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Theorem 1idsr 10572
 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 10530 . 2 R = ((P × P) / ~R )
2 oveq1 7164 . . 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 2775 . 2 ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → (([⟨𝑥, 𝑦⟩] ~R ·R 1R) = [⟨𝑥, 𝑦⟩] ~R ↔ (𝐴 ·R 1R) = 𝐴))
5 df-1r 10535 . . . 4 1R = [⟨(1P +P 1P), 1P⟩] ~R
65oveq2i 7168 . . 3 ([⟨𝑥, 𝑦⟩] ~R ·R 1R) = ([⟨𝑥, 𝑦⟩] ~R ·R [⟨(1P +P 1P), 1P⟩] ~R )
7 1pr 10489 . . . . . 6 1PP
8 addclpr 10492 . . . . . 6 ((1PP ∧ 1PP) → (1P +P 1P) ∈ P)
97, 7, 8mp2an 691 . . . . 5 (1P +P 1P) ∈ P
10 mulsrpr 10550 . . . . 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 704 . . . 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 10502 . . . . . . . 8 (𝑥 ·P (1P +P 1P)) = ((𝑥 ·P 1P) +P (𝑥 ·P 1P))
13 1idpr 10503 . . . . . . . . 9 (𝑥P → (𝑥 ·P 1P) = 𝑥)
1413oveq1d 7172 . . . . . . . 8 (𝑥P → ((𝑥 ·P 1P) +P (𝑥 ·P 1P)) = (𝑥 +P (𝑥 ·P 1P)))
1512, 14syl5req 2807 . . . . . . 7 (𝑥P → (𝑥 +P (𝑥 ·P 1P)) = (𝑥 ·P (1P +P 1P)))
16 distrpr 10502 . . . . . . . 8 (𝑦 ·P (1P +P 1P)) = ((𝑦 ·P 1P) +P (𝑦 ·P 1P))
17 1idpr 10503 . . . . . . . . 9 (𝑦P → (𝑦 ·P 1P) = 𝑦)
1817oveq1d 7172 . . . . . . . 8 (𝑦P → ((𝑦 ·P 1P) +P (𝑦 ·P 1P)) = (𝑦 +P (𝑦 ·P 1P)))
1916, 18syl5eq 2806 . . . . . . 7 (𝑦P → (𝑦 ·P (1P +P 1P)) = (𝑦 +P (𝑦 ·P 1P)))
2015, 19oveqan12d 7176 . . . . . 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 10496 . . . . . 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 7190 . . . . . . 7 (𝑥 ·P (1P +P 1P)) ∈ V
23 vex 3414 . . . . . . 7 𝑦 ∈ V
24 ovex 7190 . . . . . . 7 (𝑦 ·P 1P) ∈ V
25 addcompr 10495 . . . . . . 7 (𝑧 +P 𝑤) = (𝑤 +P 𝑧)
26 addasspr 10496 . . . . . . 7 ((𝑧 +P 𝑤) +P 𝑣) = (𝑧 +P (𝑤 +P 𝑣))
2722, 23, 24, 25, 26caov12 7379 . . . . . 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 2817 . . . . 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 10494 . . . . . . . . . 10 ((𝑥P ∧ (1P +P 1P) ∈ P) → (𝑥 ·P (1P +P 1P)) ∈ P)
309, 29mpan2 690 . . . . . . . . 9 (𝑥P → (𝑥 ·P (1P +P 1P)) ∈ P)
31 mulclpr 10494 . . . . . . . . . 10 ((𝑦P ∧ 1PP) → (𝑦 ·P 1P) ∈ P)
327, 31mpan2 690 . . . . . . . . 9 (𝑦P → (𝑦 ·P 1P) ∈ P)
33 addclpr 10492 . . . . . . . . 9 (((𝑥 ·P (1P +P 1P)) ∈ P ∧ (𝑦 ·P 1P) ∈ P) → ((𝑥 ·P (1P +P 1P)) +P (𝑦 ·P 1P)) ∈ P)
3430, 32, 33syl2an 598 . . . . . . . 8 ((𝑥P𝑦P) → ((𝑥 ·P (1P +P 1P)) +P (𝑦 ·P 1P)) ∈ P)
35 mulclpr 10494 . . . . . . . . . 10 ((𝑥P ∧ 1PP) → (𝑥 ·P 1P) ∈ P)
367, 35mpan2 690 . . . . . . . . 9 (𝑥P → (𝑥 ·P 1P) ∈ P)
37 mulclpr 10494 . . . . . . . . . 10 ((𝑦P ∧ (1P +P 1P) ∈ P) → (𝑦 ·P (1P +P 1P)) ∈ P)
