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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > zrzeroorngc | Structured version Visualization version GIF version |
Description: The zero ring is a zero object in the category of non-unital rings. (Contributed by AV, 18-Apr-2020.) |
Ref | Expression |
---|---|
zrinitorngc.u | ⊢ (𝜑 → 𝑈 ∈ 𝑉) |
zrinitorngc.c | ⊢ 𝐶 = (RngCat‘𝑈) |
zrinitorngc.z | ⊢ (𝜑 → 𝑍 ∈ (Ring ∖ NzRing)) |
zrinitorngc.e | ⊢ (𝜑 → 𝑍 ∈ 𝑈) |
Ref | Expression |
---|---|
zrzeroorngc | ⊢ (𝜑 → 𝑍 ∈ (ZeroO‘𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zrinitorngc.u | . . 3 ⊢ (𝜑 → 𝑈 ∈ 𝑉) | |
2 | zrinitorngc.c | . . 3 ⊢ 𝐶 = (RngCat‘𝑈) | |
3 | zrinitorngc.z | . . 3 ⊢ (𝜑 → 𝑍 ∈ (Ring ∖ NzRing)) | |
4 | zrinitorngc.e | . . 3 ⊢ (𝜑 → 𝑍 ∈ 𝑈) | |
5 | 1, 2, 3, 4 | zrinitorngc 42525 | . 2 ⊢ (𝜑 → 𝑍 ∈ (InitO‘𝐶)) |
6 | 1, 2, 3, 4 | zrtermorngc 42526 | . 2 ⊢ (𝜑 → 𝑍 ∈ (TermO‘𝐶)) |
7 | eqid 2771 | . . 3 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
8 | eqid 2771 | . . 3 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
9 | 2 | rngccat 42503 | . . . 4 ⊢ (𝑈 ∈ 𝑉 → 𝐶 ∈ Cat) |
10 | 1, 9 | syl 17 | . . 3 ⊢ (𝜑 → 𝐶 ∈ Cat) |
11 | 3 | eldifad 3735 | . . . . . 6 ⊢ (𝜑 → 𝑍 ∈ Ring) |
12 | ringrng 42404 | . . . . . 6 ⊢ (𝑍 ∈ Ring → 𝑍 ∈ Rng) | |
13 | 11, 12 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑍 ∈ Rng) |
14 | 4, 13 | elind 3949 | . . . 4 ⊢ (𝜑 → 𝑍 ∈ (𝑈 ∩ Rng)) |
15 | 2, 7, 1 | rngcbas 42490 | . . . 4 ⊢ (𝜑 → (Base‘𝐶) = (𝑈 ∩ Rng)) |
16 | 14, 15 | eleqtrrd 2853 | . . 3 ⊢ (𝜑 → 𝑍 ∈ (Base‘𝐶)) |
17 | 7, 8, 10, 16 | iszeroo 16859 | . 2 ⊢ (𝜑 → (𝑍 ∈ (ZeroO‘𝐶) ↔ (𝑍 ∈ (InitO‘𝐶) ∧ 𝑍 ∈ (TermO‘𝐶)))) |
18 | 5, 6, 17 | mpbir2and 692 | 1 ⊢ (𝜑 → 𝑍 ∈ (ZeroO‘𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1631 ∈ wcel 2145 ∖ cdif 3720 ∩ cin 3722 ‘cfv 6030 Basecbs 16064 Hom chom 16160 Catccat 16532 InitOcinito 16845 TermOctermo 16846 ZeroOczeroo 16847 Ringcrg 18755 NzRingcnzr 19472 Rngcrng 42399 RngCatcrngc 42482 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4905 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035 ax-un 7100 ax-cnex 10198 ax-resscn 10199 ax-1cn 10200 ax-icn 10201 ax-addcl 10202 ax-addrcl 10203 ax-mulcl 10204 ax-mulrcl 10205 ax-mulcom 10206 ax-addass 10207 ax-mulass 10208 ax-distr 10209 ax-i2m1 10210 ax-1ne0 10211 ax-1rid 10212 ax-rnegex 10213 ax-rrecex 10214 ax-cnre 10215 ax-pre-lttri 10216 ax-pre-lttrn 10217 ax-pre-ltadd 10218 ax-pre-mulgt0 10219 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-fal 1637 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-tp 4322 df-op 4324 df-uni 4576 df-int 4613 df-iun 4657 df-br 4788 df-opab 4848 df-mpt 4865 df-tr 4888 df-id 5158 df-eprel 5163 df-po 5171 df-so 5172 df-fr 5209 df-we 5211 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-rn 5261 df-res 5262 df-ima 5263 df-pred 5822 df-ord 5868 df-on 5869 df-lim 5870 df-suc 5871 df-iota 5993 df-fun 6032 df-fn 6033 df-f 6034 df-f1 6035 df-fo 6036 df-f1o 6037 df-fv 6038 df-riota 6757 df-ov 6799 df-oprab 6800 df-mpt2 6801 df-om 7217 df-1st 7319 df-2nd 7320 df-wrecs 7563 df-recs 7625 df-rdg 7663 df-1o 7717 df-oadd 7721 df-er 7900 df-map 8015 df-pm 8016 df-ixp 8067 df-en 8114 df-dom 8115 df-sdom 8116 df-fin 8117 df-card 8969 df-cda 9196 df-pnf 10282 df-mnf 10283 df-xr 10284 df-ltxr 10285 df-le 10286 df-sub 10474 df-neg 10475 df-nn 11227 df-2 11285 df-3 11286 df-4 11287 df-5 11288 df-6 11289 df-7 11290 df-8 11291 df-9 11292 df-n0 11500 df-xnn0 11571 df-z 11585 df-dec 11701 df-uz 11894 df-fz 12534 df-hash 13322 df-struct 16066 df-ndx 16067 df-slot 16068 df-base 16070 df-sets 16071 df-ress 16072 df-plusg 16162 df-hom 16174 df-cco 16175 df-0g 16310 df-cat 16536 df-cid 16537 df-homf 16538 df-ssc 16677 df-resc 16678 df-subc 16679 df-inito 16848 df-termo 16849 df-zeroo 16850 df-estrc 16970 df-mgm 17450 df-sgrp 17492 df-mnd 17503 df-mhm 17543 df-grp 17633 df-minusg 17634 df-ghm 17866 df-cmn 18402 df-abl 18403 df-mgp 18698 df-ur 18710 df-ring 18757 df-nzr 19473 df-mgmhm 42304 df-rng0 42400 df-rnghomo 42412 df-rngc 42484 |
This theorem is referenced by: (None) |
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