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Theorem 1idsr 9957
Description: 1 is an identity element for multiplication. (Contributed by NM, 2-May-1996.) (New usage is discouraged.)
Assertion
Ref Expression
1idsr (𝐴R → (𝐴 ·R 1R) = 𝐴)

Proof of Theorem 1idsr
Dummy variables 𝑥 𝑦 𝑧 𝑤 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-nr 9916 . 2 R = ((P × P) / ~R )
2 oveq1 6697 . . 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)
42, 3eqeq12d 2666 . 2 ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → (([⟨𝑥, 𝑦⟩] ~R ·R 1R) = [⟨𝑥, 𝑦⟩] ~R ↔ (𝐴 ·R 1R) = 𝐴))
5 df-1r 9921 . . . 4 1R = [⟨(1P +P 1P), 1P⟩] ~R
65oveq2i 6701 . . 3 ([⟨𝑥, 𝑦⟩] ~R ·R 1R) = ([⟨𝑥, 𝑦⟩] ~R ·R [⟨(1P +P 1P), 1P⟩] ~R )
7 1pr 9875 . . . . . 6 1PP
8 addclpr 9878 . . . . . 6 ((1PP ∧ 1PP) → (1P +P 1P) ∈ P)
97, 7, 8mp2an 708 . . . . 5 (1P +P 1P) ∈ P
10 mulsrpr 9935 . . . . 5 (((𝑥P𝑦P) ∧ ((1P +P 1P) ∈ P ∧ 1PP)) → ([⟨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 )
119, 7, 10mpanr12 721 . . . 4 ((𝑥P𝑦P) → ([⟨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 9888 . . . . . . . 8 (𝑥 ·P (1P +P 1P)) = ((𝑥 ·P 1P) +P (𝑥 ·P 1P))
13 1idpr 9889 . . . . . . . . 9 (𝑥P → (𝑥 ·P 1P) = 𝑥)
1413oveq1d 6705 . . . . . . . 8 (𝑥P → ((𝑥 ·P 1P) +P (𝑥 ·P 1P)) = (𝑥 +P (𝑥 ·P 1P)))
1512, 14syl5req 2698 . . . . . . 7 (𝑥P → (𝑥 +P (𝑥 ·P 1P)) = (𝑥 ·P (1P +P 1P)))
16 distrpr 9888 . . . . . . . 8 (𝑦 ·P (1P +P 1P)) = ((𝑦 ·P 1P) +P (𝑦 ·P 1P))
17 1idpr 9889 . . . . . . . . 9 (𝑦P → (𝑦 ·P 1P) = 𝑦)
1817oveq1d 6705 . . . . . . . 8 (𝑦P → ((𝑦 ·P 1P) +P (𝑦 ·P 1P)) = (𝑦 +P (𝑦 ·P 1P)))
1916, 18syl5eq 2697 . . . . . . 7 (𝑦P → (𝑦 ·P (1P +P 1P)) = (𝑦 +P (𝑦 ·P 1P)))
2015, 19oveqan12d 6709 . . . . . 6 ((𝑥P𝑦P) → ((𝑥 +P (𝑥 ·P 1P)) +P (𝑦 ·P (1P +P 1P))) = ((𝑥 ·P (1P +P 1P)) +P (𝑦 +P (𝑦 ·P 1P))))
21 addasspr 9882 . . . . . 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 6718 . . . . . . 7 (𝑥 ·P (1P +P 1P)) ∈ V
23 vex 3234 . . . . . . 7 𝑦 ∈ V
24 ovex 6718 . . . . . . 7 (𝑦 ·P 1P) ∈ V
25 addcompr 9881 . . . . . . 7 (𝑧 +P 𝑤) = (𝑤 +P 𝑧)
26 addasspr 9882 . . . . . . 7 ((𝑧 +P 𝑤) +P 𝑣) = (𝑧 +P (𝑤 +P 𝑣))
2722, 23, 24, 25, 26caov12 6904 . . . . . 6 ((𝑥 ·P (1P +P 1P)) +P (𝑦 +P (𝑦 ·P 1P))) = (𝑦 +P ((𝑥 ·P (1P +P 1P)) +P (𝑦 ·P 1P)))
2820, 21, 273eqtr3g 2708 . . . . 5 ((𝑥P𝑦P) → (𝑥 +P ((𝑥 ·P 1P) +P (𝑦 ·P (1P +P 1P)))) = (𝑦 +P ((𝑥 ·P (1P +P 1P)) +P (𝑦 ·P 1P))))
29 mulclpr 9880 . . . . . . . . . 10 ((𝑥P ∧ (1P +P 1P) ∈ P) → (𝑥 ·P (1P +P 1P)) ∈ P)
