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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 6687 | . . 3 ⊢ 𝐵 ∈ V |
3 | hashen1 13734 | . . 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 20047 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (♯‘𝐵) = 1) → 0 = 1 ) |
8 | 7 | eqcomd 2830 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ (♯‘𝐵) = 1) → 1 = 0 ) |
9 | 8 | ex 415 | . . 3 ⊢ (𝑅 ∈ Ring → ((♯‘𝐵) = 1 → 1 = 0 )) |
10 | eqcom 2831 | . . . 4 ⊢ ( 1 = 0 ↔ 0 = 1 ) | |
11 | 1, 5, 6 | 01eq0ring 20048 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 0 = 1 ) → 𝐵 = { 0 }) |
12 | fveq2 6673 | . . . . . . 7 ⊢ (𝐵 = { 0 } → (♯‘𝐵) = (♯‘{ 0 })) | |
13 | 5 | fvexi 6687 | . . . . . . . 8 ⊢ 0 ∈ V |
14 | hashsng 13733 | . . . . . . . 8 ⊢ ( 0 ∈ V → (♯‘{ 0 }) = 1) | |
15 | 13, 14 | mp1i 13 | . . . . . . 7 ⊢ (𝐵 = { 0 } → (♯‘{ 0 }) = 1) |
16 | 12, 15 | eqtrd 2859 | . . . . . 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 1536 ∈ wcel 2113 Vcvv 3497 {csn 4570 class class class wbr 5069 ‘cfv 6358 1oc1o 8098 ≈ cen 8509 1c1 10541 ♯chash 13693 Basecbs 16486 0gc0g 16716 1rcur 19254 Ringcrg 19300 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-card 9371 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 df-2 11703 df-n0 11901 df-xnn0 11971 df-z 11985 df-uz 12247 df-fz 12896 df-hash 13694 df-ndx 16489 df-slot 16490 df-base 16492 df-sets 16493 df-plusg 16581 df-0g 16718 df-mgm 17855 df-sgrp 17904 df-mnd 17915 df-grp 18109 df-minusg 18110 df-mgp 19243 df-ur 19255 df-ring 19302 |
This theorem is referenced by: (None) |
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