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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 2779 | . . . 4 ⊢ (𝜑 → (𝑈 ∩ Rng) = (𝑈 ∩ Rng)) | |
4 | eqidd 2779 | . . . 4 ⊢ (𝜑 → ( RngHomo ↾ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng))) = ( RngHomo ↾ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)))) | |
5 | eqidd 2779 | . . . 4 ⊢ (𝜑 → (comp‘(ExtStrCat‘𝑈)) = (comp‘(ExtStrCat‘𝑈))) | |
6 | 1, 2, 3, 4, 5 | dfrngc2 43608 | . . 3 ⊢ (𝜑 → 𝐶 = {〈(Base‘ndx), (𝑈 ∩ Rng)〉, 〈(Hom ‘ndx), ( RngHomo ↾ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)))〉, 〈(comp‘ndx), (comp‘(ExtStrCat‘𝑈))〉}) |
7 | inex1g 5081 | . . . 4 ⊢ (𝑈 ∈ 𝑉 → (𝑈 ∩ Rng) ∈ V) | |
8 | 2, 7 | syl 17 | . . 3 ⊢ (𝜑 → (𝑈 ∩ Rng) ∈ V) |
9 | rnghmfn 43526 | . . . . 5 ⊢ RngHomo Fn (Rng × Rng) | |
10 | fnfun 6288 | . . . . 5 ⊢ ( RngHomo Fn (Rng × Rng) → Fun RngHomo ) | |
11 | 9, 10 | mp1i 13 | . . . 4 ⊢ (𝜑 → Fun RngHomo ) |
12 | sqxpexg 7296 | . . . . 5 ⊢ ((𝑈 ∩ Rng) ∈ V → ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)) ∈ V) | |
13 | 8, 12 | syl 17 | . . . 4 ⊢ (𝜑 → ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)) ∈ V) |
14 | resfunexg 6806 | . . . 4 ⊢ ((Fun RngHomo ∧ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)) ∈ V) → ( RngHomo ↾ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng))) ∈ V) | |
15 | 11, 13, 14 | syl2anc 576 | . . 3 ⊢ (𝜑 → ( RngHomo ↾ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng))) ∈ V) |
16 | fvexd 6516 | . . 3 ⊢ (𝜑 → (comp‘(ExtStrCat‘𝑈)) ∈ V) | |
17 | rhmsubcrngc.h | . . . 4 ⊢ (𝜑 → 𝐻 = ( RingHom ↾ (𝐵 × 𝐵))) | |
18 | rhmfn 43554 | . . . . . 6 ⊢ RingHom Fn (Ring × Ring) | |
19 | fnfun 6288 | . . . . . 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 4068 | . . . . . . . 8 ⊢ (Ring ∩ 𝑈) = (𝑈 ∩ Ring) | |
23 | 21, 22 | syl6eq 2830 | . . . . . . 7 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Ring)) |
24 | inex1g 5081 | . . . . . . . 8 ⊢ (𝑈 ∈ 𝑉 → (𝑈 ∩ Ring) ∈ V) | |
25 | 2, 24 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (𝑈 ∩ Ring) ∈ V) |
26 | 23, 25 | eqeltrd 2866 | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ V) |
27 | sqxpexg 7296 | . . . . . 6 ⊢ (𝐵 ∈ V → (𝐵 × 𝐵) ∈ V) | |
28 | 26, 27 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝐵 × 𝐵) ∈ V) |
29 | resfunexg 6806 | . . . . 5 ⊢ ((Fun RingHom ∧ (𝐵 × 𝐵) ∈ V) → ( RingHom ↾ (𝐵 × 𝐵)) ∈ V) | |
30 | 20, 28, 29 | syl2anc 576 | . . . 4 ⊢ (𝜑 → ( RingHom ↾ (𝐵 × 𝐵)) ∈ V) |
31 | 17, 30 | eqeltrd 2866 | . . 3 ⊢ (𝜑 → 𝐻 ∈ V) |
32 | ringrng 43515 | . . . . . . 7 ⊢ (𝑟 ∈ Ring → 𝑟 ∈ Rng) | |
33 | 32 | a1i 11 | . . . . . 6 ⊢ (𝜑 → (𝑟 ∈ Ring → 𝑟 ∈ Rng)) |
34 | 33 | ssrdv 3866 | . . . . 5 ⊢ (𝜑 → Ring ⊆ Rng) |
35 | 34 | ssrind 4101 | . . . 4 ⊢ (𝜑 → (Ring ∩ 𝑈) ⊆ (Rng ∩ 𝑈)) |
36 | incom 4068 | . . . . 5 ⊢ (𝑈 ∩ Rng) = (Rng ∩ 𝑈) | |
37 | 36 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝑈 ∩ Rng) = (Rng ∩ 𝑈)) |
38 | 35, 21, 37 | 3sstr4d 3906 | . . 3 ⊢ (𝜑 → 𝐵 ⊆ (𝑈 ∩ Rng)) |
39 | 6, 8, 15, 16, 31, 38 | estrres 17250 | . 2 ⊢ (𝜑 → ((𝐶 ↾s 𝐵) sSet 〈(Hom ‘ndx), 𝐻〉) = {〈(Base‘ndx), 𝐵〉, 〈(Hom ‘ndx), 𝐻〉, 〈(comp‘ndx), (comp‘(ExtStrCat‘𝑈))〉}) |
40 | eqid 2778 | . . 3 ⊢ (𝐶 ↾cat 𝐻) = (𝐶 ↾cat 𝐻) | |
