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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 45979 for converting a monoid to a category. Lemma for bj-endmnd 35172. (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 1135 | . . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑋) ∧ 𝑔 ∈ (𝑋𝐻𝑋)) → 𝐶 ∈ Cat) |
8 | endmndlem.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
9 | 8 | 3ad2ant1 1135 | . . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑋) ∧ 𝑔 ∈ (𝑋𝐻𝑋)) → 𝑋 ∈ 𝐵) |
10 | simp3 1140 | . . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑋) ∧ 𝑔 ∈ (𝑋𝐻𝑋)) → 𝑔 ∈ (𝑋𝐻𝑋)) | |
11 | simp2 1139 | . . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑋) ∧ 𝑔 ∈ (𝑋𝐻𝑋)) → 𝑓 ∈ (𝑋𝐻𝑋)) | |
12 | 3, 4, 5, 7, 9, 9, 9, 10, 11 | catcocl 17142 | . 2 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑋) ∧ 𝑔 ∈ (𝑋𝐻𝑋)) → (𝑓(〈𝑋, 𝑋〉 · 𝑋)𝑔) ∈ (𝑋𝐻𝑋)) |
13 | 6 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑋) ∧ 𝑔 ∈ (𝑋𝐻𝑋) ∧ 𝑘 ∈ (𝑋𝐻𝑋))) → 𝐶 ∈ Cat) |
14 | 8 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑋) ∧ 𝑔 ∈ (𝑋𝐻𝑋) ∧ 𝑘 ∈ (𝑋𝐻𝑋))) → 𝑋 ∈ 𝐵) |
15 | simpr3 1198 | . . 3 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑋) ∧ 𝑔 ∈ (𝑋𝐻𝑋) ∧ 𝑘 ∈ (𝑋𝐻𝑋))) → 𝑘 ∈ (𝑋𝐻𝑋)) | |
16 | simpr2 1197 | . . 3 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑋) ∧ 𝑔 ∈ (𝑋𝐻𝑋) ∧ 𝑘 ∈ (𝑋𝐻𝑋))) → 𝑔 ∈ (𝑋𝐻𝑋)) | |
17 | simpr1 1196 | . . 3 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑋) ∧ 𝑔 ∈ (𝑋𝐻𝑋) ∧ 𝑘 ∈ (𝑋𝐻𝑋))) → 𝑓 ∈ (𝑋𝐻𝑋)) | |
18 | 3, 4, 5, 13, 14, 14, 14, 15, 16, 14, 17 | catass 17143 | . 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 2736 | . . 3 ⊢ (Id‘𝐶) = (Id‘𝐶) | |
20 | 3, 4, 19, 6, 8 | catidcl 17139 | . 2 ⊢ (𝜑 → ((Id‘𝐶)‘𝑋) ∈ (𝑋𝐻𝑋)) |
21 | 6 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑋)) → 𝐶 ∈ Cat) |
22 | 8 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑋)) → 𝑋 ∈ 𝐵) |
23 | simpr 488 | . . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑋)) → 𝑓 ∈ (𝑋𝐻𝑋)) | |
24 | 3, 4, 19, 21, 22, 5, 22, 23 | catlid 17140 | . 2 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑋)) → (((Id‘𝐶)‘𝑋)(〈𝑋, 𝑋〉 · 𝑋)𝑓) = 𝑓) |
25 | 3, 4, 19, 21, 22, 5, 22, 23 | catrid 17141 | . 2 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑋)) → (𝑓(〈𝑋, 𝑋〉 · 𝑋)((Id‘𝐶)‘𝑋)) = 𝑓) |
26 | 1, 2, 12, 18, 20, 24, 25 | ismndd 18149 | 1 ⊢ (𝜑 → 𝑀 ∈ Mnd) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2112 〈cop 4533 ‘cfv 6358 (class class class)co 7191 Basecbs 16666 +gcplusg 16749 Hom chom 16760 compcco 16761 Catccat 17121 Idccid 17122 Mndcmnd 18127 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-cat 17125 df-cid 17126 df-mgm 18068 df-sgrp 18117 df-mnd 18128 |
This theorem is referenced by: (None) |
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