![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > 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 2727 | . . . 4 ⊢ (𝜑 → (𝑈 ∩ Rng) = (𝑈 ∩ Rng)) | |
4 | eqidd 2727 | . . . 4 ⊢ (𝜑 → ( RngHom ↾ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng))) = ( RngHom ↾ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)))) | |
5 | eqidd 2727 | . . . 4 ⊢ (𝜑 → (comp‘(ExtStrCat‘𝑈)) = (comp‘(ExtStrCat‘𝑈))) | |
6 | 1, 2, 3, 4, 5 | dfrngc2 20602 | . . 3 ⊢ (𝜑 → 𝐶 = {〈(Base‘ndx), (𝑈 ∩ Rng)〉, 〈(Hom ‘ndx), ( RngHom ↾ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)))〉, 〈(comp‘ndx), (comp‘(ExtStrCat‘𝑈))〉}) |
7 | inex1g 5316 | . . . 4 ⊢ (𝑈 ∈ 𝑉 → (𝑈 ∩ Rng) ∈ V) | |
8 | 2, 7 | syl 17 | . . 3 ⊢ (𝜑 → (𝑈 ∩ Rng) ∈ V) |
9 | rnghmfn 20417 | . . . . 5 ⊢ RngHom Fn (Rng × Rng) | |
10 | fnfun 6652 | . . . . 5 ⊢ ( RngHom Fn (Rng × Rng) → Fun RngHom ) | |
11 | 9, 10 | mp1i 13 | . . . 4 ⊢ (𝜑 → Fun RngHom ) |
12 | sqxpexg 7755 | . . . . 5 ⊢ ((𝑈 ∩ Rng) ∈ V → ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)) ∈ V) | |
13 | 8, 12 | syl 17 | . . . 4 ⊢ (𝜑 → ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)) ∈ V) |
14 | resfunexg 7224 | . . . 4 ⊢ ((Fun RngHom ∧ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)) ∈ V) → ( RngHom ↾ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng))) ∈ V) | |
15 | 11, 13, 14 | syl2anc 582 | . . 3 ⊢ (𝜑 → ( RngHom ↾ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng))) ∈ V) |
16 | fvexd 6908 | . . 3 ⊢ (𝜑 → (comp‘(ExtStrCat‘𝑈)) ∈ V) | |
17 | rhmsubcrngc.h | . . . 4 ⊢ (𝜑 → 𝐻 = ( RingHom ↾ (𝐵 × 𝐵))) | |
18 | rhmfn 20477 | . . . . . 6 ⊢ RingHom Fn (Ring × Ring) | |
19 | fnfun 6652 | . . . . . 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 4199 | . . . . . . . 8 ⊢ (Ring ∩ 𝑈) = (𝑈 ∩ Ring) | |
23 | 21, 22 | eqtrdi 2782 | . . . . . . 7 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Ring)) |
24 | inex1g 5316 | . . . . . . . 8 ⊢ (𝑈 ∈ 𝑉 → (𝑈 ∩ Ring) ∈ V) | |
25 | 2, 24 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (𝑈 ∩ Ring) ∈ V) |
26 | 23, 25 | eqeltrd 2826 | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ V) |
27 | sqxpexg 7755 | . . . . . 6 ⊢ (𝐵 ∈ V → (𝐵 × 𝐵) ∈ V) | |
28 | 26, 27 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝐵 × 𝐵) ∈ V) |
29 | resfunexg 7224 | . . . . 5 ⊢ ((Fun RingHom ∧ (𝐵 × 𝐵) ∈ V) → ( RingHom ↾ (𝐵 × 𝐵)) ∈ V) | |
30 | 20, 28, 29 | syl2anc 582 | . . . 4 ⊢ (𝜑 → ( RingHom ↾ (𝐵 × 𝐵)) ∈ V) |
31 | 17, 30 | eqeltrd 2826 | . . 3 ⊢ (𝜑 → 𝐻 ∈ V) |
32 | ringrng 20260 | . . . . . . 7 ⊢ (𝑟 ∈ Ring → 𝑟 ∈ Rng) | |
33 | 32 | a1i 11 | . . . . . 6 ⊢ (𝜑 → (𝑟 ∈ Ring → 𝑟 ∈ Rng)) |
34 | 33 | ssrdv 3984 | . . . . 5 ⊢ (𝜑 → Ring ⊆ Rng) |
35 | 34 | ssrind 4234 | . . . 4 ⊢ (𝜑 → (Ring ∩ 𝑈) ⊆ (Rng ∩ 𝑈)) |
36 | incom 4199 | . . . . 5 ⊢ (𝑈 ∩ Rng) = (Rng ∩ 𝑈) | |
37 | 36 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝑈 ∩ Rng) = (Rng ∩ 𝑈)) |
