Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > irinitoringc | Structured version Visualization version GIF version |
Description: The ring of integers is an initial object in the category of unital rings (within a universe containing the ring of integers). Example 7.2 (6) of [Adamek] p. 101 , and example in [Lang] p. 58. (Contributed by AV, 3-Apr-2020.) |
Ref | Expression |
---|---|
irinitoringc.u | ⊢ (𝜑 → 𝑈 ∈ 𝑉) |
irinitoringc.z | ⊢ (𝜑 → ℤring ∈ 𝑈) |
irinitoringc.c | ⊢ 𝐶 = (RingCat‘𝑈) |
Ref | Expression |
---|---|
irinitoringc | ⊢ (𝜑 → ℤring ∈ (InitO‘𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zex 11979 | . . . . . 6 ⊢ ℤ ∈ V | |
2 | 1 | mptex 6978 | . . . . 5 ⊢ (𝑧 ∈ ℤ ↦ (𝑧(.g‘𝑟)(1r‘𝑟))) ∈ V |
3 | irinitoringc.c | . . . . . . . . 9 ⊢ 𝐶 = (RingCat‘𝑈) | |
4 | eqid 2821 | . . . . . . . . 9 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
5 | irinitoringc.u | . . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ 𝑉) | |
6 | eqid 2821 | . . . . . . . . 9 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
7 | 3, 4, 5, 6 | ringchomfval 44181 | . . . . . . . 8 ⊢ (𝜑 → (Hom ‘𝐶) = ( RingHom ↾ ((Base‘𝐶) × (Base‘𝐶)))) |
8 | 7 | adantr 481 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → (Hom ‘𝐶) = ( RingHom ↾ ((Base‘𝐶) × (Base‘𝐶)))) |
9 | 8 | oveqd 7162 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → (ℤring(Hom ‘𝐶)𝑟) = (ℤring( RingHom ↾ ((Base‘𝐶) × (Base‘𝐶)))𝑟)) |
10 | irinitoringc.z | . . . . . . . . . 10 ⊢ (𝜑 → ℤring ∈ 𝑈) | |
11 | id 22 | . . . . . . . . . . 11 ⊢ (ℤring ∈ 𝑈 → ℤring ∈ 𝑈) | |
12 | zringring 20550 | . . . . . . . . . . . 12 ⊢ ℤring ∈ Ring | |
13 | 12 | a1i 11 | . . . . . . . . . . 11 ⊢ (ℤring ∈ 𝑈 → ℤring ∈ Ring) |
14 | 11, 13 | elind 4170 | . . . . . . . . . 10 ⊢ (ℤring ∈ 𝑈 → ℤring ∈ (𝑈 ∩ Ring)) |
15 | 10, 14 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → ℤring ∈ (𝑈 ∩ Ring)) |
16 | 3, 4, 5 | ringcbas 44180 | . . . . . . . . 9 ⊢ (𝜑 → (Base‘𝐶) = (𝑈 ∩ Ring)) |
17 | 15, 16 | eleqtrrd 2916 | . . . . . . . 8 ⊢ (𝜑 → ℤring ∈ (Base‘𝐶)) |
18 | 17 | adantr 481 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → ℤring ∈ (Base‘𝐶)) |
19 | simpr 485 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → 𝑟 ∈ (Base‘𝐶)) | |
20 | 18, 19 | ovresd 7304 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → (ℤring( RingHom ↾ ((Base‘𝐶) × (Base‘𝐶)))𝑟) = (ℤring RingHom 𝑟)) |
21 | 16 | eleq2d 2898 | . . . . . . . . 9 ⊢ (𝜑 → (𝑟 ∈ (Base‘𝐶) ↔ 𝑟 ∈ (𝑈 ∩ Ring))) |
22 | elin 4168 | . . . . . . . . . 10 ⊢ (𝑟 ∈ (𝑈 ∩ Ring) ↔ (𝑟 ∈ 𝑈 ∧ 𝑟 ∈ Ring)) | |
23 | 22 | simprbi 497 | . . . . . . . . 9 ⊢ (𝑟 ∈ (𝑈 ∩ Ring) → 𝑟 ∈ Ring) |
24 | 21, 23 | syl6bi 254 | . . . . . . . 8 ⊢ (𝜑 → (𝑟 ∈ (Base‘𝐶) → 𝑟 ∈ Ring)) |
25 | 24 | imp 407 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → 𝑟 ∈ Ring) |
26 | eqid 2821 | . . . . . . . 8 ⊢ (.g‘𝑟) = (.g‘𝑟) | |
27 | eqid 2821 | . . . . . . . 8 ⊢ (𝑧 ∈ ℤ ↦ (𝑧(.g‘𝑟)(1r‘𝑟))) = (𝑧 ∈ ℤ ↦ (𝑧(.g‘𝑟)(1r‘𝑟))) | |
28 | eqid 2821 | . . . . . . . 8 ⊢ (1r‘𝑟) = (1r‘𝑟) | |
29 | 26, 27, 28 | mulgrhm2 20576 | . . . . . . 7 ⊢ (𝑟 ∈ Ring → (ℤring RingHom 𝑟) = {(𝑧 ∈ ℤ ↦ (𝑧(.g‘𝑟)(1r‘𝑟)))}) |
30 | 25, 29 | syl 17 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → (ℤring RingHom 𝑟) = {(𝑧 ∈ ℤ ↦ (𝑧(.g‘𝑟)(1r‘𝑟)))}) |
