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Mirrors > Home > MPE Home > Th. List > 0ring01eqbi | Structured version Visualization version GIF version |
Description: In a unital ring the zero equals the unity iff the ring is the zero ring. (Contributed by FL, 14-Feb-2010.) (Revised by AV, 23-Jan-2020.) |
Ref | Expression |
---|---|
0ring.b | ⊢ 𝐵 = (Base‘𝑅) |
0ring.0 | ⊢ 0 = (0g‘𝑅) |
0ring01eq.1 | ⊢ 1 = (1r‘𝑅) |
Ref | Expression |
---|---|
0ring01eqbi | ⊢ (𝑅 ∈ Ring → (𝐵 ≈ 1o ↔ 1 = 0 )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ring.b | . . . 4 ⊢ 𝐵 = (Base‘𝑅) | |
2 | 1 | fvexi 6683 | . . 3 ⊢ 𝐵 ∈ V |
3 | hashen1 13730 | . . 3 ⊢ (𝐵 ∈ V → ((♯‘𝐵) = 1 ↔ 𝐵 ≈ 1o)) | |
4 | 2, 3 | mp1i 13 | . 2 ⊢ (𝑅 ∈ Ring → ((♯‘𝐵) = 1 ↔ 𝐵 ≈ 1o)) |
5 | 0ring.0 | . . . . . 6 ⊢ 0 = (0g‘𝑅) | |
6 | 0ring01eq.1 | . . . . . 6 ⊢ 1 = (1r‘𝑅) | |
7 | 1, 5, 6 | 0ring01eq 20043 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (♯‘𝐵) = 1) → 0 = 1 ) |
8 | 7 | eqcomd 2827 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ (♯‘𝐵) = 1) → 1 = 0 ) |
9 | 8 | ex 415 | . . 3 ⊢ (𝑅 ∈ Ring → ((♯‘𝐵) = 1 → 1 = 0 )) |
10 | eqcom 2828 | . . . 4 ⊢ ( 1 = 0 ↔ 0 = 1 ) | |
11 | 1, 5, 6 | 01eq0ring 20044 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 0 = 1 ) → 𝐵 = { 0 }) |
12 | fveq2 6669 | . . . . . . 7 ⊢ (𝐵 = { 0 } → (♯‘𝐵) = (♯‘{ 0 })) | |
13 | 5 | fvexi 6683 | . . . . . . . 8 ⊢ 0 ∈ V |
14 | hashsng 13729 | . . . . . . . 8 ⊢ ( 0 ∈ V → (♯‘{ 0 }) = 1) | |
15 | 13, 14 | mp1i 13 | . . . . . . 7 ⊢ (𝐵 = { 0 } → (♯‘{ 0 }) = 1) |
16 | 12, 15 | eqtrd 2856 | . . . . . 6 ⊢ (𝐵 = { 0 } → (♯‘𝐵) = 1) |
17 | 11, 16 | syl 17 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 0 = 1 ) → (♯‘𝐵) = 1) |
18 | 17 | ex 415 | . . . 4 ⊢ (𝑅 ∈ Ring → ( 0 = 1 → (♯‘𝐵) = 1)) |
19 | 10, 18 | syl5bi 244 | . . 3 ⊢ (𝑅 ∈ Ring → ( 1 = 0 → (♯‘𝐵) = 1)) |
20 | 9, 19 | impbid 214 | . 2 ⊢ (𝑅 ∈ Ring → ((♯‘𝐵) = 1 ↔ 1 = 0 )) |
21 | 4, 20 | bitr3d 283 | 1 ⊢ (𝑅 ∈ Ring → (𝐵 ≈ 1o ↔ 1 = 0 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 Vcvv 3494 {csn 4566 class class class wbr 5065 ‘cfv 6354 1oc1o 8094 ≈ cen 8505 1c1 10537 ♯chash 13689 Basecbs 16482 0gc0g 16712 1rcur 19250 Ringcrg 19296 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-1st 7688 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-card 9367 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-nn 11638 df-2 11699 df-n0 11897 df-xnn0 11967 df-z 11981 df-uz 12243 df-fz 12892 df-hash 13690 df-ndx 16485 df-slot 16486 df-base 16488 df-sets 16489 df-plusg 16577 df-0g 16714 df-mgm 17851 df-sgrp 17900 df-mnd 17911 df-grp 18105 df-minusg 18106 df-mgp 19239 df-ur 19251 df-ring 19298 |
This theorem is referenced by: (None) |
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