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Mirrors > Home > MPE Home > Th. List > Mathboxes > mndtcco | Structured version Visualization version GIF version |
Description: The composition of the category built from a monoid is the monoid operation. (Contributed by Zhi Wang, 22-Sep-2024.) |
Ref | Expression |
---|---|
mndtcbas.c | ⊢ (𝜑 → 𝐶 = (MndToCat‘𝑀)) |
mndtcbas.m | ⊢ (𝜑 → 𝑀 ∈ Mnd) |
mndtcbas.b | ⊢ (𝜑 → 𝐵 = (Base‘𝐶)) |
mndtchom.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
mndtchom.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
mndtcco.z | ⊢ (𝜑 → 𝑍 ∈ 𝐵) |
mndtcco.o | ⊢ (𝜑 → · = (comp‘𝐶)) |
Ref | Expression |
---|---|
mndtcco | ⊢ (𝜑 → (〈𝑋, 𝑌〉 · 𝑍) = (+g‘𝑀)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mndtcco.o | . . . 4 ⊢ (𝜑 → · = (comp‘𝐶)) | |
2 | mndtcbas.c | . . . . . 6 ⊢ (𝜑 → 𝐶 = (MndToCat‘𝑀)) | |
3 | mndtcbas.m | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ Mnd) | |
4 | 2, 3 | mndtcval 45804 | . . . . 5 ⊢ (𝜑 → 𝐶 = {〈(Base‘ndx), {𝑀}〉, 〈(Hom ‘ndx), {〈𝑀, 𝑀, (Base‘𝑀)〉}〉, 〈(comp‘ndx), {〈〈𝑀, 𝑀, 𝑀〉, (+g‘𝑀)〉}〉}) |
5 | catstr 17325 | . . . . 5 ⊢ {〈(Base‘ndx), {𝑀}〉, 〈(Hom ‘ndx), {〈𝑀, 𝑀, (Base‘𝑀)〉}〉, 〈(comp‘ndx), {〈〈𝑀, 𝑀, 𝑀〉, (+g‘𝑀)〉}〉} Struct 〈1, ;15〉 | |
6 | ccoid 16786 | . . . . 5 ⊢ comp = Slot (comp‘ndx) | |
7 | snsstp3 4703 | . . . . 5 ⊢ {〈(comp‘ndx), {〈〈𝑀, 𝑀, 𝑀〉, (+g‘𝑀)〉}〉} ⊆ {〈(Base‘ndx), {𝑀}〉, 〈(Hom ‘ndx), {〈𝑀, 𝑀, (Base‘𝑀)〉}〉, 〈(comp‘ndx), {〈〈𝑀, 𝑀, 𝑀〉, (+g‘𝑀)〉}〉} | |
8 | snex 5295 | . . . . . 6 ⊢ {〈〈𝑀, 𝑀, 𝑀〉, (+g‘𝑀)〉} ∈ V | |
9 | 8 | a1i 11 | . . . . 5 ⊢ (𝜑 → {〈〈𝑀, 𝑀, 𝑀〉, (+g‘𝑀)〉} ∈ V) |
10 | eqid 2738 | . . . . 5 ⊢ (comp‘𝐶) = (comp‘𝐶) | |
11 | 4, 5, 6, 7, 9, 10 | strfv3 16628 | . . . 4 ⊢ (𝜑 → (comp‘𝐶) = {〈〈𝑀, 𝑀, 𝑀〉, (+g‘𝑀)〉}) |
12 | 1, 11 | eqtrd 2773 | . . 3 ⊢ (𝜑 → · = {〈〈𝑀, 𝑀, 𝑀〉, (+g‘𝑀)〉}) |
13 | mndtcbas.b | . . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘𝐶)) | |
14 | mndtchom.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
15 | 2, 3, 13, 14 | mndtcob 45807 | . . . 4 ⊢ (𝜑 → 𝑋 = 𝑀) |
