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Mirrors > Home > MPE Home > Th. List > domneq0 | Structured version Visualization version GIF version |
Description: In a domain, a product is zero iff it has a zero factor. (Contributed by Mario Carneiro, 28-Mar-2015.) |
Ref | Expression |
---|---|
domneq0.b | ⊢ 𝐵 = (Base‘𝑅) |
domneq0.t | ⊢ · = (.r‘𝑅) |
domneq0.z | ⊢ 0 = (0g‘𝑅) |
Ref | Expression |
---|---|
domneq0 | ⊢ ((𝑅 ∈ Domn ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 · 𝑌) = 0 ↔ (𝑋 = 0 ∨ 𝑌 = 0 ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3simpc 1131 | . . 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 19801 | . . . . 5 ⊢ (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 0 → (𝑥 = 0 ∨ 𝑦 = 0 )))) |
6 | 5 | simprbi 489 | . . . 4 ⊢ (𝑅 ∈ Domn → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 0 → (𝑥 = 0 ∨ 𝑦 = 0 ))) |
7 | 6 | 3ad2ant1 1114 | . . 3 ⊢ ((𝑅 ∈ Domn ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 0 → (𝑥 = 0 ∨ 𝑦 = 0 ))) |
8 | oveq1 6982 | . . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑥 · 𝑦) = (𝑋 · 𝑦)) | |
9 | 8 | eqeq1d 2775 | . . . . 5 ⊢ (𝑥 = 𝑋 → ((𝑥 · 𝑦) = 0 ↔ (𝑋 · 𝑦) = 0 )) |
10 | eqeq1 2777 | . . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑥 = 0 ↔ 𝑋 = 0 )) | |
11 | 10 | orbi1d 901 | . . . . 5 ⊢ (𝑥 = 𝑋 → ((𝑥 = 0 ∨ 𝑦 = 0 ) ↔ (𝑋 = 0 ∨ 𝑦 = 0 ))) |
12 | 9, 11 | imbi12d 337 | . . . 4 ⊢ (𝑥 = 𝑋 → (((𝑥 · 𝑦) = 0 → (𝑥 = 0 ∨ 𝑦 = 0 )) ↔ ((𝑋 · 𝑦) = 0 → (𝑋 = 0 ∨ 𝑦 = 0 )))) |
13 | oveq2 6983 | . . . . . 6 ⊢ (𝑦 = 𝑌 → (𝑋 · 𝑦) = (𝑋 · 𝑌)) | |
14 | 13 | eqeq1d 2775 | . . . . 5 ⊢ (𝑦 = 𝑌 → ((𝑋 · 𝑦) = 0 ↔ (𝑋 · 𝑌) = 0 )) |
15 | eqeq1 2777 | . . . . . 6 ⊢ (𝑦 = 𝑌 → (𝑦 = 0 ↔ 𝑌 = 0 )) | |
16 | 15 | orbi2d 900 | . . . . 5 ⊢ (𝑦 = 𝑌 → ((𝑋 = 0 ∨ 𝑦 = 0 ) ↔ (𝑋 = 0 ∨ 𝑌 = 0 ))) |
17 | 14, 16 | imbi12d 337 | . . . 4 ⊢ (𝑦 = 𝑌 → (((𝑋 · 𝑦) = 0 → (𝑋 = 0 ∨ 𝑦 = 0 )) ↔ ((𝑋 · 𝑌) = 0 → (𝑋 = 0 ∨ 𝑌 = 0 )))) |
18 | 12, 17 | rspc2va 3544 | . . 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 576 | . 2 ⊢ ((𝑅 ∈ Domn ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 · 𝑌) = 0 → (𝑋 = 0 ∨ 𝑌 = 0 ))) |
20 | domnring 19803 | . . . . . 6 ⊢ (𝑅 ∈ Domn → 𝑅 ∈ Ring) | |
21 | 20 | 3ad2ant1 1114 | . . . . 5 ⊢ ((𝑅 ∈ Domn ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑅 ∈ Ring) |
22 | simp3 1119 | . . . . 5 ⊢ ((𝑅 ∈ Domn ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌 ∈ 𝐵) | |
23 | 2, 3, 4 | ringlz 19073 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵) → ( 0 · 𝑌) = 0 ) |
24 | 21, 22, 23 | syl2anc 576 | . . . 4 ⊢ ((𝑅 ∈ Domn ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( 0 · 𝑌) = 0 ) |
25 | oveq1 6982 | . . . . 5 ⊢ (𝑋 = 0 → (𝑋 · 𝑌) = ( 0 · 𝑌)) | |
26 | 25 | eqeq1d 2775 | . . . 4 ⊢ (𝑋 = 0 → ((𝑋 · 𝑌) = 0 ↔ ( 0 · 𝑌) = 0 )) |
27 | 24, 26 | syl5ibrcom 239 | . . 3 ⊢ ((𝑅 ∈ Domn ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 = 0 → (𝑋 · 𝑌) = 0 )) |
