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Theorem 1idsr 7730
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 7689 . 2 R = ((P × P) / ~R )
2 oveq1 5860 . . 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 2185 . 2 ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → (([⟨𝑥, 𝑦⟩] ~R ·R 1R) = [⟨𝑥, 𝑦⟩] ~R ↔ (𝐴 ·R 1R) = 𝐴))
5 df-1r 7694 . . . 4 1R = [⟨(1P +P 1P), 1P⟩] ~R
65oveq2i 5864 . . 3 ([⟨𝑥, 𝑦⟩] ~R ·R 1R) = ([⟨𝑥, 𝑦⟩] ~R ·R [⟨(1P +P 1P), 1P⟩] ~R )
7 1pr 7516 . . . . . 6 1PP
8 addclpr 7499 . . . . . 6 ((1PP ∧ 1PP) → (1P +P 1P) ∈ P)
97, 7, 8mp2an 424 . . . . 5 (1P +P 1P) ∈ P
10 mulsrpr 7708 . . . . 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 437 . . . 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 7550 . . . . . . . . 9 ((𝑥P ∧ 1PP ∧ 1PP) → (𝑥 ·P (1P +P 1P)) = ((𝑥 ·P 1P) +P (𝑥 ·P 1P)))
137, 7, 12mp3an23 1324 . . . . . . . 8 (𝑥P → (𝑥 ·P (1P +P 1P)) = ((𝑥 ·P 1P) +P (𝑥 ·P 1P)))
14 1idpr 7554 . . . . . . . . 9 (𝑥P → (𝑥 ·P 1P) = 𝑥)
1514oveq1d 5868 . . . . . . . 8 (𝑥P → ((𝑥 ·P 1P) +P (𝑥 ·P 1P)) = (𝑥 +P (𝑥 ·P 1P)))
1613, 15eqtr2d 2204 . . . . . . 7 (𝑥P → (𝑥 +P (𝑥 ·P 1P)) = (𝑥 ·P (1P +P 1P)))
17 distrprg 7550 . . . . . . . . 9 ((𝑦P ∧ 1PP ∧ 1PP) → (𝑦 ·P (1P +P 1P)) = ((𝑦 ·P 1P) +P (𝑦 ·P 1P)))
187, 7, 17mp3an23 1324 . . . . . . . 8 (𝑦P → (𝑦 ·P (1P +P 1P)) = ((𝑦 ·P 1P) +P (𝑦 ·P 1P)))
19 1idpr 7554 . . . . . . . . 9 (𝑦P → (𝑦 ·P 1P) = 𝑦)
2019oveq1d 5868 . . . . . . . 8 (𝑦P → ((𝑦 ·P 1P) +P (𝑦 ·P 1P)) = (𝑦 +P (𝑦 ·P 1P)))
2118, 20eqtrd 2203 . . . . . . 7 (𝑦P → (𝑦 ·P (1P +P 1P)) = (𝑦 +P (𝑦 ·P 1P)))
2216, 21oveqan12d 5872 . . . . . 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 108 . . . . . . 7 ((𝑥P𝑦P) → 𝑥P)
24 mulclpr 7534 . . . . . . . 8 ((𝑥P ∧ 1PP) → (𝑥 ·P 1P) ∈ P)
2523, 7, 24sylancl 411 . . . . . . 7 ((𝑥P𝑦P) → (𝑥 ·P 1P) ∈ P)
26 mulclpr 7534 . . . . . . . . 9 ((𝑦P ∧ (1P +P 1P) ∈ P) → (𝑦 ·P (1P +P 1P)) ∈ P)
279, 26mpan2 423 . . . . . . . 8 (𝑦P → (𝑦 ·P (1P +P 1P)) ∈ P)
2827adantl 275 . . . . . . 7 ((𝑥P𝑦P) → (𝑦 ·P (1P +P 1P)) ∈ P)
29 addassprg 7541 . . . . . . 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 1233 . . . . . 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 7534 . . . . . . . 8 ((𝑥P ∧ (1P +P 1P) ∈ P) → (𝑥 ·P (1P +P 1P)) ∈ P)
3223, 9, 31sylancl 411 . . . . . . 7 ((𝑥P𝑦P) → (𝑥 ·P (1P +P 1P)) ∈ P)
33 simpr 109 . . . . . . 7 ((𝑥P𝑦P) → 𝑦P)
34 mulclpr 7534 . . . . . . . 8 ((𝑦P ∧ 1PP) → (𝑦 ·P 1P) ∈ P)
3533, 7, 34sylancl 411 . . . . . . 7 ((𝑥P𝑦P) → (𝑦 ·P 1P) ∈ P)
