Theorem 1idsr 11021

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 10979	. 2 ⊢ R = ((P × P) / ~R )
2		oveq1 7374	. . 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 2752	. 2 ⊢ ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → (([⟨𝑥, 𝑦⟩] ~R ·R 1R) = [⟨𝑥, 𝑦⟩] ~R ↔ (𝐴 ·R 1R) = 𝐴))
5		df-1r 10984	. . . 4 ⊢ 1R = [⟨(1P +P 1P), 1P⟩] ~R
6	5	oveq2i 7378	. . 3 ⊢ ([⟨𝑥, 𝑦⟩] ~R ·R 1R) = ([⟨𝑥, 𝑦⟩] ~R ·R [⟨(1P +P 1P), 1P⟩] ~R )
7		1pr 10938	. . . . . 6 ⊢ 1P ∈ P
8		addclpr 10941	. . . . . 6 ⊢ ((1P ∈ P ∧ 1P ∈ P) → (1P +P 1P) ∈ P)
9	7, 7, 8	mp2an 693	. . . . 5 ⊢ (1P +P 1P) ∈ P
10		mulsrpr 10999	. . . . 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 706	. . . 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 10951	. . . . . . . 8 ⊢ (𝑥 ·P (1P +P 1P)) = ((𝑥 ·P 1P) +P (𝑥 ·P 1P))
13		1idpr 10952	. . . . . . . . 9 ⊢ (𝑥 ∈ P → (𝑥 ·P 1P) = 𝑥)
14	13	oveq1d 7382	. . . . . . . 8 ⊢ (𝑥 ∈ P → ((𝑥 ·P 1P) +P (𝑥 ·P 1P)) = (𝑥 +P (𝑥 ·P 1P)))
15	12, 14	eqtr2id 2784	. . . . . . 7 ⊢ (𝑥 ∈ P → (𝑥 +P (𝑥 ·P 1P)) = (𝑥 ·P (1P +P 1P)))
16		distrpr 10951	. . . . . . . 8 ⊢ (𝑦 ·P (1P +P 1P)) = ((𝑦 ·P 1P) +P (𝑦 ·P 1P))
17		1idpr 10952	. . . . . . . . 9 ⊢ (𝑦 ∈ P → (𝑦 ·P 1P) = 𝑦)
18	17	oveq1d 7382	. . . . . . . 8 ⊢ (𝑦 ∈ P → ((𝑦 ·P 1P) +P (𝑦 ·P 1P)) = (𝑦 +P (𝑦 ·P 1P)))
19	16, 18	eqtrid 2783	. . . . . . 7 ⊢ (𝑦 ∈ P → (𝑦 ·P (1P +P 1P)) = (𝑦 +P (𝑦 ·P 1P)))
20	15, 19	oveqan12d 7386	. . . . . 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 10945	. . . . . 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 7400	. . . . . . 7 ⊢ (𝑥 ·P (1P +P 1P)) ∈ V
23		vex 3433	. . . . . . 7 ⊢ 𝑦 ∈ V
24		ovex 7400	. . . . . . 7 ⊢ (𝑦 ·P 1P) ∈ V
25		addcompr 10944	. . . . . . 7 ⊢ (𝑧 +P 𝑤) = (𝑤 +P 𝑧)
26		addasspr 10945	. . . . . . 7 ⊢ ((𝑧 +P 𝑤) +P 𝑣) = (𝑧 +P (𝑤 +P 𝑣))
27	22, 23, 24, 25, 26	caov12 7595	. . . . . 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 2794	. . . . 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 10943	. . . . . . . . . 10 ⊢ ((𝑥 ∈ P ∧ (1P +P 1P) ∈ P) → (𝑥 ·P (1P +P 1P)) ∈ P)
30	9, 29	mpan2 692	. . . . . . . . 9 ⊢ (𝑥 ∈ P → (𝑥 ·P (1P +P 1P)) ∈ P)
31		mulclpr 10943	. . . . . . . . . 10 ⊢ ((𝑦 ∈ P ∧ 1P ∈ P) → (𝑦 ·P 1P) ∈ P)
32	7, 31	mpan2 692	. . . . . . . . 9 ⊢ (𝑦 ∈ P → (𝑦 ·P 1P) ∈ P)
33		addclpr 10941	. . . . . . . . 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 597	. . . . . . . 8 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → ((𝑥 ·P (1P +P 1P)) +P (𝑦 ·P 1P)) ∈ P)
35		mulclpr 10943	. . . . . . . . . 10 ⊢ ((𝑥 ∈ P ∧ 1P ∈ P) → (𝑥 ·P 1P) ∈ P)
36	7, 35	mpan2 692	. . . . . . . . 9 ⊢ (𝑥 ∈ P → (𝑥 ·P 1P) ∈ P)
37		mulclpr 10943	. . . . . . . . . 10 ⊢ ((𝑦 ∈ P ∧ (1P +P 1P) ∈ P) → (𝑦 ·P (1P +P 1P)) ∈ P)
38	9, 37	mpan2 692	. . . . . . . . 9 ⊢ (𝑦 ∈ P → (𝑦 ·P (1P +P 1P)) ∈ P)
39		addclpr 10941	. . . . . . . . 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 597	. . . . . . . 8 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → ((𝑥 ·P 1P) +P (𝑦 ·P (1P +P 1P))) ∈ P)
41	34, 40	anim12i 614	. . . . . . 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 10989	. . . . . . 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 592	. . . . . 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 2774	. . 3 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → ([⟨𝑥, 𝑦⟩] ~R ·R [⟨(1P +P 1P), 1P⟩] ~R ) = [⟨𝑥, 𝑦⟩] ~R )
47	6, 46	eqtrid 2783	. 2 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → ([⟨𝑥, 𝑦⟩] ~R ·R 1R) = [⟨𝑥, 𝑦⟩] ~R )
48	1, 4, 47	ecoptocl 8754	1 ⊢ (𝐴 ∈ R → (𝐴 ·R 1R) = 𝐴)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ⟨cop 4573  (class class class)co 7367  [cec 8641  Pcnp 10782  1_Pc1p 10783   +_P cpp 10784   ·_P cmp 10785   ~_R cer 10787  Rcnr 10788  1_Rc1r 10790   ·_R cmr 10793

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-inf2 9562

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-oadd 8409  df-omul 8410  df-er 8643  df-ec 8645  df-qs 8649  df-ni 10795  df-pli 10796  df-mi 10797  df-lti 10798  df-plpq 10831  df-mpq 10832  df-ltpq 10833  df-enq 10834  df-nq 10835  df-erq 10836  df-plq 10837  df-mq 10838  df-1nq 10839  df-rq 10840  df-ltnq 10841  df-np 10904  df-1p 10905  df-plp 10906  df-mp 10907  df-ltp 10908  df-enr 10978  df-nr 10979  df-mr 10981  df-1r 10984

This theorem is referenced by:  pn0sr  11024  sqgt0sr  11029  axi2m1  11082  ax1rid  11084  axcnre  11087