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| Mirrors > Home > MPE Home > Th. List > 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 RngHom 𝑦)))) |
| Ref | Expression |
|---|---|
| rngcifuestrc | ⊢ (𝜑 → 𝐹(𝑅 Func 𝐸)𝐺) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eqid 2737 | . . . 4 ⊢ (ExtStrCat‘𝑈) = (ExtStrCat‘𝑈) | |
| 2 | rngcifuestrc.u | . . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑉) | |
| 3 | rngcifuestrc.r | . . . . . 6 ⊢ 𝑅 = (RngCat‘𝑈) | |
| 4 | rngcifuestrc.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝑅) | |
| 5 | 3, 4, 2 | rngcbas 20621 | . . . . 5 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Rng)) |
| 6 | incom 4209 | . . . . 5 ⊢ (𝑈 ∩ Rng) = (Rng ∩ 𝑈) | |
| 7 | 5, 6 | eqtrdi 2793 | . . . 4 ⊢ (𝜑 → 𝐵 = (Rng ∩ 𝑈)) |
| 8 | eqid 2737 | . . . . 5 ⊢ (Hom ‘𝑅) = (Hom ‘𝑅) | |
| 9 | 3, 4, 2, 8 | rngchomfval 20622 | . . . 4 ⊢ (𝜑 → (Hom ‘𝑅) = ( RngHom ↾ (𝐵 × 𝐵))) |
| 10 | 1, 2, 7, 9 | rnghmsubcsetc 20633 | . . 3 ⊢ (𝜑 → (Hom ‘𝑅) ∈ (Subcat‘(ExtStrCat‘𝑈))) |
| 11 | eqid 2737 | . . 3 ⊢ ((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅)) = ((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅)) | |
| 12 | eqid 2737 | . . 3 ⊢ (Base‘((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅))) = (Base‘((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅))) | |
| 13 | rngcifuestrc.f | . . . 4 ⊢ (𝜑 → 𝐹 = ( I ↾ 𝐵)) | |
| 14 | 3, 2, 5, 9 | rngcval 20618 | . . . . . . 7 ⊢ (𝜑 → 𝑅 = ((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅))) |
| 15 | 14 | fveq2d 6910 | . . . . . 6 ⊢ (𝜑 → (Base‘𝑅) = (Base‘((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅)))) |
| 16 | 4, 15 | eqtrid 2789 | . . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅)))) |
| 17 | 16 | reseq2d 5997 | . . . 4 ⊢ (𝜑 → ( I ↾ 𝐵) = ( I ↾ (Base‘((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅))))) |
| 18 | 13, 17 | eqtrd 2777 | . . 3 ⊢ (𝜑 → 𝐹 = ( I ↾ (Base‘((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅))))) |
| 19 | rngcifuestrc.g | . . . 4 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥 RngHom 𝑦)))) | |
| 20 | 16 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐵 = (Base‘((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅)))) |
| 21 | 9 | oveqdr 7459 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(Hom ‘𝑅)𝑦) = (𝑥( RngHom ↾ (𝐵 × 𝐵))𝑦)) |
| 22 | ovres 7599 | . . . . . . . 8 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥( RngHom ↾ (𝐵 × 𝐵))𝑦) = (𝑥 RngHom 𝑦)) | |
| 23 | 22 | adantl 481 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥( RngHom ↾ (𝐵 × 𝐵))𝑦) = (𝑥 RngHom 𝑦)) |
| 24 | 21, 23 | eqtr2d 2778 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 RngHom 𝑦) = (𝑥(Hom ‘𝑅)𝑦)) |
| 25 | 24 | reseq2d 5997 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ( I ↾ (𝑥 RngHom 𝑦)) = ( I ↾ (𝑥(Hom ‘𝑅)𝑦))) |
| 26 | 16, 20, 25 | mpoeq123dva 7507 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥 RngHom 𝑦))) = (𝑥 ∈ (Base‘((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅))), 𝑦 ∈ (Base‘((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅))) ↦ ( I ↾ (𝑥(Hom ‘𝑅)𝑦)))) |
| 27 | 19, 26 | eqtrd 2777 | . . 3 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ (Base‘((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅))), 𝑦 ∈ (Base‘((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅))) ↦ ( I ↾ (𝑥(Hom ‘𝑅)𝑦)))) |
| 28 | 10, 11, 12, 18, 27 | inclfusubc 17988 | . 2 ⊢ (𝜑 → 𝐹(((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅)) Func (ExtStrCat‘𝑈))𝐺) |
| 29 | rngcifuestrc.e | . . . . 5 ⊢ 𝐸 = (ExtStrCat‘𝑈) | |
| 30 | 29 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝐸 = (ExtStrCat‘𝑈)) |
| 31 | 14, 30 | oveq12d 7449 | . . 3 ⊢ (𝜑 → (𝑅 Func 𝐸) = (((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅)) Func (ExtStrCat‘𝑈))) |
| 32 | 31 | breqd 5154 | . 2 ⊢ (𝜑 → (𝐹(𝑅 Func 𝐸)𝐺 ↔ 𝐹(((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅)) Func (ExtStrCat‘𝑈))𝐺)) |
| 33 | 28, 32 | mpbird 257 | 1 ⊢ (𝜑 → 𝐹(𝑅 Func 𝐸)𝐺) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∩ cin 3950 class class class wbr 5143 I cid 5577 × cxp 5683 ↾ cres 5687 ‘cfv 6561 (class class class)co 7431 ∈ cmpo 7433 Basecbs 17247 Hom chom 17308 ↾cat cresc 17852 Func cfunc 17899 ExtStrCatcestrc 18166 Rngcrng 20149 RngHom crnghm 20434 RngCatcrngc 20616 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-map 8868 df-pm 8869 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-fz 13548 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-hom 17321 df-cco 17322 df-0g 17486 df-cat 17711 df-cid 17712 df-homf 17713 df-ssc 17854 df-resc 17855 df-subc 17856 df-func 17903 df-idfu 17904 df-full 17951 df-fth 17952 df-estrc 18167 df-mgm 18653 df-mgmhm 18705 df-sgrp 18732 df-mnd 18748 df-mhm 18796 df-grp 18954 df-ghm 19231 df-abl 19801 df-mgp 20138 df-rng 20150 df-rnghm 20436 df-rngc 20617 |
| This theorem is referenced by: funcrngcsetcALT 20641 |
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