![]() |
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 2736 | . . . 4 ⊢ (𝜑 → (𝑈 ∩ Rng) = (𝑈 ∩ Rng)) | |
4 | eqidd 2736 | . . . 4 ⊢ (𝜑 → ( RngHom ↾ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng))) = ( RngHom ↾ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)))) | |
5 | eqidd 2736 | . . . 4 ⊢ (𝜑 → (comp‘(ExtStrCat‘𝑈)) = (comp‘(ExtStrCat‘𝑈))) | |
6 | 1, 2, 3, 4, 5 | dfrngc2 20645 | . . 3 ⊢ (𝜑 → 𝐶 = {〈(Base‘ndx), (𝑈 ∩ Rng)〉, 〈(Hom ‘ndx), ( RngHom ↾ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)))〉, 〈(comp‘ndx), (comp‘(ExtStrCat‘𝑈))〉}) |
7 | inex1g 5325 | . . . 4 ⊢ (𝑈 ∈ 𝑉 → (𝑈 ∩ Rng) ∈ V) | |
8 | 2, 7 | syl 17 | . . 3 ⊢ (𝜑 → (𝑈 ∩ Rng) ∈ V) |
9 | rnghmfn 20456 | . . . . 5 ⊢ RngHom Fn (Rng × Rng) | |
10 | fnfun 6669 | . . . . 5 ⊢ ( RngHom Fn (Rng × Rng) → Fun RngHom ) | |
11 | 9, 10 | mp1i 13 | . . . 4 ⊢ (𝜑 → Fun RngHom ) |
12 | sqxpexg 7774 | . . . . 5 ⊢ ((𝑈 ∩ Rng) ∈ V → ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)) ∈ V) | |
13 | 8, 12 | syl 17 | . . . 4 ⊢ (𝜑 → ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)) ∈ V) |
14 | resfunexg 7235 | . . . 4 ⊢ ((Fun RngHom ∧ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)) ∈ V) → ( RngHom ↾ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng))) ∈ V) | |
15 | 11, 13, 14 | syl2anc 584 | . . 3 ⊢ (𝜑 → ( RngHom ↾ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng))) ∈ V) |
16 | fvexd 6922 | . . 3 ⊢ (𝜑 → (comp‘(ExtStrCat‘𝑈)) ∈ V) | |
17 | rhmsubcrngc.h | . . . 4 ⊢ (𝜑 → 𝐻 = ( RingHom ↾ (𝐵 × 𝐵))) | |
18 | rhmfn 20516 | . . . . . 6 ⊢ RingHom Fn (Ring × Ring) | |
19 | fnfun 6669 | . . . . . 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 4217 | . . . . . . . 8 ⊢ (Ring ∩ 𝑈) = (𝑈 ∩ Ring) | |
23 | 21, 22 | eqtrdi 2791 | . . . . . . 7 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Ring)) |
24 | inex1g 5325 | . . . . . . . 8 ⊢ (𝑈 ∈ 𝑉 → (𝑈 ∩ Ring) ∈ V) | |
25 | 2, 24 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (𝑈 ∩ Ring) ∈ V) |
26 | 23, 25 | eqeltrd 2839 | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ V) |
27 | sqxpexg 7774 | . . . . . 6 ⊢ (𝐵 ∈ V → (𝐵 × 𝐵) ∈ V) | |
28 | 26, 27 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝐵 × 𝐵) ∈ V) |
