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Mirrors > Home > MPE Home > Th. List > rng1nnzr | Structured version Visualization version GIF version |
Description: The (smallest) structure representing a zero ring is not a nonzero ring. (Contributed by AV, 29-Apr-2019.) |
Ref | Expression |
---|---|
rng1nnzr.m | ⊢ 𝑀 = {〈(Base‘ndx), {𝑍}〉, 〈(+g‘ndx), {〈〈𝑍, 𝑍〉, 𝑍〉}〉, 〈(.r‘ndx), {〈〈𝑍, 𝑍〉, 𝑍〉}〉} |
Ref | Expression |
---|---|
rng1nnzr | ⊢ (𝑍 ∈ 𝑉 → 𝑀 ∉ NzRing) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snex 5332 | . . . . . . 7 ⊢ {𝑍} ∈ V | |
2 | rng1nnzr.m | . . . . . . . 8 ⊢ 𝑀 = {〈(Base‘ndx), {𝑍}〉, 〈(+g‘ndx), {〈〈𝑍, 𝑍〉, 𝑍〉}〉, 〈(.r‘ndx), {〈〈𝑍, 𝑍〉, 𝑍〉}〉} | |
3 | 2 | rngbase 16620 | . . . . . . 7 ⊢ ({𝑍} ∈ V → {𝑍} = (Base‘𝑀)) |
4 | 1, 3 | mp1i 13 | . . . . . 6 ⊢ (𝑍 ∈ 𝑉 → {𝑍} = (Base‘𝑀)) |
5 | 4 | eqcomd 2827 | . . . . 5 ⊢ (𝑍 ∈ 𝑉 → (Base‘𝑀) = {𝑍}) |
6 | 5 | fveq2d 6674 | . . . 4 ⊢ (𝑍 ∈ 𝑉 → (♯‘(Base‘𝑀)) = (♯‘{𝑍})) |
7 | hashsng 13731 | . . . 4 ⊢ (𝑍 ∈ 𝑉 → (♯‘{𝑍}) = 1) | |
8 | 6, 7 | eqtrd 2856 | . . 3 ⊢ (𝑍 ∈ 𝑉 → (♯‘(Base‘𝑀)) = 1) |
9 | 2 | ring1 19352 | . . . 4 ⊢ (𝑍 ∈ 𝑉 → 𝑀 ∈ Ring) |
10 | 0ringnnzr 20042 | . . . 4 ⊢ (𝑀 ∈ Ring → ((♯‘(Base‘𝑀)) = 1 ↔ ¬ 𝑀 ∈ NzRing)) | |
11 | 9, 10 | syl 17 | . . 3 ⊢ (𝑍 ∈ 𝑉 → ((♯‘(Base‘𝑀)) = 1 ↔ ¬ 𝑀 ∈ NzRing)) |
12 | 8, 11 | mpbid 234 | . 2 ⊢ (𝑍 ∈ 𝑉 → ¬ 𝑀 ∈ NzRing) |
13 | df-nel 3124 | . 2 ⊢ (𝑀 ∉ NzRing ↔ ¬ 𝑀 ∈ NzRing) | |
14 | 12, 13 | sylibr 236 | 1 ⊢ (𝑍 ∈ 𝑉 → 𝑀 ∉ NzRing) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 = wceq 1537 ∈ wcel 2114 ∉ wnel 3123 Vcvv 3494 {csn 4567 {ctp 4571 〈cop 4573 ‘cfv 6355 1c1 10538 ♯chash 13691 ndxcnx 16480 Basecbs 16483 +gcplusg 16565 .rcmulr 16566 Ringcrg 19297 NzRingcnzr 20030 |
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-rep 5190 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-oadd 8106 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-dju 9330 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-3 11702 df-n0 11899 df-xnn0 11969 df-z 11983 df-uz 12245 df-fz 12894 df-hash 13692 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-plusg 16578 df-mulr 16579 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 df-nzr 20031 |
This theorem is referenced by: rng1nfld 20051 lmod1zrnlvec 44569 |
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