Theorem endmndlem 49176

Description: A diagonal hom-set in a category equipped with the restriction of the composition has a structure of monoid. See also df-mndtc 49739 for converting a monoid to a category. Lemma for bj-endmnd 37435. (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 17599	. 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 17600	. 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 2733	. . 3 ⊢ (Id‘𝐶) = (Id‘𝐶)
20	3, 4, 19, 6, 8	catidcl 17596	. 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 17597	. 2 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑋)) → (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑋⟩ · 𝑋)𝑓) = 𝑓)
25	3, 4, 19, 21, 22, 5, 22, 23	catrid 17598	. 2 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑋)) → (𝑓(⟨𝑋, 𝑋⟩ · 𝑋)((Id‘𝐶)‘𝑋)) = 𝑓)
26	1, 2, 12, 18, 20, 24, 25	ismndd 18672	1 ⊢ (𝜑 → 𝑀 ∈ Mnd)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  ⟨cop 4583  ‘cfv 6489  (class class class)co 7355  Basecbs 17127  +_gcplusg 17168  Hom chom 17179  compcco 17180  Catccat 17578  Idccid 17579  Mndcmnd 18650

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pr 5374

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 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rmo 3347  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-riota 7312  df-ov 7358  df-cat 17582  df-cid 17583  df-mgm 18556  df-sgrp 18635  df-mnd 18651

This theorem is referenced by: (None)