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Theorem 1idsr 7258
Description: 1 is an identity element for multiplication. (Contributed by Jim Kingdon, 5-Jan-2020.)
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 7217 . 2 R = ((P × P) / ~R )
2 oveq1 5620 . . 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)
42, 3eqeq12d 2099 . 2 ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → (([⟨𝑥, 𝑦⟩] ~R ·R 1R) = [⟨𝑥, 𝑦⟩] ~R ↔ (𝐴 ·R 1R) = 𝐴))
5 df-1r 7222 . . . 4 1R = [⟨(1P +P 1P), 1P⟩] ~R
65oveq2i 5624 . . 3 ([⟨𝑥, 𝑦⟩] ~R ·R 1R) = ([⟨𝑥, 𝑦⟩] ~R ·R [⟨(1P +P 1P), 1P⟩] ~R )
7 1pr 7057 . . . . . 6 1PP
8 addclpr 7040 . . . . . 6 ((1PP ∧ 1PP) → (1P +P 1P) ∈ P)
97, 7, 8mp2an 417 . . . . 5 (1P +P 1P) ∈ P
10 mulsrpr 7236 . . . . 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 430 . . . 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 distrprg 7091 . . . . . . . . 9 ((𝑥P ∧ 1PP ∧ 1PP) → (𝑥 ·P (1P +P 1P)) = ((𝑥 ·P 1P) +P (𝑥 ·P 1P)))
137, 7, 12mp3an23 1263 . . . . . . . 8 (𝑥P → (𝑥 ·P (1P +P 1P)) = ((𝑥 ·P 1P) +P (𝑥 ·P 1P)))
14 1idpr 7095 . . . . . . . . 9 (𝑥P → (𝑥 ·P 1P) = 𝑥)
1514oveq1d 5628 . . . . . . . 8 (𝑥P → ((𝑥 ·P 1P) +P (𝑥 ·P 1P)) = (𝑥 +P (𝑥 ·P 1P)))
1613, 15eqtr2d 2118 . . . . . . 7 (𝑥P → (𝑥 +P (𝑥 ·P 1P)) = (𝑥 ·P (1P +P 1P)))
17 distrprg 7091 . . . . . . . . 9 ((𝑦P ∧ 1PP ∧ 1PP) → (𝑦 ·P (1P +P 1P)) = ((𝑦 ·P 1P) +P (𝑦 ·P 1P)))
187, 7, 17mp3an23 1263 . . . . . . . 8 (𝑦P → (𝑦 ·P (1P +P 1P)) = ((𝑦 ·P 1P) +P (𝑦 ·P 1P)))
19 1idpr 7095 . . . . . . . . 9 (𝑦P → (𝑦 ·P 1P) = 𝑦)
2019oveq1d 5628 . . . . . . . 8 (𝑦P → ((𝑦 ·P 1P) +P (𝑦 ·P 1P)) = (𝑦 +P (𝑦 ·P 1P)))
2118, 20eqtrd 2117 . . . . . . 7 (𝑦P → (𝑦 ·P (1P +P 1P)) = (𝑦 +P (𝑦 ·P 1P)))
2216, 21oveqan12d 5632 . . . . . 6 ((𝑥P𝑦P) → ((𝑥 +P (𝑥 ·P 1P)) +P (𝑦 ·P (1P +P 1P))) = ((𝑥 ·P (1P +P 1P)) +P (𝑦 +P (𝑦 ·P 1P))))
23 simpl 107 . . . . . . 7 ((𝑥P𝑦P) → 𝑥P)
24 mulclpr 7075 . . . . . . . 8 ((𝑥P ∧ 1PP) → (𝑥 ·P 1P) ∈ P)
2523, 7, 24sylancl 404 . . . . . . 7 ((𝑥P𝑦P) → (𝑥 ·P 1P) ∈ P)
26 mulclpr 7075 . . . . . . . . 9 ((𝑦P ∧ (1P +P 1P) ∈ P) → (𝑦 ·P (1P +P 1P)) ∈ P)
279, 26mpan2 416 . . . . . . . 8 (𝑦P → (𝑦 ·P (1P +P 1P)) ∈ P)
