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Theorem dmnnzd 34004
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 2651 . . . . . 6 (GId‘𝐻) = (GId‘𝐻)
61, 2, 3, 4, 5isdmn3 34003 . . . . 5 (𝑅 ∈ Dmn ↔ (𝑅 ∈ CRingOps ∧ (GId‘𝐻) ≠ 𝑍 ∧ ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) = 𝑍 → (𝑎 = 𝑍𝑏 = 𝑍))))
76simp3bi 1098 . . . 4 (𝑅 ∈ Dmn → ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) = 𝑍 → (𝑎 = 𝑍𝑏 = 𝑍)))
8 oveq1 6697 . . . . . . 7 (𝑎 = 𝐴 → (𝑎𝐻𝑏) = (𝐴𝐻𝑏))
98eqeq1d 2653 . . . . . 6 (𝑎 = 𝐴 → ((𝑎𝐻𝑏) = 𝑍 ↔ (𝐴𝐻𝑏) = 𝑍))
10 eqeq1 2655 . . . . . . 7 (𝑎 = 𝐴 → (𝑎 = 𝑍𝐴 = 𝑍))
1110orbi1d 739 . . . . . 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 6698 . . . . . . 7 (𝑏 = 𝐵 → (𝐴𝐻𝑏) = (𝐴𝐻𝐵))
1413eqeq1d 2653 . . . . . 6 (𝑏 = 𝐵 → ((𝐴𝐻𝑏) = 𝑍 ↔ (𝐴𝐻𝐵) = 𝑍))
15 eqeq1 2655 . . . . . . 7 (𝑏 = 𝐵 → (𝑏 = 𝑍𝐵 = 𝑍))
1615orbi2d 738 . . . . . 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 3353 . . . 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 1304 1 ((𝑅 ∈ Dmn ∧ (𝐴𝑋𝐵𝑋 ∧ (𝐴𝐻𝐵) = 𝑍)) → (𝐴 = 𝑍𝐵 = 𝑍))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 382  wa 383  w3a 1054   = wceq 1523  wcel 2030  wne 2823  wral 2941  ran crn 5144  cfv 5926  (class class class)co 6690  1st c1st 7208  2nd c2nd 7209  GIdcgi 27472  CRingOpsccring 33922  Dmncdmn 33976
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-1o 7605  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-grpo 27475  df-gid 27476  df-ginv 27477  df-ablo 27527  df-ass 33772  df-exid 33774  df-mgmOLD 33778  df-sgrOLD 33790  df-mndo 33796  df-rngo 33824  df-com2 33919  df-crngo 33923  df-idl 33939  df-pridl 33940  df-prrngo 33977  df-dmn 33978  df-igen 33989
This theorem is referenced by:  dmncan1  34005
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