Theorem bj-endmnd 36503

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 36501	. . 3 ⊢ (𝜑 → (Base‘((End ‘𝐶)‘𝑋)) = (𝑋(Hom ‘𝐶)𝑋))
4	3	eqcomd 2737	. 2 ⊢ (𝜑 → (𝑋(Hom ‘𝐶)𝑋) = (Base‘((End ‘𝐶)‘𝑋)))
5	1, 2	bj-endcomp 36502	. . 3 ⊢ (𝜑 → (+g‘((End ‘𝐶)‘𝑋)) = (⟨𝑋, 𝑋⟩(comp‘𝐶)𝑋))
6	5	eqcomd 2737	. 2 ⊢ (𝜑 → (⟨𝑋, 𝑋⟩(comp‘𝐶)𝑋) = (+g‘((End ‘𝐶)‘𝑋)))
7		eqid 2731	. . 3 ⊢ (Base‘𝐶) = (Base‘𝐶)
8		eqid 2731	. . 3 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
9		eqid 2731	. . 3 ⊢ (comp‘𝐶) = (comp‘𝐶)
10	1	3ad2ant1 1132	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(Hom ‘𝐶)𝑋) ∧ 𝑦 ∈ (𝑋(Hom ‘𝐶)𝑋)) → 𝐶 ∈ Cat)
11	2	3ad2ant1 1132	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(Hom ‘𝐶)𝑋) ∧ 𝑦 ∈ (𝑋(Hom ‘𝐶)𝑋)) → 𝑋 ∈ (Base‘𝐶))
12		simp3 1137	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(Hom ‘𝐶)𝑋) ∧ 𝑦 ∈ (𝑋(Hom ‘𝐶)𝑋)) → 𝑦 ∈ (𝑋(Hom ‘𝐶)𝑋))
13		simp2 1136	. . 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 17634	. 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 480	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑋(Hom ‘𝐶)𝑋) ∧ 𝑦 ∈ (𝑋(Hom ‘𝐶)𝑋) ∧ 𝑧 ∈ (𝑋(Hom ‘𝐶)𝑋))) → 𝐶 ∈ Cat)
16	2	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑋(Hom ‘𝐶)𝑋) ∧ 𝑦 ∈ (𝑋(Hom ‘𝐶)𝑋) ∧ 𝑧 ∈ (𝑋(Hom ‘𝐶)𝑋))) → 𝑋 ∈ (Base‘𝐶))
17		simpr 484	. . . 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 1137	. . . 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 17	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑋(Hom ‘𝐶)𝑋) ∧ 𝑦 ∈ (𝑋(Hom ‘𝐶)𝑋) ∧ 𝑧 ∈ (𝑋(Hom ‘𝐶)𝑋))) → 𝑧 ∈ (𝑋(Hom ‘𝐶)𝑋))
20		simp2 1136	. . . 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 17	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑋(Hom ‘𝐶)𝑋) ∧ 𝑦 ∈ (𝑋(Hom ‘𝐶)𝑋) ∧ 𝑧 ∈ (𝑋(Hom ‘𝐶)𝑋))) → 𝑦 ∈ (𝑋(Hom ‘𝐶)𝑋))
22		simp1 1135	. . . 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 17	. . 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 17635	. 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 2731	. . 3 ⊢ (Id‘𝐶) = (Id‘𝐶)
26	7, 8, 25, 1, 2	catidcl 17631	. 2 ⊢ (𝜑 → ((Id‘𝐶)‘𝑋) ∈ (𝑋(Hom ‘𝐶)𝑋))
27	1	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(Hom ‘𝐶)𝑋)) → 𝐶 ∈ Cat)
28	2	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(Hom ‘𝐶)𝑋)) → 𝑋 ∈ (Base‘𝐶))
29		simpr 484	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(Hom ‘𝐶)𝑋)) → 𝑥 ∈ (𝑋(Hom ‘𝐶)𝑋))
30	7, 8, 25, 27, 28, 9, 28, 29	catlid 17632	. 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 17633	. 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 18682	1 ⊢ (𝜑 → ((End ‘𝐶)‘𝑋) ∈ Mnd)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   ∈ wcel 2105  ⟨cop 4634  ‘cfv 6543  (class class class)co 7412  Basecbs 17149  +_gcplusg 17202  Hom chom 17213  compcco 17214  Catccat 17613  Idccid 17614  Mndcmnd 18660  End cend 36498

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-cnex 11170  ax-resscn 11171  ax-1cn 11172  ax-icn 11173  ax-addcl 11174  ax-addrcl 11175  ax-mulcl 11176  ax-mulrcl 11177  ax-mulcom 11178  ax-addass 11179  ax-mulass 11180  ax-distr 11181  ax-i2m1 11182  ax-1ne0 11183  ax-1rid 11184  ax-rnegex 11185  ax-rrecex 11186  ax-cnre 11187  ax-pre-lttri 11188  ax-pre-lttrn 11189  ax-pre-ltadd 11190  ax-pre-mulgt0 11191

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7860  df-2nd 7980  df-frecs 8270  df-wrecs 8301  df-recs 8375  df-rdg 8414  df-er 8707  df-en 8944  df-dom 8945  df-sdom 8946  df-pnf 11255  df-mnf 11256  df-xr 11257  df-ltxr 11258  df-le 11259  df-sub 11451  df-neg 11452  df-nn 12218  df-2 12280  df-slot 17120  df-ndx 17132  df-base 17150  df-plusg 17215  df-cat 17617  df-cid 17618  df-mgm 18566  df-sgrp 18645  df-mnd 18661  df-bj-end 36499

This theorem is referenced by: (None)