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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 5436 | . . . . . . 7 ⊢ {𝑍} ∈ V | |
| 2 | rng1nnzr.m | . . . . . . . 8 ⊢ 𝑀 = {〈(Base‘ndx), {𝑍}〉, 〈(+g‘ndx), {〈〈𝑍, 𝑍〉, 𝑍〉}〉, 〈(.r‘ndx), {〈〈𝑍, 𝑍〉, 𝑍〉}〉} | |
| 3 | 2 | rngbase 17343 | . . . . . . 7 ⊢ ({𝑍} ∈ V → {𝑍} = (Base‘𝑀)) | 
| 4 | 1, 3 | mp1i 13 | . . . . . 6 ⊢ (𝑍 ∈ 𝑉 → {𝑍} = (Base‘𝑀)) | 
| 5 | 4 | eqcomd 2743 | . . . . 5 ⊢ (𝑍 ∈ 𝑉 → (Base‘𝑀) = {𝑍}) | 
| 6 | 5 | fveq2d 6910 | . . . 4 ⊢ (𝑍 ∈ 𝑉 → (♯‘(Base‘𝑀)) = (♯‘{𝑍})) | 
| 7 | hashsng 14408 | . . . 4 ⊢ (𝑍 ∈ 𝑉 → (♯‘{𝑍}) = 1) | |
| 8 | 6, 7 | eqtrd 2777 | . . 3 ⊢ (𝑍 ∈ 𝑉 → (♯‘(Base‘𝑀)) = 1) | 
| 9 | 2 | ring1 20307 | . . . 4 ⊢ (𝑍 ∈ 𝑉 → 𝑀 ∈ Ring) | 
| 10 | 0ringnnzr 20525 | . . . 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 232 | . 2 ⊢ (𝑍 ∈ 𝑉 → ¬ 𝑀 ∈ NzRing) | 
| 13 | df-nel 3047 | . 2 ⊢ (𝑀 ∉ NzRing ↔ ¬ 𝑀 ∈ NzRing) | |
| 14 | 12, 13 | sylibr 234 | 1 ⊢ (𝑍 ∈ 𝑉 → 𝑀 ∉ NzRing) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 = wceq 1540 ∈ wcel 2108 ∉ wnel 3046 Vcvv 3480 {csn 4626 {ctp 4630 〈cop 4632 ‘cfv 6561 1c1 11156 ♯chash 14369 ndxcnx 17230 Basecbs 17247 +gcplusg 17297 .rcmulr 17298 Ringcrg 20230 NzRingcnzr 20512 | 
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-oadd 8510 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-dju 9941 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-xnn0 12600 df-z 12614 df-uz 12879 df-fz 13548 df-hash 14370 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-plusg 17310 df-mulr 17311 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-minusg 18955 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-nzr 20513 | 
| This theorem is referenced by: rng1nfld 20780 lmod1zrnlvec 48411 | 
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