Theorem bj-endmnd 37850

Description: The monoid of endomorphisms on an object of a category is a monoid. (Contributed by BJ, 5-Apr-2024.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
bj-endval.c	⊢ (𝜑 → 𝐶 ∈ Cat)
bj-endval.x	⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))

Assertion

Ref	Expression
bj-endmnd	⊢ (𝜑 → ((End ‘𝐶)‘𝑋) ∈ Mnd)

Proof of Theorem bj-endmnd

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		bj-endval.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
2		bj-endval.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))
3	1, 2	bj-endbase 37848	. . 3 ⊢ (𝜑 → (Base‘((End ‘𝐶)‘𝑋)) = (𝑋(Hom ‘𝐶)𝑋))
4	3	eqcomd 2775	. 2 ⊢ (𝜑 → (𝑋(Hom ‘𝐶)𝑋) = (Base‘((End ‘𝐶)‘𝑋)))
5	1, 2	bj-endcomp 37849	. . 3 ⊢ (𝜑 → (+g‘((End ‘𝐶)‘𝑋)) = (⟨𝑋, 𝑋⟩(comp‘𝐶)𝑋))
6	5	eqcomd 2775	. 2 ⊢ (𝜑 → (⟨𝑋, 𝑋⟩(comp‘𝐶)𝑋) = (+g‘((End ‘𝐶)‘𝑋)))
7		eqid 2769	. . 3 ⊢ (Base‘𝐶) = (Base‘𝐶)
8		eqid 2769	. . 3 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
9		eqid 2769	. . 3 ⊢ (comp‘𝐶) = (comp‘𝐶)
10	1	3ad2ant1 1149	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(Hom ‘𝐶)𝑋) ∧ 𝑦 ∈ (𝑋(Hom ‘𝐶)𝑋)) → 𝐶 ∈ Cat)
11	2	3ad2ant1 1149	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(Hom ‘𝐶)𝑋) ∧ 𝑦 ∈ (𝑋(Hom ‘𝐶)𝑋)) → 𝑋 ∈ (Base‘𝐶))
12		simp3 1154	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(Hom ‘𝐶)𝑋) ∧ 𝑦 ∈ (𝑋(Hom ‘𝐶)𝑋)) → 𝑦 ∈ (𝑋(Hom ‘𝐶)𝑋))
13		simp2 1153	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(Hom ‘𝐶)𝑋) ∧ 𝑦 ∈ (𝑋(Hom ‘𝐶)𝑋)) → 𝑥 ∈ (𝑋(Hom ‘𝐶)𝑋))
14	7, 8, 9, 10, 11, 11, 11, 12, 13	catcocl 17741	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(Hom ‘𝐶)𝑋) ∧ 𝑦 ∈ (𝑋(Hom ‘𝐶)𝑋)) → (𝑥(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑋)𝑦) ∈ (𝑋(Hom ‘𝐶)𝑋))
15	1	adantr 485	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑋(Hom ‘𝐶)𝑋) ∧ 𝑦 ∈ (𝑋(Hom ‘𝐶)𝑋) ∧ 𝑧 ∈ (𝑋(Hom ‘𝐶)𝑋))) → 𝐶 ∈ Cat)
16	2	adantr 485	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑋(Hom ‘𝐶)𝑋) ∧ 𝑦 ∈ (𝑋(Hom ‘𝐶)𝑋) ∧ 𝑧 ∈ (𝑋(Hom ‘𝐶)𝑋))) → 𝑋 ∈ (Base‘𝐶))
17		simpr 489	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑋(Hom ‘𝐶)𝑋) ∧ 𝑦 ∈ (𝑋(Hom ‘𝐶)𝑋) ∧ 𝑧 ∈ (𝑋(Hom ‘𝐶)𝑋))) → (𝑥 ∈ (𝑋(Hom ‘𝐶)𝑋) ∧ 𝑦 ∈ (𝑋(Hom ‘𝐶)𝑋) ∧ 𝑧 ∈ (𝑋(Hom ‘𝐶)𝑋)))
18		simp3 1154	. . . 4 ⊢ ((𝑥 ∈ (𝑋(Hom ‘𝐶)𝑋) ∧ 𝑦 ∈ (𝑋(Hom ‘𝐶)𝑋) ∧ 𝑧 ∈ (𝑋(Hom ‘𝐶)𝑋)) → 𝑧 ∈ (𝑋(Hom ‘𝐶)𝑋))
19	17, 18	syl 18	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑋(Hom ‘𝐶)𝑋) ∧ 𝑦 ∈ (𝑋(Hom ‘𝐶)𝑋) ∧ 𝑧 ∈ (𝑋(Hom ‘𝐶)𝑋))) → 𝑧 ∈ (𝑋(Hom ‘𝐶)𝑋))
20		simp2 1153	. . . 4 ⊢ ((𝑥 ∈ (𝑋(Hom ‘𝐶)𝑋) ∧ 𝑦 ∈ (𝑋(Hom ‘𝐶)𝑋) ∧ 𝑧 ∈ (𝑋(Hom ‘𝐶)𝑋)) → 𝑦 ∈ (𝑋(Hom ‘𝐶)𝑋))
21	17, 20	syl 18	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑋(Hom ‘𝐶)𝑋) ∧ 𝑦 ∈ (𝑋(Hom ‘𝐶)𝑋) ∧ 𝑧 ∈ (𝑋(Hom ‘𝐶)𝑋))) → 𝑦 ∈ (𝑋(Hom ‘𝐶)𝑋))
22		simp1 1152	. . . 4 ⊢ ((𝑥 ∈ (𝑋(Hom ‘𝐶)𝑋) ∧ 𝑦 ∈ (𝑋(Hom ‘𝐶)𝑋) ∧ 𝑧 ∈ (𝑋(Hom ‘𝐶)𝑋)) → 𝑥 ∈ (𝑋(Hom ‘𝐶)𝑋))
23	17, 22	syl 18	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑋(Hom ‘𝐶)𝑋) ∧ 𝑦 ∈ (𝑋(Hom ‘𝐶)𝑋) ∧ 𝑧 ∈ (𝑋(Hom ‘𝐶)𝑋))) → 𝑥 ∈ (𝑋(Hom ‘𝐶)𝑋))
24	7, 8, 9, 15, 16, 16, 16, 19, 21, 16, 23	catass 17742	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑋(Hom ‘𝐶)𝑋) ∧ 𝑦 ∈ (𝑋(Hom ‘𝐶)𝑋) ∧ 𝑧 ∈ (𝑋(Hom ‘𝐶)𝑋))) → ((𝑥(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑋)𝑦)(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑋)𝑧) = (𝑥(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑋)(𝑦(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑋)𝑧)))
25		eqid 2769	. . 3 ⊢ (Id‘𝐶) = (Id‘𝐶)
26	7, 8, 25, 1, 2	catidcl 17738	. 2 ⊢ (𝜑 → ((Id‘𝐶)‘𝑋) ∈ (𝑋(Hom ‘𝐶)𝑋))
27	1	adantr 485	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(Hom ‘𝐶)𝑋)) → 𝐶 ∈ Cat)
28	2	adantr 485	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(Hom ‘𝐶)𝑋)) → 𝑋 ∈ (Base‘𝐶))
29		simpr 489	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(Hom ‘𝐶)𝑋)) → 𝑥 ∈ (𝑋(Hom ‘𝐶)𝑋))
30	7, 8, 25, 27, 28, 9, 28, 29	catlid 17739	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(Hom ‘𝐶)𝑋)) → (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑋)𝑥) = 𝑥)
31	7, 8, 25, 27, 28, 9, 28, 29	catrid 17740	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(Hom ‘𝐶)𝑋)) → (𝑥(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑋)((Id‘𝐶)‘𝑋)) = 𝑥)
32	4, 6, 14, 24, 26, 30, 31	ismndd 18814	1 ⊢ (𝜑 → ((End ‘𝐶)‘𝑋) ∈ Mnd)

