Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rngcifuestrc | Structured version Visualization version GIF version |
Description: The "inclusion functor" from the category of non-unital rings into the category of extensible structures. (Contributed by AV, 30-Mar-2020.) |
Ref | Expression |
---|---|
rngcifuestrc.r | ⊢ 𝑅 = (RngCat‘𝑈) |
rngcifuestrc.e | ⊢ 𝐸 = (ExtStrCat‘𝑈) |
rngcifuestrc.b | ⊢ 𝐵 = (Base‘𝑅) |
rngcifuestrc.u | ⊢ (𝜑 → 𝑈 ∈ 𝑉) |
rngcifuestrc.f | ⊢ (𝜑 → 𝐹 = ( I ↾ 𝐵)) |
rngcifuestrc.g | ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥 RngHomo 𝑦)))) |
Ref | Expression |
---|---|
rngcifuestrc | ⊢ (𝜑 → 𝐹(𝑅 Func 𝐸)𝐺) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2821 | . . . . 5 ⊢ (ExtStrCat‘𝑈) = (ExtStrCat‘𝑈) | |
2 | rngcifuestrc.u | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ 𝑉) | |
3 | rngcifuestrc.r | . . . . . . 7 ⊢ 𝑅 = (RngCat‘𝑈) | |
4 | rngcifuestrc.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝑅) | |
5 | 3, 4, 2 | rngcbas 44230 | . . . . . 6 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Rng)) |
6 | incom 4177 | . . . . . 6 ⊢ (𝑈 ∩ Rng) = (Rng ∩ 𝑈) | |
7 | 5, 6 | syl6eq 2872 | . . . . 5 ⊢ (𝜑 → 𝐵 = (Rng ∩ 𝑈)) |
8 | eqid 2821 | . . . . . 6 ⊢ (Hom ‘𝑅) = (Hom ‘𝑅) | |
9 | 3, 4, 2, 8 | rngchomfval 44231 | . . . . 5 ⊢ (𝜑 → (Hom ‘𝑅) = ( RngHomo ↾ (𝐵 × 𝐵))) |
10 | 1, 2, 7, 9 | rnghmsubcsetc 44242 | . . . 4 ⊢ (𝜑 → (Hom ‘𝑅) ∈ (Subcat‘(ExtStrCat‘𝑈))) |
11 | 10 | idi 1 | . . 3 ⊢ (𝜑 → (Hom ‘𝑅) ∈ (Subcat‘(ExtStrCat‘𝑈))) |
12 | eqid 2821 | . . 3 ⊢ ((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅)) = ((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅)) | |
13 | eqid 2821 | . . 3 ⊢ (Base‘((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅))) = (Base‘((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅))) | |
14 | rngcifuestrc.f | . . . 4 ⊢ (𝜑 → 𝐹 = ( I ↾ 𝐵)) | |
15 | 3, 2, 5, 9 | rngcval 44227 | . . . . . . 7 ⊢ (𝜑 → 𝑅 = ((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅))) |
16 | 15 | fveq2d 6668 | . . . . . 6 ⊢ (𝜑 → (Base‘𝑅) = (Base‘((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅)))) |
17 | 4, 16 | syl5eq 2868 | . . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅)))) |
18 | 17 | reseq2d 5847 | . . . 4 ⊢ (𝜑 → ( I ↾ 𝐵) = ( I ↾ (Base‘((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅))))) |
19 | 14, 18 | eqtrd 2856 | . . 3 ⊢ (𝜑 → 𝐹 = ( I ↾ (Base‘((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅))))) |
20 | rngcifuestrc.g | . . . 4 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥 RngHomo 𝑦)))) | |
21 | 17 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐵 = (Base‘((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅)))) |
22 | 9 | oveqdr 7178 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(Hom ‘𝑅)𝑦) = (𝑥( RngHomo ↾ (𝐵 × 𝐵))𝑦)) |
23 | ovres 7308 | . . . . . . . 8 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥( RngHomo ↾ (𝐵 × 𝐵))𝑦) = (𝑥 RngHomo 𝑦)) | |
24 | 23 | adantl 484 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥( RngHomo ↾ (𝐵 × 𝐵))𝑦) = (𝑥 RngHomo 𝑦)) |
25 | 22, 24 | eqtr2d 2857 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 RngHomo 𝑦) = (𝑥(Hom ‘𝑅)𝑦)) |
26 | 25 | reseq2d 5847 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ( I ↾ (𝑥 RngHomo 𝑦)) = ( I ↾ (𝑥(Hom ‘𝑅)𝑦))) |
27 | 17, 21, 26 | mpoeq123dva 7222 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥 RngHomo 𝑦))) = (𝑥 ∈ (Base‘((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅))), 𝑦 ∈ (Base‘((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅))) ↦ ( I ↾ (𝑥(Hom ‘𝑅)𝑦)))) |
28 | 20, 27 | eqtrd 2856 | . . 3 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ (Base‘((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅))), 𝑦 ∈ (Base‘((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅))) ↦ ( I ↾ (𝑥(Hom ‘𝑅)𝑦)))) |
29 | 11, 12, 13, 19, 28 | inclfusubc 44132 | . 2 ⊢ (𝜑 → 𝐹(((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅)) Func (ExtStrCat‘𝑈))𝐺) |
30 | rngcifuestrc.e | . . . . 5 ⊢ 𝐸 = (ExtStrCat‘𝑈) | |
31 | 30 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝐸 = (ExtStrCat‘𝑈)) |
32 | 15, 31 | oveq12d 7168 | . . 3 ⊢ (𝜑 → (𝑅 Func 𝐸) = (((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅)) Func (ExtStrCat‘𝑈))) |
33 | 32 | breqd 5069 | . 2 ⊢ (𝜑 → (𝐹(𝑅 Func 𝐸)𝐺 ↔ 𝐹(((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅)) Func (ExtStrCat‘𝑈))𝐺)) |
34 | 29, 33 | mpbird 259 | 1 ⊢ (𝜑 → 𝐹(𝑅 Func 𝐸)𝐺) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∩ cin 3934 class class class wbr 5058 I cid 5453 × cxp 5547 ↾ cres 5551 ‘cfv 6349 (class class class)co 7150 ∈ cmpo 7152 Basecbs 16477 Hom chom 16570 ↾cat cresc 17072 Subcatcsubc 17073 Func cfunc 17118 ExtStrCatcestrc 17366 Rngcrng 44139 RngHomo crngh 44150 RngCatcrngc 44222 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-map 8402 df-pm 8403 df-ixp 8456 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-fz 12887 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-hom 16583 df-cco 16584 df-0g 16709 df-cat 16933 df-cid 16934 df-homf 16935 df-ssc 17074 df-resc 17075 df-subc 17076 df-func 17122 df-idfu 17123 df-full 17168 df-fth 17169 df-estrc 17367 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-mhm 17950 df-grp 18100 df-ghm 18350 df-abl 18903 df-mgp 19234 df-mgmhm 44040 df-rng0 44140 df-rnghomo 44152 df-rngc 44224 |
This theorem is referenced by: funcrngcsetcALT 44264 |
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