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Mirrors > Home > MPE Home > Th. List > Mathboxes > 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 2821 | . . . . . . . . . . 11 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
4 | zrtermoringc.u | . . . . . . . . . . . 12 ⊢ (𝜑 → 𝑈 ∈ 𝑉) | |
5 | 4 | ad2antrr 724 | . . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ 𝑟 ∈ NzRing) → 𝑈 ∈ 𝑉) |
6 | eqid 2821 | . . . . . . . . . . 11 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
7 | zrtermoringc.e | . . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑍 ∈ 𝑈) | |
8 | zrtermoringc.z | . . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑍 ∈ (Ring ∖ NzRing)) | |
9 | 8 | eldifad 3948 | . . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑍 ∈ Ring) |
10 | 7, 9 | elind 4171 | . . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑍 ∈ (𝑈 ∩ Ring)) |
11 | 2, 3, 4 | ringcbas 44302 | . . . . . . . . . . . . 13 ⊢ (𝜑 → (Base‘𝐶) = (𝑈 ∩ Ring)) |
12 | 10, 11 | eleqtrrd 2916 | . . . . . . . . . . . 12 ⊢ (𝜑 → 𝑍 ∈ (Base‘𝐶)) |
13 | 12 | ad2antrr 724 | . . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ 𝑟 ∈ NzRing) → 𝑍 ∈ (Base‘𝐶)) |
14 | simplr 767 | . . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ 𝑟 ∈ NzRing) → 𝑟 ∈ (Base‘𝐶)) | |
15 | 2, 3, 5, 6, 13, 14 | ringchom 44304 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ 𝑟 ∈ NzRing) → (𝑍(Hom ‘𝐶)𝑟) = (𝑍 RingHom 𝑟)) |
16 | 8 | adantr 483 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → 𝑍 ∈ (Ring ∖ NzRing)) |
17 | nrhmzr 44164 | . . . . . . . . . . 11 ⊢ ((𝑍 ∈ (Ring ∖ NzRing) ∧ 𝑟 ∈ NzRing) → (𝑍 RingHom 𝑟) = ∅) | |
18 | 16, 17 | sylan 582 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ 𝑟 ∈ NzRing) → (𝑍 RingHom 𝑟) = ∅) |
19 | 15, 18 | eqtrd 2856 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ 𝑟 ∈ NzRing) → (𝑍(Hom ‘𝐶)𝑟) = ∅) |
20 | eq0 4308 | . . . . . . . . 9 ⊢ ((𝑍(Hom ‘𝐶)𝑟) = ∅ ↔ ∀ℎ ¬ ℎ ∈ (𝑍(Hom ‘𝐶)𝑟)) | |
21 | 19, 20 | sylib 220 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ 𝑟 ∈ NzRing) → ∀ℎ ¬ ℎ ∈ (𝑍(Hom ‘𝐶)𝑟)) |
22 | alnex 1782 | . . . . . . . 8 ⊢ (∀ℎ ¬ ℎ ∈ (𝑍(Hom ‘𝐶)𝑟) ↔ ¬ ∃ℎ ℎ ∈ (𝑍(Hom ‘𝐶)𝑟)) | |
23 | 21, 22 | sylib 220 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ 𝑟 ∈ NzRing) → ¬ ∃ℎ ℎ ∈ (𝑍(Hom ‘𝐶)𝑟)) |
24 | euex 2662 | . . . . . . 7 ⊢ (∃!ℎ ℎ ∈ (𝑍(Hom ‘𝐶)𝑟) → ∃ℎ ℎ ∈ (𝑍(Hom ‘𝐶)𝑟)) | |
25 | 23, 24 | nsyl 142 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) ∧ 𝑟 ∈ NzRing) → ¬ ∃!ℎ ℎ ∈ (𝑍(Hom ‘𝐶)𝑟)) |
26 | 25 | ex 415 | . . . . 5 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → (𝑟 ∈ NzRing → ¬ ∃!ℎ ℎ ∈ (𝑍(Hom ‘𝐶)𝑟))) |
27 | 26 | reximdva 3274 | . . . 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 3238 | . . 3 ⊢ (∃𝑟 ∈ (Base‘𝐶) ¬ ∃!ℎ ℎ ∈ (𝑍(Hom ‘𝐶)𝑟) ↔ ¬ ∀𝑟 ∈ (Base‘𝐶)∃!ℎ ℎ ∈ (𝑍(Hom ‘𝐶)𝑟)) | |
30 | 28, 29 | sylib 220 | . 2 ⊢ (𝜑 → ¬ ∀𝑟 ∈ (Base‘𝐶)∃!ℎ ℎ ∈ (𝑍(Hom ‘𝐶)𝑟)) |
31 | df-nel 3124 | . . 3 ⊢ (𝑍 ∉ (InitO‘𝐶) ↔ ¬ 𝑍 ∈ (InitO‘𝐶)) | |
32 | 2 | ringccat 44315 | . . . . . 6 ⊢ (𝑈 ∈ 𝑉 → 𝐶 ∈ Cat) |
33 | 4, 32 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ Cat) |
34 | 3, 6, 33, 12 | isinito 17260 | . . . 4 ⊢ (𝜑 → (𝑍 ∈ (InitO‘𝐶) ↔ ∀𝑟 ∈ (Base‘𝐶)∃!ℎ ℎ ∈ (𝑍(Hom ‘𝐶)𝑟))) |
35 | 34 | notbid 320 | . . 3 ⊢ (𝜑 → (¬ 𝑍 ∈ (InitO‘𝐶) ↔ ¬ ∀𝑟 ∈ (Base‘𝐶)∃!ℎ ℎ ∈ (𝑍(Hom ‘𝐶)𝑟))) |
36 | 31, 35 | syl5bb 285 | . 2 ⊢ (𝜑 → (𝑍 ∉ (InitO‘𝐶) ↔ ¬ ∀𝑟 ∈ (Base‘𝐶)∃!ℎ ℎ ∈ (𝑍(Hom ‘𝐶)𝑟))) |
37 | 30, 36 | mpbird 259 | 1 ⊢ (𝜑 → 𝑍 ∉ (InitO‘𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ∀wal 1535 = wceq 1537 ∃wex 1780 ∈ wcel 2114 ∃!weu 2653 ∉ wnel 3123 ∀wral 3138 ∃wrex 3139 ∖ cdif 3933 ∩ cin 3935 ∅c0 4291 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 Hom chom 16576 Catccat 16935 InitOcinito 17248 Ringcrg 19297 RingHom crh 19464 NzRingcnzr 20030 RingCatcringc 44294 |
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-fal 1550 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-map 8408 df-pm 8409 df-ixp 8462 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-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-xnn0 11969 df-z 11983 df-dec 12100 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-ress 16491 df-plusg 16578 df-hom 16589 df-cco 16590 df-0g 16715 df-cat 16939 df-cid 16940 df-homf 16941 df-ssc 17080 df-resc 17081 df-subc 17082 df-inito 17251 df-estrc 17373 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-mhm 17956 df-grp 18106 df-minusg 18107 df-ghm 18356 df-mgp 19240 df-ur 19252 df-ring 19299 df-rnghom 19467 df-nzr 20031 df-ringc 44296 |
This theorem is referenced by: (None) |
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