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Mirrors > Home > MPE Home > Th. List > Mathboxes > mndtcbasval | Structured version Visualization version GIF version |
Description: The base set of the category built from a monoid. (Contributed by Zhi Wang, 22-Sep-2024.) (New usage is discouraged.) |
Ref | Expression |
---|---|
mndtcbas.c | ⊢ (𝜑 → 𝐶 = (MndToCat‘𝑀)) |
mndtcbas.m | ⊢ (𝜑 → 𝑀 ∈ Mnd) |
mndtcbas.b | ⊢ (𝜑 → 𝐵 = (Base‘𝐶)) |
Ref | Expression |
---|---|
mndtcbasval | ⊢ (𝜑 → 𝐵 = {𝑀}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mndtcbas.c | . . . 4 ⊢ (𝜑 → 𝐶 = (MndToCat‘𝑀)) | |
2 | mndtcbas.m | . . . 4 ⊢ (𝜑 → 𝑀 ∈ Mnd) | |
3 | 1, 2 | mndtcval 45867 | . . 3 ⊢ (𝜑 → 𝐶 = {〈(Base‘ndx), {𝑀}〉, 〈(Hom ‘ndx), {〈𝑀, 𝑀, (Base‘𝑀)〉}〉, 〈(comp‘ndx), {〈〈𝑀, 𝑀, 𝑀〉, (+g‘𝑀)〉}〉}) |
4 | 3 | fveq2d 6690 | . 2 ⊢ (𝜑 → (Base‘𝐶) = (Base‘{〈(Base‘ndx), {𝑀}〉, 〈(Hom ‘ndx), {〈𝑀, 𝑀, (Base‘𝑀)〉}〉, 〈(comp‘ndx), {〈〈𝑀, 𝑀, 𝑀〉, (+g‘𝑀)〉}〉})) |
5 | mndtcbas.b | . 2 ⊢ (𝜑 → 𝐵 = (Base‘𝐶)) | |
6 | snex 5308 | . . 3 ⊢ {𝑀} ∈ V | |
7 | catstr 17344 | . . . 4 ⊢ {〈(Base‘ndx), {𝑀}〉, 〈(Hom ‘ndx), {〈𝑀, 𝑀, (Base‘𝑀)〉}〉, 〈(comp‘ndx), {〈〈𝑀, 𝑀, 𝑀〉, (+g‘𝑀)〉}〉} Struct 〈1, ;15〉 | |
8 | baseid 16658 | . . . 4 ⊢ Base = Slot (Base‘ndx) | |
9 | snsstp1 4714 | . . . 4 ⊢ {〈(Base‘ndx), {𝑀}〉} ⊆ {〈(Base‘ndx), {𝑀}〉, 〈(Hom ‘ndx), {〈𝑀, 𝑀, (Base‘𝑀)〉}〉, 〈(comp‘ndx), {〈〈𝑀, 𝑀, 𝑀〉, (+g‘𝑀)〉}〉} | |
10 | 7, 8, 9 | strfv 16646 | . . 3 ⊢ ({𝑀} ∈ V → {𝑀} = (Base‘{〈(Base‘ndx), {𝑀}〉, 〈(Hom ‘ndx), {〈𝑀, 𝑀, (Base‘𝑀)〉}〉, 〈(comp‘ndx), {〈〈𝑀, 𝑀, 𝑀〉, (+g‘𝑀)〉}〉})) |
11 | 6, 10 | mp1i 13 | . 2 ⊢ (𝜑 → {𝑀} = (Base‘{〈(Base‘ndx), {𝑀}〉, 〈(Hom ‘ndx), {〈𝑀, 𝑀, (Base‘𝑀)〉}〉, 〈(comp‘ndx), {〈〈𝑀, 𝑀, 𝑀〉, (+g‘𝑀)〉}〉})) |
12 | 4, 5, 11 | 3eqtr4d 2784 | 1 ⊢ (𝜑 → 𝐵 = {𝑀}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 Vcvv 3400 {csn 4526 {ctp 4530 〈cop 4532 〈cotp 4534 ‘cfv 6349 1c1 10628 5c5 11786 ;cdc 12191 ndxcnx 16595 Basecbs 16598 +gcplusg 16680 Hom chom 16691 compcco 16692 Mndcmnd 18039 MndToCatcmndtc 45865 |
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 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7491 ax-cnex 10683 ax-resscn 10684 ax-1cn 10685 ax-icn 10686 ax-addcl 10687 ax-addrcl 10688 ax-mulcl 10689 ax-mulrcl 10690 ax-mulcom 10691 ax-addass 10692 ax-mulass 10693 ax-distr 10694 ax-i2m1 10695 ax-1ne0 10696 ax-1rid 10697 ax-rnegex 10698 ax-rrecex 10699 ax-cnre 10700 ax-pre-lttri 10701 ax-pre-lttrn 10702 ax-pre-ltadd 10703 ax-pre-mulgt0 10704 |
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 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-tp 4531 df-op 4533 df-ot 4535 df-uni 4807 df-iun 4893 df-br 5041 df-opab 5103 df-mpt 5121 df-tr 5147 df-id 5439 df-eprel 5444 df-po 5452 df-so 5453 df-fr 5493 df-we 5495 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-pred 6139 df-ord 6185 df-on 6186 df-lim 6187 df-suc 6188 df-iota 6307 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7139 df-ov 7185 df-oprab 7186 df-mpo 7187 df-om 7612 df-1st 7726 df-2nd 7727 df-wrecs 7988 df-recs 8049 df-rdg 8087 df-1o 8143 df-er 8332 df-en 8568 df-dom 8569 df-sdom 8570 df-fin 8571 df-pnf 10767 df-mnf 10768 df-xr 10769 df-ltxr 10770 df-le 10771 df-sub 10962 df-neg 10963 df-nn 11729 df-2 11791 df-3 11792 df-4 11793 df-5 11794 df-6 11795 df-7 11796 df-8 11797 df-9 11798 df-n0 11989 df-z 12075 df-dec 12192 df-uz 12337 df-fz 12994 df-struct 16600 df-ndx 16601 df-slot 16602 df-base 16604 df-hom 16704 df-cco 16705 df-mndtc 45866 |
This theorem is referenced by: mndtcbas 45869 mndtcob 45870 |
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