38 | 35, 21, 37 | 3sstr4d 4026 | . . 3 ⊢ (𝜑 → 𝐵 ⊆ (𝑈 ∩ Rng)) |
39 | 6, 8, 15, 16, 31, 38 | estrres 18158 | . 2 ⊢ (𝜑 → ((𝐶 ↾s 𝐵) sSet 〈(Hom ‘ndx), 𝐻〉) = {〈(Base‘ndx), 𝐵〉, 〈(Hom ‘ndx), 𝐻〉, 〈(comp‘ndx), (comp‘(ExtStrCat‘𝑈))〉}) |
40 | eqid 2726 | . . 3 ⊢ (𝐶 ↾cat 𝐻) = (𝐶 ↾cat 𝐻) | |
41 | fvexd 6908 | . . . 4 ⊢ (𝜑 → (RngCat‘𝑈) ∈ V) | |
42 | 1, 41 | eqeltrid 2830 | . . 3 ⊢ (𝜑 → 𝐶 ∈ V) |
43 | 23, 17 | rhmresfn 20622 | . . 3 ⊢ (𝜑 → 𝐻 Fn (𝐵 × 𝐵)) |
44 | 40, 42, 26, 43 | rescval2 17839 | . 2 ⊢ (𝜑 → (𝐶 ↾cat 𝐻) = ((𝐶 ↾s 𝐵) sSet 〈(Hom ‘ndx), 𝐻〉)) |
45 | eqid 2726 | . . 3 ⊢ (RingCat‘𝑈) = (RingCat‘𝑈) | |
46 | 45, 2, 23, 17, 5 | dfringc2 20631 | . 2 ⊢ (𝜑 → (RingCat‘𝑈) = {〈(Base‘ndx), 𝐵〉, 〈(Hom ‘ndx), 𝐻〉, 〈(comp‘ndx), (comp‘(ExtStrCat‘𝑈))〉}) |
47 | 39, 44, 46 | 3eqtr4d 2776 | 1 ⊢ (𝜑 → (𝐶 ↾cat 𝐻) = (RingCat‘𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 Vcvv 3462 ∩ cin 3945 {ctp 4627 〈cop 4629 × cxp 5672 ↾ cres 5676 Fun wfun 6540 Fn wfn 6541 ‘cfv 6546 (class class class)co 7416 sSet csts 17160 ndxcnx 17190 Basecbs 17208 ↾s cress 17237 Hom chom 17272 compcco 17273 ↾cat cresc 17819 ExtStrCatcestrc 18140 Rngcrng 20131 Ringcrg 20212 RngHom crnghm 20412 RingHom crh 20447 RngCatcrngc 20590 RingCatcringc 20619 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5282 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 ax-cnex 11205 ax-resscn 11206 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-addrcl 11210 ax-mulcl 11211 ax-mulrcl 11212 ax-mulcom 11213 ax-addass 11214 ax-mulass 11215 ax-distr 11216 ax-i2m1 11217 ax-1ne0 11218 ax-1rid 11219 ax-rnegex 11220 ax-rrecex 11221 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 ax-pre-ltadd 11225 ax-pre-mulgt0 11226 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4906 df-iun 4995 df-br 5146 df-opab 5208 df-mpt 5229 df-tr 5263 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6304 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7995 df-2nd 7996 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-1o 8488 df-er 8726 df-map 8849 df-en 8967 df-dom 8968 df-sdom 8969 df-fin 8970 df-pnf 11291 df-mnf 11292 df-xr 11293 df-ltxr 11294 df-le 11295 df-sub 11487 df-neg 11488 df-nn 12259 df-2 12321 df-3 12322 df-4 12323 df-5 12324 df-6 12325 df-7 12326 df-8 12327 df-9 12328 df-n0 12519 df-z 12605 df-dec 12724 df-uz 12869 df-fz 13533 df-struct 17144 df-sets 17161 df-slot 17179 df-ndx 17191 df-base 17209 df-ress 17238 df-plusg 17274 df-hom 17285 df-cco 17286 df-0g 17451 df-resc 17822 df-estrc 18141 df-mgm 18628 df-sgrp 18707 df-mnd 18723 df-mhm 18768 df-grp 18926 df-minusg 18927 df-ghm 19203 df-cmn 19776 df-abl 19777 df-mgp 20114 df-rng 20132 df-ur 20161 df-ring 20214 df-rnghm 20414 df-rhm 20450 df-rngc 20591 df-ringc 20620 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |