Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dfrngc2 | Structured version Visualization version GIF version |
Description: Alternate definition of the category of non-unital rings (in a universe). (Contributed by AV, 16-Mar-2020.) |
Ref | Expression |
---|---|
dfrngc2.c | ⊢ 𝐶 = (RngCat‘𝑈) |
dfrngc2.u | ⊢ (𝜑 → 𝑈 ∈ 𝑉) |
dfrngc2.b | ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Rng)) |
dfrngc2.h | ⊢ (𝜑 → 𝐻 = ( RngHomo ↾ (𝐵 × 𝐵))) |
dfrngc2.o | ⊢ (𝜑 → · = (comp‘(ExtStrCat‘𝑈))) |
Ref | Expression |
---|---|
dfrngc2 | ⊢ (𝜑 → 𝐶 = {〈(Base‘ndx), 𝐵〉, 〈(Hom ‘ndx), 𝐻〉, 〈(comp‘ndx), · 〉}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfrngc2.c | . . 3 ⊢ 𝐶 = (RngCat‘𝑈) | |
2 | dfrngc2.u | . . 3 ⊢ (𝜑 → 𝑈 ∈ 𝑉) | |
3 | dfrngc2.b | . . 3 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Rng)) | |
4 | dfrngc2.h | . . 3 ⊢ (𝜑 → 𝐻 = ( RngHomo ↾ (𝐵 × 𝐵))) | |
5 | 1, 2, 3, 4 | rngcval 41962 | . 2 ⊢ (𝜑 → 𝐶 = ((ExtStrCat‘𝑈) ↾cat 𝐻)) |
6 | eqid 2622 | . . . 4 ⊢ ((ExtStrCat‘𝑈) ↾cat 𝐻) = ((ExtStrCat‘𝑈) ↾cat 𝐻) | |
7 | fvexd 6203 | . . . 4 ⊢ (𝜑 → (ExtStrCat‘𝑈) ∈ V) | |
8 | inex1g 4801 | . . . . . 6 ⊢ (𝑈 ∈ 𝑉 → (𝑈 ∩ Rng) ∈ V) | |
9 | 2, 8 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝑈 ∩ Rng) ∈ V) |
10 | 3, 9 | eqeltrd 2701 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ V) |
11 | 3, 4 | rnghmresfn 41963 | . . . 4 ⊢ (𝜑 → 𝐻 Fn (𝐵 × 𝐵)) |
12 | 6, 7, 10, 11 | rescval2 16488 | . . 3 ⊢ (𝜑 → ((ExtStrCat‘𝑈) ↾cat 𝐻) = (((ExtStrCat‘𝑈) ↾s 𝐵) sSet 〈(Hom ‘ndx), 𝐻〉)) |
13 | eqid 2622 | . . . . 5 ⊢ (ExtStrCat‘𝑈) = (ExtStrCat‘𝑈) | |
14 | eqidd 2623 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) = (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))) | |
15 | dfrngc2.o | . . . . . 6 ⊢ (𝜑 → · = (comp‘(ExtStrCat‘𝑈))) | |
16 | eqid 2622 | . . . . . . 7 ⊢ (comp‘(ExtStrCat‘𝑈)) = (comp‘(ExtStrCat‘𝑈)) | |
17 | 13, 2, 16 | estrccofval 16769 | . . . . . 6 ⊢ (𝜑 → (comp‘(ExtStrCat‘𝑈)) = (𝑣 ∈ (𝑈 × 𝑈), 𝑧 ∈ 𝑈 ↦ (𝑔 ∈ ((Base‘𝑧) ↑𝑚 (Base‘(2nd ‘𝑣))), 𝑓 ∈ ((Base‘(2nd ‘𝑣)) ↑𝑚 (Base‘(1st ‘𝑣))) ↦ (𝑔 ∘ 𝑓)))) |
18 | 15, 17 | eqtrd 2656 | . . . . 5 ⊢ (𝜑 → · = (𝑣 ∈ (𝑈 × 𝑈), 𝑧 ∈ 𝑈 ↦ (𝑔 ∈ ((Base‘𝑧) ↑𝑚 (Base‘(2nd ‘𝑣))), 𝑓 ∈ ((Base‘(2nd ‘𝑣)) ↑𝑚 (Base‘(1st ‘𝑣))) ↦ (𝑔 ∘ 𝑓)))) |
19 | 13, 2, 14, 18 | estrcval 16764 | . . . 4 ⊢ (𝜑 → (ExtStrCat‘𝑈) = {〈(Base‘ndx), 𝑈〉, 〈(Hom ‘ndx), (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))〉, 〈(comp‘ndx), · 〉}) |
20 | 2, 2 | jca 554 | . . . . 5 ⊢ (𝜑 → (𝑈 ∈ 𝑉 ∧ 𝑈 ∈ 𝑉)) |
