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Mirrors > Home > MPE Home > Th. List > 01eq0ring | Structured version Visualization version GIF version |
Description: If the zero and the identity element of a ring are the same, the ring is the zero ring. (Contributed by AV, 16-Apr-2019.) |
Ref | Expression |
---|---|
0ring.b | ⊢ 𝐵 = (Base‘𝑅) |
0ring.0 | ⊢ 0 = (0g‘𝑅) |
0ring01eq.1 | ⊢ 1 = (1r‘𝑅) |
Ref | Expression |
---|---|
01eq0ring | ⊢ ((𝑅 ∈ Ring ∧ 0 = 1 ) → 𝐵 = { 0 }) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ring.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝑅) | |
2 | 1 | fvexi 6861 | . . . . . 6 ⊢ 𝐵 ∈ V |
3 | hashv01gt1 14252 | . . . . . 6 ⊢ (𝐵 ∈ V → ((♯‘𝐵) = 0 ∨ (♯‘𝐵) = 1 ∨ 1 < (♯‘𝐵))) | |
4 | 2, 3 | ax-mp 5 | . . . . 5 ⊢ ((♯‘𝐵) = 0 ∨ (♯‘𝐵) = 1 ∨ 1 < (♯‘𝐵)) |
5 | hasheq0 14270 | . . . . . . . . 9 ⊢ (𝐵 ∈ V → ((♯‘𝐵) = 0 ↔ 𝐵 = ∅)) | |
6 | 2, 5 | ax-mp 5 | . . . . . . . 8 ⊢ ((♯‘𝐵) = 0 ↔ 𝐵 = ∅) |
7 | ne0i 4299 | . . . . . . . . 9 ⊢ ( 0 ∈ 𝐵 → 𝐵 ≠ ∅) | |
8 | eqneqall 2955 | . . . . . . . . 9 ⊢ (𝐵 = ∅ → (𝐵 ≠ ∅ → ((♯‘𝐵) ≠ 1 → 0 ≠ 1 ))) | |
9 | 7, 8 | syl5com 31 | . . . . . . . 8 ⊢ ( 0 ∈ 𝐵 → (𝐵 = ∅ → ((♯‘𝐵) ≠ 1 → 0 ≠ 1 ))) |
10 | 6, 9 | biimtrid 241 | . . . . . . 7 ⊢ ( 0 ∈ 𝐵 → ((♯‘𝐵) = 0 → ((♯‘𝐵) ≠ 1 → 0 ≠ 1 ))) |
11 | 0ring.0 | . . . . . . . 8 ⊢ 0 = (0g‘𝑅) | |
12 | 1, 11 | ring0cl 19997 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → 0 ∈ 𝐵) |
13 | 10, 12 | syl11 33 | . . . . . 6 ⊢ ((♯‘𝐵) = 0 → (𝑅 ∈ Ring → ((♯‘𝐵) ≠ 1 → 0 ≠ 1 ))) |
14 | eqneqall 2955 | . . . . . . 7 ⊢ ((♯‘𝐵) = 1 → ((♯‘𝐵) ≠ 1 → 0 ≠ 1 )) | |
15 | 14 | a1d 25 | . . . . . 6 ⊢ ((♯‘𝐵) = 1 → (𝑅 ∈ Ring → ((♯‘𝐵) ≠ 1 → 0 ≠ 1 ))) |
16 | 0ring01eq.1 | . . . . . . . . . . 11 ⊢ 1 = (1r‘𝑅) | |
17 | 1, 16, 11 | ring1ne0 20022 | . . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 1 < (♯‘𝐵)) → 1 ≠ 0 ) |
18 | 17 | necomd 3000 | . . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 1 < (♯‘𝐵)) → 0 ≠ 1 ) |
19 | 18 | ex 414 | . . . . . . . 8 ⊢ (𝑅 ∈ Ring → (1 < (♯‘𝐵) → 0 ≠ 1 )) |
20 | 19 | a1i 11 | . . . . . . 7 ⊢ ((♯‘𝐵) ≠ 1 → (𝑅 ∈ Ring → (1 < (♯‘𝐵) → 0 ≠ 1 ))) |
21 | 20 | com13 88 | . . . . . 6 ⊢ (1 < (♯‘𝐵) → (𝑅 ∈ Ring → ((♯‘𝐵) ≠ 1 → 0 ≠ 1 ))) |
22 | 13, 15, 21 | 3jaoi 1428 | . . . . 5 ⊢ (((♯‘𝐵) = 0 ∨ (♯‘𝐵) = 1 ∨ 1 < (♯‘𝐵)) → (𝑅 ∈ Ring → ((♯‘𝐵) ≠ 1 → 0 ≠ 1 ))) |
23 | 4, 22 | ax-mp 5 | . . . 4 ⊢ (𝑅 ∈ Ring → ((♯‘𝐵) ≠ 1 → 0 ≠ 1 )) |
24 | 23 | necon4d 2968 | . . 3 ⊢ (𝑅 ∈ Ring → ( 0 = 1 → (♯‘𝐵) = 1)) |
25 | 24 | imp 408 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 0 = 1 ) → (♯‘𝐵) = 1) |
26 | 1, 11 | 0ring 20756 | . 2 ⊢ ((𝑅 ∈ Ring ∧ (♯‘𝐵) = 1) → 𝐵 = { 0 }) |
27 | 25, 26 | syldan 592 | 1 ⊢ ((𝑅 ∈ Ring ∧ 0 = 1 ) → 𝐵 = { 0 }) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∨ w3o 1087 = wceq 1542 ∈ wcel 2107 ≠ wne 2944 Vcvv 3448 ∅c0 4287 {csn 4591 class class class wbr 5110 ‘cfv 6501 0cc0 11058 1c1 11059 < clt 11196 ♯chash 14237 Basecbs 17090 0gc0g 17328 1rcur 19920 Ringcrg 19971 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-1st 7926 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-card 9882 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-nn 12161 df-2 12223 df-n0 12421 df-xnn0 12493 df-z 12507 df-uz 12771 df-fz 13432 df-hash 14238 df-sets 17043 df-slot 17061 df-ndx 17073 df-base 17091 df-plusg 17153 df-0g 17330 df-mgm 18504 df-sgrp 18553 df-mnd 18564 df-grp 18758 df-minusg 18759 df-mgp 19904 df-ur 19921 df-ring 19973 |
This theorem is referenced by: 0ring01eqbi 20759 zarcmplem 32502 imadrhmcl 40745 ldepspr 46628 |
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