Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
||
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 41766 | . . 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 2610 | . . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
7 | 5, 1, 6 | estrchomfeqhom 16599 | . . . 4 ⊢ (𝜑 → (Homf ‘𝐶) = (Hom ‘𝐶)) |
8 | 5, 1, 6 | estrchomfval 16589 | . . . 4 ⊢ (𝜑 → (Hom ‘𝐶) = (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))) |
9 | 7, 8 | eqtrd 2644 | . . 3 ⊢ (𝜑 → (Homf ‘𝐶) = (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))) |
10 | 3, 4, 9 | 3brtr4d 4615 | . 2 ⊢ (𝜑 → 𝐻 ⊆cat (Homf ‘𝐶)) |
11 | 5, 1, 2, 4 | rnghmsubcsetclem1 41767 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((Id‘𝐶)‘𝑥) ∈ (𝑥𝐻𝑥)) |
12 | 5, 1, 2, 4 | rnghmsubcsetclem2 41768 | . . . 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 2949 | . 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 2610 | . . 3 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶) | |
16 | eqid 2610 | . . 3 ⊢ (Id‘𝐶) = (Id‘𝐶) | |
17 | eqid 2610 | . . 3 ⊢ (comp‘𝐶) = (comp‘𝐶) | |
18 | 5 | estrccat 16596 | . . . 4 ⊢ (𝑈 ∈ 𝑉 → 𝐶 ∈ Cat) |
19 | 1, 18 | syl 17 | . . 3 ⊢ (𝜑 → 𝐶 ∈ Cat) |
20 | incom 3767 | . . . . 5 ⊢ (Rng ∩ 𝑈) = (𝑈 ∩ Rng) | |
21 | 2, 20 | syl6eq 2660 | . . . 4 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Rng)) |
22 | 21, 4 | rnghmresfn 41755 | . . 3 ⊢ (𝜑 → 𝐻 Fn (𝐵 × 𝐵)) |
23 | 15, 16, 17, 19, 22 | issubc2 16319 | . 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 959 | 1 ⊢ (𝜑 → 𝐻 ∈ (Subcat‘𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ∩ cin 3539 〈cop 4131 class class class wbr 4583 × cxp 5036 ↾ cres 5040 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 ↑𝑚 cmap 7744 Basecbs 15695 Hom chom 15779 compcco 15780 Catccat 16148 Idccid 16149 Homf chomf 16150 ⊆cat cssc 16290 Subcatcsubc 16292 ExtStrCatcestrc 16585 Rngcrng 41664 RngHomo crngh 41675 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-fz 12198 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-hom 15793 df-cco 15794 df-0g 15925 df-cat 16152 df-cid 16153 df-homf 16154 df-ssc 16293 df-resc 16294 df-subc 16295 df-estrc 16586 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-mhm 17158 df-grp 17248 df-ghm 17481 df-abl 18019 df-mgp 18313 df-mgmhm 41569 df-rng0 41665 df-rnghomo 41677 df-rngc 41751 |
This theorem is referenced by: rngccat 41770 rngcid 41771 rngcifuestrc 41789 funcrngcsetc 41790 |
Copyright terms: Public domain | W3C validator |