Theorem domneq0 14292

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 1022	. . 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 14289	. . . . 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 1044	. . 3 ⊢ ((𝑅 ∈ Domn ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 0 → (𝑥 = 0 ∨ 𝑦 = 0 )))
8		oveq1 6025	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑥 · 𝑦) = (𝑋 · 𝑦))
9	8	eqeq1d 2240	. . . . 5 ⊢ (𝑥 = 𝑋 → ((𝑥 · 𝑦) = 0 ↔ (𝑋 · 𝑦) = 0 ))
10		eqeq1 2238	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑥 = 0 ↔ 𝑋 = 0 ))
11	10	orbi1d 798	. . . . 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 6026	. . . . . 6 ⊢ (𝑦 = 𝑌 → (𝑋 · 𝑦) = (𝑋 · 𝑌))
14	13	eqeq1d 2240	. . . . 5 ⊢ (𝑦 = 𝑌 → ((𝑋 · 𝑦) = 0 ↔ (𝑋 · 𝑌) = 0 ))
15		eqeq1 2238	. . . . . 6 ⊢ (𝑦 = 𝑌 → (𝑦 = 0 ↔ 𝑌 = 0 ))
16	15	orbi2d 797	. . . . 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 2924	. . 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 14291	. . . . . 6 ⊢ (𝑅 ∈ Domn → 𝑅 ∈ Ring)
21	20	3ad2ant1 1044	. . . . 5 ⊢ ((𝑅 ∈ Domn ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑅 ∈ Ring)
22		simp3 1025	. . . . 5 ⊢ ((𝑅 ∈ Domn ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌 ∈ 𝐵)
23	2, 3, 4	ringlz 14062	. . . . 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 6025	. . . . 5 ⊢ (𝑋 = 0 → (𝑋 · 𝑌) = ( 0 · 𝑌))
26	25	eqeq1d 2240	. . . 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 1024	. . . . 5 ⊢ ((𝑅 ∈ Domn ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋 ∈ 𝐵)
29	2, 3, 4	ringrz 14063	. . . . 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 6026	. . . . 5 ⊢ (𝑌 = 0 → (𝑋 · 𝑌) = (𝑋 · 0 ))
32	31	eqeq1d 2240	. . . 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 724	. 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 715   ∧ w3a 1004   = wceq 1397   ∈ wcel 2202  ∀wral 2510  ‘cfv 5326  (class class class)co 6018  Basecbs 13087  ._rcmulr 13166  0_gc0g 13344  Ringcrg 14015  NzRingcnzr 14199  Domncdomn 14276

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-addcom 8132  ax-addass 8134  ax-i2m1 8137  ax-0lt1 8138  ax-0id 8140  ax-rnegex 8141  ax-pre-ltirr 8144  ax-pre-ltadd 8148

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-pnf 8216  df-mnf 8217  df-ltxr 8219  df-inn 9144  df-2 9202  df-3 9203  df-ndx 13090  df-slot 13091  df-base 13093  df-sets 13094  df-plusg 13178  df-mulr 13179  df-0g 13346  df-mgm 13444  df-sgrp 13490  df-mnd 13505  df-grp 13591  df-minusg 13592  df-mgp 13940  df-ring 14017  df-nzr 14200  df-domn 14279

This theorem is referenced by:  domnmuln0  14293  znidomb  14678