Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rngcresringcat | Structured version Visualization version GIF version |
Description: The restriction of the category of non-unital rings to the set of unital ring homomorphisms is the category of unital rings. (Contributed by AV, 16-Mar-2020.) |
Ref | Expression |
---|---|
rhmsubcrngc.c | ⊢ 𝐶 = (RngCat‘𝑈) |
rhmsubcrngc.u | ⊢ (𝜑 → 𝑈 ∈ 𝑉) |
rhmsubcrngc.b | ⊢ (𝜑 → 𝐵 = (Ring ∩ 𝑈)) |
rhmsubcrngc.h | ⊢ (𝜑 → 𝐻 = ( RingHom ↾ (𝐵 × 𝐵))) |
Ref | Expression |
---|---|
rngcresringcat | ⊢ (𝜑 → (𝐶 ↾cat 𝐻) = (RingCat‘𝑈)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rhmsubcrngc.c | . . . 4 ⊢ 𝐶 = (RngCat‘𝑈) | |
2 | rhmsubcrngc.u | . . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑉) | |
3 | eqidd 2735 | . . . 4 ⊢ (𝜑 → (𝑈 ∩ Rng) = (𝑈 ∩ Rng)) | |
4 | eqidd 2735 | . . . 4 ⊢ (𝜑 → ( RngHomo ↾ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng))) = ( RngHomo ↾ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)))) | |
5 | eqidd 2735 | . . . 4 ⊢ (𝜑 → (comp‘(ExtStrCat‘𝑈)) = (comp‘(ExtStrCat‘𝑈))) | |
6 | 1, 2, 3, 4, 5 | dfrngc2 45157 | . . 3 ⊢ (𝜑 → 𝐶 = {〈(Base‘ndx), (𝑈 ∩ Rng)〉, 〈(Hom ‘ndx), ( RngHomo ↾ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)))〉, 〈(comp‘ndx), (comp‘(ExtStrCat‘𝑈))〉}) |
7 | inex1g 5201 | . . . 4 ⊢ (𝑈 ∈ 𝑉 → (𝑈 ∩ Rng) ∈ V) | |
8 | 2, 7 | syl 17 | . . 3 ⊢ (𝜑 → (𝑈 ∩ Rng) ∈ V) |
9 | rnghmfn 45075 | . . . . 5 ⊢ RngHomo Fn (Rng × Rng) | |
10 | fnfun 6468 | . . . . 5 ⊢ ( RngHomo Fn (Rng × Rng) → Fun RngHomo ) | |
11 | 9, 10 | mp1i 13 | . . . 4 ⊢ (𝜑 → Fun RngHomo ) |
12 | sqxpexg 7529 | . . . . 5 ⊢ ((𝑈 ∩ Rng) ∈ V → ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)) ∈ V) | |
13 | 8, 12 | syl 17 | . . . 4 ⊢ (𝜑 → ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)) ∈ V) |
14 | resfunexg 7020 | . . . 4 ⊢ ((Fun RngHomo ∧ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)) ∈ V) → ( RngHomo ↾ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng))) ∈ V) | |
15 | 11, 13, 14 | syl2anc 587 | . . 3 ⊢ (𝜑 → ( RngHomo ↾ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng))) ∈ V) |
16 | fvexd 6721 | . . 3 ⊢ (𝜑 → (comp‘(ExtStrCat‘𝑈)) ∈ V) | |
17 | rhmsubcrngc.h | . . . 4 ⊢ (𝜑 → 𝐻 = ( RingHom ↾ (𝐵 × 𝐵))) | |
18 | rhmfn 45103 | . . . . . 6 ⊢ RingHom Fn (Ring × Ring) | |
19 | fnfun 6468 | . . . . . 6 ⊢ ( RingHom Fn (Ring × Ring) → Fun RingHom ) | |
20 | 18, 19 | mp1i 13 | . . . . 5 ⊢ (𝜑 → Fun RingHom ) |
21 | rhmsubcrngc.b | . . . . . . . 8 ⊢ (𝜑 → 𝐵 = (Ring ∩ 𝑈)) | |
22 | incom 4105 | . . . . . . . 8 ⊢ (Ring ∩ 𝑈) = (𝑈 ∩ Ring) | |
23 | 21, 22 | eqtrdi 2790 | . . . . . . 7 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Ring)) |
24 | inex1g 5201 | . . . . . . . 8 ⊢ (𝑈 ∈ 𝑉 → (𝑈 ∩ Ring) ∈ V) | |
25 | 2, 24 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (𝑈 ∩ Ring) ∈ V) |
26 | 23, 25 | eqeltrd 2834 | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ V) |
