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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 6857 | . . 3 ⊢ 𝐵 ∈ V |
3 | hashen1 14276 | . . 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 20757 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (♯‘𝐵) = 1) → 0 = 1 ) |
8 | 7 | eqcomd 2739 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ (♯‘𝐵) = 1) → 1 = 0 ) |
9 | 8 | ex 414 | . . 3 ⊢ (𝑅 ∈ Ring → ((♯‘𝐵) = 1 → 1 = 0 )) |
10 | eqcom 2740 | . . . 4 ⊢ ( 1 = 0 ↔ 0 = 1 ) | |
11 | 1, 5, 6 | 01eq0ring 20758 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 0 = 1 ) → 𝐵 = { 0 }) |
12 | fveq2 6843 | . . . . . . 7 ⊢ (𝐵 = { 0 } → (♯‘𝐵) = (♯‘{ 0 })) | |
13 | 5 | fvexi 6857 | . . . . . . . 8 ⊢ 0 ∈ V |
14 | hashsng 14275 | . . . . . . . 8 ⊢ ( 0 ∈ V → (♯‘{ 0 }) = 1) | |
15 | 13, 14 | mp1i 13 | . . . . . . 7 ⊢ (𝐵 = { 0 } → (♯‘{ 0 }) = 1) |
16 | 12, 15 | eqtrd 2773 | . . . . . 6 ⊢ (𝐵 = { 0 } → (♯‘𝐵) = 1) |
17 | 11, 16 | syl 17 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 0 = 1 ) → (♯‘𝐵) = 1) |
18 | 17 | ex 414 | . . . 4 ⊢ (𝑅 ∈ Ring → ( 0 = 1 → (♯‘𝐵) = 1)) |
19 | 10, 18 | biimtrid 241 | . . 3 ⊢ (𝑅 ∈ Ring → ( 1 = 0 → (♯‘𝐵) = 1)) |
20 | 9, 19 | impbid 211 | . 2 ⊢ (𝑅 ∈ Ring → ((♯‘𝐵) = 1 ↔ 1 = 0 )) |
21 | 4, 20 | bitr3d 281 | 1 ⊢ (𝑅 ∈ Ring → (𝐵 ≈ 1o ↔ 1 = 0 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 Vcvv 3444 {csn 4587 class class class wbr 5106 ‘cfv 6497 1oc1o 8406 ≈ cen 8883 1c1 11057 ♯chash 14236 Basecbs 17088 0gc0g 17326 1rcur 19918 Ringcrg 19969 |
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 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-card 9880 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-2 12221 df-n0 12419 df-xnn0 12491 df-z 12505 df-uz 12769 df-fz 13431 df-hash 14237 df-sets 17041 df-slot 17059 df-ndx 17071 df-base 17089 df-plusg 17151 df-0g 17328 df-mgm 18502 df-sgrp 18551 df-mnd 18562 df-grp 18756 df-minusg 18757 df-mgp 19902 df-ur 19919 df-ring 19971 |
This theorem is referenced by: (None) |
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