Theorem endmndlem 49008

Description: A diagonal hom-set in a category equipped with the restriction of the composition has a structure of monoid. See also df-mndtc 49571 for converting a monoid to a category. Lemma for bj-endmnd 37313. (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 17653	. 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 17654	. 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 17650	. 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 17651	. 2 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑋)) → (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑋⟩ · 𝑋)𝑓) = 𝑓)
25	3, 4, 19, 21, 22, 5, 22, 23	catrid 17652	. 2 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑋)) → (𝑓(⟨𝑋, 𝑋⟩ · 𝑋)((Id‘𝐶)‘𝑋)) = 𝑓)
26	1, 2, 12, 18, 20, 24, 25	ismndd 18690	1 ⊢ (𝜑 → 𝑀 ∈ Mnd)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ⟨cop 4598  ‘cfv 6514  (class class class)co 7390  Basecbs 17186  +_gcplusg 17227  Hom chom 17238  compcco 17239  Catccat 17632  Idccid 17633  Mndcmnd 18668

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  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 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-cat 17636  df-cid 17637  df-mgm 18574  df-sgrp 18653  df-mnd 18669

This theorem is referenced by: (None)