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Mirrors > Home > MPE Home > Th. List > zrninitoringc | Structured version Visualization version GIF version |
Description: The zero ring is not an initial object in the category of unital rings (if the universe contains at least one unital ring different from the zero ring). (Contributed by AV, 18-Apr-2020.) |
Ref | Expression |
---|---|
zrtermoringc.u | ⊢ (𝜑 → 𝑈 ∈ 𝑉) |
zrtermoringc.c | ⊢ 𝐶 = (RingCat‘𝑈) |
zrtermoringc.z | ⊢ (𝜑 → 𝑍 ∈ (Ring ∖ NzRing)) |
zrtermoringc.e | ⊢ (𝜑 → 𝑍 ∈ 𝑈) |
zrninitoringc.e | ⊢ (𝜑 → ∃𝑟 ∈ (Base‘𝐶)𝑟 ∈ NzRing) |
Ref | Expression |
---|---|
zrninitoringc | ⊢ (𝜑 → 𝑍 ∉ (InitO‘𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zrninitoringc.e | . . . 4 ⊢ (𝜑 → ∃𝑟 ∈ (Base‘𝐶)𝑟 ∈ NzRing) | |
2 | zrtermoringc.c | . . . . . . . . . . 11 ⊢ 𝐶 = (RingCat‘𝑈) | |
3 | eqid 2724 | . . . . . . . . . . 11 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
4 | zrtermoringc.u | . . . . . . . . . . . 12 ⊢ (𝜑 → 𝑈 ∈ 𝑉) | |
5 | 4 | ad2antrr 723 | . . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ 𝑟 ∈ NzRing) → 𝑈 ∈ 𝑉) |
6 | eqid 2724 | . . . . . . . . . . 11 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
7 | zrtermoringc.e | . . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑍 ∈ 𝑈) | |
8 | zrtermoringc.z | . . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑍 ∈ (Ring ∖ NzRing)) | |
9 | 8 | eldifad 3952 | . . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑍 ∈ Ring) |
10 | 7, 9 | elind 4186 | . . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑍 ∈ (𝑈 ∩ Ring)) |
11 | 2, 3, 4 | ringcbas 20531 | . . . . . . . . . . . . 13 ⊢ (𝜑 → (Base‘𝐶) = (𝑈 ∩ Ring)) |
12 | 10, 11 | eleqtrrd 2828 | . . . . . . . . . . . 12 ⊢ (𝜑 → 𝑍 ∈ (Base‘𝐶)) |
13 | 12 | ad2antrr 723 | . . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ 𝑟 ∈ NzRing) → 𝑍 ∈ (Base‘𝐶)) |
14 | simplr 766 | . . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ 𝑟 ∈ NzRing) → 𝑟 ∈ (Base‘𝐶)) | |
15 | 2, 3, 5, 6, 13, 14 | ringchom 20533 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ 𝑟 ∈ NzRing) → (𝑍(Hom ‘𝐶)𝑟) = (𝑍 RingHom 𝑟)) |
16 | 8 | adantr 480 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → 𝑍 ∈ (Ring ∖ NzRing)) |
17 | nrhmzr 20422 | . . . . . . . . . . 11 ⊢ ((𝑍 ∈ (Ring ∖ NzRing) ∧ 𝑟 ∈ NzRing) → (𝑍 RingHom 𝑟) = ∅) | |
18 | 16, 17 | sylan 579 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ 𝑟 ∈ NzRing) → (𝑍 RingHom 𝑟) = ∅) |
19 | 15, 18 | eqtrd 2764 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ 𝑟 ∈ NzRing) → (𝑍(Hom ‘𝐶)𝑟) = ∅) |
20 | eq0 4335 | . . . . . . . . 9 ⊢ ((𝑍(Hom ‘𝐶)𝑟) = ∅ ↔ ∀ℎ ¬ ℎ ∈ (𝑍(Hom ‘𝐶)𝑟)) | |
21 | 19, 20 | sylib 217 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ 𝑟 ∈ NzRing) → ∀ℎ ¬ ℎ ∈ (𝑍(Hom ‘𝐶)𝑟)) |
