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Theorem 1idsr 7758
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 7717 . 2 R = ((P × P) / ~R )
2 oveq1 5876 . . 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 2192 . 2 ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → (([⟨𝑥, 𝑦⟩] ~R ·R 1R) = [⟨𝑥, 𝑦⟩] ~R ↔ (𝐴 ·R 1R) = 𝐴))
5 df-1r 7722 . . . 4 1R = [⟨(1P +P 1P), 1P⟩] ~R
65oveq2i 5880 . . 3 ([⟨𝑥, 𝑦⟩] ~R ·R 1R) = ([⟨𝑥, 𝑦⟩] ~R ·R [⟨(1P +P 1P), 1P⟩] ~R )
7 1pr 7544 . . . . . 6 1PP
8 addclpr 7527 . . . . . 6 ((1PP ∧ 1PP) → (1P +P 1P) ∈ P)
97, 7, 8mp2an 426 . . . . 5 (1P +P 1P) ∈ P
10 mulsrpr 7736 . . . . 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 439 . . . 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 7578 . . . . . . . . 9 ((𝑥P ∧ 1PP ∧ 1PP) → (𝑥 ·P (1P +P 1P)) = ((𝑥 ·P 1P) +P (𝑥 ·P 1P)))
137, 7, 12mp3an23 1329 . . . . . . . 8 (𝑥P → (𝑥 ·P (1P +P 1P)) = ((𝑥 ·P 1P) +P (𝑥 ·P 1P)))
14 1idpr 7582 . . . . . . . . 9 (𝑥P → (𝑥 ·P 1P) = 𝑥)
1514oveq1d 5884 . . . . . . . 8 (𝑥P → ((𝑥 ·P 1P) +P (𝑥 ·P 1P)) = (𝑥 +P (𝑥 ·P 1P)))
1613, 15eqtr2d 2211 . . . . . . 7 (𝑥P → (𝑥 +P (𝑥 ·P 1P)) = (𝑥 ·P (1P +P 1P)))
17 distrprg 7578 . . . . . . . . 9 ((𝑦P ∧ 1PP ∧ 1PP) → (𝑦 ·P (1P +P 1P)) = ((𝑦 ·P 1P) +P (𝑦 ·P 1P)))
187, 7, 17mp3an23 1329 . . . . . . . 8 (𝑦P → (𝑦 ·P (1P +P 1P)) = ((𝑦 ·P 1P) +P (𝑦 ·P 1P)))
19 1idpr 7582 . . . . . . . . 9 (𝑦P → (𝑦 ·P 1P) = 𝑦)
2019oveq1d 5884 . . . . . . . 8 (𝑦P → ((𝑦 ·P 1P) +P (𝑦 ·P 1P)) = (𝑦 +P (𝑦 ·P 1P)))
2118, 20eqtrd 2210 . . . . . . 7 (𝑦P → (𝑦 ·P (1P +P 1P)) = (𝑦 +P (𝑦 ·P 1P)))
2216, 21oveqan12d 5888 . . . . . 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 109 . . . . . . 7 ((𝑥P𝑦P) → 𝑥P)
24 mulclpr 7562 . . . . . . . 8 ((𝑥P ∧ 1PP) → (𝑥 ·P 1P) ∈ P)
2523, 7, 24sylancl 413 . . . . . . 7 ((𝑥P𝑦P) → (𝑥 ·P 1P) ∈ P)
26 mulclpr 7562 . . . . . . . . 9 ((𝑦P ∧ (1P +P 1P) ∈ P) → (𝑦 ·P (1P +P 1P)) ∈ P)
279, 26mpan2 425 . . . . . . . 8 (𝑦P → (𝑦 ·P (1P +P 1P)) ∈ P)
2827adantl 277 . . . . . . 7 ((𝑥P𝑦P) → (𝑦 ·P (1P +P 1P)) ∈ P)
29 addassprg 7569 . . . . . . 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 1238 . . . . . 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 7562 . . . . . . . 8 ((𝑥P ∧ (1P +P 1P) ∈ P) → (𝑥 ·P (1P +P 1P)) ∈ P)
3223, 9, 31sylancl 413 . . . . . . 7 ((𝑥P𝑦P) → (𝑥 ·P (1P +P 1P)) ∈ P)
33 simpr 110 . . . . . . 7 ((𝑥P𝑦P) → 𝑦P)
34 mulclpr 7562 . . . . . . . 8 ((𝑦P ∧ 1PP) → (𝑦 ·P 1P) ∈ P)
3533, 7, 34sylancl 413 . . . . . . 7 ((𝑥P𝑦P) → (𝑦 ·P 1P) ∈ P)
36 addcomprg 7568 . . . . . . . 8 ((𝑧P𝑤P) → (𝑧 +P 𝑤) = (𝑤 +P 𝑧))
