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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 2735 | . . . 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 20638 | . . . . 5 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Rng)) |
6 | incom 4217 | . . . . 5 ⊢ (𝑈 ∩ Rng) = (Rng ∩ 𝑈) | |
7 | 5, 6 | eqtrdi 2791 | . . . 4 ⊢ (𝜑 → 𝐵 = (Rng ∩ 𝑈)) |
8 | eqid 2735 | . . . . 5 ⊢ (Hom ‘𝑅) = (Hom ‘𝑅) | |
9 | 3, 4, 2, 8 | rngchomfval 20639 | . . . 4 ⊢ (𝜑 → (Hom ‘𝑅) = ( RngHom ↾ (𝐵 × 𝐵))) |
10 | 1, 2, 7, 9 | rnghmsubcsetc 20650 | . . 3 ⊢ (𝜑 → (Hom ‘𝑅) ∈ (Subcat‘(ExtStrCat‘𝑈))) |
11 | eqid 2735 | . . 3 ⊢ ((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅)) = ((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅)) | |
12 | eqid 2735 | . . 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 20635 | . . . . . . 7 ⊢ (𝜑 → 𝑅 = ((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅))) |
15 | 14 | fveq2d 6911 | . . . . . 6 ⊢ (𝜑 → (Base‘𝑅) = (Base‘((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅)))) |
16 | 4, 15 | eqtrid 2787 | . . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅)))) |
17 | 16 | reseq2d 6000 | . . . 4 ⊢ (𝜑 → ( I ↾ 𝐵) = ( I ↾ (Base‘((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅))))) |
18 | 13, 17 | eqtrd 2775 | . . 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 2776 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 RngHom 𝑦) = (𝑥(Hom ‘𝑅)𝑦)) |
25 | 24 | reseq2d 6000 | . . . . 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 2775 | . . 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 17995 | . 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 5159 | . 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 1537 ∈ wcel 2106 ∩ cin 3962 class class class wbr 5148 I cid 5582 × cxp 5687 ↾ cres 5691 ‘cfv 6563 (class class class)co 7431 ∈ cmpo 7433 Basecbs 17245 Hom chom 17309 ↾cat cresc 17856 Func cfunc 17905 ExtStrCatcestrc 18177 Rngcrng 20170 RngHom crnghm 20451 RngCatcrngc 20633 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-map 8867 df-pm 8868 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-fz 13545 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-hom 17322 df-cco 17323 df-0g 17488 df-cat 17713 df-cid 17714 df-homf 17715 df-ssc 17858 df-resc 17859 df-subc 17860 df-func 17909 df-idfu 17910 df-full 17958 df-fth 17959 df-estrc 18178 df-mgm 18666 df-mgmhm 18718 df-sgrp 18745 df-mnd 18761 df-mhm 18809 df-grp 18967 df-ghm 19244 df-abl 19816 df-mgp 20153 df-rng 20171 df-rnghm 20453 df-rngc 20634 |
This theorem is referenced by: funcrngcsetcALT 20658 |
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