Theorem domneq0 13906

Description: In a domain, a product is zero iff it has a zero factor. (Contributed by Mario Carneiro, 28-Mar-2015.)

Hypotheses

Ref	Expression
domneq0.b	⊢ 𝐵 = (Base‘𝑅)
domneq0.t	⊢ · = (.r‘𝑅)
domneq0.z	⊢ 0 = (0g‘𝑅)

Assertion

Ref	Expression
domneq0	⊢ ((𝑅 ∈ Domn ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 · 𝑌) = 0 ↔ (𝑋 = 0 ∨ 𝑌 = 0 )))

Proof of Theorem domneq0

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		3simpc 998	. . 3 ⊢ ((𝑅 ∈ Domn ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵))
2		domneq0.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑅)
3		domneq0.t	. . . . . 6 ⊢ · = (.r‘𝑅)
4		domneq0.z	. . . . . 6 ⊢ 0 = (0g‘𝑅)
5	2, 3, 4	isdomn 13903	. . . . 5 ⊢ (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 0 → (𝑥 = 0 ∨ 𝑦 = 0 ))))
6	5	simprbi 275	. . . 4 ⊢ (𝑅 ∈ Domn → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 0 → (𝑥 = 0 ∨ 𝑦 = 0 )))
7	6	3ad2ant1 1020	. . 3 ⊢ ((𝑅 ∈ Domn ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 0 → (𝑥 = 0 ∨ 𝑦 = 0 )))
8		oveq1 5932	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑥 · 𝑦) = (𝑋 · 𝑦))
9	8	eqeq1d 2205	. . . . 5 ⊢ (𝑥 = 𝑋 → ((𝑥 · 𝑦) = 0 ↔ (𝑋 · 𝑦) = 0 ))
10		eqeq1 2203	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑥 = 0 ↔ 𝑋 = 0 ))
11	10	orbi1d 792	. . . . 5 ⊢ (𝑥 = 𝑋 → ((𝑥 = 0 ∨ 𝑦 = 0 ) ↔ (𝑋 = 0 ∨ 𝑦 = 0 )))
12	9, 11	imbi12d 234	. . . 4 ⊢ (𝑥 = 𝑋 → (((𝑥 · 𝑦) = 0 → (𝑥 = 0 ∨ 𝑦 = 0 )) ↔ ((𝑋 · 𝑦) = 0 → (𝑋 = 0 ∨ 𝑦 = 0 ))))
13		oveq2 5933	. . . . . 6 ⊢ (𝑦 = 𝑌 → (𝑋 · 𝑦) = (𝑋 · 𝑌))
14	13	eqeq1d 2205	. . . . 5 ⊢ (𝑦 = 𝑌 → ((𝑋 · 𝑦) = 0 ↔ (𝑋 · 𝑌) = 0 ))
15		eqeq1 2203	. . . . . 6 ⊢ (𝑦 = 𝑌 → (𝑦 = 0 ↔ 𝑌 = 0 ))
16	15	orbi2d 791	. . . . 5 ⊢ (𝑦 = 𝑌 → ((𝑋 = 0 ∨ 𝑦 = 0 ) ↔ (𝑋 = 0 ∨ 𝑌 = 0 )))
17	14, 16	imbi12d 234	. . . 4 ⊢ (𝑦 = 𝑌 → (((𝑋 · 𝑦) = 0 → (𝑋 = 0 ∨ 𝑦 = 0 )) ↔ ((𝑋 · 𝑌) = 0 → (𝑋 = 0 ∨ 𝑌 = 0 ))))
18	12, 17	rspc2va 2882	. . 3 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 0 → (𝑥 = 0 ∨ 𝑦 = 0 ))) → ((𝑋 · 𝑌) = 0 → (𝑋 = 0 ∨ 𝑌 = 0 )))
19	1, 7, 18	syl2anc 411	. 2 ⊢ ((𝑅 ∈ Domn ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 · 𝑌) = 0 → (𝑋 = 0 ∨ 𝑌 = 0 )))
20		domnring 13905	. . . . . 6 ⊢ (𝑅 ∈ Domn → 𝑅 ∈ Ring)
21	20	3ad2ant1 1020	. . . . 5 ⊢ ((𝑅 ∈ Domn ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑅 ∈ Ring)
22		simp3 1001	. . . . 5 ⊢ ((𝑅 ∈ Domn ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌 ∈ 𝐵)
23	2, 3, 4	ringlz 13677	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵) → ( 0 · 𝑌) = 0 )
24	21, 22, 23	syl2anc 411	. . . 4 ⊢ ((𝑅 ∈ Domn ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( 0 · 𝑌) = 0 )
25		oveq1 5932	. . . . 5 ⊢ (𝑋 = 0 → (𝑋 · 𝑌) = ( 0 · 𝑌))
26	25	eqeq1d 2205	. . . 4 ⊢ (𝑋 = 0 → ((𝑋 · 𝑌) = 0 ↔ ( 0 · 𝑌) = 0 ))
27	24, 26	syl5ibrcom 157	. . 3 ⊢ ((𝑅 ∈ Domn ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 = 0 → (𝑋 · 𝑌) = 0 ))
28		simp2 1000	. . . . 5 ⊢ ((𝑅 ∈ Domn ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋 ∈ 𝐵)
29	2, 3, 4	ringrz 13678	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋 · 0 ) = 0 )
30	21, 28, 29	syl2anc 411	. . . 4 ⊢ ((𝑅 ∈ Domn ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 · 0 ) = 0 )
31		oveq2 5933	. . . . 5 ⊢ (𝑌 = 0 → (𝑋 · 𝑌) = (𝑋 · 0 ))
32	31	eqeq1d 2205	. . . 4 ⊢ (𝑌 = 0 → ((𝑋 · 𝑌) = 0 ↔ (𝑋 · 0 ) = 0 ))
33	30, 32	syl5ibrcom 157	. . 3 ⊢ ((𝑅 ∈ Domn ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑌 = 0 → (𝑋 · 𝑌) = 0 ))
34	27, 33	jaod 718	. 2 ⊢ ((𝑅 ∈ Domn ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 = 0 ∨ 𝑌 = 0 ) → (𝑋 · 𝑌) = 0 ))
35	19, 34	impbid 129	1 ⊢ ((𝑅 ∈ Domn ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 · 𝑌) = 0 ↔ (𝑋 = 0 ∨ 𝑌 = 0 )))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 709   ∧ w3a 980   = wceq 1364   ∈ wcel 2167  ∀wral 2475  ‘cfv 5259  (class class class)co 5925  Basecbs 12705  ._rcmulr 12783  0_gc0g 12960  Ringcrg 13630  NzRingcnzr 13813  Domncdomn 13890

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7989  ax-resscn 7990  ax-1cn 7991  ax-1re 7992  ax-icn 7993  ax-addcl 7994  ax-addrcl 7995  ax-mulcl 7996  ax-addcom 7998  ax-addass 8000  ax-i2m1 8003  ax-0lt1 8004  ax-0id 8006  ax-rnegex 8007  ax-pre-ltirr 8010  ax-pre-ltadd 8014

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-pnf 8082  df-mnf 8083  df-ltxr 8085  df-inn 9010  df-2 9068  df-3 9069  df-ndx 12708  df-slot 12709  df-base 12711  df-sets 12712  df-plusg 12795  df-mulr 12796  df-0g 12962  df-mgm 13060  df-sgrp 13106  df-mnd 13121  df-grp 13207  df-minusg 13208  df-mgp 13555  df-ring 13632  df-nzr 13814  df-domn 13893

This theorem is referenced by:  domnmuln0  13907  znidomb  14292