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101   ∈ wcel 2149  ⟨cop 4600  ‘cfv 6537  (class class class)co 7411  Basecbs 17269  +_gcplusg 17310  Hom chom 17321  compcco 17322  Catccat 17720  Idccid 17721  Mndcmnd 18792  End cend 37845

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-cnex 11156  ax-resscn 11157  ax-1cn 11158  ax-icn 11159  ax-addcl 11160  ax-addrcl 11161  ax-mulcl 11162  ax-mulrcl 11163  ax-mulcom 11164  ax-addass 11165  ax-mulass 11166  ax-distr 11167  ax-i2m1 11168  ax-1ne0 11169  ax-1rid 11170  ax-rnegex 11171  ax-rrecex 11172  ax-cnre 11173  ax-pre-lttri 11174  ax-pre-lttrn 11175  ax-pre-ltadd 11176  ax-pre-mulgt0 11177

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7863  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-er 8694  df-en 8944  df-dom 8945  df-sdom 8946  df-pnf 11245  df-mnf 11246  df-xr 11247  df-ltxr 11248  df-le 11249  df-sub 11443  df-neg 11444  df-nn 12234  df-2 12303  df-slot 17242  df-ndx 17254  df-base 17270  df-plusg 17323  df-cat 17724  df-cid 17725  df-mgm 18698  df-sgrp 18777  df-mnd 18793  df-bj-end 37846

This theorem is referenced by: (None)