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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 19267 | . . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑎 ∈ 𝐵) → (𝑎 · 0 ) = 0 ) |
7 | 2, 6 | sylan 580 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝑎 · 0 ) = 0 ) |
8 | eqeq12 2832 | . . . . . . . 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 19251 | . . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋 · 1 ) = 𝑋) |
17 | 3, 4, 5 | ringrz 19267 | . . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋 · 0 ) = 0 ) |
18 | 16, 17 | eqeq12d 2834 | . . . . . . . 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 3286 | . 2 ⊢ (𝜑 → (𝑋 = 0 ↔ 1 = 0 )) |
26 | 25 | necon3bid 3057 | 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 3013 ∃wrex 3136 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 .rcmulr 16554 0gc0g 16701 1rcur 19180 Ringcrg 19226 |
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 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
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 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-plusg 16566 df-0g 16703 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-grp 18044 df-mgp 19169 df-ur 19181 df-ring 19228 |
This theorem is referenced by: (None) |
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