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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 6684 | . . . . . 6 ⊢ 𝐵 ∈ V |
3 | hashv01gt1 13706 | . . . . . 6 ⊢ (𝐵 ∈ V → ((♯‘𝐵) = 0 ∨ (♯‘𝐵) = 1 ∨ 1 < (♯‘𝐵))) | |
4 | 2, 3 | ax-mp 5 | . . . . 5 ⊢ ((♯‘𝐵) = 0 ∨ (♯‘𝐵) = 1 ∨ 1 < (♯‘𝐵)) |
5 | hasheq0 13725 | . . . . . . . . 9 ⊢ (𝐵 ∈ V → ((♯‘𝐵) = 0 ↔ 𝐵 = ∅)) | |
6 | 2, 5 | ax-mp 5 | . . . . . . . 8 ⊢ ((♯‘𝐵) = 0 ↔ 𝐵 = ∅) |
7 | ne0i 4300 | . . . . . . . . 9 ⊢ ( 0 ∈ 𝐵 → 𝐵 ≠ ∅) | |
8 | eqneqall 3027 | . . . . . . . . 9 ⊢ (𝐵 = ∅ → (𝐵 ≠ ∅ → ((♯‘𝐵) ≠ 1 → 0 ≠ 1 ))) | |
9 | 7, 8 | syl5com 31 | . . . . . . . 8 ⊢ ( 0 ∈ 𝐵 → (𝐵 = ∅ → ((♯‘𝐵) ≠ 1 → 0 ≠ 1 ))) |
10 | 6, 9 | syl5bi 244 | . . . . . . 7 ⊢ ( 0 ∈ 𝐵 → ((♯‘𝐵) = 0 → ((♯‘𝐵) ≠ 1 → 0 ≠ 1 ))) |
11 | 0ring.0 | . . . . . . . 8 ⊢ 0 = (0g‘𝑅) | |
12 | 1, 11 | ring0cl 19319 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → 0 ∈ 𝐵) |
13 | 10, 12 | syl11 33 | . . . . . 6 ⊢ ((♯‘𝐵) = 0 → (𝑅 ∈ Ring → ((♯‘𝐵) ≠ 1 → 0 ≠ 1 ))) |
14 | eqneqall 3027 | . . . . . . 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 19341 | . . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 1 < (♯‘𝐵)) → 1 ≠ 0 ) |
18 | 17 | necomd 3071 | . . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 1 < (♯‘𝐵)) → 0 ≠ 1 ) |
19 | 18 | ex 415 | . . . . . . . 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 1423 | . . . . 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 3040 | . . 3 ⊢ (𝑅 ∈ Ring → ( 0 = 1 → (♯‘𝐵) = 1)) |
25 | 24 | imp 409 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 0 = 1 ) → (♯‘𝐵) = 1) |
26 | 1, 11 | 0ring 20043 | . 2 ⊢ ((𝑅 ∈ Ring ∧ (♯‘𝐵) = 1) → 𝐵 = { 0 }) |
27 | 25, 26 | syldan 593 | 1 ⊢ ((𝑅 ∈ Ring ∧ 0 = 1 ) → 𝐵 = { 0 }) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∨ w3o 1082 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 Vcvv 3494 ∅c0 4291 {csn 4567 class class class wbr 5066 ‘cfv 6355 0cc0 10537 1c1 10538 < clt 10675 ♯chash 13691 Basecbs 16483 0gc0g 16713 1rcur 19251 Ringcrg 19297 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-n0 11899 df-xnn0 11969 df-z 11983 df-uz 12245 df-fz 12894 df-hash 13692 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-plusg 16578 df-0g 16715 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-grp 18106 df-minusg 18107 df-mgp 19240 df-ur 19252 df-ring 19299 |
This theorem is referenced by: 0ring01eqbi 20046 ldepspr 44548 |
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