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Mirrors > Home > MPE Home > Th. List > rrgeq0 | Structured version Visualization version GIF version |
Description: Left-multiplication by a left regular element does not change zeroness. (Contributed by Stefan O'Rear, 28-Mar-2015.) |
Ref | Expression |
---|---|
rrgval.e | ⊢ 𝐸 = (RLReg‘𝑅) |
rrgval.b | ⊢ 𝐵 = (Base‘𝑅) |
rrgval.t | ⊢ · = (.r‘𝑅) |
rrgval.z | ⊢ 0 = (0g‘𝑅) |
Ref | Expression |
---|---|
rrgeq0 | ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐸 ∧ 𝑌 ∈ 𝐵) → ((𝑋 · 𝑌) = 0 ↔ 𝑌 = 0 )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rrgval.e | . . . 4 ⊢ 𝐸 = (RLReg‘𝑅) | |
2 | rrgval.b | . . . 4 ⊢ 𝐵 = (Base‘𝑅) | |
3 | rrgval.t | . . . 4 ⊢ · = (.r‘𝑅) | |
4 | rrgval.z | . . . 4 ⊢ 0 = (0g‘𝑅) | |
5 | 1, 2, 3, 4 | rrgeq0i 20351 | . . 3 ⊢ ((𝑋 ∈ 𝐸 ∧ 𝑌 ∈ 𝐵) → ((𝑋 · 𝑌) = 0 → 𝑌 = 0 )) |
6 | 5 | 3adant1 1132 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐸 ∧ 𝑌 ∈ 𝐵) → ((𝑋 · 𝑌) = 0 → 𝑌 = 0 )) |
7 | simp1 1138 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐸 ∧ 𝑌 ∈ 𝐵) → 𝑅 ∈ Ring) | |
8 | 1, 2, 3, 4 | rrgval 20349 | . . . . . 6 ⊢ 𝐸 = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 0 → 𝑦 = 0 )} |
9 | 8 | ssrab3 4009 | . . . . 5 ⊢ 𝐸 ⊆ 𝐵 |
10 | simp2 1139 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐸 ∧ 𝑌 ∈ 𝐵) → 𝑋 ∈ 𝐸) | |
11 | 9, 10 | sselid 3912 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐸 ∧ 𝑌 ∈ 𝐵) → 𝑋 ∈ 𝐵) |
12 | 2, 3, 4 | ringrz 19630 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋 · 0 ) = 0 ) |
13 | 7, 11, 12 | syl2anc 587 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐸 ∧ 𝑌 ∈ 𝐵) → (𝑋 · 0 ) = 0 ) |
14 | oveq2 7239 | . . . 4 ⊢ (𝑌 = 0 → (𝑋 · 𝑌) = (𝑋 · 0 )) | |
15 | 14 | eqeq1d 2740 | . . 3 ⊢ (𝑌 = 0 → ((𝑋 · 𝑌) = 0 ↔ (𝑋 · 0 ) = 0 )) |
16 | 13, 15 | syl5ibrcom 250 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐸 ∧ 𝑌 ∈ 𝐵) → (𝑌 = 0 → (𝑋 · 𝑌) = 0 )) |
17 | 6, 16 | impbid 215 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐸 ∧ 𝑌 ∈ 𝐵) → ((𝑋 · 𝑌) = 0 ↔ 𝑌 = 0 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ w3a 1089 = wceq 1543 ∈ wcel 2111 ∀wral 3062 ‘cfv 6397 (class class class)co 7231 Basecbs 16784 .rcmulr 16827 0gc0g 16968 Ringcrg 19586 RLRegcrlreg 20341 |
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 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-sep 5206 ax-nul 5213 ax-pow 5272 ax-pr 5336 ax-un 7541 ax-cnex 10809 ax-resscn 10810 ax-1cn 10811 ax-icn 10812 ax-addcl 10813 ax-addrcl 10814 ax-mulcl 10815 ax-mulrcl 10816 ax-mulcom 10817 ax-addass 10818 ax-mulass 10819 ax-distr 10820 ax-i2m1 10821 ax-1ne0 10822 ax-1rid 10823 ax-rnegex 10824 ax-rrecex 10825 ax-cnre 10826 ax-pre-lttri 10827 ax-pre-lttrn 10828 ax-pre-ltadd 10829 ax-pre-mulgt0 10830 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3067 df-rex 3068 df-reu 3069 df-rmo 3070 df-rab 3071 df-v 3422 df-sbc 3709 df-csb 3826 df-dif 3883 df-un 3885 df-in 3887 df-ss 3897 df-pss 3899 df-nul 4252 df-if 4454 df-pw 4529 df-sn 4556 df-pr 4558 df-tp 4560 df-op 4562 df-uni 4834 df-iun 4920 df-br 5068 df-opab 5130 df-mpt 5150 df-tr 5176 df-id 5469 df-eprel 5474 df-po 5482 df-so 5483 df-fr 5523 df-we 5525 df-xp 5571 df-rel 5572 df-cnv 5573 df-co 5574 df-dm 5575 df-rn 5576 df-res 5577 df-ima 5578 df-pred 6175 df-ord 6233 df-on 6234 df-lim 6235 df-suc 6236 df-iota 6355 df-fun 6399 df-fn 6400 df-f 6401 df-f1 6402 df-fo 6403 df-f1o 6404 df-fv 6405 df-riota 7188 df-ov 7234 df-oprab 7235 df-mpo 7236 df-om 7663 df-wrecs 8067 df-recs 8128 df-rdg 8166 df-er 8411 df-en 8647 df-dom 8648 df-sdom 8649 df-pnf 10893 df-mnf 10894 df-xr 10895 df-ltxr 10896 df-le 10897 df-sub 11088 df-neg 11089 df-nn 11855 df-2 11917 df-sets 16741 df-slot 16759 df-ndx 16769 df-base 16785 df-plusg 16839 df-0g 16970 df-mgm 18138 df-sgrp 18187 df-mnd 18198 df-grp 18392 df-mgp 19529 df-ring 19588 df-rlreg 20345 |
This theorem is referenced by: rrgsupp 20353 |
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