Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rngcresringcat | Structured version Visualization version GIF version |
Description: The restriction of the category of non-unital rings to the set of unital ring homomorphisms is the category of unital rings. (Contributed by AV, 16-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 |
---|---|
rngcresringcat | ⊢ (𝜑 → (𝐶 ↾cat 𝐻) = (RingCat‘𝑈)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rhmsubcrngc.c | . . . 4 ⊢ 𝐶 = (RngCat‘𝑈) | |
2 | rhmsubcrngc.u | . . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑉) | |
3 | eqidd 2611 | . . . 4 ⊢ (𝜑 → (𝑈 ∩ Rng) = (𝑈 ∩ Rng)) | |
4 | eqidd 2611 | . . . 4 ⊢ (𝜑 → ( RngHomo ↾ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng))) = ( RngHomo ↾ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)))) | |
5 | eqidd 2611 | . . . 4 ⊢ (𝜑 → (comp‘(ExtStrCat‘𝑈)) = (comp‘(ExtStrCat‘𝑈))) | |
6 | 1, 2, 3, 4, 5 | dfrngc2 41764 | . . 3 ⊢ (𝜑 → 𝐶 = {〈(Base‘ndx), (𝑈 ∩ Rng)〉, 〈(Hom ‘ndx), ( RngHomo ↾ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)))〉, 〈(comp‘ndx), (comp‘(ExtStrCat‘𝑈))〉}) |
7 | inex1g 4729 | . . . 4 ⊢ (𝑈 ∈ 𝑉 → (𝑈 ∩ Rng) ∈ V) | |
8 | 2, 7 | syl 17 | . . 3 ⊢ (𝜑 → (𝑈 ∩ Rng) ∈ V) |
9 | rnghmfn 41680 | . . . . 5 ⊢ RngHomo Fn (Rng × Rng) | |
10 | fnfun 5902 | . . . . 5 ⊢ ( RngHomo Fn (Rng × Rng) → Fun RngHomo ) | |
11 | 9, 10 | mp1i 13 | . . . 4 ⊢ (𝜑 → Fun RngHomo ) |
12 | sqxpexg 6861 | . . . . 5 ⊢ ((𝑈 ∩ Rng) ∈ V → ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)) ∈ V) | |
13 | 8, 12 | syl 17 | . . . 4 ⊢ (𝜑 → ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)) ∈ V) |
14 | resfunexg 6384 | . . . 4 ⊢ ((Fun RngHomo ∧ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)) ∈ V) → ( RngHomo ↾ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng))) ∈ V) | |
15 | 11, 13, 14 | syl2anc 691 | . . 3 ⊢ (𝜑 → ( RngHomo ↾ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng))) ∈ V) |
16 | fvex 6113 | . . . 4 ⊢ (comp‘(ExtStrCat‘𝑈)) ∈ V | |
17 | 16 | a1i 11 | . . 3 ⊢ (𝜑 → (comp‘(ExtStrCat‘𝑈)) ∈ V) |
18 | rhmsubcrngc.b | . . . . 5 ⊢ (𝜑 → 𝐵 = (Ring ∩ 𝑈)) | |
19 | incom 3767 | . . . . 5 ⊢ (Ring ∩ 𝑈) = (𝑈 ∩ Ring) | |
20 | 18, 19 | syl6eq 2660 | . . . 4 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Ring)) |
21 | inex1g 4729 | . . . . 5 ⊢ (𝑈 ∈ 𝑉 → (𝑈 ∩ Ring) ∈ V) | |
22 | 2, 21 | syl 17 | . . . 4 ⊢ (𝜑 → (𝑈 ∩ Ring) ∈ V) |
23 | 20, 22 | eqeltrd 2688 | . . 3 ⊢ (𝜑 → 𝐵 ∈ V) |
24 | rhmsubcrngc.h | . . . 4 ⊢ (𝜑 → 𝐻 = ( RingHom ↾ (𝐵 × 𝐵))) | |
25 | rhmfn 41708 | . . . . . 6 ⊢ RingHom Fn (Ring × Ring) | |
26 | fnfun 5902 | . . . . . 6 ⊢ ( RingHom Fn (Ring × Ring) → Fun RingHom ) | |