3736adantl 277 . . . . . . 7 (((𝑥P𝑦P) ∧ (𝑧P𝑤P)) → (𝑧 +P 𝑤) = (𝑤 +P 𝑧))
38 addassprg 7569 . . . . . . . 8 ((𝑧P𝑤P𝑣P) → ((𝑧 +P 𝑤) +P 𝑣) = (𝑧 +P (𝑤 +P 𝑣)))
3938adantl 277 . . . . . . 7 (((𝑥P𝑦P) ∧ (𝑧P𝑤P𝑣P)) → ((𝑧 +P 𝑤) +P 𝑣) = (𝑧 +P (𝑤 +P 𝑣)))
4032, 33, 35, 37, 39caov12d 6050 . . . . . 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 2218 . . . . 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 425 . . . . . . . . 9 (𝑥P → (𝑥 ·P (1P +P 1P)) ∈ P)
437, 34mpan2 425 . . . . . . . . 9 (𝑦P → (𝑦 ·P 1P) ∈ P)
44 addclpr 7527 . . . . . . . . 9 (((𝑥 ·P (1P +P 1P)) ∈ P ∧ (𝑦 ·P 1P) ∈ P) → ((𝑥 ·P (1P +P 1P)) +P (𝑦 ·P 1P)) ∈ P)
4542, 43, 44syl2an 289 . . . . . . . 8 ((𝑥P𝑦P) → ((𝑥 ·P (1P +P 1P)) +P (𝑦 ·P 1P)) ∈ P)
467, 24mpan2 425 . . . . . . . . 9 (𝑥P → (𝑥 ·P 1P) ∈ P)
47 addclpr 7527 . . . . . . . . 9 (((𝑥 ·P 1P) ∈ P ∧ (𝑦 ·P (1P +P 1P)) ∈ P) → ((𝑥 ·P 1P) +P (𝑦 ·P (1P +P 1P))) ∈ P)
4846, 27, 47syl2an 289 . . . . . . . 8 ((𝑥P𝑦P) → ((𝑥 ·P 1P) +P (𝑦 ·P (1P +P 1P))) ∈ P)
4945, 48anim12i 338 . . . . . . 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 7726 . . . . . . 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 282 . . . . . 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 397 . . . . 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 167 . . . 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 2213 . . 3 ((𝑥P𝑦P) → ([⟨𝑥, 𝑦⟩] ~R ·R [⟨(1P +P 1P), 1P⟩] ~R ) = [⟨𝑥, 𝑦⟩] ~R )
556, 54eqtrid 2222 . 2 ((𝑥P𝑦P) → ([⟨𝑥, 𝑦⟩] ~R ·R 1R) = [⟨𝑥, 𝑦⟩] ~R )
561, 4, 55ecoptocl 6616 1 (𝐴R → (𝐴 ·R 1R) = 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  w3a 978   = wceq 1353  wcel 2148  cop 3594  (class class class)co 5869  [cec 6527  Pcnp 7281  1Pc1p 7282   +P cpp 7283   ·P cmp 7284   ~R cer 7286  Rcnr 7287  1Rc1r 7289   ·R cmr 7292
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-eprel 4286  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-irdg 6365  df-1o 6411  df-2o 6412  df-oadd 6415  df-omul 6416  df-er 6529  df-ec 6531  df-qs 6535  df-ni 7294  df-pli 7295  df-mi 7296  df-lti 7297  df-plpq 7334  df-mpq 7335  df-enq 7337  df-nqqs 7338  df-plqqs 7339  df-mqqs 7340  df-1nqqs 7341  df-rq 7342  df-ltnqqs 7343  df-enq0 7414  df-nq0 7415  df-0nq0 7416  df-plq0 7417  df-mq0 7418  df-inp 7456  df-i1p 7457  df-iplp 7458  df-imp 7459  df-enr 7716  df-nr 7717  df-mr 7719  df-1r 7722
This theorem is referenced by:  pn0sr  7761  axi2m1  7865  ax1rid  7867  axcnre  7871
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