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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-endcomp | Structured version Visualization version GIF version |
Description: Composition law of the monoid of endomorphisms on an object of a category. (Contributed by BJ, 5-Apr-2024.) |
Ref | Expression |
---|---|
bj-endval.c | ⊢ (𝜑 → 𝐶 ∈ Cat) |
bj-endval.x | ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶)) |
Ref | Expression |
---|---|
bj-endcomp | ⊢ (𝜑 → (+g‘((End ‘𝐶)‘𝑋)) = (〈𝑋, 𝑋〉(comp‘𝐶)𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-endval.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
2 | bj-endval.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶)) | |
3 | 1, 2 | bj-endval 35095 | . . 3 ⊢ (𝜑 → ((End ‘𝐶)‘𝑋) = {〈(Base‘ndx), (𝑋(Hom ‘𝐶)𝑋)〉, 〈(+g‘ndx), (〈𝑋, 𝑋〉(comp‘𝐶)𝑋)〉}) |
4 | 3 | fveq1d 6670 | . 2 ⊢ (𝜑 → (((End ‘𝐶)‘𝑋)‘(+g‘ndx)) = ({〈(Base‘ndx), (𝑋(Hom ‘𝐶)𝑋)〉, 〈(+g‘ndx), (〈𝑋, 𝑋〉(comp‘𝐶)𝑋)〉}‘(+g‘ndx))) |
5 | fvexd 6683 | . . 3 ⊢ (𝜑 → ((End ‘𝐶)‘𝑋) ∈ V) | |
6 | df-plusg 16674 | . . 3 ⊢ +g = Slot 2 | |
7 | 2nn 11782 | . . 3 ⊢ 2 ∈ ℕ | |
8 | 5, 6, 7 | strndxid 16606 | . 2 ⊢ (𝜑 → (((End ‘𝐶)‘𝑋)‘(+g‘ndx)) = (+g‘((End ‘𝐶)‘𝑋))) |
9 | basendxnplusgndx 16704 | . . 3 ⊢ (Base‘ndx) ≠ (+g‘ndx) | |
10 | fvex 6681 | . . . 4 ⊢ (+g‘ndx) ∈ V | |
11 | ovex 7197 | . . . 4 ⊢ (〈𝑋, 𝑋〉(comp‘𝐶)𝑋) ∈ V | |
12 | 10, 11 | fvpr2 6957 | . . 3 ⊢ ((Base‘ndx) ≠ (+g‘ndx) → ({〈(Base‘ndx), (𝑋(Hom ‘𝐶)𝑋)〉, 〈(+g‘ndx), (〈𝑋, 𝑋〉(comp‘𝐶)𝑋)〉}‘(+g‘ndx)) = (〈𝑋, 𝑋〉(comp‘𝐶)𝑋)) |
13 | 9, 12 | mp1i 13 | . 2 ⊢ (𝜑 → ({〈(Base‘ndx), (𝑋(Hom ‘𝐶)𝑋)〉, 〈(+g‘ndx), (〈𝑋, 𝑋〉(comp‘𝐶)𝑋)〉}‘(+g‘ndx)) = (〈𝑋, 𝑋〉(comp‘𝐶)𝑋)) |
14 | 4, 8, 13 | 3eqtr3d 2781 | 1 ⊢ (𝜑 → (+g‘((End ‘𝐶)‘𝑋)) = (〈𝑋, 𝑋〉(comp‘𝐶)𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2113 ≠ wne 2934 Vcvv 3397 {cpr 4515 〈cop 4519 ‘cfv 6333 (class class class)co 7164 2c2 11764 ndxcnx 16576 Basecbs 16579 +gcplusg 16661 Hom chom 16672 compcco 16673 Catccat 17031 End cend 35093 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-rep 5151 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 ax-cnex 10664 ax-resscn 10665 ax-1cn 10666 ax-icn 10667 ax-addcl 10668 ax-addrcl 10669 ax-mulcl 10670 ax-mulrcl 10671 ax-mulcom 10672 ax-addass 10673 ax-mulass 10674 ax-distr 10675 ax-i2m1 10676 ax-1ne0 10677 ax-1rid 10678 ax-rnegex 10679 ax-rrecex 10680 ax-cnre 10681 ax-pre-lttri 10682 ax-pre-lttrn 10683 ax-pre-ltadd 10684 ax-pre-mulgt0 10685 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-pss 3860 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-tp 4518 df-op 4520 df-uni 4794 df-iun 4880 df-br 5028 df-opab 5090 df-mpt 5108 df-tr 5134 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6123 df-ord 6169 df-on 6170 df-lim 6171 df-suc 6172 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-riota 7121 df-ov 7167 df-oprab 7168 df-mpo 7169 df-om 7594 df-wrecs 7969 df-recs 8030 df-rdg 8068 df-er 8313 df-en 8549 df-dom 8550 df-sdom 8551 df-pnf 10748 df-mnf 10749 df-xr 10750 df-ltxr 10751 df-le 10752 df-sub 10943 df-neg 10944 df-nn 11710 df-2 11772 df-ndx 16582 df-slot 16583 df-base 16585 df-plusg 16674 df-bj-end 35094 |
This theorem is referenced by: bj-endmnd 35098 |
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