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Mirrors > Home > MPE Home > Th. List > ringinvnz1ne0 | Structured version Visualization version GIF version |
Description: In a unital 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 7365 | . . . . 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 20012 | . . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑎 ∈ 𝐵) → (𝑎 · 0 ) = 0 ) |
7 | 2, 6 | sylan 580 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝑎 · 0 ) = 0 ) |
8 | eqeq12 2753 | . . . . . . . 8 ⊢ (((𝑎 · 𝑋) = 1 ∧ (𝑎 · 0 ) = 0 ) → ((𝑎 · 𝑋) = (𝑎 · 0 ) ↔ 1 = 0 )) | |
9 | 8 | biimpd 228 | . . . . . . 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 7365 | . . . . 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 19993 | . . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋 · 1 ) = 𝑋) |
17 | 3, 4, 5 | ringrz 20012 | . . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋 · 0 ) = 0 ) |
18 | 16, 17 | eqeq12d 2752 | . . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → ((𝑋 · 1 ) = (𝑋 · 0 ) ↔ 𝑋 = 0 )) |
19 | 18 | biimpd 228 | . . . . . . 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 724 | . . . . 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 211 | . . 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 3159 | . 2 ⊢ (𝜑 → (𝑋 = 0 ↔ 1 = 0 )) |
26 | 25 | necon3bid 2988 | 1 ⊢ (𝜑 → (𝑋 ≠ 0 ↔ 1 ≠ 0 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 ∃wrex 3073 ‘cfv 6496 (class class class)co 7357 Basecbs 17083 .rcmulr 17134 0gc0g 17321 1rcur 19913 Ringcrg 19964 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-er 8648 df-en 8884 df-dom 8885 df-sdom 8886 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-nn 12154 df-2 12216 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-plusg 17146 df-0g 17323 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-grp 18751 df-mgp 19897 df-ur 19914 df-ring 19966 |
This theorem is referenced by: (None) |
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