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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 6659 | . . . . . 6 ⊢ 𝐵 ∈ V |
3 | hashv01gt1 13701 | . . . . . 6 ⊢ (𝐵 ∈ V → ((♯‘𝐵) = 0 ∨ (♯‘𝐵) = 1 ∨ 1 < (♯‘𝐵))) | |
4 | 2, 3 | ax-mp 5 | . . . . 5 ⊢ ((♯‘𝐵) = 0 ∨ (♯‘𝐵) = 1 ∨ 1 < (♯‘𝐵)) |
5 | hasheq0 13720 | . . . . . . . . 9 ⊢ (𝐵 ∈ V → ((♯‘𝐵) = 0 ↔ 𝐵 = ∅)) | |
6 | 2, 5 | ax-mp 5 | . . . . . . . 8 ⊢ ((♯‘𝐵) = 0 ↔ 𝐵 = ∅) |
7 | ne0i 4250 | . . . . . . . . 9 ⊢ ( 0 ∈ 𝐵 → 𝐵 ≠ ∅) | |
8 | eqneqall 2998 | . . . . . . . . 9 ⊢ (𝐵 = ∅ → (𝐵 ≠ ∅ → ((♯‘𝐵) ≠ 1 → 0 ≠ 1 ))) | |
9 | 7, 8 | syl5com 31 | . . . . . . . 8 ⊢ ( 0 ∈ 𝐵 → (𝐵 = ∅ → ((♯‘𝐵) ≠ 1 → 0 ≠ 1 ))) |
10 | 6, 9 | syl5bi 245 | . . . . . . 7 ⊢ ( 0 ∈ 𝐵 → ((♯‘𝐵) = 0 → ((♯‘𝐵) ≠ 1 → 0 ≠ 1 ))) |
11 | 0ring.0 | . . . . . . . 8 ⊢ 0 = (0g‘𝑅) | |
12 | 1, 11 | ring0cl 19315 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → 0 ∈ 𝐵) |
13 | 10, 12 | syl11 33 | . . . . . 6 ⊢ ((♯‘𝐵) = 0 → (𝑅 ∈ Ring → ((♯‘𝐵) ≠ 1 → 0 ≠ 1 ))) |
14 | eqneqall 2998 | . . . . . . 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 19337 | . . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 1 < (♯‘𝐵)) → 1 ≠ 0 ) |
18 | 17 | necomd 3042 | . . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 1 < (♯‘𝐵)) → 0 ≠ 1 ) |
19 | 18 | ex 416 | . . . . . . . 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 1424 | . . . . 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 3011 | . . 3 ⊢ (𝑅 ∈ Ring → ( 0 = 1 → (♯‘𝐵) = 1)) |
25 | 24 | imp 410 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 0 = 1 ) → (♯‘𝐵) = 1) |
26 | 1, 11 | 0ring 20036 | . 2 ⊢ ((𝑅 ∈ Ring ∧ (♯‘𝐵) = 1) → 𝐵 = { 0 }) |
27 | 25, 26 | syldan 594 | 1 ⊢ ((𝑅 ∈ Ring ∧ 0 = 1 ) → 𝐵 = { 0 }) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∨ w3o 1083 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 Vcvv 3441 ∅c0 4243 {csn 4525 class class class wbr 5030 ‘cfv 6324 0cc0 10526 1c1 10527 < clt 10664 ♯chash 13686 Basecbs 16475 0gc0g 16705 1rcur 19244 Ringcrg 19290 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-n0 11886 df-xnn0 11956 df-z 11970 df-uz 12232 df-fz 12886 df-hash 13687 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-plusg 16570 df-0g 16707 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-grp 18098 df-minusg 18099 df-mgp 19233 df-ur 19245 df-ring 19292 |
This theorem is referenced by: 0ring01eqbi 20039 zarcmplem 31234 ldepspr 44882 |
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