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Mirrors > Home > MPE Home > Th. List > ringinvnz1ne0 | Structured version Visualization version GIF version |
Description: In a unitary ring, a left invertible element is different from zero iff 1 ≠ 0. (Contributed by FL, 18-Apr-2010.) (Revised by AV, 24-Aug-2021.) |
Ref | Expression |
---|---|
ringinvnzdiv.b | ⊢ 𝐵 = (Base‘𝑅) |
ringinvnzdiv.t | ⊢ · = (.r‘𝑅) |
ringinvnzdiv.u | ⊢ 1 = (1r‘𝑅) |
ringinvnzdiv.z | ⊢ 0 = (0g‘𝑅) |
ringinvnzdiv.r | ⊢ (𝜑 → 𝑅 ∈ Ring) |
ringinvnzdiv.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
ringinvnzdiv.a | ⊢ (𝜑 → ∃𝑎 ∈ 𝐵 (𝑎 · 𝑋) = 1 ) |
Ref | Expression |
---|---|
ringinvnz1ne0 | ⊢ (𝜑 → (𝑋 ≠ 0 ↔ 1 ≠ 0 )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 7153 | . . . . 5 ⊢ (𝑋 = 0 → (𝑎 · 𝑋) = (𝑎 · 0 )) | |
2 | ringinvnzdiv.r | . . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
3 | ringinvnzdiv.b | . . . . . . . 8 ⊢ 𝐵 = (Base‘𝑅) | |
4 | ringinvnzdiv.t | . . . . . . . 8 ⊢ · = (.r‘𝑅) | |
5 | ringinvnzdiv.z | . . . . . . . 8 ⊢ 0 = (0g‘𝑅) | |
6 | 3, 4, 5 | ringrz 19269 | . . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑎 ∈ 𝐵) → (𝑎 · 0 ) = 0 ) |
7 | 2, 6 | sylan 580 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝑎 · 0 ) = 0 ) |
8 | eqeq12 2835 | . . . . . . . 8 ⊢ (((𝑎 · 𝑋) = 1 ∧ (𝑎 · 0 ) = 0 ) → ((𝑎 · 𝑋) = (𝑎 · 0 ) ↔ 1 = 0 )) | |
9 | 8 | biimpd 230 | . . . . . . 7 ⊢ (((𝑎 · 𝑋) = 1 ∧ (𝑎 · 0 ) = 0 ) → ((𝑎 · 𝑋) = (𝑎 · 0 ) → 1 = 0 )) |
10 | 9 | ex 413 | . . . . . 6 ⊢ ((𝑎 · 𝑋) = 1 → ((𝑎 · 0 ) = 0 → ((𝑎 · 𝑋) = (𝑎 · 0 ) → 1 = 0 ))) |
11 | 7, 10 | mpan9 507 | . . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ (𝑎 · 𝑋) = 1 ) → ((𝑎 · 𝑋) = (𝑎 · 0 ) → 1 = 0 )) |
12 | 1, 11 | syl5 34 | . . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ (𝑎 · 𝑋) = 1 ) → (𝑋 = 0 → 1 = 0 )) |
13 | oveq2 7153 | . . . . 5 ⊢ ( 1 = 0 → (𝑋 · 1 ) = (𝑋 · 0 )) | |
14 | ringinvnzdiv.x | . . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
15 | ringinvnzdiv.u | . . . . . . . . . 10 ⊢ 1 = (1r‘𝑅) | |
16 | 3, 4, 15 | ringridm 19253 | . . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋 · 1 ) = 𝑋) |
17 | 3, 4, 5 | ringrz 19269 | . . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋 · 0 ) = 0 ) |
18 | 16, 17 | eqeq12d 2837 | . . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → ((𝑋 · 1 ) = (𝑋 · 0 ) ↔ 𝑋 = 0 )) |
19 | 18 | biimpd 230 | . . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → ((𝑋 · 1 ) = (𝑋 · 0 ) → 𝑋 = 0 )) |
20 | 2, 14, 19 | syl2anc 584 | . . . . . 6 ⊢ (𝜑 → ((𝑋 · 1 ) = (𝑋 · 0 ) → 𝑋 = 0 )) |
21 | 20 | ad2antrr 722 | . . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ (𝑎 · 𝑋) = 1 ) → ((𝑋 · 1 ) = (𝑋 · 0 ) → 𝑋 = 0 )) |
22 | 13, 21 | syl5 34 | . . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ (𝑎 · 𝑋) = 1 ) → ( 1 = 0 → 𝑋 = 0 )) |
23 | 12, 22 | impbid 213 | . . 3 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ (𝑎 · 𝑋) = 1 ) → (𝑋 = 0 ↔ 1 = 0 )) |
24 | ringinvnzdiv.a | . . 3 ⊢ (𝜑 → ∃𝑎 ∈ 𝐵 (𝑎 · 𝑋) = 1 ) | |
25 | 23, 24 | r19.29a 3289 | . 2 ⊢ (𝜑 → (𝑋 = 0 ↔ 1 = 0 )) |
26 | 25 | necon3bid 3060 | 1 ⊢ (𝜑 → (𝑋 ≠ 0 ↔ 1 ≠ 0 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ≠ wne 3016 ∃wrex 3139 ‘cfv 6349 (class class class)co 7145 Basecbs 16473 .rcmulr 16556 0gc0g 16703 1rcur 19182 Ringcrg 19228 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4833 df-iun 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7569 df-wrecs 7938 df-recs 7999 df-rdg 8037 df-er 8279 df-en 8499 df-dom 8500 df-sdom 8501 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11628 df-2 11689 df-ndx 16476 df-slot 16477 df-base 16479 df-sets 16480 df-plusg 16568 df-0g 16705 df-mgm 17842 df-sgrp 17891 df-mnd 17902 df-grp 18046 df-mgp 19171 df-ur 19183 df-ring 19230 |
This theorem is referenced by: (None) |
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