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Mirrors > Home > MPE Home > Th. List > 0unit | Structured version Visualization version GIF version |
Description: The additive identity is a unit if and only if 1 = 0, i.e. we are in the zero ring. (Contributed by Mario Carneiro, 4-Dec-2014.) |
Ref | Expression |
---|---|
0unit.1 | ⊢ 𝑈 = (Unit‘𝑅) |
0unit.2 | ⊢ 0 = (0g‘𝑅) |
0unit.3 | ⊢ 1 = (1r‘𝑅) |
Ref | Expression |
---|---|
0unit | ⊢ (𝑅 ∈ Ring → ( 0 ∈ 𝑈 ↔ 1 = 0 )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0unit.1 | . . . 4 ⊢ 𝑈 = (Unit‘𝑅) | |
2 | eqid 2735 | . . . 4 ⊢ (invr‘𝑅) = (invr‘𝑅) | |
3 | eqid 2735 | . . . 4 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
4 | 0unit.3 | . . . 4 ⊢ 1 = (1r‘𝑅) | |
5 | 1, 2, 3, 4 | unitrinv 20411 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 0 ∈ 𝑈) → ( 0 (.r‘𝑅)((invr‘𝑅)‘ 0 )) = 1 ) |
6 | eqid 2735 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
7 | 1, 2, 6 | ringinvcl 20409 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 0 ∈ 𝑈) → ((invr‘𝑅)‘ 0 ) ∈ (Base‘𝑅)) |
8 | 0unit.2 | . . . . 5 ⊢ 0 = (0g‘𝑅) | |
9 | 6, 3, 8 | ringlz 20307 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ ((invr‘𝑅)‘ 0 ) ∈ (Base‘𝑅)) → ( 0 (.r‘𝑅)((invr‘𝑅)‘ 0 )) = 0 ) |
10 | 7, 9 | syldan 591 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 0 ∈ 𝑈) → ( 0 (.r‘𝑅)((invr‘𝑅)‘ 0 )) = 0 ) |
11 | 5, 10 | eqtr3d 2777 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 0 ∈ 𝑈) → 1 = 0 ) |
12 | simpr 484 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 1 = 0 ) → 1 = 0 ) | |
13 | 1, 4 | 1unit 20391 | . . . 4 ⊢ (𝑅 ∈ Ring → 1 ∈ 𝑈) |
14 | 13 | adantr 480 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 1 = 0 ) → 1 ∈ 𝑈) |
15 | 12, 14 | eqeltrrd 2840 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 1 = 0 ) → 0 ∈ 𝑈) |
16 | 11, 15 | impbida 801 | 1 ⊢ (𝑅 ∈ Ring → ( 0 ∈ 𝑈 ↔ 1 = 0 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 .rcmulr 17299 0gc0g 17486 1rcur 20199 Ringcrg 20251 Unitcui 20372 invrcinvr 20404 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-tpos 8250 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-0g 17488 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-grp 18967 df-minusg 18968 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-oppr 20351 df-dvdsr 20374 df-unit 20375 df-invr 20405 |
This theorem is referenced by: nzrunit 20541 fidomndrng 20791 imadrhmcl 20815 gzrngunitlem 21468 unitnz 33229 isdrng4 33279 |
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