41 | fvexd 6516 | . . . 4 ⊢ (𝜑 → (RngCat‘𝑈) ∈ V) | |
42 | 1, 41 | syl5eqel 2870 | . . 3 ⊢ (𝜑 → 𝐶 ∈ V) |
43 | 23, 17 | rhmresfn 43645 | . . 3 ⊢ (𝜑 → 𝐻 Fn (𝐵 × 𝐵)) |
44 | 40, 42, 26, 43 | rescval2 16959 | . 2 ⊢ (𝜑 → (𝐶 ↾cat 𝐻) = ((𝐶 ↾s 𝐵) sSet 〈(Hom ‘ndx), 𝐻〉)) |
45 | eqid 2778 | . . 3 ⊢ (RingCat‘𝑈) = (RingCat‘𝑈) | |
46 | 45, 2, 23, 17, 5 | dfringc2 43654 | . 2 ⊢ (𝜑 → (RingCat‘𝑈) = {〈(Base‘ndx), 𝐵〉, 〈(Hom ‘ndx), 𝐻〉, 〈(comp‘ndx), (comp‘(ExtStrCat‘𝑈))〉}) |
47 | 39, 44, 46 | 3eqtr4d 2824 | 1 ⊢ (𝜑 → (𝐶 ↾cat 𝐻) = (RingCat‘𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1507 ∈ wcel 2050 Vcvv 3415 ∩ cin 3830 {ctp 4446 〈cop 4448 × cxp 5406 ↾ cres 5410 Fun wfun 6184 Fn wfn 6185 ‘cfv 6190 (class class class)co 6978 ndxcnx 16339 sSet csts 16340 Basecbs 16342 ↾s cress 16343 Hom chom 16435 compcco 16436 ↾cat cresc 16939 ExtStrCatcestrc 17233 Ringcrg 19023 RingHom crh 19190 Rngcrng 43510 RngHomo crngh 43521 RngCatcrngc 43593 RingCatcringc 43639 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-rep 5050 ax-sep 5061 ax-nul 5068 ax-pow 5120 ax-pr 5187 ax-un 7281 ax-cnex 10393 ax-resscn 10394 ax-1cn 10395 ax-icn 10396 ax-addcl 10397 ax-addrcl 10398 ax-mulcl 10399 ax-mulrcl 10400 ax-mulcom 10401 ax-addass 10402 ax-mulass 10403 ax-distr 10404 ax-i2m1 10405 ax-1ne0 10406 ax-1rid 10407 ax-rnegex 10408 ax-rrecex 10409 ax-cnre 10410 ax-pre-lttri 10411 ax-pre-lttrn 10412 ax-pre-ltadd 10413 ax-pre-mulgt0 10414 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-fal 1520 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2583 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rmo 3096 df-rab 3097 df-v 3417 df-sbc 3684 df-csb 3789 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-pss 3847 df-nul 4181 df-if 4352 df-pw 4425 df-sn 4443 df-pr 4445 df-tp 4447 df-op 4449 df-uni 4714 df-int 4751 df-iun 4795 df-br 4931 df-opab 4993 df-mpt 5010 df-tr 5032 df-id 5313 df-eprel 5318 df-po 5327 df-so 5328 df-fr 5367 df-we 5369 df-xp 5414 df-rel 5415 df-cnv 5416 df-co 5417 df-dm 5418 df-rn 5419 df-res 5420 df-ima 5421 df-pred 5988 df-ord 6034 df-on 6035 df-lim 6036 df-suc 6037 df-iota 6154 df-fun 6192 df-fn 6193 df-f 6194 df-f1 6195 df-fo 6196 df-f1o 6197 df-fv 6198 df-riota 6939 df-ov 6981 df-oprab 6982 df-mpo 6983 df-om 7399 df-1st 7503 df-2nd 7504 df-wrecs 7752 df-recs 7814 df-rdg 7852 df-1o 7907 df-oadd 7911 df-er 8091 df-map 8210 df-en 8309 df-dom 8310 df-sdom 8311 df-fin 8312 df-pnf 10478 df-mnf 10479 df-xr 10480 df-ltxr 10481 df-le 10482 df-sub 10674 df-neg 10675 df-nn 11442 df-2 11506 df-3 11507 df-4 11508 df-5 11509 df-6 11510 df-7 11511 df-8 11512 df-9 11513 df-n0 11711 df-z 11797 df-dec 11915 df-uz 12062 df-fz 12712 df-struct 16344 df-ndx 16345 df-slot 16346 df-base 16348 df-sets 16349 df-ress 16350 df-plusg 16437 df-hom 16448 df-cco 16449 df-0g 16574 df-resc 16942 df-estrc 17234 df-mgm 17713 df-sgrp 17755 df-mnd 17766 df-mhm 17806 df-grp 17897 df-minusg 17898 df-ghm 18130 df-cmn 18671 df-abl 18672 df-mgp 18966 df-ur 18978 df-ring 19025 df-rnghom 19193 df-rng0 43511 df-rnghomo 43523 df-rngc 43595 df-ringc 43641 |
This theorem is referenced by: (None) |
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