27 | sqxpexg 7529 | . . . . . 6 ⊢ (𝐵 ∈ V → (𝐵 × 𝐵) ∈ V) | |
28 | 26, 27 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝐵 × 𝐵) ∈ V) |
29 | resfunexg 7020 | . . . . 5 ⊢ ((Fun RingHom ∧ (𝐵 × 𝐵) ∈ V) → ( RingHom ↾ (𝐵 × 𝐵)) ∈ V) | |
30 | 20, 28, 29 | syl2anc 587 | . . . 4 ⊢ (𝜑 → ( RingHom ↾ (𝐵 × 𝐵)) ∈ V) |
31 | 17, 30 | eqeltrd 2834 | . . 3 ⊢ (𝜑 → 𝐻 ∈ V) |
32 | ringrng 45064 | . . . . . . 7 ⊢ (𝑟 ∈ Ring → 𝑟 ∈ Rng) | |
33 | 32 | a1i 11 | . . . . . 6 ⊢ (𝜑 → (𝑟 ∈ Ring → 𝑟 ∈ Rng)) |
34 | 33 | ssrdv 3897 | . . . . 5 ⊢ (𝜑 → Ring ⊆ Rng) |
35 | 34 | ssrind 4140 | . . . 4 ⊢ (𝜑 → (Ring ∩ 𝑈) ⊆ (Rng ∩ 𝑈)) |
36 | incom 4105 | . . . . 5 ⊢ (𝑈 ∩ Rng) = (Rng ∩ 𝑈) | |
37 | 36 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝑈 ∩ Rng) = (Rng ∩ 𝑈)) |
38 | 35, 21, 37 | 3sstr4d 3938 | . . 3 ⊢ (𝜑 → 𝐵 ⊆ (𝑈 ∩ Rng)) |
39 | 6, 8, 15, 16, 31, 38 | estrres 17618 | . 2 ⊢ (𝜑 → ((𝐶 ↾s 𝐵) sSet 〈(Hom ‘ndx), 𝐻〉) = {〈(Base‘ndx), 𝐵〉, 〈(Hom ‘ndx), 𝐻〉, 〈(comp‘ndx), (comp‘(ExtStrCat‘𝑈))〉}) |
40 | eqid 2734 | . . 3 ⊢ (𝐶 ↾cat 𝐻) = (𝐶 ↾cat 𝐻) | |
41 | fvexd 6721 | . . . 4 ⊢ (𝜑 → (RngCat‘𝑈) ∈ V) | |
42 | 1, 41 | eqeltrid 2838 | . . 3 ⊢ (𝜑 → 𝐶 ∈ V) |
43 | 23, 17 | rhmresfn 45194 | . . 3 ⊢ (𝜑 → 𝐻 Fn (𝐵 × 𝐵)) |
44 | 40, 42, 26, 43 | rescval2 17305 | . 2 ⊢ (𝜑 → (𝐶 ↾cat 𝐻) = ((𝐶 ↾s 𝐵) sSet 〈(Hom ‘ndx), 𝐻〉)) |
45 | eqid 2734 | . . 3 ⊢ (RingCat‘𝑈) = (RingCat‘𝑈) | |
46 | 45, 2, 23, 17, 5 | dfringc2 45203 | . 2 ⊢ (𝜑 → (RingCat‘𝑈) = {〈(Base‘ndx), 𝐵〉, 〈(Hom ‘ndx), 𝐻〉, 〈(comp‘ndx), (comp‘(ExtStrCat‘𝑈))〉}) |
47 | 39, 44, 46 | 3eqtr4d 2784 | 1 ⊢ (𝜑 → (𝐶 ↾cat 𝐻) = (RingCat‘𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2110 Vcvv 3401 ∩ cin 3856 {ctp 4535 〈cop 4537 × cxp 5538 ↾ cres 5542 Fun wfun 6363 Fn wfn 6364 ‘cfv 6369 (class class class)co 7202 ndxcnx 16681 sSet csts 16682 Basecbs 16684 ↾s cress 16685 Hom chom 16778 compcco 16779 ↾cat cresc 17285 ExtStrCatcestrc 17601 Ringcrg 19534 RingHom crh 19704 Rngcrng 45059 RngHomo crngh 45070 RngCatcrngc 45142 RingCatcringc 45188 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-om 7634 df-1st 7750 df-2nd 7751 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-1o 8191 df-er 8380 df-map 8499 df-en 8616 df-dom 8617 df-sdom 8618 df-fin 8619 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-nn 11814 df-2 11876 df-3 11877 df-4 11878 df-5 11879 df-6 11880 df-7 11881 df-8 11882 df-9 11883 df-n0 12074 df-z 12160 df-dec 12277 df-uz 12422 df-fz 13079 df-struct 16686 df-ndx 16687 df-slot 16688 df-base 16690 df-sets 16691 df-ress 16692 df-plusg 16780 df-hom 16791 df-cco 16792 df-0g 16918 df-resc 17288 df-estrc 17602 df-mgm 18086 df-sgrp 18135 df-mnd 18146 df-mhm 18190 df-grp 18340 df-minusg 18341 df-ghm 18592 df-cmn 19144 df-abl 19145 df-mgp 19477 df-ur 19489 df-ring 19536 df-rnghom 19707 df-rng0 45060 df-rnghomo 45072 df-rngc 45144 df-ringc 45190 |
This theorem is referenced by: (None) |
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