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Theorem dmnnzd 32838
Description: A domain has no zero-divisors (besides zero). (Contributed by Jeff Madsen, 19-Jun-2010.)
Hypotheses
Ref Expression
dmnnzd.1 𝐺 = (1st𝑅)
dmnnzd.2 𝐻 = (2nd𝑅)
dmnnzd.3 𝑋 = ran 𝐺
dmnnzd.4 𝑍 = (GId‘𝐺)
Assertion
Ref Expression
dmnnzd ((𝑅 ∈ Dmn ∧ (𝐴𝑋𝐵𝑋 ∧ (𝐴𝐻𝐵) = 𝑍)) → (𝐴 = 𝑍𝐵 = 𝑍))

Proof of Theorem dmnnzd
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dmnnzd.1 . . . . . 6 𝐺 = (1st𝑅)
2 dmnnzd.2 . . . . . 6 𝐻 = (2nd𝑅)
3 dmnnzd.3 . . . . . 6 𝑋 = ran 𝐺
4 dmnnzd.4 . . . . . 6 𝑍 = (GId‘𝐺)
5 eqid 2610 . . . . . 6 (GId‘𝐻) = (GId‘𝐻)
61, 2, 3, 4, 5isdmn3 32837 . . . . 5 (𝑅 ∈ Dmn ↔ (𝑅 ∈ CRingOps ∧ (GId‘𝐻) ≠ 𝑍 ∧ ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) = 𝑍 → (𝑎 = 𝑍𝑏 = 𝑍))))
76simp3bi 1071 . . . 4 (𝑅 ∈ Dmn → ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) = 𝑍 → (𝑎 = 𝑍𝑏 = 𝑍)))
8 oveq1 6534 . . . . . . 7 (𝑎 = 𝐴 → (𝑎𝐻𝑏) = (𝐴𝐻𝑏))
98eqeq1d 2612 . . . . . 6 (𝑎 = 𝐴 → ((𝑎𝐻𝑏) = 𝑍 ↔ (𝐴𝐻𝑏) = 𝑍))
10 eqeq1 2614 . . . . . . 7 (𝑎 = 𝐴 → (𝑎 = 𝑍𝐴 = 𝑍))
1110orbi1d 735 . . . . . 6 (𝑎 = 𝐴 → ((𝑎 = 𝑍𝑏 = 𝑍) ↔ (𝐴 = 𝑍𝑏 = 𝑍)))
129, 11imbi12d 333 . . . . 5 (𝑎 = 𝐴 → (((𝑎𝐻𝑏) = 𝑍 → (𝑎 = 𝑍𝑏 = 𝑍)) ↔ ((𝐴𝐻𝑏) = 𝑍 → (𝐴 = 𝑍𝑏 = 𝑍))))
13 oveq2 6535 . . . . . . 7 (𝑏 = 𝐵 → (𝐴𝐻𝑏) = (𝐴𝐻𝐵))
1413eqeq1d 2612 . . . . . 6 (𝑏 = 𝐵 → ((𝐴𝐻𝑏) = 𝑍 ↔ (𝐴𝐻𝐵) = 𝑍))
15 eqeq1 2614 . . . . . . 7 (𝑏 = 𝐵 → (𝑏 = 𝑍𝐵 = 𝑍))
1615orbi2d 734 . . . . . 6 (𝑏 = 𝐵 → ((𝐴 = 𝑍𝑏 = 𝑍) ↔ (𝐴 = 𝑍𝐵 = 𝑍)))
1714, 16imbi12d 333 . . . . 5 (𝑏 = 𝐵 → (((𝐴𝐻𝑏) = 𝑍 → (𝐴 = 𝑍𝑏 = 𝑍)) ↔ ((𝐴𝐻𝐵) = 𝑍 → (𝐴 = 𝑍𝐵 = 𝑍))))
1812, 17rspc2v 3293 . . . 4 ((𝐴𝑋𝐵𝑋) → (∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) = 𝑍 → (𝑎 = 𝑍𝑏 = 𝑍)) → ((𝐴𝐻𝐵) = 𝑍 → (𝐴 = 𝑍𝐵 = 𝑍))))
197, 18syl5com 31 . . 3 (𝑅 ∈ Dmn → ((𝐴𝑋𝐵𝑋) → ((𝐴𝐻𝐵) = 𝑍 → (𝐴 = 𝑍𝐵 = 𝑍))))
2019expd 451 . 2 (𝑅 ∈ Dmn → (𝐴𝑋 → (𝐵𝑋 → ((𝐴𝐻𝐵) = 𝑍 → (𝐴 = 𝑍𝐵 = 𝑍)))))
21203imp2 1274 1 ((𝑅 ∈ Dmn ∧ (𝐴𝑋𝐵𝑋 ∧ (𝐴𝐻𝐵) = 𝑍)) → (𝐴 = 𝑍𝐵 = 𝑍))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 382  wa 383  w3a 1031   = wceq 1475  wcel 1977  wne 2780  wral 2896  ran crn 5029  cfv 5790  (class class class)co 6527  1st c1st 7035  2nd c2nd 7036  GIdcgi 26522  CRingOpsccring 32756  Dmncdmn 32810
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4694  ax-sep 4704  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4368  df-int 4406  df-iun 4452  df-br 4579  df-opab 4639  df-mpt 4640  df-tr 4676  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6936  df-1st 7037  df-2nd 7038  df-1o 7425  df-er 7607  df-en 7820  df-dom 7821  df-sdom 7822  df-fin 7823  df-grpo 26525  df-gid 26526  df-ginv 26527  df-ablo 26577  df-ass 32606  df-exid 32608  df-mgmOLD 32612  df-sgrOLD 32624  df-mndo 32630  df-rngo 32658  df-com2 32753  df-crngo 32757  df-idl 32773  df-pridl 32774  df-prrngo 32811  df-dmn 32812  df-igen 32823
This theorem is referenced by:  dmncan1  32839
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