Theorem dmnnzd 38065

Description: A domain has no zero-divisors (besides zero). (Contributed by Jeff Madsen, 19-Jun-2010.)

Hypotheses

Ref	Expression
dmnnzd.1	⊢ 𝐺 = (1st ‘𝑅)
dmnnzd.2	⊢ 𝐻 = (2nd ‘𝑅)
dmnnzd.3	⊢ 𝑋 = ran 𝐺
dmnnzd.4	⊢ 𝑍 = (GId‘𝐺)

Assertion

Ref	Expression
dmnnzd	⊢ ((𝑅 ∈ Dmn ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ (𝐴𝐻𝐵) = 𝑍)) → (𝐴 = 𝑍 ∨ 𝐵 = 𝑍))

Proof of Theorem dmnnzd

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dmnnzd.1	. . . . . 6 ⊢ 𝐺 = (1st ‘𝑅)
2		dmnnzd.2	. . . . . 6 ⊢ 𝐻 = (2nd ‘𝑅)
3		dmnnzd.3	. . . . . 6 ⊢ 𝑋 = ran 𝐺
4		dmnnzd.4	. . . . . 6 ⊢ 𝑍 = (GId‘𝐺)
5		eqid 2729	. . . . . 6 ⊢ (GId‘𝐻) = (GId‘𝐻)
6	1, 2, 3, 4, 5	isdmn3 38064	. . . . 5 ⊢ (𝑅 ∈ Dmn ↔ (𝑅 ∈ CRingOps ∧ (GId‘𝐻) ≠ 𝑍 ∧ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) = 𝑍 → (𝑎 = 𝑍 ∨ 𝑏 = 𝑍))))
7	6	simp3bi 1147	. . . 4 ⊢ (𝑅 ∈ Dmn → ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) = 𝑍 → (𝑎 = 𝑍 ∨ 𝑏 = 𝑍)))
8		oveq1 7356	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (𝑎𝐻𝑏) = (𝐴𝐻𝑏))
9	8	eqeq1d 2731	. . . . . 6 ⊢ (𝑎 = 𝐴 → ((𝑎𝐻𝑏) = 𝑍 ↔ (𝐴𝐻𝑏) = 𝑍))
10		eqeq1 2733	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (𝑎 = 𝑍 ↔ 𝐴 = 𝑍))
11	10	orbi1d 916	. . . . . 6 ⊢ (𝑎 = 𝐴 → ((𝑎 = 𝑍 ∨ 𝑏 = 𝑍) ↔ (𝐴 = 𝑍 ∨ 𝑏 = 𝑍)))
12	9, 11	imbi12d 344	. . . . 5 ⊢ (𝑎 = 𝐴 → (((𝑎𝐻𝑏) = 𝑍 → (𝑎 = 𝑍 ∨ 𝑏 = 𝑍)) ↔ ((𝐴𝐻𝑏) = 𝑍 → (𝐴 = 𝑍 ∨ 𝑏 = 𝑍))))
13		oveq2 7357	. . . . . . 7 ⊢ (𝑏 = 𝐵 → (𝐴𝐻𝑏) = (𝐴𝐻𝐵))
14	13	eqeq1d 2731	. . . . . 6 ⊢ (𝑏 = 𝐵 → ((𝐴𝐻𝑏) = 𝑍 ↔ (𝐴𝐻𝐵) = 𝑍))
15		eqeq1 2733	. . . . . . 7 ⊢ (𝑏 = 𝐵 → (𝑏 = 𝑍 ↔ 𝐵 = 𝑍))
16	15	orbi2d 915	. . . . . 6 ⊢ (𝑏 = 𝐵 → ((𝐴 = 𝑍 ∨ 𝑏 = 𝑍) ↔ (𝐴 = 𝑍 ∨ 𝐵 = 𝑍)))
17	14, 16	imbi12d 344	. . . . 5 ⊢ (𝑏 = 𝐵 → (((𝐴𝐻𝑏) = 𝑍 → (𝐴 = 𝑍 ∨ 𝑏 = 𝑍)) ↔ ((𝐴𝐻𝐵) = 𝑍 → (𝐴 = 𝑍 ∨ 𝐵 = 𝑍))))
18	12, 17	rspc2v 3588	. . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) = 𝑍 → (𝑎 = 𝑍 ∨ 𝑏 = 𝑍)) → ((𝐴𝐻𝐵) = 𝑍 → (𝐴 = 𝑍 ∨ 𝐵 = 𝑍))))
19	7, 18	syl5com 31	. . 3 ⊢ (𝑅 ∈ Dmn → ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴𝐻𝐵) = 𝑍 → (𝐴 = 𝑍 ∨ 𝐵 = 𝑍))))
20	19	expd 415	. 2 ⊢ (𝑅 ∈ Dmn → (𝐴 ∈ 𝑋 → (𝐵 ∈ 𝑋 → ((𝐴𝐻𝐵) = 𝑍 → (𝐴 = 𝑍 ∨ 𝐵 = 𝑍)))))
21	20	3imp2 1350	1 ⊢ ((𝑅 ∈ Dmn ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ (𝐴𝐻𝐵) = 𝑍)) → (𝐴 = 𝑍 ∨ 𝐵 = 𝑍))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044  ran crn 5620  ‘cfv 6482  (class class class)co 7349  1^st c1st 7922  2^nd c2nd 7923  GIdcgi 30434  CRingOpsccring 37983  Dmncdmn 38037

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-int 4897  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-1st 7924  df-2nd 7925  df-1o 8388  df-en 8873  df-grpo 30437  df-gid 30438  df-ginv 30439  df-ablo 30489  df-ass 37833  df-exid 37835  df-mgmOLD 37839  df-sgrOLD 37851  df-mndo 37857  df-rngo 37885  df-com2 37980  df-crngo 37984  df-idl 38000  df-pridl 38001  df-prrngo 38038  df-dmn 38039  df-igen 38050

This theorem is referenced by:  dmncan1  38066