16 | mndtchom.y | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
17 | 2, 3, 13, 16 | mndtcob 45807 | . . . 4 ⊢ (𝜑 → 𝑌 = 𝑀) |
18 | 15, 17 | opeq12d 4766 | . . 3 ⊢ (𝜑 → 〈𝑋, 𝑌〉 = 〈𝑀, 𝑀〉) |
19 | mndtcco.z | . . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝐵) | |
20 | 2, 3, 13, 19 | mndtcob 45807 | . . 3 ⊢ (𝜑 → 𝑍 = 𝑀) |
21 | 12, 18, 20 | oveq123d 7185 | . 2 ⊢ (𝜑 → (〈𝑋, 𝑌〉 · 𝑍) = (〈𝑀, 𝑀〉{〈〈𝑀, 𝑀, 𝑀〉, (+g‘𝑀)〉}𝑀)) |
22 | df-ov 7167 | . . 3 ⊢ (〈𝑀, 𝑀〉{〈〈𝑀, 𝑀, 𝑀〉, (+g‘𝑀)〉}𝑀) = ({〈〈𝑀, 𝑀, 𝑀〉, (+g‘𝑀)〉}‘〈〈𝑀, 𝑀〉, 𝑀〉) | |
23 | df-ot 4522 | . . . 4 ⊢ 〈𝑀, 𝑀, 𝑀〉 = 〈〈𝑀, 𝑀〉, 𝑀〉 | |
24 | 23 | fveq2i 6671 | . . 3 ⊢ ({〈〈𝑀, 𝑀, 𝑀〉, (+g‘𝑀)〉}‘〈𝑀, 𝑀, 𝑀〉) = ({〈〈𝑀, 𝑀, 𝑀〉, (+g‘𝑀)〉}‘〈〈𝑀, 𝑀〉, 𝑀〉) |
25 | otex 5320 | . . . 4 ⊢ 〈𝑀, 𝑀, 𝑀〉 ∈ V | |
26 | fvex 6681 | . . . 4 ⊢ (+g‘𝑀) ∈ V | |
27 | 25, 26 | fvsn 6947 | . . 3 ⊢ ({〈〈𝑀, 𝑀, 𝑀〉, (+g‘𝑀)〉}‘〈𝑀, 𝑀, 𝑀〉) = (+g‘𝑀) |
28 | 22, 24, 27 | 3eqtr2i 2767 | . 2 ⊢ (〈𝑀, 𝑀〉{〈〈𝑀, 𝑀, 𝑀〉, (+g‘𝑀)〉}𝑀) = (+g‘𝑀) |
29 | 21, 28 | eqtrdi 2789 | 1 ⊢ (𝜑 → (〈𝑋, 𝑌〉 · 𝑍) = (+g‘𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2113 Vcvv 3397 {csn 4513 {ctp 4517 〈cop 4519 〈cotp 4521 ‘cfv 6333 (class class class)co 7164 1c1 10609 5c5 11767 ;cdc 12172 ndxcnx 16576 Basecbs 16579 +gcplusg 16661 Hom chom 16672 compcco 16673 Mndcmnd 18020 MndToCatcmndtc 45802 |
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-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-ot 4522 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-1st 7707 df-2nd 7708 df-wrecs 7969 df-recs 8030 df-rdg 8068 df-1o 8124 df-er 8313 df-en 8549 df-dom 8550 df-sdom 8551 df-fin 8552 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-3 11773 df-4 11774 df-5 11775 df-6 11776 df-7 11777 df-8 11778 df-9 11779 df-n0 11970 df-z 12056 df-dec 12173 df-uz 12318 df-fz 12975 df-struct 16581 df-ndx 16582 df-slot 16583 df-base 16585 df-hom 16685 df-cco 16686 df-mndtc 45803 |
This theorem is referenced by: mndtcco2 45810 mndtccatid 45811 |
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