309, 29mpan2 707 . . . . . . . . 9 (𝑥P → (𝑥 ·P (1P +P 1P)) ∈ P)
31 mulclpr 9880 . . . . . . . . . 10 ((𝑦P ∧ 1PP) → (𝑦 ·P 1P) ∈ P)
327, 31mpan2 707 . . . . . . . . 9 (𝑦P → (𝑦 ·P 1P) ∈ P)
33 addclpr 9878 . . . . . . . . 9 (((𝑥 ·P (1P +P 1P)) ∈ P ∧ (𝑦 ·P 1P) ∈ P) → ((𝑥 ·P (1P +P 1P)) +P (𝑦 ·P 1P)) ∈ P)
3430, 32, 33syl2an 493 . . . . . . . 8 ((𝑥P𝑦P) → ((𝑥 ·P (1P +P 1P)) +P (𝑦 ·P 1P)) ∈ P)
35 mulclpr 9880 . . . . . . . . . 10 ((𝑥P ∧ 1PP) → (𝑥 ·P 1P) ∈ P)
367, 35mpan2 707 . . . . . . . . 9 (𝑥P → (𝑥 ·P 1P) ∈ P)
37 mulclpr 9880 . . . . . . . . . 10 ((𝑦P ∧ (1P +P 1P) ∈ P) → (𝑦 ·P (1P +P 1P)) ∈ P)
389, 37mpan2 707 . . . . . . . . 9 (𝑦P → (𝑦 ·P (1P +P 1P)) ∈ P)
39 addclpr 9878 . . . . . . . . 9 (((𝑥 ·P 1P) ∈ P ∧ (𝑦 ·P (1P +P 1P)) ∈ P) → ((𝑥 ·P 1P) +P (𝑦 ·P (1P +P 1P))) ∈ P)
4036, 38, 39syl2an 493 . . . . . . . 8 ((𝑥P𝑦P) → ((𝑥 ·P 1P) +P (𝑦 ·P (1P +P 1P))) ∈ P)
4134, 40anim12i 589 . . . . . . 7 (((𝑥P𝑦P) ∧ (𝑥P𝑦P)) → (((𝑥 ·P (1P +P 1P)) +P (𝑦 ·P 1P)) ∈ P ∧ ((𝑥 ·P 1P) +P (𝑦 ·P (1P +P 1P))) ∈ P))
42 enreceq 9925 . . . . . . 7 (((𝑥P𝑦P) ∧ (((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)))))
4341, 42syldan 486 . . . . . 6 (((𝑥P𝑦P) ∧ (𝑥P𝑦P)) → ([⟨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)))))
4443anidms 678 . . . . 5 ((𝑥P𝑦P) → ([⟨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)))))
4528, 44mpbird 247 . . . 4 ((𝑥P𝑦P) → [⟨𝑥, 𝑦⟩] ~R = [⟨((𝑥 ·P (1P +P 1P)) +P (𝑦 ·P 1P)), ((𝑥 ·P 1P) +P (𝑦 ·P (1P +P 1P)))⟩] ~R )
4611, 45eqtr4d 2688 . . 3 ((𝑥P𝑦P) → ([⟨𝑥, 𝑦⟩] ~R ·R [⟨(1P +P 1P), 1P⟩] ~R ) = [⟨𝑥, 𝑦⟩] ~R )
476, 46syl5eq 2697 . 2 ((𝑥P𝑦P) → ([⟨𝑥, 𝑦⟩] ~R ·R 1R) = [⟨𝑥, 𝑦⟩] ~R )
481, 4, 47ecoptocl 7880 1 (𝐴R → (𝐴 ·R 1R) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  cop 4216  (class class class)co 6690  [cec 7785  Pcnp 9719  1Pc1p 9720   +P cpp 9721   ·P cmp 9722   ~R cer 9724  Rcnr 9725  1Rc1r 9727   ·R cmr 9730
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-omul 7610  df-er 7787  df-ec 7789  df-qs 7793  df-ni 9732  df-pli 9733  df-mi 9734  df-lti 9735  df-plpq 9768  df-mpq 9769  df-ltpq 9770  df-enq 9771  df-nq 9772  df-erq 9773  df-plq 9774  df-mq 9775  df-1nq 9776  df-rq 9777  df-ltnq 9778  df-np 9841  df-1p 9842  df-plp 9843  df-mp 9844  df-ltp 9845  df-enr 9915  df-nr 9916  df-mr 9918  df-1r 9921
This theorem is referenced by:  pn0sr  9960  sqgt0sr  9965  axi2m1  10018  ax1rid  10020  axcnre  10023
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