27 | 25, 26 | mp1i 13 | . . . . 5 ⊢ (𝜑 → Fun RingHom ) |
28 | sqxpexg 6861 | . . . . . 6 ⊢ (𝐵 ∈ V → (𝐵 × 𝐵) ∈ V) | |
29 | 23, 28 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝐵 × 𝐵) ∈ V) |
30 | resfunexg 6384 | . . . . 5 ⊢ ((Fun RingHom ∧ (𝐵 × 𝐵) ∈ V) → ( RingHom ↾ (𝐵 × 𝐵)) ∈ V) | |
31 | 27, 29, 30 | syl2anc 691 | . . . 4 ⊢ (𝜑 → ( RingHom ↾ (𝐵 × 𝐵)) ∈ V) |
32 | 24, 31 | eqeltrd 2688 | . . 3 ⊢ (𝜑 → 𝐻 ∈ V) |
33 | ringrng 41669 | . . . . . . 7 ⊢ (𝑟 ∈ Ring → 𝑟 ∈ Rng) | |
34 | 33 | a1i 11 | . . . . . 6 ⊢ (𝜑 → (𝑟 ∈ Ring → 𝑟 ∈ Rng)) |
35 | 34 | ssrdv 3574 | . . . . 5 ⊢ (𝜑 → Ring ⊆ Rng) |
36 | ssrin 3800 | . . . . 5 ⊢ (Ring ⊆ Rng → (Ring ∩ 𝑈) ⊆ (Rng ∩ 𝑈)) | |
37 | 35, 36 | syl 17 | . . . 4 ⊢ (𝜑 → (Ring ∩ 𝑈) ⊆ (Rng ∩ 𝑈)) |
38 | incom 3767 | . . . . 5 ⊢ (𝑈 ∩ Rng) = (Rng ∩ 𝑈) | |
39 | 38 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝑈 ∩ Rng) = (Rng ∩ 𝑈)) |
40 | 37, 18, 39 | 3sstr4d 3611 | . . 3 ⊢ (𝜑 → 𝐵 ⊆ (𝑈 ∩ Rng)) |
41 | 6, 8, 15, 17, 23, 32, 40 | estrres 16602 | . 2 ⊢ (𝜑 → ((𝐶 ↾s 𝐵) sSet 〈(Hom ‘ndx), 𝐻〉) = {〈(Base‘ndx), 𝐵〉, 〈(Hom ‘ndx), 𝐻〉, 〈(comp‘ndx), (comp‘(ExtStrCat‘𝑈))〉}) |
42 | eqid 2610 | . . 3 ⊢ (𝐶 ↾cat 𝐻) = (𝐶 ↾cat 𝐻) | |
43 | 1 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝐶 = (RngCat‘𝑈)) |
44 | fvex 6113 | . . . . 5 ⊢ (RngCat‘𝑈) ∈ V | |
45 | 44 | a1i 11 | . . . 4 ⊢ (𝜑 → (RngCat‘𝑈) ∈ V) |
46 | 43, 45 | eqeltrd 2688 | . . 3 ⊢ (𝜑 → 𝐶 ∈ V) |
47 | 20, 24 | rhmresfn 41801 | . . 3 ⊢ (𝜑 → 𝐻 Fn (𝐵 × 𝐵)) |
48 | 42, 46, 23, 47 | rescval2 16311 | . 2 ⊢ (𝜑 → (𝐶 ↾cat 𝐻) = ((𝐶 ↾s 𝐵) sSet 〈(Hom ‘ndx), 𝐻〉)) |
49 | eqid 2610 | . . 3 ⊢ (RingCat‘𝑈) = (RingCat‘𝑈) | |
50 | 49, 2, 20, 24, 5 | dfringc2 41810 | . 2 ⊢ (𝜑 → (RingCat‘𝑈) = {〈(Base‘ndx), 𝐵〉, 〈(Hom ‘ndx), 𝐻〉, 〈(comp‘ndx), (comp‘(ExtStrCat‘𝑈))〉}) |
51 | 41, 48, 50 | 3eqtr4d 2654 | 1 ⊢ (𝜑 → (𝐶 ↾cat 𝐻) = (RingCat‘𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∩ cin 3539 ⊆ wss 3540 {ctp 4129 〈cop 4131 × cxp 5036 ↾ cres 5040 Fun wfun 5798 Fn wfn 5799 ‘cfv 5804 (class class class)co 6549 ndxcnx 15692 sSet csts 15693 Basecbs 15695 ↾s cress 15696 Hom chom 15779 compcco 15780 ↾cat cresc 16291 ExtStrCatcestrc 16585 Ringcrg 18370 RingHom crh 18535 Rngcrng 41664 RngHomo crngh 41675 RngCatcrngc 41749 RingCatcringc 41795 |
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-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-resc 16294 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-rng0 41665 df-rnghomo 41677 df-rngc 41751 df-ringc 41797 |
This theorem is referenced by: (None) |
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