Theorem endmndlem 49821

Description: A diagonal hom-set in a category equipped with the restriction of the composition has a structure of monoid. See also df-mndtc 50384 for converting a monoid to a category. Lemma for bj-endmnd 37990. (Contributed by Zhi Wang, 25-Sep-2024.)

Hypotheses

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))

Assertion

Ref	Expression
endmndlem	⊢ (𝜑 → 𝑀 ∈ Mnd)

Proof of Theorem endmndlem

Dummy variables 𝑓 𝑔 𝑘 are mutually distinct and distinct from all other variables.

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 1150	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑋) ∧ 𝑔 ∈ (𝑋𝐻𝑋)) → 𝐶 ∈ Cat)
8		endmndlem.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
9	8	3ad2ant1 1150	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑋) ∧ 𝑔 ∈ (𝑋𝐻𝑋)) → 𝑋 ∈ 𝐵)
10		simp3 1155	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑋) ∧ 𝑔 ∈ (𝑋𝐻𝑋)) → 𝑔 ∈ (𝑋𝐻𝑋))
11		simp2 1154	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑋) ∧ 𝑔 ∈ (𝑋𝐻𝑋)) → 𝑓 ∈ (𝑋𝐻𝑋))
12	3, 4, 5, 7, 9, 9, 9, 10, 11	catcocl 17747	. 2 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑋) ∧ 𝑔 ∈ (𝑋𝐻𝑋)) → (𝑓(⟨𝑋, 𝑋⟩ · 𝑋)𝑔) ∈ (𝑋𝐻𝑋))
13	6	adantr 485	. . 3 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑋) ∧ 𝑔 ∈ (𝑋𝐻𝑋) ∧ 𝑘 ∈ (𝑋𝐻𝑋))) → 𝐶 ∈ Cat)
14	8	adantr 485	. . 3 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑋) ∧ 𝑔 ∈ (𝑋𝐻𝑋) ∧ 𝑘 ∈ (𝑋𝐻𝑋))) → 𝑋 ∈ 𝐵)
15		simpr3 1214	. . 3 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑋) ∧ 𝑔 ∈ (𝑋𝐻𝑋) ∧ 𝑘 ∈ (𝑋𝐻𝑋))) → 𝑘 ∈ (𝑋𝐻𝑋))
16		simpr2 1213	. . 3 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑋) ∧ 𝑔 ∈ (𝑋𝐻𝑋) ∧ 𝑘 ∈ (𝑋𝐻𝑋))) → 𝑔 ∈ (𝑋𝐻𝑋))
17		simpr1 1212	. . 3 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑋) ∧ 𝑔 ∈ (𝑋𝐻𝑋) ∧ 𝑘 ∈ (𝑋𝐻𝑋))) → 𝑓 ∈ (𝑋𝐻𝑋))
18	3, 4, 5, 13, 14, 14, 14, 15, 16, 14, 17	catass 17748	. 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 2762	. . 3 ⊢ (Id‘𝐶) = (Id‘𝐶)
20	3, 4, 19, 6, 8	catidcl 17744	. 2 ⊢ (𝜑 → ((Id‘𝐶)‘𝑋) ∈ (𝑋𝐻𝑋))
21	6	adantr 485	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑋)) → 𝐶 ∈ Cat)
22	8	adantr 485	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑋)) → 𝑋 ∈ 𝐵)
23		simpr 489	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑋)) → 𝑓 ∈ (𝑋𝐻𝑋))
24	3, 4, 19, 21, 22, 5, 22, 23	catlid 17745	. 2 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑋)) → (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑋⟩ · 𝑋)𝑓) = 𝑓)
25	3, 4, 19, 21, 22, 5, 22, 23	catrid 17746	. 2 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑋)) → (𝑓(⟨𝑋, 𝑋⟩ · 𝑋)((Id‘𝐶)‘𝑋)) = 𝑓)
26	1, 2, 12, 18, 20, 24, 25	ismndd 18820	1 ⊢ (𝜑 → 𝑀 ∈ Mnd)

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   ∧ w3a 1102   = wceq 1569   ∈ wcel 2142  ⟨cop 4594  ‘cfv 6536  (class class class)co 7412  Basecbs 17275  +_gcplusg 17316  Hom chom 17327  compcco 17328  Catccat 17726  Idccid 17727  Mndcmnd 18798

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7369  df-ov 7415  df-cat 17730  df-cid 17731  df-mgm 18704  df-sgrp 18783  df-mnd 18799

This theorem is used by: (None)