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Theorem dmnnzd 36274
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 2736 . . . . . 6 (GId‘𝐻) = (GId‘𝐻)
61, 2, 3, 4, 5isdmn3 36273 . . . . 5 (𝑅 ∈ Dmn ↔ (𝑅 ∈ CRingOps ∧ (GId‘𝐻) ≠ 𝑍 ∧ ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) = 𝑍 → (𝑎 = 𝑍𝑏 = 𝑍))))
76simp3bi 1147 . . . 4 (𝑅 ∈ Dmn → ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) = 𝑍 → (𝑎 = 𝑍𝑏 = 𝑍)))
8 oveq1 7310 . . . . . . 7 (𝑎 = 𝐴 → (𝑎𝐻𝑏) = (𝐴𝐻𝑏))
98eqeq1d 2738 . . . . . 6 (𝑎 = 𝐴 → ((𝑎𝐻𝑏) = 𝑍 ↔ (𝐴𝐻𝑏) = 𝑍))
10 eqeq1 2740 . . . . . . 7 (𝑎 = 𝐴 → (𝑎 = 𝑍𝐴 = 𝑍))
1110orbi1d 915 . . . . . 6 (𝑎 = 𝐴 → ((𝑎 = 𝑍𝑏 = 𝑍) ↔ (𝐴 = 𝑍𝑏 = 𝑍)))
129, 11imbi12d 346 . . . . 5 (𝑎 = 𝐴 → (((𝑎𝐻𝑏) = 𝑍 → (𝑎 = 𝑍𝑏 = 𝑍)) ↔ ((𝐴𝐻𝑏) = 𝑍 → (𝐴 = 𝑍𝑏 = 𝑍))))
13 oveq2 7311 . . . . . . 7 (𝑏 = 𝐵 → (𝐴𝐻𝑏) = (𝐴𝐻𝐵))
1413eqeq1d 2738 . . . . . 6 (𝑏 = 𝐵 → ((𝐴𝐻𝑏) = 𝑍 ↔ (𝐴𝐻𝐵) = 𝑍))
15 eqeq1 2740 . . . . . . 7 (𝑏 = 𝐵 → (𝑏 = 𝑍𝐵 = 𝑍))
1615orbi2d 914 . . . . . 6 (𝑏 = 𝐵 → ((𝐴 = 𝑍𝑏 = 𝑍) ↔ (𝐴 = 𝑍𝐵 = 𝑍)))
1714, 16imbi12d 346 . . . . 5 (𝑏 = 𝐵 → (((𝐴𝐻𝑏) = 𝑍 → (𝐴 = 𝑍𝑏 = 𝑍)) ↔ ((𝐴𝐻𝐵) = 𝑍 → (𝐴 = 𝑍𝐵 = 𝑍))))
1812, 17rspc2v 3575 . . . 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 417 . 2 (𝑅 ∈ Dmn → (𝐴𝑋 → (𝐵𝑋 → ((𝐴𝐻𝐵) = 𝑍 → (𝐴 = 𝑍𝐵 = 𝑍)))))
21203imp2 1349 1 ((𝑅 ∈ Dmn ∧ (𝐴𝑋𝐵𝑋 ∧ (𝐴𝐻𝐵) = 𝑍)) → (𝐴 = 𝑍𝐵 = 𝑍))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wo 845  w3a 1087   = wceq 1539  wcel 2104  wne 2941  wral 3062  ran crn 5597  cfv 6454  (class class class)co 7303  1st c1st 7857  2nd c2nd 7858  GIdcgi 28893  CRingOpsccring 36192  Dmncdmn 36246
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-rep 5218  ax-sep 5232  ax-nul 5239  ax-pow 5297  ax-pr 5361  ax-un 7616
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3or 1088  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3285  df-reu 3286  df-rab 3287  df-v 3439  df-sbc 3722  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-int 4887  df-iun 4933  df-br 5082  df-opab 5144  df-mpt 5165  df-tr 5199  df-id 5496  df-eprel 5502  df-po 5510  df-so 5511  df-fr 5551  df-we 5553  df-xp 5602  df-rel 5603  df-cnv 5604  df-co 5605  df-dm 5606  df-rn 5607  df-res 5608  df-ima 5609  df-ord 6280  df-on 6281  df-lim 6282  df-suc 6283  df-iota 6406  df-fun 6456  df-fn 6457  df-f 6458  df-f1 6459  df-fo 6460  df-f1o 6461  df-fv 6462  df-riota 7260  df-ov 7306  df-oprab 7307  df-mpo 7308  df-om 7741  df-1st 7859  df-2nd 7860  df-1o 8324  df-er 8525  df-en 8761  df-dom 8762  df-sdom 8763  df-fin 8764  df-grpo 28896  df-gid 28897  df-ginv 28898  df-ablo 28948  df-ass 36042  df-exid 36044  df-mgmOLD 36048  df-sgrOLD 36060  df-mndo 36066  df-rngo 36094  df-com2 36189  df-crngo 36193  df-idl 36209  df-pridl 36210  df-prrngo 36247  df-dmn 36248  df-igen 36259
This theorem is referenced by:  dmncan1  36275
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