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Mirrors > Home > MPE Home > Th. List > Mathboxes > rhmsubcALTV | Structured version Visualization version GIF version |
Description: According to df-subc 17860, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 (see subcssc 17891 and subcss2 17894). Therefore, the set of unital ring homomorphisms is a "subcategory" of the category of non-unital rings. (Contributed by AV, 2-Mar-2020.) (New usage is discouraged.) |
Ref | Expression |
---|---|
rngcrescrhmALTV.u | ⊢ (𝜑 → 𝑈 ∈ 𝑉) |
rngcrescrhmALTV.c | ⊢ 𝐶 = (RngCatALTV‘𝑈) |
rngcrescrhmALTV.r | ⊢ (𝜑 → 𝑅 = (Ring ∩ 𝑈)) |
rngcrescrhmALTV.h | ⊢ 𝐻 = ( RingHom ↾ (𝑅 × 𝑅)) |
Ref | Expression |
---|---|
rhmsubcALTV | ⊢ (𝜑 → 𝐻 ∈ (Subcat‘(RngCatALTV‘𝑈))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rngcrescrhmALTV.u | . . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑉) | |
2 | rngcrescrhmALTV.r | . . . 4 ⊢ (𝜑 → 𝑅 = (Ring ∩ 𝑈)) | |
3 | eqidd 2736 | . . . 4 ⊢ (𝜑 → (Rng ∩ 𝑈) = (Rng ∩ 𝑈)) | |
4 | 1, 2, 3 | rhmsscrnghm 20682 | . . 3 ⊢ (𝜑 → ( RingHom ↾ (𝑅 × 𝑅)) ⊆cat ( RngHom ↾ ((Rng ∩ 𝑈) × (Rng ∩ 𝑈)))) |
5 | rngcrescrhmALTV.h | . . . 4 ⊢ 𝐻 = ( RingHom ↾ (𝑅 × 𝑅)) | |
6 | 5 | a1i 11 | . . 3 ⊢ (𝜑 → 𝐻 = ( RingHom ↾ (𝑅 × 𝑅))) |
7 | eqid 2735 | . . . 4 ⊢ (RngCatALTV‘𝑈) = (RngCatALTV‘𝑈) | |
8 | eqid 2735 | . . . 4 ⊢ (Rng ∩ 𝑈) = (Rng ∩ 𝑈) | |
9 | eqid 2735 | . . . 4 ⊢ (Homf ‘(RngCatALTV‘𝑈)) = (Homf ‘(RngCatALTV‘𝑈)) | |
10 | 7, 8, 1, 9 | rngchomrnghmresALTV 48123 | . . 3 ⊢ (𝜑 → (Homf ‘(RngCatALTV‘𝑈)) = ( RngHom ↾ ((Rng ∩ 𝑈) × (Rng ∩ 𝑈)))) |
11 | 4, 6, 10 | 3brtr4d 5180 | . 2 ⊢ (𝜑 → 𝐻 ⊆cat (Homf ‘(RngCatALTV‘𝑈))) |
12 | rngcrescrhmALTV.c | . . . . 5 ⊢ 𝐶 = (RngCatALTV‘𝑈) | |
13 | 1, 12, 2, 5 | rhmsubcALTVlem3 48127 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑅) → ((Id‘(RngCatALTV‘𝑈))‘𝑥) ∈ (𝑥𝐻𝑥)) |
14 | 1, 12, 2, 5 | rhmsubcALTVlem4 48128 | . . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑅) ∧ (𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑅)) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧))) → (𝑔(〈𝑥, 𝑦〉(comp‘(RngCatALTV‘𝑈))𝑧)𝑓) ∈ (𝑥𝐻𝑧)) |
15 | 14 | ralrimivva 3200 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑅) ∧ (𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑅)) → ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(〈𝑥, 𝑦〉(comp‘(RngCatALTV‘𝑈))𝑧)𝑓) ∈ (𝑥𝐻𝑧)) |
