Theorem dmnnzd 38929

Description: Obsolete theorem, use idomnzd 49365 instead. A domain has no zero-divisors (besides zero). (Contributed by Jeff Madsen, 19-Jun-2010.) (New usage is discouraged.) (Proof modification is discouraged.)

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 2760	. . . . . 6 ⊢ (GId‘𝐻) = (GId‘𝐻)
6	1, 2, 3, 4, 5	isdmn3 38928	. . . . 5 ⊢ (𝑅 ∈ Dmn ↔ (𝑅 ∈ CRingOps ∧ (GId‘𝐻) ≠ 𝑍 ∧ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) = 𝑍 → (𝑎 = 𝑍 ∨ 𝑏 = 𝑍))))
7	6	simp3bi 1165	. . . 4 ⊢ (𝑅 ∈ Dmn → ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) = 𝑍 → (𝑎 = 𝑍 ∨ 𝑏 = 𝑍)))
8		oveq1 7415	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (𝑎𝐻𝑏) = (𝐴𝐻𝑏))
9	8	eqeq1d 2762	. . . . . 6 ⊢ (𝑎 = 𝐴 → ((𝑎𝐻𝑏) = 𝑍 ↔ (𝐴𝐻𝑏) = 𝑍))
10		eqeq1 2764	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (𝑎 = 𝑍 ↔ 𝐴 = 𝑍))
11	10	orbi1d 930	. . . . . 6 ⊢ (𝑎 = 𝐴 → ((𝑎 = 𝑍 ∨ 𝑏 = 𝑍) ↔ (𝐴 = 𝑍 ∨ 𝑏 = 𝑍)))
12	9, 11	imbi12d 347	. . . . 5 ⊢ (𝑎 = 𝐴 → (((𝑎𝐻𝑏) = 𝑍 → (𝑎 = 𝑍 ∨ 𝑏 = 𝑍)) ↔ ((𝐴𝐻𝑏) = 𝑍 → (𝐴 = 𝑍 ∨ 𝑏 = 𝑍))))
13		oveq2 7416	. . . . . . 7 ⊢ (𝑏 = 𝐵 → (𝐴𝐻𝑏) = (𝐴𝐻𝐵))
14	13	eqeq1d 2762	. . . . . 6 ⊢ (𝑏 = 𝐵 → ((𝐴𝐻𝑏) = 𝑍 ↔ (𝐴𝐻𝐵) = 𝑍))
15		eqeq1 2764	. . . . . . 7 ⊢ (𝑏 = 𝐵 → (𝑏 = 𝑍 ↔ 𝐵 = 𝑍))
16	15	orbi2d 929	. . . . . 6 ⊢ (𝑏 = 𝐵 → ((𝐴 = 𝑍 ∨ 𝑏 = 𝑍) ↔ (𝐴 = 𝑍 ∨ 𝐵 = 𝑍)))
17	14, 16	imbi12d 347	. . . . 5 ⊢ (𝑏 = 𝐵 → (((𝐴𝐻𝑏) = 𝑍 → (𝐴 = 𝑍 ∨ 𝑏 = 𝑍)) ↔ ((𝐴𝐻𝐵) = 𝑍 → (𝐴 = 𝑍 ∨ 𝐵 = 𝑍))))
18	12, 17	rspc2v 3586	. . . 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 32	. . 3 ⊢ (𝑅 ∈ Dmn → ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴𝐻𝐵) = 𝑍 → (𝐴 = 𝑍 ∨ 𝐵 = 𝑍))))
20	19	expd 421	. 2 ⊢ (𝑅 ∈ Dmn → (𝐴 ∈ 𝑋 → (𝐵 ∈ 𝑋 → ((𝐴𝐻𝐵) = 𝑍 → (𝐴 = 𝑍 ∨ 𝐵 = 𝑍)))))
21	20	3imp2 1368	1 ⊢ ((𝑅 ∈ Dmn ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ (𝐴𝐻𝐵) = 𝑍)) → (𝐴 = 𝑍 ∨ 𝐵 = 𝑍))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2955  ∀wral 3076  ran crn 5648  ‘cfv 6527  (class class class)co 7408  1^st c1st 7982  2^nd c2nd 7983  GIdcgi 31025  CRingOpsccring 38847  Dmncdmn 38901

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-suc 6357  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-riota 7365  df-ov 7411  df-oprab 7412  df-mpo 7413  df-1st 7984  df-2nd 7985  df-1o 8454  df-en 8952  df-grpo 31028  df-gid 31029  df-ginv 31030  df-ablo 31080  df-ass 38697  df-exid 38699  df-mgmOLD 38703  df-sgrOLD 38715  df-mndo 38721  df-rngo 38749  df-com2 38844  df-crngo 38848  df-idl 38864  df-pridl 38865  df-prrngo 38902  df-dmn 38903  df-igen 38914

This theorem is used by:  dmncan1  38930