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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 48887 for converting a monoid to a category. Lemma for bj-endmnd 37301. (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 1132 | . . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑋) ∧ 𝑔 ∈ (𝑋𝐻𝑋)) → 𝐶 ∈ Cat) |
8 | endmndlem.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
9 | 8 | 3ad2ant1 1132 | . . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑋) ∧ 𝑔 ∈ (𝑋𝐻𝑋)) → 𝑋 ∈ 𝐵) |
10 | simp3 1137 | . . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑋) ∧ 𝑔 ∈ (𝑋𝐻𝑋)) → 𝑔 ∈ (𝑋𝐻𝑋)) | |
11 | simp2 1136 | . . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑋) ∧ 𝑔 ∈ (𝑋𝐻𝑋)) → 𝑓 ∈ (𝑋𝐻𝑋)) | |
12 | 3, 4, 5, 7, 9, 9, 9, 10, 11 | catcocl 17730 | . 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 1195 | . . 3 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑋) ∧ 𝑔 ∈ (𝑋𝐻𝑋) ∧ 𝑘 ∈ (𝑋𝐻𝑋))) → 𝑘 ∈ (𝑋𝐻𝑋)) | |
16 | simpr2 1194 | . . 3 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑋) ∧ 𝑔 ∈ (𝑋𝐻𝑋) ∧ 𝑘 ∈ (𝑋𝐻𝑋))) → 𝑔 ∈ (𝑋𝐻𝑋)) | |
17 | simpr1 1193 | . . 3 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑋) ∧ 𝑔 ∈ (𝑋𝐻𝑋) ∧ 𝑘 ∈ (𝑋𝐻𝑋))) → 𝑓 ∈ (𝑋𝐻𝑋)) | |
18 | 3, 4, 5, 13, 14, 14, 14, 15, 16, 14, 17 | catass 17731 | . 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 2735 | . . 3 ⊢ (Id‘𝐶) = (Id‘𝐶) | |
20 | 3, 4, 19, 6, 8 | catidcl 17727 | . 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 17728 | . 2 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑋)) → (((Id‘𝐶)‘𝑋)(〈𝑋, 𝑋〉 · 𝑋)𝑓) = 𝑓) |
25 | 3, 4, 19, 21, 22, 5, 22, 23 | catrid 17729 | . 2 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑋)) → (𝑓(〈𝑋, 𝑋〉 · 𝑋)((Id‘𝐶)‘𝑋)) = 𝑓) |
26 | 1, 2, 12, 18, 20, 24, 25 | ismndd 18782 | 1 ⊢ (𝜑 → 𝑀 ∈ Mnd) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 〈cop 4637 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 +gcplusg 17298 Hom chom 17309 compcco 17310 Catccat 17709 Idccid 17710 Mndcmnd 18760 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-cat 17713 df-cid 17714 df-mgm 18666 df-sgrp 18745 df-mnd 18761 |
This theorem is referenced by: (None) |
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