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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ringccofval | Structured version Visualization version GIF version |
Description: Composition in the category of unital rings. (Contributed by AV, 14-Feb-2020.) (Revised by AV, 8-Mar-2020.) |
Ref | Expression |
---|---|
ringcco.c | ⊢ 𝐶 = (RingCat‘𝑈) |
ringcco.u | ⊢ (𝜑 → 𝑈 ∈ 𝑉) |
ringcco.o | ⊢ · = (comp‘𝐶) |
Ref | Expression |
---|---|
ringccofval | ⊢ (𝜑 → · = (comp‘(ExtStrCat‘𝑈))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ringcco.c | . . . 4 ⊢ 𝐶 = (RingCat‘𝑈) | |
2 | ringcco.u | . . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑉) | |
3 | eqid 2779 | . . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
4 | 1, 3, 2 | ringcbas 43644 | . . . 4 ⊢ (𝜑 → (Base‘𝐶) = (𝑈 ∩ Ring)) |
5 | eqid 2779 | . . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
6 | 1, 3, 2, 5 | ringchomfval 43645 | . . . 4 ⊢ (𝜑 → (Hom ‘𝐶) = ( RingHom ↾ ((Base‘𝐶) × (Base‘𝐶)))) |
7 | 1, 2, 4, 6 | ringcval 43641 | . . 3 ⊢ (𝜑 → 𝐶 = ((ExtStrCat‘𝑈) ↾cat (Hom ‘𝐶))) |
8 | 7 | fveq2d 6503 | . 2 ⊢ (𝜑 → (comp‘𝐶) = (comp‘((ExtStrCat‘𝑈) ↾cat (Hom ‘𝐶)))) |
9 | ringcco.o | . . 3 ⊢ · = (comp‘𝐶) | |
10 | 9 | a1i 11 | . 2 ⊢ (𝜑 → · = (comp‘𝐶)) |
11 | eqid 2779 | . . 3 ⊢ ((ExtStrCat‘𝑈) ↾cat (Hom ‘𝐶)) = ((ExtStrCat‘𝑈) ↾cat (Hom ‘𝐶)) | |
12 | eqid 2779 | . . 3 ⊢ (Base‘(ExtStrCat‘𝑈)) = (Base‘(ExtStrCat‘𝑈)) | |
13 | fvexd 6514 | . . 3 ⊢ (𝜑 → (ExtStrCat‘𝑈) ∈ V) | |
14 | 4, 6 | rhmresfn 43642 | . . 3 ⊢ (𝜑 → (Hom ‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶))) |
15 | inss1 4093 | . . . . 5 ⊢ (𝑈 ∩ Ring) ⊆ 𝑈 | |
16 | 15 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝑈 ∩ Ring) ⊆ 𝑈) |
17 | eqid 2779 | . . . . . 6 ⊢ (ExtStrCat‘𝑈) = (ExtStrCat‘𝑈) | |
18 | 17, 2 | estrcbas 17233 | . . . . 5 ⊢ (𝜑 → 𝑈 = (Base‘(ExtStrCat‘𝑈))) |
19 | 18 | eqcomd 2785 | . . . 4 ⊢ (𝜑 → (Base‘(ExtStrCat‘𝑈)) = 𝑈) |
20 | 16, 4, 19 | 3sstr4d 3905 | . . 3 ⊢ (𝜑 → (Base‘𝐶) ⊆ (Base‘(ExtStrCat‘𝑈))) |
21 | eqid 2779 | . . 3 ⊢ (comp‘(ExtStrCat‘𝑈)) = (comp‘(ExtStrCat‘𝑈)) | |
22 | 11, 12, 13, 14, 20, 21 | rescco 16960 | . 2 ⊢ (𝜑 → (comp‘(ExtStrCat‘𝑈)) = (comp‘((ExtStrCat‘𝑈) ↾cat (Hom ‘𝐶)))) |
23 | 8, 10, 22 | 3eqtr4d 2825 | 1 ⊢ (𝜑 → · = (comp‘(ExtStrCat‘𝑈))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1507 ∈ wcel 2050 Vcvv 3416 ∩ cin 3829 ⊆ wss 3830 ‘cfv 6188 (class class class)co 6976 Basecbs 16339 Hom chom 16432 compcco 16433 ↾cat cresc 16936 ExtStrCatcestrc 17230 Ringcrg 19020 RingCatcringc 43636 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2751 ax-rep 5049 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 ax-cnex 10391 ax-resscn 10392 ax-1cn 10393 ax-icn 10394 ax-addcl 10395 ax-addrcl 10396 ax-mulcl 10397 ax-mulrcl 10398 ax-mulcom 10399 ax-addass 10400 ax-mulass 10401 ax-distr 10402 ax-i2m1 10403 ax-1ne0 10404 ax-1rid 10405 ax-rnegex 10406 ax-rrecex 10407 ax-cnre 10408 ax-pre-lttri 10409 ax-pre-lttrn 10410 ax-pre-ltadd 10411 ax-pre-mulgt0 10412 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2760 df-cleq 2772 df-clel 2847 df-nfc 2919 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3418 df-sbc 3683 df-csb 3788 df-dif 3833 df-un 3835 df-in 3837 df-ss 3844 df-pss 3846 df-nul 4180 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-tp 4446 df-op 4448 df-uni 4713 df-int 4750 df-iun 4794 df-br 4930 df-opab 4992 df-mpt 5009 df-tr 5031 df-id 5312 df-eprel 5317 df-po 5326 df-so 5327 df-fr 5366 df-we 5368 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-pred 5986 df-ord 6032 df-on 6033 df-lim 6034 df-suc 6035 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-riota 6937 df-ov 6979 df-oprab 6980 df-mpo 6981 df-om 7397 df-1st 7501 df-2nd 7502 df-wrecs 7750 df-recs 7812 df-rdg 7850 df-1o 7905 df-oadd 7909 df-er 8089 df-map 8208 df-en 8307 df-dom 8308 df-sdom 8309 df-fin 8310 df-pnf 10476 df-mnf 10477 df-xr 10478 df-ltxr 10479 df-le 10480 df-sub 10672 df-neg 10673 df-nn 11440 df-2 11503 df-3 11504 df-4 11505 df-5 11506 df-6 11507 df-7 11508 df-8 11509 df-9 11510 df-n0 11708 df-z 11794 df-dec 11912 df-uz 12059 df-fz 12709 df-struct 16341 df-ndx 16342 df-slot 16343 df-base 16345 df-sets 16346 df-ress 16347 df-plusg 16434 df-hom 16445 df-cco 16446 df-0g 16571 df-resc 16939 df-estrc 17231 df-mhm 17803 df-ghm 18127 df-mgp 18963 df-ur 18975 df-ring 19022 df-rnghom 19190 df-ringc 43638 |
This theorem is referenced by: ringcco 43650 |
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