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Mirrors > Home > MPE Home > Th. List > Mathboxes > rnghmsubcsetc | Structured version Visualization version GIF version |
Description: The non-unital ring homomorphisms between non-unital rings (in a universe) are a subcategory of the category of extensible structures. (Contributed by AV, 9-Mar-2020.) |
Ref | Expression |
---|---|
rnghmsubcsetc.c | ⊢ 𝐶 = (ExtStrCat‘𝑈) |
rnghmsubcsetc.u | ⊢ (𝜑 → 𝑈 ∈ 𝑉) |
rnghmsubcsetc.b | ⊢ (𝜑 → 𝐵 = (Rng ∩ 𝑈)) |
rnghmsubcsetc.h | ⊢ (𝜑 → 𝐻 = ( RngHomo ↾ (𝐵 × 𝐵))) |
Ref | Expression |
---|---|
rnghmsubcsetc | ⊢ (𝜑 → 𝐻 ∈ (Subcat‘𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rnghmsubcsetc.u | . . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑉) | |
2 | rnghmsubcsetc.b | . . . 4 ⊢ (𝜑 → 𝐵 = (Rng ∩ 𝑈)) | |
3 | 1, 2 | rnghmsscmap 42299 | . . 3 ⊢ (𝜑 → ( RngHomo ↾ (𝐵 × 𝐵)) ⊆cat (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))) |
4 | rnghmsubcsetc.h | . . 3 ⊢ (𝜑 → 𝐻 = ( RngHomo ↾ (𝐵 × 𝐵))) | |
5 | rnghmsubcsetc.c | . . . . 5 ⊢ 𝐶 = (ExtStrCat‘𝑈) | |
6 | eqid 2651 | . . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
7 | 5, 1, 6 | estrchomfeqhom 16823 | . . . 4 ⊢ (𝜑 → (Homf ‘𝐶) = (Hom ‘𝐶)) |
8 | 5, 1, 6 | estrchomfval 16813 | . . . 4 ⊢ (𝜑 → (Hom ‘𝐶) = (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))) |
9 | 7, 8 | eqtrd 2685 | . . 3 ⊢ (𝜑 → (Homf ‘𝐶) = (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))) |
10 | 3, 4, 9 | 3brtr4d 4717 | . 2 ⊢ (𝜑 → 𝐻 ⊆cat (Homf ‘𝐶)) |
11 | 5, 1, 2, 4 | rnghmsubcsetclem1 42300 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((Id‘𝐶)‘𝑥) ∈ (𝑥𝐻𝑥)) |
12 | 5, 1, 2, 4 | rnghmsubcsetclem2 42301 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(〈𝑥, 𝑦〉(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐻𝑧)) |
13 | 11, 12 | jca 553 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (((Id‘𝐶)‘𝑥) ∈ (𝑥𝐻𝑥) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(〈𝑥, 𝑦〉(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐻𝑧))) |
14 | 13 | ralrimiva 2995 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 (((Id‘𝐶)‘𝑥) ∈ (𝑥𝐻𝑥) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(〈𝑥, 𝑦〉(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐻𝑧))) |
15 | eqid 2651 | . . 3 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶) | |
16 | eqid 2651 | . . 3 ⊢ (Id‘𝐶) = (Id‘𝐶) | |
17 | eqid 2651 | . . 3 ⊢ (comp‘𝐶) = (comp‘𝐶) | |
18 | 5 | estrccat 16820 | . . . 4 ⊢ (𝑈 ∈ 𝑉 → 𝐶 ∈ Cat) |
19 | 1, 18 | syl 17 | . . 3 ⊢ (𝜑 → 𝐶 ∈ Cat) |
20 | incom 3838 | . . . . 5 ⊢ (Rng ∩ 𝑈) = (𝑈 ∩ Rng) | |
21 | 2, 20 | syl6eq 2701 | . . . 4 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Rng)) |
22 | 21, 4 | rnghmresfn 42288 | . . 3 ⊢ (𝜑 → 𝐻 Fn (𝐵 × 𝐵)) |
23 | 15, 16, 17, 19, 22 | issubc2 16543 | . 2 ⊢ (𝜑 → (𝐻 ∈ (Subcat‘𝐶) ↔ (𝐻 ⊆cat (Homf ‘𝐶) ∧ ∀𝑥 ∈ 𝐵 (((Id‘𝐶)‘𝑥) ∈ (𝑥𝐻𝑥) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(〈𝑥, 𝑦〉(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐻𝑧))))) |
24 | 10, 14, 23 | mpbir2and 977 | 1 ⊢ (𝜑 → 𝐻 ∈ (Subcat‘𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1523 ∈ wcel 2030 ∀wral 2941 ∩ cin 3606 〈cop 4216 class class class wbr 4685 × cxp 5141 ↾ cres 5145 ‘cfv 5926 (class class class)co 6690 ↦ cmpt2 6692 ↑𝑚 cmap 7899 Basecbs 15904 Hom chom 15999 compcco 16000 Catccat 16372 Idccid 16373 Homf chomf 16374 ⊆cat cssc 16514 Subcatcsubc 16516 ExtStrCatcestrc 16809 Rngcrng 42199 RngHomo crngh 42210 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-fal 1529 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-1st 7210 df-2nd 7211 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-oadd 7609 df-er 7787 df-map 7901 df-pm 7902 df-ixp 7951 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-nn 11059 df-2 11117 df-3 11118 df-4 11119 df-5 11120 df-6 11121 df-7 11122 df-8 11123 df-9 11124 df-n0 11331 df-z 11416 df-dec 11532 df-uz 11726 df-fz 12365 df-struct 15906 df-ndx 15907 df-slot 15908 df-base 15910 df-sets 15911 df-ress 15912 df-plusg 16001 df-hom 16013 df-cco 16014 df-0g 16149 df-cat 16376 df-cid 16377 df-homf 16378 df-ssc 16517 df-resc 16518 df-subc 16519 df-estrc 16810 df-mgm 17289 df-sgrp 17331 df-mnd 17342 df-mhm 17382 df-grp 17472 df-ghm 17705 df-abl 18242 df-mgp 18536 df-mgmhm 42104 df-rng0 42200 df-rnghomo 42212 df-rngc 42284 |
This theorem is referenced by: rngccat 42303 rngcid 42304 rngcifuestrc 42322 funcrngcsetc 42323 |
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