21 | eqid 2622 | . . . . . 6 ⊢ (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) = (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) | |
22 | 21 | mpt2exg 7245 | . . . . 5 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑈 ∈ 𝑉) → (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) ∈ V) |
23 | 20, 22 | syl 17 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) ∈ V) |
24 | fvexd 6203 | . . . . 5 ⊢ (𝜑 → (comp‘(ExtStrCat‘𝑈)) ∈ V) | |
25 | 15, 24 | eqeltrd 2701 | . . . 4 ⊢ (𝜑 → · ∈ V) |
26 | rnghmfn 41890 | . . . . . . 7 ⊢ RngHomo Fn (Rng × Rng) | |
27 | fnfun 5988 | . . . . . . 7 ⊢ ( RngHomo Fn (Rng × Rng) → Fun RngHomo ) | |
28 | 26, 27 | mp1i 13 | . . . . . 6 ⊢ (𝜑 → Fun RngHomo ) |
29 | sqxpexg 6963 | . . . . . . 7 ⊢ (𝐵 ∈ V → (𝐵 × 𝐵) ∈ V) | |
30 | 10, 29 | syl 17 | . . . . . 6 ⊢ (𝜑 → (𝐵 × 𝐵) ∈ V) |
31 | resfunexg 6479 | . . . . . 6 ⊢ ((Fun RngHomo ∧ (𝐵 × 𝐵) ∈ V) → ( RngHomo ↾ (𝐵 × 𝐵)) ∈ V) | |
32 | 28, 30, 31 | syl2anc 693 | . . . . 5 ⊢ (𝜑 → ( RngHomo ↾ (𝐵 × 𝐵)) ∈ V) |
33 | 4, 32 | eqeltrd 2701 | . . . 4 ⊢ (𝜑 → 𝐻 ∈ V) |
34 | inss1 3833 | . . . . . 6 ⊢ (𝑈 ∩ Rng) ⊆ 𝑈 | |
35 | 34 | a1i 11 | . . . . 5 ⊢ (𝜑 → (𝑈 ∩ Rng) ⊆ 𝑈) |
36 | 3 | sseq1d 3632 | . . . . 5 ⊢ (𝜑 → (𝐵 ⊆ 𝑈 ↔ (𝑈 ∩ Rng) ⊆ 𝑈)) |
37 | 35, 36 | mpbird 247 | . . . 4 ⊢ (𝜑 → 𝐵 ⊆ 𝑈) |
38 | 19, 2, 23, 25, 10, 33, 37 | estrres 16779 | . . 3 ⊢ (𝜑 → (((ExtStrCat‘𝑈) ↾s 𝐵) sSet 〈(Hom ‘ndx), 𝐻〉) = {〈(Base‘ndx), 𝐵〉, 〈(Hom ‘ndx), 𝐻〉, 〈(comp‘ndx), · 〉}) |
39 | 12, 38 | eqtrd 2656 | . 2 ⊢ (𝜑 → ((ExtStrCat‘𝑈) ↾cat 𝐻) = {〈(Base‘ndx), 𝐵〉, 〈(Hom ‘ndx), 𝐻〉, 〈(comp‘ndx), · 〉}) |
40 | 5, 39 | eqtrd 2656 | 1 ⊢ (𝜑 → 𝐶 = {〈(Base‘ndx), 𝐵〉, 〈(Hom ‘ndx), 𝐻〉, 〈(comp‘ndx), · 〉}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ∩ cin 3573 ⊆ wss 3574 {ctp 4181 〈cop 4183 × cxp 5112 ↾ cres 5116 ∘ ccom 5118 Fun wfun 5882 Fn wfn 5883 ‘cfv 5888 (class class class)co 6650 ↦ cmpt2 6652 1st c1st 7166 2nd c2nd 7167 ↑𝑚 cmap 7857 ndxcnx 15854 sSet csts 15855 Basecbs 15857 ↾s cress 15858 Hom chom 15952 compcco 15953 ↾cat cresc 16468 ExtStrCatcestrc 16762 Rngcrng 41874 RngHomo crngh 41885 RngCatcrngc 41957 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-fz 12327 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-hom 15966 df-cco 15967 df-resc 16471 df-estrc 16763 df-rnghomo 41887 df-rngc 41959 |
This theorem is referenced by: rngcresringcat 42030 |
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