Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rhmsubcsetc | Structured version Visualization version GIF version |
Description: The unital ring homomorphisms between unital rings (in a universe) are a subcategory of the category of extensible structures. (Contributed by AV, 9-Mar-2020.) |
Ref | Expression |
---|---|
rhmsubcsetc.c | ⊢ 𝐶 = (ExtStrCat‘𝑈) |
rhmsubcsetc.u | ⊢ (𝜑 → 𝑈 ∈ 𝑉) |
rhmsubcsetc.b | ⊢ (𝜑 → 𝐵 = (Ring ∩ 𝑈)) |
rhmsubcsetc.h | ⊢ (𝜑 → 𝐻 = ( RingHom ↾ (𝐵 × 𝐵))) |
Ref | Expression |
---|---|
rhmsubcsetc | ⊢ (𝜑 → 𝐻 ∈ (Subcat‘𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rhmsubcsetc.u | . . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑉) | |
2 | rhmsubcsetc.b | . . . 4 ⊢ (𝜑 → 𝐵 = (Ring ∩ 𝑈)) | |
3 | 1, 2 | rhmsscmap 44311 | . . 3 ⊢ (𝜑 → ( RingHom ↾ (𝐵 × 𝐵)) ⊆cat (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑m (Base‘𝑥)))) |
4 | rhmsubcsetc.h | . . 3 ⊢ (𝜑 → 𝐻 = ( RingHom ↾ (𝐵 × 𝐵))) | |
5 | rhmsubcsetc.c | . . . . 5 ⊢ 𝐶 = (ExtStrCat‘𝑈) | |
6 | eqid 2821 | . . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
7 | 5, 1, 6 | estrchomfeqhom 17386 | . . . 4 ⊢ (𝜑 → (Homf ‘𝐶) = (Hom ‘𝐶)) |
8 | 5, 1, 6 | estrchomfval 17376 | . . . 4 ⊢ (𝜑 → (Hom ‘𝐶) = (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑m (Base‘𝑥)))) |
9 | 7, 8 | eqtrd 2856 | . . 3 ⊢ (𝜑 → (Homf ‘𝐶) = (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑m (Base‘𝑥)))) |
10 | 3, 4, 9 | 3brtr4d 5098 | . 2 ⊢ (𝜑 → 𝐻 ⊆cat (Homf ‘𝐶)) |
11 | 5, 1, 2, 4 | rhmsubcsetclem1 44312 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((Id‘𝐶)‘𝑥) ∈ (𝑥𝐻𝑥)) |
12 | 5, 1, 2, 4 | rhmsubcsetclem2 44313 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(〈𝑥, 𝑦〉(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐻𝑧)) |
13 | 11, 12 | jca 514 | . . 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 3182 | . 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 2821 | . . 3 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶) | |
16 | eqid 2821 | . . 3 ⊢ (Id‘𝐶) = (Id‘𝐶) | |
17 | eqid 2821 | . . 3 ⊢ (comp‘𝐶) = (comp‘𝐶) | |
18 | 5 | estrccat 17383 | . . . 4 ⊢ (𝑈 ∈ 𝑉 → 𝐶 ∈ Cat) |
19 | 1, 18 | syl 17 | . . 3 ⊢ (𝜑 → 𝐶 ∈ Cat) |
20 | incom 4178 | . . . . 5 ⊢ (Ring ∩ 𝑈) = (𝑈 ∩ Ring) | |
21 | 2, 20 | syl6eq 2872 | . . . 4 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Ring)) |
22 | 21, 4 | rhmresfn 44300 | . . 3 ⊢ (𝜑 → 𝐻 Fn (𝐵 × 𝐵)) |
23 | 15, 16, 17, 19, 22 | issubc2 17106 | . 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 711 | 1 ⊢ (𝜑 → 𝐻 ∈ (Subcat‘𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3138 ∩ cin 3935 〈cop 4573 class class class wbr 5066 × cxp 5553 ↾ cres 5557 ‘cfv 6355 (class class class)co 7156 ∈ cmpo 7158 ↑m cmap 8406 Basecbs 16483 Hom chom 16576 compcco 16577 Catccat 16935 Idccid 16936 Homf chomf 16937 ⊆cat cssc 17077 Subcatcsubc 17079 ExtStrCatcestrc 17372 Ringcrg 19297 RingHom crh 19464 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-map 8408 df-pm 8409 df-ixp 8462 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-fz 12894 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-hom 16589 df-cco 16590 df-0g 16715 df-cat 16939 df-cid 16940 df-homf 16941 df-ssc 17080 df-resc 17081 df-subc 17082 df-estrc 17373 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-mhm 17956 df-grp 18106 df-ghm 18356 df-mgp 19240 df-ur 19252 df-ring 19299 df-rnghom 19467 df-ringc 44296 |
This theorem is referenced by: ringccat 44315 ringcid 44316 funcringcsetc 44326 |
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