![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > 0ringdif | Structured version Visualization version GIF version |
Description: A zero ring is a ring which is not a nonzero ring. (Contributed by AV, 17-Apr-2020.) |
Ref | Expression |
---|---|
0ring.b | ⊢ 𝐵 = (Base‘𝑅) |
0ring.0 | ⊢ 0 = (0g‘𝑅) |
Ref | Expression |
---|---|
0ringdif | ⊢ (𝑅 ∈ (Ring ∖ NzRing) ↔ (𝑅 ∈ Ring ∧ 𝐵 = { 0 })) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldif 3953 | . 2 ⊢ (𝑅 ∈ (Ring ∖ NzRing) ↔ (𝑅 ∈ Ring ∧ ¬ 𝑅 ∈ NzRing)) | |
2 | 0ring.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝑅) | |
3 | 2 | a1i 11 | . . . . 5 ⊢ (𝑅 ∈ Ring → 𝐵 = (Base‘𝑅)) |
4 | 3 | fveqeq2d 6893 | . . . 4 ⊢ (𝑅 ∈ Ring → ((♯‘𝐵) = 1 ↔ (♯‘(Base‘𝑅)) = 1)) |
5 | 0ring.0 | . . . . . . 7 ⊢ 0 = (0g‘𝑅) | |
6 | 2, 5 | 0ring 20426 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ (♯‘𝐵) = 1) → 𝐵 = { 0 }) |
7 | 6 | ex 412 | . . . . 5 ⊢ (𝑅 ∈ Ring → ((♯‘𝐵) = 1 → 𝐵 = { 0 })) |
8 | fveq2 6885 | . . . . . 6 ⊢ (𝐵 = { 0 } → (♯‘𝐵) = (♯‘{ 0 })) | |
9 | 5 | fvexi 6899 | . . . . . . 7 ⊢ 0 ∈ V |
10 | hashsng 14334 | . . . . . . 7 ⊢ ( 0 ∈ V → (♯‘{ 0 }) = 1) | |
11 | 9, 10 | ax-mp 5 | . . . . . 6 ⊢ (♯‘{ 0 }) = 1 |
12 | 8, 11 | eqtrdi 2782 | . . . . 5 ⊢ (𝐵 = { 0 } → (♯‘𝐵) = 1) |
13 | 7, 12 | impbid1 224 | . . . 4 ⊢ (𝑅 ∈ Ring → ((♯‘𝐵) = 1 ↔ 𝐵 = { 0 })) |
14 | 0ringnnzr 20425 | . . . 4 ⊢ (𝑅 ∈ Ring → ((♯‘(Base‘𝑅)) = 1 ↔ ¬ 𝑅 ∈ NzRing)) | |
15 | 4, 13, 14 | 3bitr3rd 310 | . . 3 ⊢ (𝑅 ∈ Ring → (¬ 𝑅 ∈ NzRing ↔ 𝐵 = { 0 })) |
16 | 15 | pm5.32i 574 | . 2 ⊢ ((𝑅 ∈ Ring ∧ ¬ 𝑅 ∈ NzRing) ↔ (𝑅 ∈ Ring ∧ 𝐵 = { 0 })) |
17 | 1, 16 | bitri 275 | 1 ⊢ (𝑅 ∈ (Ring ∖ NzRing) ↔ (𝑅 ∈ Ring ∧ 𝐵 = { 0 })) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 Vcvv 3468 ∖ cdif 3940 {csn 4623 ‘cfv 6537 1c1 11113 ♯chash 14295 Basecbs 17153 0gc0g 17394 Ringcrg 20138 NzRingcnzr 20414 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 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 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-oadd 8471 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-dju 9898 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-n0 12477 df-xnn0 12549 df-z 12563 df-uz 12827 df-fz 13491 df-hash 14296 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-plusg 17219 df-0g 17396 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-grp 18866 df-minusg 18867 df-cmn 19702 df-abl 19703 df-mgp 20040 df-rng 20058 df-ur 20087 df-ring 20140 df-nzr 20415 |
This theorem is referenced by: 0ringbas 20428 |
Copyright terms: Public domain | W3C validator |