31 | 9, 20, 30 | 3eqtrd 2860 | . . . . 5 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → (ℤring(Hom ‘𝐶)𝑟) = {(𝑧 ∈ ℤ ↦ (𝑧(.g‘𝑟)(1r‘𝑟)))}) |
32 | sneq 4569 | . . . . . . 7 ⊢ (𝑓 = (𝑧 ∈ ℤ ↦ (𝑧(.g‘𝑟)(1r‘𝑟))) → {𝑓} = {(𝑧 ∈ ℤ ↦ (𝑧(.g‘𝑟)(1r‘𝑟)))}) | |
33 | 32 | eqeq2d 2832 | . . . . . 6 ⊢ (𝑓 = (𝑧 ∈ ℤ ↦ (𝑧(.g‘𝑟)(1r‘𝑟))) → ((ℤring(Hom ‘𝐶)𝑟) = {𝑓} ↔ (ℤring(Hom ‘𝐶)𝑟) = {(𝑧 ∈ ℤ ↦ (𝑧(.g‘𝑟)(1r‘𝑟)))})) |
34 | 33 | spcegv 3597 | . . . . 5 ⊢ ((𝑧 ∈ ℤ ↦ (𝑧(.g‘𝑟)(1r‘𝑟))) ∈ V → ((ℤring(Hom ‘𝐶)𝑟) = {(𝑧 ∈ ℤ ↦ (𝑧(.g‘𝑟)(1r‘𝑟)))} → ∃𝑓(ℤring(Hom ‘𝐶)𝑟) = {𝑓})) |
35 | 2, 31, 34 | mpsyl 68 | . . . 4 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → ∃𝑓(ℤring(Hom ‘𝐶)𝑟) = {𝑓}) |
36 | eusn 4660 | . . . 4 ⊢ (∃!𝑓 𝑓 ∈ (ℤring(Hom ‘𝐶)𝑟) ↔ ∃𝑓(ℤring(Hom ‘𝐶)𝑟) = {𝑓}) | |
37 | 35, 36 | sylibr 235 | . . 3 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝐶)) → ∃!𝑓 𝑓 ∈ (ℤring(Hom ‘𝐶)𝑟)) |
38 | 37 | ralrimiva 3182 | . 2 ⊢ (𝜑 → ∀𝑟 ∈ (Base‘𝐶)∃!𝑓 𝑓 ∈ (ℤring(Hom ‘𝐶)𝑟)) |
39 | 3 | ringccat 44193 | . . . 4 ⊢ (𝑈 ∈ 𝑉 → 𝐶 ∈ Cat) |
40 | 5, 39 | syl 17 | . . 3 ⊢ (𝜑 → 𝐶 ∈ Cat) |
41 | 12 | a1i 11 | . . . . 5 ⊢ (𝜑 → ℤring ∈ Ring) |
42 | 10, 41 | elind 4170 | . . . 4 ⊢ (𝜑 → ℤring ∈ (𝑈 ∩ Ring)) |
43 | 42, 16 | eleqtrrd 2916 | . . 3 ⊢ (𝜑 → ℤring ∈ (Base‘𝐶)) |
44 | 4, 6, 40, 43 | isinito 17250 | . 2 ⊢ (𝜑 → (ℤring ∈ (InitO‘𝐶) ↔ ∀𝑟 ∈ (Base‘𝐶)∃!𝑓 𝑓 ∈ (ℤring(Hom ‘𝐶)𝑟))) |
45 | 38, 44 | mpbird 258 | 1 ⊢ (𝜑 → ℤring ∈ (InitO‘𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∃wex 1771 ∈ wcel 2105 ∃!weu 2649 ∀wral 3138 Vcvv 3495 ∩ cin 3934 {csn 4559 ↦ cmpt 5138 × cxp 5547 ↾ cres 5551 ‘cfv 6349 (class class class)co 7145 ℤcz 11970 Basecbs 16473 Hom chom 16566 Catccat 16925 InitOcinito 17238 .gcmg 18164 1rcur 19182 Ringcrg 19228 RingHom crh 19395 ℤringzring 20547 RingCatcringc 44172 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-addf 10605 ax-mulf 10606 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 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 3497 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4833 df-int 4870 df-iun 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7569 df-1st 7680 df-2nd 7681 df-wrecs 7938 df-recs 7999 df-rdg 8037 df-1o 8093 df-oadd 8097 df-er 8279 df-map 8398 df-pm 8399 df-ixp 8451 df-en 8499 df-dom 8500 df-sdom 8501 df-fin 8502 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11628 df-2 11689 df-3 11690 df-4 11691 df-5 11692 df-6 11693 df-7 11694 df-8 11695 df-9 11696 df-n0 11887 df-z 11971 df-dec 12088 df-uz 12233 df-fz 12883 df-seq 13360 df-struct 16475 df-ndx 16476 df-slot 16477 df-base 16479 df-sets 16480 df-ress 16481 df-plusg 16568 df-mulr 16569 df-starv 16570 df-tset 16574 df-ple 16575 df-ds 16577 df-unif 16578 df-hom 16579 df-cco 16580 df-0g 16705 df-cat 16929 df-cid 16930 df-homf 16931 df-ssc 17070 df-resc 17071 df-subc 17072 df-inito 17241 df-estrc 17363 df-mgm 17842 df-sgrp 17891 df-mnd 17902 df-mhm 17946 df-grp 18046 df-minusg 18047 df-mulg 18165 df-subg 18216 df-ghm 18296 df-cmn 18839 df-mgp 19171 df-ur 19183 df-ring 19230 df-cring 19231 df-rnghom 19398 df-subrg 19464 df-cnfld 20476 df-zring 20548 df-ringc 44174 |
This theorem is referenced by: nzerooringczr 44241 |
Copyright terms: Public domain | W3C validator |