2827adantl 271 . . . . . . 7 ((𝑥P𝑦P) → (𝑦 ·P (1P +P 1P)) ∈ P)
29 addassprg 7082 . . . . . . 7 ((𝑥P ∧ (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)))))
3023, 25, 28, 29syl3anc 1172 . . . . . 6 ((𝑥P𝑦P) → ((𝑥 +P (𝑥 ·P 1P)) +P (𝑦 ·P (1P +P 1P))) = (𝑥 +P ((𝑥 ·P 1P) +P (𝑦 ·P (1P +P 1P)))))
31 mulclpr 7075 . . . . . . . 8 ((𝑥P ∧ (1P +P 1P) ∈ P) → (𝑥 ·P (1P +P 1P)) ∈ P)
3223, 9, 31sylancl 404 . . . . . . 7 ((𝑥P𝑦P) → (𝑥 ·P (1P +P 1P)) ∈ P)
33 simpr 108 . . . . . . 7 ((𝑥P𝑦P) → 𝑦P)
34 mulclpr 7075 . . . . . . . 8 ((𝑦P ∧ 1PP) → (𝑦 ·P 1P) ∈ P)
3533, 7, 34sylancl 404 . . . . . . 7 ((𝑥P𝑦P) → (𝑦 ·P 1P) ∈ P)
36 addcomprg 7081 . . . . . . . 8 ((𝑧P𝑤P) → (𝑧 +P 𝑤) = (𝑤 +P 𝑧))
3736adantl 271 . . . . . . 7 (((𝑥P𝑦P) ∧ (𝑧P𝑤P)) → (𝑧 +P 𝑤) = (𝑤 +P 𝑧))
38 addassprg 7082 . . . . . . . 8 ((𝑧P𝑤P𝑣P) → ((𝑧 +P 𝑤) +P 𝑣) = (𝑧 +P (𝑤 +P 𝑣)))
3938adantl 271 . . . . . . 7 (((𝑥P𝑦P) ∧ (𝑧P𝑤P𝑣P)) → ((𝑧 +P 𝑤) +P 𝑣) = (𝑧 +P (𝑤 +P 𝑣)))
4032, 33, 35, 37, 39caov12d 5783 . . . . . 6 ((𝑥P𝑦P) → ((𝑥 ·P (1P +P 1P)) +P (𝑦 +P (𝑦 ·P 1P))) = (𝑦 +P ((𝑥 ·P (1P +P 1P)) +P (𝑦 ·P 1P))))
4122, 30, 403eqtr3d 2125 . . . . 5 ((𝑥P𝑦P) → (𝑥 +P ((𝑥 ·P 1P) +P (𝑦 ·P (1P +P 1P)))) = (𝑦 +P ((𝑥 ·P (1P +P 1P)) +P (𝑦 ·P 1P))))
429, 31mpan2 416 . . . . . . . . 9 (𝑥P → (𝑥 ·P (1P +P 1P)) ∈ P)
437, 34mpan2 416 . . . . . . . . 9 (𝑦P → (𝑦 ·P 1P) ∈ P)
44 addclpr 7040 . . . . . . . . 9 (((𝑥 ·P (1P +P 1P)) ∈ P ∧ (𝑦 ·P 1P) ∈ P) → ((𝑥 ·P (1P +P 1P)) +P (𝑦 ·P 1P)) ∈ P)
4542, 43, 44syl2an 283 . . . . . . . 8 ((𝑥P𝑦P) → ((𝑥 ·P (1P +P 1P)) +P (𝑦 ·P 1P)) ∈ P)
467, 24mpan2 416 . . . . . . . . 9 (𝑥P → (𝑥 ·P 1P) ∈ P)
47 addclpr 7040 . . . . . . . . 9 (((𝑥 ·P 1P) ∈ P ∧ (𝑦 ·P (1P +P 1P)) ∈ P) → ((𝑥 ·P 1P) +P (𝑦 ·P (1P +P 1P))) ∈ P)
4846, 27, 47syl2an 283 . . . . . . . 8 ((𝑥P𝑦P) → ((𝑥 ·P 1P) +P (𝑦 ·P (1P +P 1P))) ∈ P)
4945, 48anim12i 331 . . . . . . 7 (((𝑥P𝑦P) ∧ (𝑥P𝑦P)) → (((𝑥 ·P (1P +P 1P)) +P (𝑦 ·P 1P)) ∈ P ∧ ((𝑥 ·P 1P) +P (𝑦 ·P (1P +P 1P))) ∈ P))
50 enreceq 7226 . . . . . . 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)))))
5149, 50syldan 276 . . . . . 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)))))
5251anidms 389 . . . . 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)))))
