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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 7024 | . . . . 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 19028 | . . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑎 ∈ 𝐵) → (𝑎 · 0 ) = 0 ) |
7 | 2, 6 | sylan 580 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝑎 · 0 ) = 0 ) |
8 | eqeq12 2808 | . . . . . . . 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 7024 | . . . . 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 19012 | . . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋 · 1 ) = 𝑋) |
17 | 3, 4, 5 | ringrz 19028 | . . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋 · 0 ) = 0 ) |
18 | 16, 17 | eqeq12d 2810 | . . . . . . . 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 3252 | . 2 ⊢ (𝜑 → (𝑋 = 0 ↔ 1 = 0 )) |
26 | 25 | necon3bid 3028 | 1 ⊢ (𝜑 → (𝑋 ≠ 0 ↔ 1 ≠ 0 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1522 ∈ wcel 2081 ≠ wne 2984 ∃wrex 3106 ‘cfv 6225 (class class class)co 7016 Basecbs 16312 .rcmulr 16395 0gc0g 16542 1rcur 18941 Ringcrg 18987 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-cnex 10439 ax-resscn 10440 ax-1cn 10441 ax-icn 10442 ax-addcl 10443 ax-addrcl 10444 ax-mulcl 10445 ax-mulrcl 10446 ax-mulcom 10447 ax-addass 10448 ax-mulass 10449 ax-distr 10450 ax-i2m1 10451 ax-1ne0 10452 ax-1rid 10453 ax-rnegex 10454 ax-rrecex 10455 ax-cnre 10456 ax-pre-lttri 10457 ax-pre-lttrn 10458 ax-pre-ltadd 10459 ax-pre-mulgt0 10460 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-pss 3876 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-tp 4477 df-op 4479 df-uni 4746 df-iun 4827 df-br 4963 df-opab 5025 df-mpt 5042 df-tr 5064 df-id 5348 df-eprel 5353 df-po 5362 df-so 5363 df-fr 5402 df-we 5404 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-pred 6023 df-ord 6069 df-on 6070 df-lim 6071 df-suc 6072 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-riota 6977 df-ov 7019 df-oprab 7020 df-mpo 7021 df-om 7437 df-wrecs 7798 df-recs 7860 df-rdg 7898 df-er 8139 df-en 8358 df-dom 8359 df-sdom 8360 df-pnf 10523 df-mnf 10524 df-xr 10525 df-ltxr 10526 df-le 10527 df-sub 10719 df-neg 10720 df-nn 11487 df-2 11548 df-ndx 16315 df-slot 16316 df-base 16318 df-sets 16319 df-plusg 16407 df-0g 16544 df-mgm 17681 df-sgrp 17723 df-mnd 17734 df-grp 17864 df-mgp 18930 df-ur 18942 df-ring 18989 |
This theorem is referenced by: (None) |
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