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| Mirrors > Home > MPE Home > Th. List > Mathboxes > endmndlem | Structured version Visualization version GIF version | ||
| Description: A diagonal hom-set in a category equipped with the restriction of the composition has a structure of monoid. See also df-mndtc 49175 for converting a monoid to a category. Lemma for bj-endmnd 37319. (Contributed by Zhi Wang, 25-Sep-2024.) | 
| Ref | Expression | 
|---|---|
| endmndlem.b | ⊢ 𝐵 = (Base‘𝐶) | 
| endmndlem.h | ⊢ 𝐻 = (Hom ‘𝐶) | 
| endmndlem.o | ⊢ · = (comp‘𝐶) | 
| endmndlem.c | ⊢ (𝜑 → 𝐶 ∈ Cat) | 
| endmndlem.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) | 
| endmndlem.m | ⊢ (𝜑 → (𝑋𝐻𝑋) = (Base‘𝑀)) | 
| endmndlem.p | ⊢ (𝜑 → (〈𝑋, 𝑋〉 · 𝑋) = (+g‘𝑀)) | 
| Ref | Expression | 
|---|---|
| endmndlem | ⊢ (𝜑 → 𝑀 ∈ Mnd) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | endmndlem.m | . 2 ⊢ (𝜑 → (𝑋𝐻𝑋) = (Base‘𝑀)) | |
| 2 | endmndlem.p | . 2 ⊢ (𝜑 → (〈𝑋, 𝑋〉 · 𝑋) = (+g‘𝑀)) | |
| 3 | endmndlem.b | . . 3 ⊢ 𝐵 = (Base‘𝐶) | |
| 4 | endmndlem.h | . . 3 ⊢ 𝐻 = (Hom ‘𝐶) | |
| 5 | endmndlem.o | . . 3 ⊢ · = (comp‘𝐶) | |
| 6 | endmndlem.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
| 7 | 6 | 3ad2ant1 1134 | . . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑋) ∧ 𝑔 ∈ (𝑋𝐻𝑋)) → 𝐶 ∈ Cat) | 
| 8 | endmndlem.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
| 9 | 8 | 3ad2ant1 1134 | . . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑋) ∧ 𝑔 ∈ (𝑋𝐻𝑋)) → 𝑋 ∈ 𝐵) | 
| 10 | simp3 1139 | . . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑋) ∧ 𝑔 ∈ (𝑋𝐻𝑋)) → 𝑔 ∈ (𝑋𝐻𝑋)) | |
| 11 | simp2 1138 | . . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑋) ∧ 𝑔 ∈ (𝑋𝐻𝑋)) → 𝑓 ∈ (𝑋𝐻𝑋)) | |
| 12 | 3, 4, 5, 7, 9, 9, 9, 10, 11 | catcocl 17728 | . 2 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑋) ∧ 𝑔 ∈ (𝑋𝐻𝑋)) → (𝑓(〈𝑋, 𝑋〉 · 𝑋)𝑔) ∈ (𝑋𝐻𝑋)) | 
| 13 | 6 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑋) ∧ 𝑔 ∈ (𝑋𝐻𝑋) ∧ 𝑘 ∈ (𝑋𝐻𝑋))) → 𝐶 ∈ Cat) | 
| 14 | 8 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑋) ∧ 𝑔 ∈ (𝑋𝐻𝑋) ∧ 𝑘 ∈ (𝑋𝐻𝑋))) → 𝑋 ∈ 𝐵) | 
| 15 | simpr3 1197 | . . 3 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑋) ∧ 𝑔 ∈ (𝑋𝐻𝑋) ∧ 𝑘 ∈ (𝑋𝐻𝑋))) → 𝑘 ∈ (𝑋𝐻𝑋)) | |
| 16 | simpr2 1196 | . . 3 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑋) ∧ 𝑔 ∈ (𝑋𝐻𝑋) ∧ 𝑘 ∈ (𝑋𝐻𝑋))) → 𝑔 ∈ (𝑋𝐻𝑋)) | |
| 17 | simpr1 1195 | . . 3 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑋) ∧ 𝑔 ∈ (𝑋𝐻𝑋) ∧ 𝑘 ∈ (𝑋𝐻𝑋))) → 𝑓 ∈ (𝑋𝐻𝑋)) | |
| 18 | 3, 4, 5, 13, 14, 14, 14, 15, 16, 14, 17 | catass 17729 | . 2 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑋) ∧ 𝑔 ∈ (𝑋𝐻𝑋) ∧ 𝑘 ∈ (𝑋𝐻𝑋))) → ((𝑓(〈𝑋, 𝑋〉 · 𝑋)𝑔)(〈𝑋, 𝑋〉 · 𝑋)𝑘) = (𝑓(〈𝑋, 𝑋〉 · 𝑋)(𝑔(〈𝑋, 𝑋〉 · 𝑋)𝑘))) | 
| 19 | eqid 2737 | . . 3 ⊢ (Id‘𝐶) = (Id‘𝐶) | |
| 20 | 3, 4, 19, 6, 8 | catidcl 17725 | . 2 ⊢ (𝜑 → ((Id‘𝐶)‘𝑋) ∈ (𝑋𝐻𝑋)) | 
| 21 | 6 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑋)) → 𝐶 ∈ Cat) | 
| 22 | 8 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑋)) → 𝑋 ∈ 𝐵) | 
| 23 | simpr 484 | . . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑋)) → 𝑓 ∈ (𝑋𝐻𝑋)) | |
| 24 | 3, 4, 19, 21, 22, 5, 22, 23 | catlid 17726 | . 2 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑋)) → (((Id‘𝐶)‘𝑋)(〈𝑋, 𝑋〉 · 𝑋)𝑓) = 𝑓) | 
| 25 | 3, 4, 19, 21, 22, 5, 22, 23 | catrid 17727 | . 2 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑋)) → (𝑓(〈𝑋, 𝑋〉 · 𝑋)((Id‘𝐶)‘𝑋)) = 𝑓) | 
| 26 | 1, 2, 12, 18, 20, 24, 25 | ismndd 18769 | 1 ⊢ (𝜑 → 𝑀 ∈ Mnd) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 〈cop 4632 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 +gcplusg 17297 Hom chom 17308 compcco 17309 Catccat 17707 Idccid 17708 Mndcmnd 18747 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-cat 17711 df-cid 17712 df-mgm 18653 df-sgrp 18732 df-mnd 18748 | 
| This theorem is referenced by: (None) | 
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