389, 37mpan2 690 . . . . . . . . 9 (𝑦P → (𝑦 ·P (1P +P 1P)) ∈ P)
39 addclpr 10492 . . . . . . . . 9 (((𝑥 ·P 1P) ∈ P ∧ (𝑦 ·P (1P +P 1P)) ∈ P) → ((𝑥 ·P 1P) +P (𝑦 ·P (1P +P 1P))) ∈ P)
4036, 38, 39syl2an 598 . . . . . . . 8 ((𝑥P𝑦P) → ((𝑥 ·P 1P) +P (𝑦 ·P (1P +P 1P))) ∈ P)
4134, 40anim12i 615 . . . . . . 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 10540 . . . . . . 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 594 . . . . . 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 570 . . . . 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 260 . . . 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 2797 . . 3 ((𝑥P𝑦P) → ([⟨𝑥, 𝑦⟩] ~R ·R [⟨(1P +P 1P), 1P⟩] ~R ) = [⟨𝑥, 𝑦⟩] ~R )
476, 46syl5eq 2806 . 2 ((𝑥P𝑦P) → ([⟨𝑥, 𝑦⟩] ~R ·R 1R) = [⟨𝑥, 𝑦⟩] ~R )
481, 4, 47ecoptocl 8404 1 (𝐴R → (𝐴 ·R 1R) = 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1539   ∈ wcel 2112  ⟨cop 4532  (class class class)co 7157  [cec 8304  Pcnp 10333  1Pc1p 10334   +P cpp 10335   ·P cmp 10336   ~R cer 10338  Rcnr 10339  1Rc1r 10341   ·R cmr 10344 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5174  ax-nul 5181  ax-pow 5239  ax-pr 5303  ax-un 7466  ax-inf2 9151 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-reu 3078  df-rmo 3079  df-rab 3080  df-v 3412  df-sbc 3700  df-csb 3809  df-dif 3864  df-un 3866  df-in 3868  df-ss 3878  df-pss 3880  df-nul 4229  df-if 4425  df-pw 4500  df-sn 4527  df-pr 4529  df-tp 4531  df-op 4533  df-uni 4803  df-int 4843  df-iun 4889  df-br 5038  df-opab 5100  df-mpt 5118  df-tr 5144  df-id 5435  df-eprel 5440  df-po 5448  df-so 5449  df-fr 5488  df-we 5490  df-xp 5535  df-rel 5536  df-cnv 5537  df-co 5538  df-dm 5539  df-rn 5540  df-res 5541  df-ima 5542  df-pred 6132  df-ord 6178  df-on 6179  df-lim 6180  df-suc 6181  df-iota 6300  df-fun 6343  df-fn 6344  df-f 6345  df-f1 6346  df-fo 6347  df-f1o 6348  df-fv 6349  df-ov 7160  df-oprab 7161  df-mpo 7162  df-om 7587  df-1st 7700  df-2nd 7701  df-wrecs 7964  df-recs 8025  df-rdg 8063  df-1o 8119  df-oadd 8123  df-omul 8124  df-er 8306  df-ec 8308  df-qs 8312  df-ni 10346  df-pli 10347  df-mi 10348  df-lti 10349  df-plpq 10382  df-mpq 10383  df-ltpq 10384  df-enq 10385  df-nq 10386  df-erq 10387  df-plq 10388  df-mq 10389  df-1nq 10390  df-rq 10391  df-ltnq 10392  df-np 10455  df-1p 10456  df-plp 10457  df-mp 10458  df-ltp 10459  df-enr 10529  df-nr 10530  df-mr 10532  df-1r 10535 This theorem is referenced by:  pn0sr  10575  sqgt0sr  10580  axi2m1  10633  ax1rid  10635  axcnre  10638
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