Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rngccofval | Structured version Visualization version GIF version |
Description: Composition in the category of non-unital rings. (Contributed by AV, 27-Feb-2020.) (Revised by AV, 8-Mar-2020.) |
Ref | Expression |
---|---|
rngcco.c | ⊢ 𝐶 = (RngCat‘𝑈) |
rngcco.u | ⊢ (𝜑 → 𝑈 ∈ 𝑉) |
rngcco.o | ⊢ · = (comp‘𝐶) |
Ref | Expression |
---|---|
rngccofval | ⊢ (𝜑 → · = (comp‘(ExtStrCat‘𝑈))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rngcco.c | . . . 4 ⊢ 𝐶 = (RngCat‘𝑈) | |
2 | rngcco.u | . . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑉) | |
3 | eqid 2736 | . . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
4 | 1, 3, 2 | rngcbas 45581 | . . . 4 ⊢ (𝜑 → (Base‘𝐶) = (𝑈 ∩ Rng)) |
5 | eqid 2736 | . . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
6 | 1, 3, 2, 5 | rngchomfval 45582 | . . . 4 ⊢ (𝜑 → (Hom ‘𝐶) = ( RngHomo ↾ ((Base‘𝐶) × (Base‘𝐶)))) |
7 | 1, 2, 4, 6 | rngcval 45578 | . . 3 ⊢ (𝜑 → 𝐶 = ((ExtStrCat‘𝑈) ↾cat (Hom ‘𝐶))) |
8 | 7 | fveq2d 6808 | . 2 ⊢ (𝜑 → (comp‘𝐶) = (comp‘((ExtStrCat‘𝑈) ↾cat (Hom ‘𝐶)))) |
9 | rngcco.o | . . 3 ⊢ · = (comp‘𝐶) | |
10 | 9 | a1i 11 | . 2 ⊢ (𝜑 → · = (comp‘𝐶)) |
11 | eqid 2736 | . . 3 ⊢ ((ExtStrCat‘𝑈) ↾cat (Hom ‘𝐶)) = ((ExtStrCat‘𝑈) ↾cat (Hom ‘𝐶)) | |
12 | eqid 2736 | . . 3 ⊢ (Base‘(ExtStrCat‘𝑈)) = (Base‘(ExtStrCat‘𝑈)) | |
13 | fvexd 6819 | . . 3 ⊢ (𝜑 → (ExtStrCat‘𝑈) ∈ V) | |
14 | 4, 6 | rnghmresfn 45579 | . . 3 ⊢ (𝜑 → (Hom ‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶))) |
15 | inss1 4168 | . . . . 5 ⊢ (𝑈 ∩ Rng) ⊆ 𝑈 | |
16 | 15 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝑈 ∩ Rng) ⊆ 𝑈) |
17 | eqid 2736 | . . . . . 6 ⊢ (ExtStrCat‘𝑈) = (ExtStrCat‘𝑈) | |
18 | 17, 2 | estrcbas 17886 | . . . . 5 ⊢ (𝜑 → 𝑈 = (Base‘(ExtStrCat‘𝑈))) |
19 | 18 | eqcomd 2742 | . . . 4 ⊢ (𝜑 → (Base‘(ExtStrCat‘𝑈)) = 𝑈) |
20 | 16, 4, 19 | 3sstr4d 3973 | . . 3 ⊢ (𝜑 → (Base‘𝐶) ⊆ (Base‘(ExtStrCat‘𝑈))) |
21 | eqid 2736 | . . 3 ⊢ (comp‘(ExtStrCat‘𝑈)) = (comp‘(ExtStrCat‘𝑈)) | |
22 | 11, 12, 13, 14, 20, 21 | rescco 17590 | . 2 ⊢ (𝜑 → (comp‘(ExtStrCat‘𝑈)) = (comp‘((ExtStrCat‘𝑈) ↾cat (Hom ‘𝐶)))) |
23 | 8, 10, 22 | 3eqtr4d 2786 | 1 ⊢ (𝜑 → · = (comp‘(ExtStrCat‘𝑈))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2104 Vcvv 3437 ∩ cin 3891 ⊆ wss 3892 ‘cfv 6458 (class class class)co 7307 Basecbs 16957 Hom chom 17018 compcco 17019 ↾cat cresc 17565 ExtStrCatcestrc 17883 Rngcrng 45490 RngCatcrngc 45573 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 ax-rep 5218 ax-sep 5232 ax-nul 5239 ax-pow 5297 ax-pr 5361 ax-un 7620 ax-cnex 10973 ax-resscn 10974 ax-1cn 10975 ax-icn 10976 ax-addcl 10977 ax-addrcl 10978 ax-mulcl 10979 ax-mulrcl 10980 ax-mulcom 10981 ax-addass 10982 ax-mulass 10983 ax-distr 10984 ax-i2m1 10985 ax-1ne0 10986 ax-1rid 10987 ax-rnegex 10988 ax-rrecex 10989 ax-cnre 10990 ax-pre-lttri 10991 ax-pre-lttrn 10992 ax-pre-ltadd 10993 ax-pre-mulgt0 10994 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3or 1088 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3286 df-rab 3287 df-v 3439 df-sbc 3722 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4566 df-pr 4568 df-tp 4570 df-op 4572 df-uni 4845 df-iun 4933 df-br 5082 df-opab 5144 df-mpt 5165 df-tr 5199 df-id 5500 df-eprel 5506 df-po 5514 df-so 5515 df-fr 5555 df-we 5557 df-xp 5606 df-rel 5607 df-cnv 5608 df-co 5609 df-dm 5610 df-rn 5611 df-res 5612 df-ima 5613 df-pred 6217 df-ord 6284 df-on 6285 df-lim 6286 df-suc 6287 df-iota 6410 df-fun 6460 df-fn 6461 df-f 6462 df-f1 6463 df-fo 6464 df-f1o 6465 df-fv 6466 df-riota 7264 df-ov 7310 df-oprab 7311 df-mpo 7312 df-om 7745 df-1st 7863 df-2nd 7864 df-frecs 8128 df-wrecs 8159 df-recs 8233 df-rdg 8272 df-1o 8328 df-er 8529 df-en 8765 df-dom 8766 df-sdom 8767 df-fin 8768 df-pnf 11057 df-mnf 11058 df-xr 11059 df-ltxr 11060 df-le 11061 df-sub 11253 df-neg 11254 df-nn 12020 df-2 12082 df-3 12083 df-4 12084 df-5 12085 df-6 12086 df-7 12087 df-8 12088 df-9 12089 df-n0 12280 df-z 12366 df-dec 12484 df-uz 12629 df-fz 13286 df-struct 16893 df-sets 16910 df-slot 16928 df-ndx 16940 df-base 16958 df-ress 16987 df-hom 17031 df-cco 17032 df-resc 17568 df-estrc 17884 df-rnghomo 45503 df-rngc 45575 |
This theorem is referenced by: rngcco 45587 |
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