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Mirrors > Home > MPE Home > Th. List > 0ring01eq | Structured version Visualization version GIF version |
Description: In a ring with only one element, i.e. a zero ring, the zero and the identity element are the same. (Contributed by AV, 14-Apr-2019.) |
Ref | Expression |
---|---|
0ring.b | ⊢ 𝐵 = (Base‘𝑅) |
0ring.0 | ⊢ 0 = (0g‘𝑅) |
0ring01eq.1 | ⊢ 1 = (1r‘𝑅) |
Ref | Expression |
---|---|
0ring01eq | ⊢ ((𝑅 ∈ Ring ∧ (♯‘𝐵) = 1) → 0 = 1 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ring.b | . . 3 ⊢ 𝐵 = (Base‘𝑅) | |
2 | 0ring.0 | . . 3 ⊢ 0 = (0g‘𝑅) | |
3 | 1, 2 | 0ring 20253 | . 2 ⊢ ((𝑅 ∈ Ring ∧ (♯‘𝐵) = 1) → 𝐵 = { 0 }) |
4 | 0ring01eq.1 | . . . . 5 ⊢ 1 = (1r‘𝑅) | |
5 | 1, 4 | ringidcl 20040 | . . . 4 ⊢ (𝑅 ∈ Ring → 1 ∈ 𝐵) |
6 | eleq2 2821 | . . . . 5 ⊢ (𝐵 = { 0 } → ( 1 ∈ 𝐵 ↔ 1 ∈ { 0 })) | |
7 | elsni 4639 | . . . . . 6 ⊢ ( 1 ∈ { 0 } → 1 = 0 ) | |
8 | 7 | eqcomd 2737 | . . . . 5 ⊢ ( 1 ∈ { 0 } → 0 = 1 ) |
9 | 6, 8 | syl6bi 252 | . . . 4 ⊢ (𝐵 = { 0 } → ( 1 ∈ 𝐵 → 0 = 1 )) |
10 | 5, 9 | syl5com 31 | . . 3 ⊢ (𝑅 ∈ Ring → (𝐵 = { 0 } → 0 = 1 )) |
11 | 10 | adantr 481 | . 2 ⊢ ((𝑅 ∈ Ring ∧ (♯‘𝐵) = 1) → (𝐵 = { 0 } → 0 = 1 )) |
12 | 3, 11 | mpd 15 | 1 ⊢ ((𝑅 ∈ Ring ∧ (♯‘𝐵) = 1) → 0 = 1 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 {csn 4622 ‘cfv 6532 1c1 11093 ♯chash 14272 Basecbs 17126 0gc0g 17367 1rcur 19963 Ringcrg 20014 |
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 2702 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 ax-cnex 11148 ax-resscn 11149 ax-1cn 11150 ax-icn 11151 ax-addcl 11152 ax-addrcl 11153 ax-mulcl 11154 ax-mulrcl 11155 ax-mulcom 11156 ax-addass 11157 ax-mulass 11158 ax-distr 11159 ax-i2m1 11160 ax-1ne0 11161 ax-1rid 11162 ax-rnegex 11163 ax-rrecex 11164 ax-cnre 11165 ax-pre-lttri 11166 ax-pre-lttrn 11167 ax-pre-ltadd 11168 ax-pre-mulgt0 11169 |
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 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6289 df-ord 6356 df-on 6357 df-lim 6358 df-suc 6359 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-riota 7349 df-ov 7396 df-oprab 7397 df-mpo 7398 df-om 7839 df-1st 7957 df-2nd 7958 df-frecs 8248 df-wrecs 8279 df-recs 8353 df-rdg 8392 df-1o 8448 df-er 8686 df-en 8923 df-dom 8924 df-sdom 8925 df-fin 8926 df-card 9916 df-pnf 11232 df-mnf 11233 df-xr 11234 df-ltxr 11235 df-le 11236 df-sub 11428 df-neg 11429 df-nn 12195 df-2 12257 df-n0 12455 df-z 12541 df-uz 12805 df-fz 13467 df-hash 14273 df-sets 17079 df-slot 17097 df-ndx 17109 df-base 17127 df-plusg 17192 df-0g 17369 df-mgm 18543 df-sgrp 18592 df-mnd 18603 df-grp 18797 df-mgp 19947 df-ur 19964 df-ring 20016 |
This theorem is referenced by: 0ring01eqbi 20257 0ringmon1p 32481 lmod0rng 46414 0ring1eq0 46418 |
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