16 | 15 | ralrimivva 3200 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑅) → ∀𝑦 ∈ 𝑅 ∀𝑧 ∈ 𝑅 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(〈𝑥, 𝑦〉(comp‘(RngCatALTV‘𝑈))𝑧)𝑓) ∈ (𝑥𝐻𝑧)) |
17 | 13, 16 | jca 511 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑅) → (((Id‘(RngCatALTV‘𝑈))‘𝑥) ∈ (𝑥𝐻𝑥) ∧ ∀𝑦 ∈ 𝑅 ∀𝑧 ∈ 𝑅 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(〈𝑥, 𝑦〉(comp‘(RngCatALTV‘𝑈))𝑧)𝑓) ∈ (𝑥𝐻𝑧))) |
18 | 17 | ralrimiva 3144 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝑅 (((Id‘(RngCatALTV‘𝑈))‘𝑥) ∈ (𝑥𝐻𝑥) ∧ ∀𝑦 ∈ 𝑅 ∀𝑧 ∈ 𝑅 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(〈𝑥, 𝑦〉(comp‘(RngCatALTV‘𝑈))𝑧)𝑓) ∈ (𝑥𝐻𝑧))) |
19 | eqid 2735 | . . 3 ⊢ (Id‘(RngCatALTV‘𝑈)) = (Id‘(RngCatALTV‘𝑈)) | |
20 | eqid 2735 | . . 3 ⊢ (comp‘(RngCatALTV‘𝑈)) = (comp‘(RngCatALTV‘𝑈)) | |
21 | 7 | rngccatALTV 48117 | . . . 4 ⊢ (𝑈 ∈ 𝑉 → (RngCatALTV‘𝑈) ∈ Cat) |
22 | 1, 21 | syl 17 | . . 3 ⊢ (𝜑 → (RngCatALTV‘𝑈) ∈ Cat) |
23 | 1, 12, 2, 5 | rhmsubcALTVlem1 48125 | . . 3 ⊢ (𝜑 → 𝐻 Fn (𝑅 × 𝑅)) |
24 | 9, 19, 20, 22, 23 | issubc2 17887 | . 2 ⊢ (𝜑 → (𝐻 ∈ (Subcat‘(RngCatALTV‘𝑈)) ↔ (𝐻 ⊆cat (Homf ‘(RngCatALTV‘𝑈)) ∧ ∀𝑥 ∈ 𝑅 (((Id‘(RngCatALTV‘𝑈))‘𝑥) ∈ (𝑥𝐻𝑥) ∧ ∀𝑦 ∈ 𝑅 ∀𝑧 ∈ 𝑅 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(〈𝑥, 𝑦〉(comp‘(RngCatALTV‘𝑈))𝑧)𝑓) ∈ (𝑥𝐻𝑧))))) |
25 | 11, 18, 24 | mpbir2and 713 | 1 ⊢ (𝜑 → 𝐻 ∈ (Subcat‘(RngCatALTV‘𝑈))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∀wral 3059 ∩ cin 3962 〈cop 4637 class class class wbr 5148 × cxp 5687 ↾ cres 5691 ‘cfv 6563 (class class class)co 7431 compcco 17310 Catccat 17709 Idccid 17710 Homf chomf 17711 ⊆cat cssc 17855 Subcatcsubc 17857 Rngcrng 20170 Ringcrg 20251 RngHom crnghm 20451 RingHom crh 20486 RngCatALTVcrngcALTV 48107 |
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-pm 8868 df-ixp 8937 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-plusg 17311 df-hom 17322 df-cco 17323 df-0g 17488 df-cat 17713 df-cid 17714 df-homf 17715 df-ssc 17858 df-subc 17860 df-mgm 18666 df-mgmhm 18718 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-rngcALTV 48108 |
This theorem is referenced by: rhmsubcALTVcat 48130 |
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