36 addcomprg 7540 . . . . . . . 8 ((𝑧P𝑤P) → (𝑧 +P 𝑤) = (𝑤 +P 𝑧))
3736adantl 275 . . . . . . 7 (((𝑥P𝑦P) ∧ (𝑧P𝑤P)) → (𝑧 +P 𝑤) = (𝑤 +P 𝑧))
38 addassprg 7541 . . . . . . . 8 ((𝑧P𝑤P𝑣P) → ((𝑧 +P 𝑤) +P 𝑣) = (𝑧 +P (𝑤 +P 𝑣)))
3938adantl 275 . . . . . . 7 (((𝑥P𝑦P) ∧ (𝑧P𝑤P𝑣P)) → ((𝑧 +P 𝑤) +P 𝑣) = (𝑧 +P (𝑤 +P 𝑣)))
4032, 33, 35, 37, 39caov12d 6034 . . . . . 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 2211 . . . . 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 423 . . . . . . . . 9 (𝑥P → (𝑥 ·P (1P +P 1P)) ∈ P)
437, 34mpan2 423 . . . . . . . . 9 (𝑦P → (𝑦 ·P 1P) ∈ P)
44 addclpr 7499 . . . . . . . . 9 (((𝑥 ·P (1P +P 1P)) ∈ P ∧ (𝑦 ·P 1P) ∈ P) → ((𝑥 ·P (1P +P 1P)) +P (𝑦 ·P 1P)) ∈ P)
4542, 43, 44syl2an 287 . . . . . . . 8 ((𝑥P𝑦P) → ((𝑥 ·P (1P +P 1P)) +P (𝑦 ·P 1P)) ∈ P)
467, 24mpan2 423 . . . . . . . . 9 (𝑥P → (𝑥 ·P 1P) ∈ P)
47 addclpr 7499 . . . . . . . . 9 (((𝑥 ·P 1P) ∈ P ∧ (𝑦 ·P (1P +P 1P)) ∈ P) → ((𝑥 ·P 1P) +P (𝑦 ·P (1P +P 1P))) ∈ P)
4846, 27, 47syl2an 287 . . . . . . . 8 ((𝑥P𝑦P) → ((𝑥 ·P 1P) +P (𝑦 ·P (1P +P 1P))) ∈ P)
4945, 48anim12i 336 . . . . . . 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 7698 . . . . . . 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 280 . . . . . 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 395 . . . . 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 166 . . . 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 2206 . . 3 ((𝑥P𝑦P) → ([⟨𝑥, 𝑦⟩] ~R ·R [⟨(1P +P 1P), 1P⟩] ~R ) = [⟨𝑥, 𝑦⟩] ~R )
556, 54eqtrid 2215 . 2 ((𝑥P𝑦P) → ([⟨𝑥, 𝑦⟩] ~R ·R 1R) = [⟨𝑥, 𝑦⟩] ~R )
561, 4, 55ecoptocl 6600 1 (𝐴R → (𝐴 ·R 1R) = 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  w3a 973   = wceq 1348  wcel 2141  cop 3586  (class class class)co 5853  [cec 6511  Pcnp 7253  1Pc1p 7254   +P cpp 7255   ·P cmp 7256   ~R cer 7258  Rcnr 7259  1Rc1r 7261   ·R cmr 7264
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-eprel 4274  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-1o 6395  df-2o 6396  df-oadd 6399  df-omul 6400  df-er 6513  df-ec 6515  df-qs 6519  df-ni 7266  df-pli 7267  df-mi 7268  df-lti 7269  df-plpq 7306  df-mpq 7307  df-enq 7309  df-nqqs 7310  df-plqqs 7311  df-mqqs 7312  df-1nqqs 7313  df-rq 7314  df-ltnqqs 7315  df-enq0 7386  df-nq0 7387  df-0nq0 7388  df-plq0 7389  df-mq0 7390  df-inp 7428  df-i1p 7429  df-iplp 7430  df-imp 7431  df-enr 7688  df-nr 7689  df-mr 7691  df-1r 7694
This theorem is referenced by:  pn0sr  7733  axi2m1  7837  ax1rid  7839  axcnre  7843
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