Theorem bj-endmnd 37523

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 37521	. . 3 ⊢ (𝜑 → (Base‘((End ‘𝐶)‘𝑋)) = (𝑋(Hom ‘𝐶)𝑋))
4	3	eqcomd 2742	. 2 ⊢ (𝜑 → (𝑋(Hom ‘𝐶)𝑋) = (Base‘((End ‘𝐶)‘𝑋)))
5	1, 2	bj-endcomp 37522	. . 3 ⊢ (𝜑 → (+g‘((End ‘𝐶)‘𝑋)) = (⟨𝑋, 𝑋⟩(comp‘𝐶)𝑋))
6	5	eqcomd 2742	. 2 ⊢ (𝜑 → (⟨𝑋, 𝑋⟩(comp‘𝐶)𝑋) = (+g‘((End ‘𝐶)‘𝑋)))
7		eqid 2736	. . 3 ⊢ (Base‘𝐶) = (Base‘𝐶)
8		eqid 2736	. . 3 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
9		eqid 2736	. . 3 ⊢ (comp‘𝐶) = (comp‘𝐶)
10	1	3ad2ant1 1133	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(Hom ‘𝐶)𝑋) ∧ 𝑦 ∈ (𝑋(Hom ‘𝐶)𝑋)) → 𝐶 ∈ Cat)
11	2	3ad2ant1 1133	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(Hom ‘𝐶)𝑋) ∧ 𝑦 ∈ (𝑋(Hom ‘𝐶)𝑋)) → 𝑋 ∈ (Base‘𝐶))
12		simp3 1138	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(Hom ‘𝐶)𝑋) ∧ 𝑦 ∈ (𝑋(Hom ‘𝐶)𝑋)) → 𝑦 ∈ (𝑋(Hom ‘𝐶)𝑋))
13		simp2 1137	. . 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 17608	. 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 1138	. . . 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 1137	. . . 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 1136	. . . 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 17609	. 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 2736	. . 3 ⊢ (Id‘𝐶) = (Id‘𝐶)
26	7, 8, 25, 1, 2	catidcl 17605	. 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 17606	. 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 17607	. 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 18681	1 ⊢ (𝜑 → ((End ‘𝐶)‘𝑋) ∈ Mnd)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   ∈ wcel 2113  ⟨cop 4586  ‘cfv 6492  (class class class)co 7358  Basecbs 17136  +_gcplusg 17177  Hom chom 17188  compcco 17189  Catccat 17587  Idccid 17588  Mndcmnd 18659  End cend 37518

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 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-cnex 11082  ax-resscn 11083  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-addrcl 11087  ax-mulcl 11088  ax-mulrcl 11089  ax-mulcom 11090  ax-addass 11091  ax-mulass 11092  ax-distr 11093  ax-i2m1 11094  ax-1ne0 11095  ax-1rid 11096  ax-rnegex 11097  ax-rrecex 11098  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101  ax-pre-ltadd 11102  ax-pre-mulgt0 11103

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-er 8635  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11168  df-mnf 11169  df-xr 11170  df-ltxr 11171  df-le 11172  df-sub 11366  df-neg 11367  df-nn 12146  df-2 12208  df-slot 17109  df-ndx 17121  df-base 17137  df-plusg 17190  df-cat 17591  df-cid 17592  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-bj-end 37519

This theorem is referenced by: (None)