Theorem endmndlem 49026

Description: A diagonal hom-set in a category equipped with the restriction of the composition has a structure of monoid. See also df-mndtc 49589 for converting a monoid to a category. Lemma for bj-endmnd 37331. (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 1133	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑋) ∧ 𝑔 ∈ (𝑋𝐻𝑋)) → 𝐶 ∈ Cat)
8		endmndlem.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
9	8	3ad2ant1 1133	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑋) ∧ 𝑔 ∈ (𝑋𝐻𝑋)) → 𝑋 ∈ 𝐵)
10		simp3 1138	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑋) ∧ 𝑔 ∈ (𝑋𝐻𝑋)) → 𝑔 ∈ (𝑋𝐻𝑋))
11		simp2 1137	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑋) ∧ 𝑔 ∈ (𝑋𝐻𝑋)) → 𝑓 ∈ (𝑋𝐻𝑋))
12	3, 4, 5, 7, 9, 9, 9, 10, 11	catcocl 17583	. 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 17584	. 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 2730	. . 3 ⊢ (Id‘𝐶) = (Id‘𝐶)
20	3, 4, 19, 6, 8	catidcl 17580	. 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 17581	. 2 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑋)) → (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑋⟩ · 𝑋)𝑓) = 𝑓)
25	3, 4, 19, 21, 22, 5, 22, 23	catrid 17582	. 2 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑋)) → (𝑓(⟨𝑋, 𝑋⟩ · 𝑋)((Id‘𝐶)‘𝑋)) = 𝑓)
26	1, 2, 12, 18, 20, 24, 25	ismndd 18656	1 ⊢ (𝜑 → 𝑀 ∈ Mnd)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2110  ⟨cop 4580  ‘cfv 6477  (class class class)co 7341  Basecbs 17112  +_gcplusg 17153  Hom chom 17164  compcco 17165  Catccat 17562  Idccid 17563  Mndcmnd 18634

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3344  df-reu 3345  df-rab 3394  df-v 3436  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4282  df-if 4474  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4858  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6433  df-fun 6479  df-fn 6480  df-f 6481  df-f1 6482  df-fo 6483  df-f1o 6484  df-fv 6485  df-riota 7298  df-ov 7344  df-cat 17566  df-cid 17567  df-mgm 18540  df-sgrp 18619  df-mnd 18635

This theorem is referenced by: (None)