Theorem dmnnzd 38457

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 2741	. . . . . 6 ⊢ (GId‘𝐻) = (GId‘𝐻)
6	1, 2, 3, 4, 5	isdmn3 38456	. . . . 5 ⊢ (𝑅 ∈ Dmn ↔ (𝑅 ∈ CRingOps ∧ (GId‘𝐻) ≠ 𝑍 ∧ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) = 𝑍 → (𝑎 = 𝑍 ∨ 𝑏 = 𝑍))))
7	6	simp3bi 1154	. . . 4 ⊢ (𝑅 ∈ Dmn → ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) = 𝑍 → (𝑎 = 𝑍 ∨ 𝑏 = 𝑍)))
8		oveq1 7367	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (𝑎𝐻𝑏) = (𝐴𝐻𝑏))
9	8	eqeq1d 2743	. . . . . 6 ⊢ (𝑎 = 𝐴 → ((𝑎𝐻𝑏) = 𝑍 ↔ (𝐴𝐻𝑏) = 𝑍))
10		eqeq1 2745	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (𝑎 = 𝑍 ↔ 𝐴 = 𝑍))
11	10	orbi1d 923	. . . . . 6 ⊢ (𝑎 = 𝐴 → ((𝑎 = 𝑍 ∨ 𝑏 = 𝑍) ↔ (𝐴 = 𝑍 ∨ 𝑏 = 𝑍)))
12	9, 11	imbi12d 346	. . . . 5 ⊢ (𝑎 = 𝐴 → (((𝑎𝐻𝑏) = 𝑍 → (𝑎 = 𝑍 ∨ 𝑏 = 𝑍)) ↔ ((𝐴𝐻𝑏) = 𝑍 → (𝐴 = 𝑍 ∨ 𝑏 = 𝑍))))
13		oveq2 7368	. . . . . . 7 ⊢ (𝑏 = 𝐵 → (𝐴𝐻𝑏) = (𝐴𝐻𝐵))
14	13	eqeq1d 2743	. . . . . 6 ⊢ (𝑏 = 𝐵 → ((𝐴𝐻𝑏) = 𝑍 ↔ (𝐴𝐻𝐵) = 𝑍))
15		eqeq1 2745	. . . . . . 7 ⊢ (𝑏 = 𝐵 → (𝑏 = 𝑍 ↔ 𝐵 = 𝑍))
16	15	orbi2d 922	. . . . . 6 ⊢ (𝑏 = 𝐵 → ((𝐴 = 𝑍 ∨ 𝑏 = 𝑍) ↔ (𝐴 = 𝑍 ∨ 𝐵 = 𝑍)))
17	14, 16	imbi12d 346	. . . . 5 ⊢ (𝑏 = 𝐵 → (((𝐴𝐻𝑏) = 𝑍 → (𝐴 = 𝑍 ∨ 𝑏 = 𝑍)) ↔ ((𝐴𝐻𝐵) = 𝑍 → (𝐴 = 𝑍 ∨ 𝐵 = 𝑍))))
18	12, 17	rspc2v 3573	. . . 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 417	. 2 ⊢ (𝑅 ∈ Dmn → (𝐴 ∈ 𝑋 → (𝐵 ∈ 𝑋 → ((𝐴𝐻𝐵) = 𝑍 → (𝐴 = 𝑍 ∨ 𝐵 = 𝑍)))))
21	20	3imp2 1357	1 ⊢ ((𝑅 ∈ Dmn ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ (𝐴𝐻𝐵) = 𝑍)) → (𝐴 = 𝑍 ∨ 𝐵 = 𝑍))

Syntax hints:   → wi 4   ∧ wa 397   ∨ wo 854   ∧ w3a 1093   = wceq 1548   ∈ wcel 2121   ≠ wne 2936  ∀wral 3055  ran crn 5622  ‘cfv 6489  (class class class)co 7360  1^st c1st 7933  2^nd c2nd 7934  GIdcgi 30583  CRingOpsccring 38375  Dmncdmn 38429

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5202  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365  ax-un 7682

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-rmo 3346  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-int 4881  df-iun 4926  df-br 5076  df-opab 5138  df-mpt 5157  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-1st 7935  df-2nd 7936  df-1o 8399  df-en 8888  df-grpo 30586  df-gid 30587  df-ginv 30588  df-ablo 30638  df-ass 38225  df-exid 38227  df-mgmOLD 38231  df-sgrOLD 38243  df-mndo 38249  df-rngo 38277  df-com2 38372  df-crngo 38376  df-idl 38392  df-pridl 38393  df-prrngo 38430  df-dmn 38431  df-igen 38442

This theorem is referenced by:  dmncan1  38458