Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rhmsubcrngc | Structured version Visualization version GIF version |
Description: The unital ring homomorphisms between unital rings (in a universe) are a subcategory of the category of non-unital rings. (Contributed by AV, 12-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 |
---|---|
rhmsubcrngc | ⊢ (𝜑 → 𝐻 ∈ (Subcat‘𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rhmsubcrngc.u | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ 𝑉) | |
2 | rhmsubcrngc.b | . . . . 5 ⊢ (𝜑 → 𝐵 = (Ring ∩ 𝑈)) | |
3 | eqid 2610 | . . . . . . 7 ⊢ (RngCat‘𝑈) = (RngCat‘𝑈) | |
4 | eqid 2610 | . . . . . . 7 ⊢ (Base‘(RngCat‘𝑈)) = (Base‘(RngCat‘𝑈)) | |
5 | 3, 4, 1 | rngcbas 41757 | . . . . . 6 ⊢ (𝜑 → (Base‘(RngCat‘𝑈)) = (𝑈 ∩ Rng)) |
6 | incom 3767 | . . . . . 6 ⊢ (𝑈 ∩ Rng) = (Rng ∩ 𝑈) | |
7 | 5, 6 | syl6eq 2660 | . . . . 5 ⊢ (𝜑 → (Base‘(RngCat‘𝑈)) = (Rng ∩ 𝑈)) |
8 | 1, 2, 7 | rhmsscrnghm 41818 | . . . 4 ⊢ (𝜑 → ( RingHom ↾ (𝐵 × 𝐵)) ⊆cat ( RngHomo ↾ ((Base‘(RngCat‘𝑈)) × (Base‘(RngCat‘𝑈))))) |
9 | rhmsubcrngc.c | . . . . . . . 8 ⊢ 𝐶 = (RngCat‘𝑈) | |
10 | 9 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → 𝐶 = (RngCat‘𝑈)) |
11 | 10 | fveq2d 6107 | . . . . . 6 ⊢ (𝜑 → (Base‘𝐶) = (Base‘(RngCat‘𝑈))) |
12 | 11 | sqxpeqd 5065 | . . . . 5 ⊢ (𝜑 → ((Base‘𝐶) × (Base‘𝐶)) = ((Base‘(RngCat‘𝑈)) × (Base‘(RngCat‘𝑈)))) |
13 | 12 | reseq2d 5317 | . . . 4 ⊢ (𝜑 → ( RngHomo ↾ ((Base‘𝐶) × (Base‘𝐶))) = ( RngHomo ↾ ((Base‘(RngCat‘𝑈)) × (Base‘(RngCat‘𝑈))))) |
14 | 8, 13 | breqtrrd 4611 | . . 3 ⊢ (𝜑 → ( RingHom ↾ (𝐵 × 𝐵)) ⊆cat ( RngHomo ↾ ((Base‘𝐶) × (Base‘𝐶)))) |
15 | rhmsubcrngc.h | . . 3 ⊢ (𝜑 → 𝐻 = ( RingHom ↾ (𝐵 × 𝐵))) | |
16 | eqid 2610 | . . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
17 | 9, 16, 1 | rngchomfeqhom 41761 | . . . 4 ⊢ (𝜑 → (Homf ‘𝐶) = (Hom ‘𝐶)) |
18 | eqid 2610 | . . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
19 | 9, 16, 1, 18 | rngchomfval 41758 | . . . 4 ⊢ (𝜑 → (Hom ‘𝐶) = ( RngHomo ↾ ((Base‘𝐶) × (Base‘𝐶)))) |
20 | 17, 19 | eqtrd 2644 | . . 3 ⊢ (𝜑 → (Homf ‘𝐶) = ( RngHomo ↾ ((Base‘𝐶) × (Base‘𝐶)))) |
21 | 14, 15, 20 | 3brtr4d 4615 | . 2 ⊢ (𝜑 → 𝐻 ⊆cat (Homf ‘𝐶)) |
22 | 9, 1, 2, 15 | rhmsubcrngclem1 41819 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((Id‘𝐶)‘𝑥) ∈ (𝑥𝐻𝑥)) |
23 | 9, 1, 2, 15 | rhmsubcrngclem2 41820 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(〈𝑥, 𝑦〉(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐻𝑧)) |
24 | 22, 23 | 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))) |
25 | 24 | 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))) |
26 | eqid 2610 | . . 3 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶) | |
27 | eqid 2610 | . . 3 ⊢ (Id‘𝐶) = (Id‘𝐶) | |
28 | eqid 2610 | . . 3 ⊢ (comp‘𝐶) = (comp‘𝐶) | |
29 | 9 | rngccat 41770 | . . . 4 ⊢ (𝑈 ∈ 𝑉 → 𝐶 ∈ Cat) |
30 | 1, 29 | syl 17 | . . 3 ⊢ (𝜑 → 𝐶 ∈ Cat) |
31 | incom 3767 | . . . . 5 ⊢ (Ring ∩ 𝑈) = (𝑈 ∩ Ring) | |
32 | 2, 31 | syl6eq 2660 | . . . 4 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Ring)) |
33 | 32, 15 | rhmresfn 41801 | . . 3 ⊢ (𝜑 → 𝐻 Fn (𝐵 × 𝐵)) |
34 | 26, 27, 28, 30, 33 | 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))))) |
35 | 21, 25, 34 | 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 Basecbs 15695 Hom chom 15779 compcco 15780 Catccat 16148 Idccid 16149 Homf chomf 16150 ⊆cat cssc 16290 Subcatcsubc 16292 Ringcrg 18370 RingHom crh 18535 Rngcrng 41664 RngHomo crngh 41675 RngCatcrngc 41749 |
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-minusg 17249 df-ghm 17481 df-cmn 18018 df-abl 18019 df-mgp 18313 df-ur 18325 df-ring 18372 df-rnghom 18538 df-mgmhm 41569 df-rng0 41665 df-rnghomo 41677 df-rngc 41751 df-ringc 41797 |
This theorem is referenced by: (None) |
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