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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 49231 | . . . . 5 ⊢ (𝜑 → 𝐶 = {〈(Base‘ndx), {𝑀}〉, 〈(Hom ‘ndx), {〈𝑀, 𝑀, (Base‘𝑀)〉}〉, 〈(comp‘ndx), {〈〈𝑀, 𝑀, 𝑀〉, (+g‘𝑀)〉}〉}) | 
| 5 | catstr 18006 | . . . . 5 ⊢ {〈(Base‘ndx), {𝑀}〉, 〈(Hom ‘ndx), {〈𝑀, 𝑀, (Base‘𝑀)〉}〉, 〈(comp‘ndx), {〈〈𝑀, 𝑀, 𝑀〉, (+g‘𝑀)〉}〉} Struct 〈1, ;15〉 | |
| 6 | ccoid 17459 | . . . . 5 ⊢ comp = Slot (comp‘ndx) | |
| 7 | snsstp3 4817 | . . . . 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 5435 | . . . . . 6 ⊢ {〈〈𝑀, 𝑀, 𝑀〉, (+g‘𝑀)〉} ∈ V | |
| 9 | 8 | a1i 11 | . . . . 5 ⊢ (𝜑 → {〈〈𝑀, 𝑀, 𝑀〉, (+g‘𝑀)〉} ∈ V) | 
| 10 | eqid 2736 | . . . . 5 ⊢ (comp‘𝐶) = (comp‘𝐶) | |
| 11 | 4, 5, 6, 7, 9, 10 | strfv3 17242 | . . . 4 ⊢ (𝜑 → (comp‘𝐶) = {〈〈𝑀, 𝑀, 𝑀〉, (+g‘𝑀)〉}) | 
| 12 | 1, 11 | eqtrd 2776 | . . 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 49234 | . . . 4 ⊢ (𝜑 → 𝑋 = 𝑀) | 
| 16 | mndtchom.y | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
| 17 | 2, 3, 13, 16 | mndtcob 49234 | . . . 4 ⊢ (𝜑 → 𝑌 = 𝑀) | 
| 18 | 15, 17 | opeq12d 4880 | . . 3 ⊢ (𝜑 → 〈𝑋, 𝑌〉 = 〈𝑀, 𝑀〉) | 
| 19 | mndtcco.z | . . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝐵) | |
| 20 | 2, 3, 13, 19 | mndtcob 49234 | . . 3 ⊢ (𝜑 → 𝑍 = 𝑀) | 
| 21 | 12, 18, 20 | oveq123d 7453 | . 2 ⊢ (𝜑 → (〈𝑋, 𝑌〉 · 𝑍) = (〈𝑀, 𝑀〉{〈〈𝑀, 𝑀, 𝑀〉, (+g‘𝑀)〉}𝑀)) | 
| 22 | df-ov 7435 | . . 3 ⊢ (〈𝑀, 𝑀〉{〈〈𝑀, 𝑀, 𝑀〉, (+g‘𝑀)〉}𝑀) = ({〈〈𝑀, 𝑀, 𝑀〉, (+g‘𝑀)〉}‘〈〈𝑀, 𝑀〉, 𝑀〉) | |
| 23 | df-ot 4634 | . . . 4 ⊢ 〈𝑀, 𝑀, 𝑀〉 = 〈〈𝑀, 𝑀〉, 𝑀〉 | |
| 24 | 23 | fveq2i 6908 | . . 3 ⊢ ({〈〈𝑀, 𝑀, 𝑀〉, (+g‘𝑀)〉}‘〈𝑀, 𝑀, 𝑀〉) = ({〈〈𝑀, 𝑀, 𝑀〉, (+g‘𝑀)〉}‘〈〈𝑀, 𝑀〉, 𝑀〉) | 
| 25 | otex 5469 | . . . 4 ⊢ 〈𝑀, 𝑀, 𝑀〉 ∈ V | |
| 26 | fvex 6918 | . . . 4 ⊢ (+g‘𝑀) ∈ V | |
| 27 | 25, 26 | fvsn 7202 | . . 3 ⊢ ({〈〈𝑀, 𝑀, 𝑀〉, (+g‘𝑀)〉}‘〈𝑀, 𝑀, 𝑀〉) = (+g‘𝑀) | 
| 28 | 22, 24, 27 | 3eqtr2i 2770 | . 2 ⊢ (〈𝑀, 𝑀〉{〈〈𝑀, 𝑀, 𝑀〉, (+g‘𝑀)〉}𝑀) = (+g‘𝑀) | 
| 29 | 21, 28 | eqtrdi 2792 | 1 ⊢ (𝜑 → (〈𝑋, 𝑌〉 · 𝑍) = (+g‘𝑀)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2107 Vcvv 3479 {csn 4625 {ctp 4629 〈cop 4631 〈cotp 4633 ‘cfv 6560 (class class class)co 7432 1c1 11157 5c5 12325 ;cdc 12735 ndxcnx 17231 Basecbs 17248 +gcplusg 17298 Hom chom 17309 compcco 17310 Mndcmnd 18748 MndToCatcmndtc 49229 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-ot 4634 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-nn 12268 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12529 df-z 12616 df-dec 12736 df-uz 12880 df-fz 13549 df-struct 17185 df-slot 17220 df-ndx 17232 df-base 17249 df-hom 17322 df-cco 17323 df-mndtc 49230 | 
| This theorem is referenced by: mndtcco2 49238 mndtccatid 49239 oppgoppcco 49243 | 
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