~R ·R 1R) = [⟨x, y⟩] ~R )
56	1, 4, 55	ecoptocl 3239	1 ⊢ (A ∈ R → (A ·R 1R) = A)

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196   = wceq 1091   ∈ wcel 1092  ⟨cop 1810  (class class class)co 3001  [cec 3198  Pcnp 3779  1_Pc1p 3780   +_P cpp 3781   ·_P cmp 3782   ~_R cer 3786  Rcnr 3787  1_Rc1r 3789   ·_R cmr 3792

This theorem is referenced by:  pn0sr 4004  sqgt0sr 4009  ax1id 4077  axrecex 4079  axi2m1 4082  axcnre 4087

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-inf 1079

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fv 2438  df-rdg 2970  df-opr 3003  df-oprab 3004  df-1o 3104  df-oadd 3106  df-omul 3107  df-er 3200  df-ec 3202  df-qs 3205  df-ni 3794  df-pli 3795  df-mi 3796  df-lti 3797  df-plpq 3829  df-mpq 3830  df-enq 3831  df-nq 3832  df-plq 3833  df-mq 3834  df-rq 3835  df-ltq 3836  df-1q 3837  df-np 3880  df-1p 3881  df-plp 3882  df-mp 3883  df-ltp 3884  df-mpr 3959  df-enr 3960  df-nr 3961  df-mr 3963  df-1r 3966

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