5341, 52mpbird 165 . . . 4 ((𝑥P𝑦P) → [⟨𝑥, 𝑦⟩] ~R = [⟨((𝑥 ·P (1P +P 1P)) +P (𝑦 ·P 1P)), ((𝑥 ·P 1P) +P (𝑦 ·P (1P +P 1P)))⟩] ~R )
5411, 53eqtr4d 2120 . . 3 ((𝑥P𝑦P) → ([⟨𝑥, 𝑦⟩] ~R ·R [⟨(1P +P 1P), 1P⟩] ~R ) = [⟨𝑥, 𝑦⟩] ~R )
556, 54syl5eq 2129 . 2 ((𝑥P𝑦P) → ([⟨𝑥, 𝑦⟩] ~R ·R 1R) = [⟨𝑥, 𝑦⟩] ~R )
561, 4, 55ecoptocl 6331 1 (𝐴R → (𝐴 ·R 1R) = 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  w3a 922   = wceq 1287  wcel 1436  cop 3434  (class class class)co 5613  [cec 6242  Pcnp 6794  1Pc1p 6795   +P cpp 6796   ·P cmp 6797   ~R cer 6799  Rcnr 6800  1Rc1r 6802   ·R cmr 6805
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-coll 3929  ax-sep 3932  ax-nul 3940  ax-pow 3984  ax-pr 4010  ax-un 4234  ax-setind 4326  ax-iinf 4376
This theorem depends on definitions:  df-bi 115  df-dc 779  df-3or 923  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-ral 2360  df-rex 2361  df-reu 2362  df-rab 2364  df-v 2617  df-sbc 2830  df-csb 2923  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-nul 3276  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3637  df-int 3672  df-iun 3715  df-br 3821  df-opab 3875  df-mpt 3876  df-tr 3912  df-eprel 4090  df-id 4094  df-po 4097  df-iso 4098  df-iord 4167  df-on 4169  df-suc 4172  df-iom 4379  df-xp 4417  df-rel 4418  df-cnv 4419  df-co 4420  df-dm 4421  df-rn 4422  df-res 4423  df-ima 4424  df-iota 4946  df-fun 4983  df-fn 4984  df-f 4985  df-f1 4986  df-fo 4987  df-f1o 4988  df-fv 4989  df-ov 5616  df-oprab 5617  df-mpt2 5618  df-1st 5868  df-2nd 5869  df-recs 6024  df-irdg 6089  df-1o 6135  df-2o 6136  df-oadd 6139  df-omul 6140  df-er 6244  df-ec 6246  df-qs 6250  df-ni 6807  df-pli 6808  df-mi 6809  df-lti 6810  df-plpq 6847  df-mpq 6848  df-enq 6850  df-nqqs 6851  df-plqqs 6852  df-mqqs 6853  df-1nqqs 6854  df-rq 6855  df-ltnqqs 6856  df-enq0 6927  df-nq0 6928  df-0nq0 6929  df-plq0 6930  df-mq0 6931  df-inp 6969  df-i1p 6970  df-iplp 6971  df-imp 6972  df-enr 7216  df-nr 7217  df-mr 7219  df-1r 7222
This theorem is referenced by:  pn0sr  7261  axi2m1  7354  ax1rid  7356  axcnre  7360
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