22 | alnex 1775 | . . . . . . . 8 ⊢ (∀ℎ ¬ ℎ ∈ (𝑍(Hom ‘𝐶)𝑟) ↔ ¬ ∃ℎ ℎ ∈ (𝑍(Hom ‘𝐶)𝑟)) | |
23 | 21, 22 | sylib 217 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ 𝑟 ∈ NzRing) → ¬ ∃ℎ ℎ ∈ (𝑍(Hom ‘𝐶)𝑟)) |
24 | euex 2563 | . . . . . . 7 ⊢ (∃!ℎ ℎ ∈ (𝑍(Hom ‘𝐶)𝑟) → ∃ℎ ℎ ∈ (𝑍(Hom ‘𝐶)𝑟)) | |
25 | 23, 24 | nsyl 140 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ 𝑟 ∈ NzRing) → ¬ ∃!ℎ ℎ ∈ (𝑍(Hom ‘𝐶)𝑟)) |
26 | 25 | ex 412 | . . . . 5 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → (𝑟 ∈ NzRing → ¬ ∃!ℎ ℎ ∈ (𝑍(Hom ‘𝐶)𝑟))) |
27 | 26 | reximdva 3160 | . . . 4 ⊢ (𝜑 → (∃𝑟 ∈ (Base‘𝐶)𝑟 ∈ NzRing → ∃𝑟 ∈ (Base‘𝐶) ¬ ∃!ℎ ℎ ∈ (𝑍(Hom ‘𝐶)𝑟))) |
28 | 1, 27 | mpd 15 | . . 3 ⊢ (𝜑 → ∃𝑟 ∈ (Base‘𝐶) ¬ ∃!ℎ ℎ ∈ (𝑍(Hom ‘𝐶)𝑟)) |
29 | rexnal 3092 | . . 3 ⊢ (∃𝑟 ∈ (Base‘𝐶) ¬ ∃!ℎ ℎ ∈ (𝑍(Hom ‘𝐶)𝑟) ↔ ¬ ∀𝑟 ∈ (Base‘𝐶)∃!ℎ ℎ ∈ (𝑍(Hom ‘𝐶)𝑟)) | |
30 | 28, 29 | sylib 217 | . 2 ⊢ (𝜑 → ¬ ∀𝑟 ∈ (Base‘𝐶)∃!ℎ ℎ ∈ (𝑍(Hom ‘𝐶)𝑟)) |
31 | df-nel 3039 | . . 3 ⊢ (𝑍 ∉ (InitO‘𝐶) ↔ ¬ 𝑍 ∈ (InitO‘𝐶)) | |
32 | 2 | ringccat 20544 | . . . . . 6 ⊢ (𝑈 ∈ 𝑉 → 𝐶 ∈ Cat) |
33 | 4, 32 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ Cat) |
34 | 3, 6, 33, 12 | isinito 17945 | . . . 4 ⊢ (𝜑 → (𝑍 ∈ (InitO‘𝐶) ↔ ∀𝑟 ∈ (Base‘𝐶)∃!ℎ ℎ ∈ (𝑍(Hom ‘𝐶)𝑟))) |
35 | 34 | notbid 318 | . . 3 ⊢ (𝜑 → (¬ 𝑍 ∈ (InitO‘𝐶) ↔ ¬ ∀𝑟 ∈ (Base‘𝐶)∃!ℎ ℎ ∈ (𝑍(Hom ‘𝐶)𝑟))) |
36 | 31, 35 | bitrid 283 | . 2 ⊢ (𝜑 → (𝑍 ∉ (InitO‘𝐶) ↔ ¬ ∀𝑟 ∈ (Base‘𝐶)∃!ℎ ℎ ∈ (𝑍(Hom ‘𝐶)𝑟))) |
37 | 30, 36 | mpbird 257 | 1 ⊢ (𝜑 → 𝑍 ∉ (InitO‘𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∀wal 1531 = wceq 1533 ∃wex 1773 ∈ wcel 2098 ∃!weu 2554 ∉ wnel 3038 ∀wral 3053 ∃wrex 3062 ∖ cdif 3937 ∩ cin 3939 ∅c0 4314 ‘cfv 6533 (class class class)co 7401 Basecbs 17140 Hom chom 17204 Catccat 17604 InitOcinito 17930 Ringcrg 20123 RingHom crh 20356 NzRingcnzr 20399 RingCatcringc 20526 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-tp 4625 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-1st 7968 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-oadd 8465 df-er 8698 df-map 8817 df-pm 8818 df-ixp 8887 df-en 8935 df-dom 8936 df-sdom 8937 df-fin 8938 df-dju 9891 df-card 9929 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-xnn0 12541 df-z 12555 df-dec 12674 df-uz 12819 df-fz 13481 df-hash 14287 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-hom 17217 df-cco 17218 df-0g 17383 df-cat 17608 df-cid 17609 df-homf 17610 df-ssc 17753 df-resc 17754 df-subc 17755 df-inito 17933 df-estrc 18073 df-mgm 18560 df-sgrp 18639 df-mnd 18655 df-mhm 18700 df-grp 18853 df-minusg 18854 df-ghm 19124 df-cmn 19687 df-abl 19688 df-mgp 20025 df-rng 20043 df-ur 20072 df-ring 20125 df-rhm 20359 df-nzr 20400 df-ringc 20527 |
This theorem is referenced by: (None) |
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