29 | resfunexg 7235 | . . . . 5 ⊢ ((Fun RingHom ∧ (𝐵 × 𝐵) ∈ V) → ( RingHom ↾ (𝐵 × 𝐵)) ∈ V) | |
30 | 20, 28, 29 | syl2anc 584 | . . . 4 ⊢ (𝜑 → ( RingHom ↾ (𝐵 × 𝐵)) ∈ V) |
31 | 17, 30 | eqeltrd 2839 | . . 3 ⊢ (𝜑 → 𝐻 ∈ V) |
32 | ringrng 20299 | . . . . . . 7 ⊢ (𝑟 ∈ Ring → 𝑟 ∈ Rng) | |
33 | 32 | a1i 11 | . . . . . 6 ⊢ (𝜑 → (𝑟 ∈ Ring → 𝑟 ∈ Rng)) |
34 | 33 | ssrdv 4001 | . . . . 5 ⊢ (𝜑 → Ring ⊆ Rng) |
35 | 34 | ssrind 4252 | . . . 4 ⊢ (𝜑 → (Ring ∩ 𝑈) ⊆ (Rng ∩ 𝑈)) |
36 | incom 4217 | . . . . 5 ⊢ (𝑈 ∩ Rng) = (Rng ∩ 𝑈) | |
37 | 36 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝑈 ∩ Rng) = (Rng ∩ 𝑈)) |
38 | 35, 21, 37 | 3sstr4d 4043 | . . 3 ⊢ (𝜑 → 𝐵 ⊆ (𝑈 ∩ Rng)) |
39 | 6, 8, 15, 16, 31, 38 | estrres 18195 | . 2 ⊢ (𝜑 → ((𝐶 ↾s 𝐵) sSet 〈(Hom ‘ndx), 𝐻〉) = {〈(Base‘ndx), 𝐵〉, 〈(Hom ‘ndx), 𝐻〉, 〈(comp‘ndx), (comp‘(ExtStrCat‘𝑈))〉}) |
40 | eqid 2735 | . . 3 ⊢ (𝐶 ↾cat 𝐻) = (𝐶 ↾cat 𝐻) | |
41 | fvexd 6922 | . . . 4 ⊢ (𝜑 → (RngCat‘𝑈) ∈ V) | |
42 | 1, 41 | eqeltrid 2843 | . . 3 ⊢ (𝜑 → 𝐶 ∈ V) |
43 | 23, 17 | rhmresfn 20665 | . . 3 ⊢ (𝜑 → 𝐻 Fn (𝐵 × 𝐵)) |
44 | 40, 42, 26, 43 | rescval2 17876 | . 2 ⊢ (𝜑 → (𝐶 ↾cat 𝐻) = ((𝐶 ↾s 𝐵) sSet 〈(Hom ‘ndx), 𝐻〉)) |
45 | eqid 2735 | . . 3 ⊢ (RingCat‘𝑈) = (RingCat‘𝑈) | |
46 | 45, 2, 23, 17, 5 | dfringc2 20674 | . 2 ⊢ (𝜑 → (RingCat‘𝑈) = {〈(Base‘ndx), 𝐵〉, 〈(Hom ‘ndx), 𝐻〉, 〈(comp‘ndx), (comp‘(ExtStrCat‘𝑈))〉}) |
47 | 39, 44, 46 | 3eqtr4d 2785 | 1 ⊢ (𝜑 → (𝐶 ↾cat 𝐻) = (RingCat‘𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ∩ cin 3962 {ctp 4635 〈cop 4637 × cxp 5687 ↾ cres 5691 Fun wfun 6557 Fn wfn 6558 ‘cfv 6563 (class class class)co 7431 sSet csts 17197 ndxcnx 17227 Basecbs 17245 ↾s cress 17274 Hom chom 17309 compcco 17310 ↾cat cresc 17856 ExtStrCatcestrc 18177 Rngcrng 20170 Ringcrg 20251 RngHom crnghm 20451 RingHom crh 20486 RngCatcrngc 20633 RingCatcringc 20662 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-fz 13545 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-hom 17322 df-cco 17323 df-0g 17488 df-resc 17859 df-estrc 18178 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-mhm 18809 df-grp 18967 df-minusg 18968 df-ghm 19244 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-rnghm 20453 df-rhm 20489 df-rngc 20634 df-ringc 20663 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |