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Theorem 1idsr 6696
Description: 1 is an identity element for multiplication. (Contributed by Jim Kingdon, 5-Jan-2020.)
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.
StepHypRef Expression
1 df-nr 6655 . 2 R = ((P × P) / ~R )
2 oveq1 5462 . . 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 2051 . 2 ([⟨x, y⟩] ~R = A → (([⟨x, y⟩] ~R ·R 1R) = [⟨x, y⟩] ~R ↔ (A ·R 1R) = A))
5 df-1r 6660 . . . 4 1R = [⟨(1P +P 1P), 1P⟩] ~R
65oveq2i 5466 . . 3 ([⟨x, y⟩] ~R ·R 1R) = ([⟨x, y⟩] ~R ·R [⟨(1P +P 1P), 1P⟩] ~R )
7 1pr 6535 . . . . . 6 1P P
8 addclpr 6520 . . . . . 6 ((1P P 1P P) → (1P +P 1P) P)
97, 7, 8mp2an 402 . . . . 5 (1P +P 1P) P
10 mulsrpr 6674 . . . . 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 )
119, 7, 10mpanr12 415 . . . 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 distrprg 6564 . . . . . . . . 9 ((x P 1P P 1P P) → (x ·P (1P +P 1P)) = ((x ·P 1P) +P (x ·P 1P)))
137, 7, 12mp3an23 1223 . . . . . . . 8 (x P → (x ·P (1P +P 1P)) = ((x ·P 1P) +P (x ·P 1P)))
14 1idpr 6568 . . . . . . . . 9 (x P → (x ·P 1P) = x)
1514oveq1d 5470 . . . . . . . 8 (x P → ((x ·P 1P) +P (x ·P 1P)) = (x +P (x ·P 1P)))
1613, 15eqtr2d 2070 . . . . . . 7 (x P → (x +P (x ·P 1P)) = (x ·P (1P +P 1P)))
17 distrprg 6564 . . . . . . . . 9 ((y P 1P P 1P P) → (y ·P (1P +P 1P)) = ((y ·P 1P) +P (y ·P 1P)))
187, 7, 17mp3an23 1223 . . . . . . . 8 (y P → (y ·P (1P +P 1P)) = ((y ·P 1P) +P (y ·P 1P)))
19 1idpr 6568 . . . . . . . . 9 (y P → (y ·P 1P) = y)
2019oveq1d 5470 . . . . . . . 8 (y P → ((y ·P 1P) +P (y ·P 1P)) = (y +P (y ·P 1P)))
2118, 20eqtrd 2069 . . . . . . 7 (y P → (y ·P (1P +P 1P)) = (y +P (y ·P 1P)))
2216, 21oveqan12d 5474 . . . . . 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))))
23 simpl 102 . . . . . . 7 ((x P y P) → x P)
24 mulclpr 6553 . . . . . . . 8 ((x P 1P P) → (x ·P 1P) P)
2523, 7, 24sylancl 392 . . . . . . 7 ((x P y P) → (x ·P 1P) P)
26 mulclpr 6553 . . . . . . . . 9 ((y P (1P +P 1P) P) → (y ·P (1P +P 1P)) P)
279, 26mpan2 401 . . . . . . . 8 (y P → (y ·P (1P +P 1P)) P)
2827adantl 262 . . . . . . 7 ((x P y P) → (y ·P (1P +P 1P)) P)
29 addassprg 6555 . . . . . . 7 ((x P (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 1134 . . . . . 6 ((x P y P) → ((x +P (x ·P 1P)) +P (y ·P (1P +P 1P))) = (x +P ((x ·P 1P) +P (y ·P (1P +P 1P)))))
31 mulclpr 6553 . . . . . . . 8 ((x P (1P +P 1P) P) → (x ·P (1P +P 1P)) P)
3223, 9, 31sylancl 392 . . . . . . 7 ((x P y P) → (x ·P (1P +P 1P)) P)
33 simpr 103 . . . . . . 7 ((x P y P) → y P)
34 mulclpr 6553 . . . . . . . 8 ((y P 1P P) → (y ·P 1P) P)
3533, 7, 34sylancl 392 . . . . . . 7 ((x P y P) → (y ·P 1P) P)
36 addcomprg 6554 . . . . . . . 8 ((z P w P) → (z +P w) = (w +P z))
