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Mirrors > Home > MPE Home > Th. List > ring1ne0 | Structured version Visualization version GIF version |
Description: If a ring has at least two elements, its one and zero are different. (Contributed by AV, 13-Apr-2019.) |
Ref | Expression |
---|---|
ring1ne0.b | ⊢ 𝐵 = (Base‘𝑅) |
ring1ne0.u | ⊢ 1 = (1r‘𝑅) |
ring1ne0.z | ⊢ 0 = (0g‘𝑅) |
Ref | Expression |
---|---|
ring1ne0 | ⊢ ((𝑅 ∈ Ring ∧ 1 < (#‘𝐵)) → 1 ≠ 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ring1ne0.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑅) | |
2 | fvex 6239 | . . . . 5 ⊢ (Base‘𝑅) ∈ V | |
3 | 1, 2 | eqeltri 2726 | . . . 4 ⊢ 𝐵 ∈ V |
4 | hashgt12el 13248 | . . . 4 ⊢ ((𝐵 ∈ V ∧ 1 < (#‘𝐵)) → ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 𝑥 ≠ 𝑦) | |
5 | 3, 4 | mpan 706 | . . 3 ⊢ (1 < (#‘𝐵) → ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 𝑥 ≠ 𝑦) |
6 | 5 | adantl 481 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 1 < (#‘𝐵)) → ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 𝑥 ≠ 𝑦) |
7 | ring1ne0.u | . . . . . . . 8 ⊢ 1 = (1r‘𝑅) | |
8 | ring1ne0.z | . . . . . . . 8 ⊢ 0 = (0g‘𝑅) | |
9 | 1, 7, 8 | ring1eq0 18636 | . . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ( 1 = 0 → 𝑥 = 𝑦)) |
10 | 9 | necon3d 2844 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 ≠ 𝑦 → 1 ≠ 0 )) |
11 | 10 | 3expib 1287 | . . . . 5 ⊢ (𝑅 ∈ Ring → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 ≠ 𝑦 → 1 ≠ 0 ))) |
12 | 11 | adantr 480 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 1 < (#‘𝐵)) → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 ≠ 𝑦 → 1 ≠ 0 ))) |
13 | 12 | com3l 89 | . . 3 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 ≠ 𝑦 → ((𝑅 ∈ Ring ∧ 1 < (#‘𝐵)) → 1 ≠ 0 ))) |
14 | 13 | rexlimivv 3065 | . 2 ⊢ (∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 𝑥 ≠ 𝑦 → ((𝑅 ∈ Ring ∧ 1 < (#‘𝐵)) → 1 ≠ 0 )) |
15 | 6, 14 | mpcom 38 | 1 ⊢ ((𝑅 ∈ Ring ∧ 1 < (#‘𝐵)) → 1 ≠ 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1054 = wceq 1523 ∈ wcel 2030 ≠ wne 2823 ∃wrex 2942 Vcvv 3231 class class class wbr 4685 ‘cfv 5926 1c1 9975 < clt 10112 #chash 13157 Basecbs 15904 0gc0g 16147 1rcur 18547 Ringcrg 18593 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-1st 7210 df-2nd 7211 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-card 8803 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-nn 11059 df-2 11117 df-n0 11331 df-xnn0 11402 df-z 11416 df-uz 11726 df-fz 12365 df-hash 13158 df-ndx 15907 df-slot 15908 df-base 15910 df-sets 15911 df-plusg 16001 df-0g 16149 df-mgm 17289 df-sgrp 17331 df-mnd 17342 df-grp 17472 df-minusg 17473 df-mgp 18536 df-ur 18548 df-ring 18595 |
This theorem is referenced by: isnzr2hash 19312 01eq0ring 19320 el0ldep 42580 |
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