28 | simp2 1118 | . . . . 5 ⊢ ((𝑅 ∈ Domn ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
29 | 2, 3, 4 | ringrz 19074 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋 · 0 ) = 0 ) |
30 | 21, 28, 29 | syl2anc 576 | . . . 4 ⊢ ((𝑅 ∈ Domn ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 · 0 ) = 0 ) |
31 | oveq2 6983 | . . . . 5 ⊢ (𝑌 = 0 → (𝑋 · 𝑌) = (𝑋 · 0 )) | |
32 | 31 | eqeq1d 2775 | . . . 4 ⊢ (𝑌 = 0 → ((𝑋 · 𝑌) = 0 ↔ (𝑋 · 0 ) = 0 )) |
33 | 30, 32 | syl5ibrcom 239 | . . 3 ⊢ ((𝑅 ∈ Domn ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑌 = 0 → (𝑋 · 𝑌) = 0 )) |
34 | 27, 33 | jaod 846 | . 2 ⊢ ((𝑅 ∈ Domn ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 = 0 ∨ 𝑌 = 0 ) → (𝑋 · 𝑌) = 0 )) |
35 | 19, 34 | impbid 204 | 1 ⊢ ((𝑅 ∈ Domn ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 · 𝑌) = 0 ↔ (𝑋 = 0 ∨ 𝑌 = 0 ))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 ∨ wo 834 ∧ w3a 1069 = wceq 1508 ∈ wcel 2051 ∀wral 3083 ‘cfv 6186 (class class class)co 6975 Basecbs 16338 .rcmulr 16421 0gc0g 16568 Ringcrg 19033 NzRingcnzr 19764 Domncdomn 19787 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2745 ax-sep 5057 ax-nul 5064 ax-pow 5116 ax-pr 5183 ax-un 7278 ax-cnex 10390 ax-resscn 10391 ax-1cn 10392 ax-icn 10393 ax-addcl 10394 ax-addrcl 10395 ax-mulcl 10396 ax-mulrcl 10397 ax-mulcom 10398 ax-addass 10399 ax-mulass 10400 ax-distr 10401 ax-i2m1 10402 ax-1ne0 10403 ax-1rid 10404 ax-rnegex 10405 ax-rrecex 10406 ax-cnre 10407 ax-pre-lttri 10408 ax-pre-lttrn 10409 ax-pre-ltadd 10410 ax-pre-mulgt0 10411 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2754 df-cleq 2766 df-clel 2841 df-nfc 2913 df-ne 2963 df-nel 3069 df-ral 3088 df-rex 3089 df-reu 3090 df-rmo 3091 df-rab 3092 df-v 3412 df-sbc 3677 df-csb 3782 df-dif 3827 df-un 3829 df-in 3831 df-ss 3838 df-pss 3840 df-nul 4174 df-if 4346 df-pw 4419 df-sn 4437 df-pr 4439 df-tp 4441 df-op 4443 df-uni 4710 df-iun 4791 df-br 4927 df-opab 4989 df-mpt 5006 df-tr 5028 df-id 5309 df-eprel 5314 df-po 5323 df-so 5324 df-fr 5363 df-we 5365 df-xp 5410 df-rel 5411 df-cnv 5412 df-co 5413 df-dm 5414 df-rn 5415 df-res 5416 df-ima 5417 df-pred 5984 df-ord 6030 df-on 6031 df-lim 6032 df-suc 6033 df-iota 6150 df-fun 6188 df-fn 6189 df-f 6190 df-f1 6191 df-fo 6192 df-f1o 6193 df-fv 6194 df-riota 6936 df-ov 6978 df-oprab 6979 df-mpo 6980 df-om 7396 df-wrecs 7749 df-recs 7811 df-rdg 7849 df-er 8088 df-en 8306 df-dom 8307 df-sdom 8308 df-pnf 10475 df-mnf 10476 df-xr 10477 df-ltxr 10478 df-le 10479 df-sub 10671 df-neg 10672 df-nn 11439 df-2 11502 df-ndx 16341 df-slot 16342 df-base 16344 df-sets 16345 df-plusg 16433 df-0g 16570 df-mgm 17723 df-sgrp 17765 df-mnd 17776 df-grp 17907 df-minusg 17908 df-mgp 18976 df-ring 19035 df-nzr 19765 df-domn 19791 |
This theorem is referenced by: domnmuln0 19805 opprdomn 19808 fidomndrnglem 19813 domnchr 20397 znidomb 20426 fta1glem2 24479 lidldomn1 43586 |
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