3736adantl 262 . . . . . . 7 (((x P y P) (z P w P)) → (z +P w) = (w +P z))
38 addassprg 6555 . . . . . . . 8 ((z P w P v P) → ((z +P w) +P v) = (z +P (w +P v)))
3938adantl 262 . . . . . . 7 (((x P y P) (z P w P v P)) → ((z +P w) +P v) = (z +P (w +P v)))
4032, 33, 35, 37, 39caov12d 5624 . . . . . 6 ((x P y P) → ((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 2077 . . . . 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))))
429, 31mpan2 401 . . . . . . . . 9 (x P → (x ·P (1P +P 1P)) P)
437, 34mpan2 401 . . . . . . . . 9 (y P → (y ·P 1P) P)
44 addclpr 6520 . . . . . . . . 9 (((x ·P (1P +P 1P)) P (y ·P 1P) P) → ((x ·P (1P +P 1P)) +P (y ·P 1P)) P)
4542, 43, 44syl2an 273 . . . . . . . 8 ((x P y P) → ((x ·P (1P +P 1P)) +P (y ·P 1P)) P)
467, 24mpan2 401 . . . . . . . . 9 (x P → (x ·P 1P) P)
47 addclpr 6520 . . . . . . . . 9 (((x ·P 1P) P (y ·P (1P +P 1P)) P) → ((x ·P 1P) +P (y ·P (1P +P 1P))) P)
4846, 27, 47syl2an 273 . . . . . . . 8 ((x P y P) → ((x ·P 1P) +P (y ·P (1P +P 1P))) P)
4945, 48anim12i 321 . . . . . . 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))
50 enreceq 6664 . . . . . . 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)))))
5149, 50syldan 266 . . . . . 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)))))
5251anidms 377 . . . . 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)))))
5341, 52mpbird 156 . . . 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 )
5411, 53eqtr4d 2072 . . 3 ((x P y P) → ([⟨x, y⟩] ~R ·R [⟨(1P +P 1P), 1P⟩] ~R ) = [⟨x, y⟩] ~R )
556, 54syl5eq 2081 . 2 ((x P y P) → ([⟨x, y⟩] ~R ·R 1R) = [⟨x, y⟩] ~R )
561, 4, 55ecoptocl 6129 1 (A R → (A ·R 1R) = A)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   w3a 884   = wceq 1242   wcel 1390  cop 3370  (class class class)co 5455  [cec 6040  Pcnp 6275  1Pc1p 6276   +P cpp 6277   ·P cmp 6278   ~R cer 6280  Rcnr 6281  1Rc1r 6283   ·R cmr 6286
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254
This theorem depends on definitions:  df-bi 110  df-dc 742  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-eprel 4017  df-id 4021  df-po 4024  df-iso 4025  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-1o 5940  df-2o 5941  df-oadd 5944  df-omul 5945  df-er 6042  df-ec 6044  df-qs 6048  df-ni 6288  df-pli 6289  df-mi 6290  df-lti 6291  df-plpq 6328  df-mpq 6329  df-enq 6331  df-nqqs 6332  df-plqqs 6333  df-mqqs 6334  df-1nqqs 6335  df-rq 6336  df-ltnqqs 6337  df-enq0 6407  df-nq0 6408  df-0nq0 6409  df-plq0 6410  df-mq0 6411  df-inp 6449  df-i1p 6450  df-iplp 6451  df-imp 6452  df-enr 6654  df-nr 6655  df-mr 6657  df-1r 6660
This theorem is referenced by:  pn0sr  6699  axi2m1  6759  ax1rid  6761  axcnre  6765
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