Theorem bj-endmnd 37573

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 37571	. . 3 ⊢ (𝜑 → (Base‘((End ‘𝐶)‘𝑋)) = (𝑋(Hom ‘𝐶)𝑋))
4	3	eqcomd 2743	. 2 ⊢ (𝜑 → (𝑋(Hom ‘𝐶)𝑋) = (Base‘((End ‘𝐶)‘𝑋)))
5	1, 2	bj-endcomp 37572	. . 3 ⊢ (𝜑 → (+g‘((End ‘𝐶)‘𝑋)) = (⟨𝑋, 𝑋⟩(comp‘𝐶)𝑋))
6	5	eqcomd 2743	. 2 ⊢ (𝜑 → (⟨𝑋, 𝑋⟩(comp‘𝐶)𝑋) = (+g‘((End ‘𝐶)‘𝑋)))
7		eqid 2737	. . 3 ⊢ (Base‘𝐶) = (Base‘𝐶)
8		eqid 2737	. . 3 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
9		eqid 2737	. . 3 ⊢ (comp‘𝐶) = (comp‘𝐶)
10	1	3ad2ant1 1134	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(Hom ‘𝐶)𝑋) ∧ 𝑦 ∈ (𝑋(Hom ‘𝐶)𝑋)) → 𝐶 ∈ Cat)
11	2	3ad2ant1 1134	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(Hom ‘𝐶)𝑋) ∧ 𝑦 ∈ (𝑋(Hom ‘𝐶)𝑋)) → 𝑋 ∈ (Base‘𝐶))
12		simp3 1139	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(Hom ‘𝐶)𝑋) ∧ 𝑦 ∈ (𝑋(Hom ‘𝐶)𝑋)) → 𝑦 ∈ (𝑋(Hom ‘𝐶)𝑋))
13		simp2 1138	. . 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 17620	. 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 1139	. . . 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 1138	. . . 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 1137	. . . 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 17621	. 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 2737	. . 3 ⊢ (Id‘𝐶) = (Id‘𝐶)
26	7, 8, 25, 1, 2	catidcl 17617	. 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 17618	. 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 17619	. 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 18693	1 ⊢ (𝜑 → ((End ‘𝐶)‘𝑋) ∈ Mnd)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   ∈ wcel 2114  ⟨cop 4588  ‘cfv 6500  (class class class)co 7368  Basecbs 17148  +_gcplusg 17189  Hom chom 17200  compcco 17201  Catccat 17599  Idccid 17600  Mndcmnd 18671  End cend 37568

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-er 8645  df-en 8896  df-dom 8897  df-sdom 8898  df-pnf 11180  df-mnf 11181  df-xr 11182  df-ltxr 11183  df-le 11184  df-sub 11378  df-neg 11379  df-nn 12158  df-2 12220  df-slot 17121  df-ndx 17133  df-base 17149  df-plusg 17202  df-cat 17603  df-cid 17604  df-mgm 18577  df-sgrp 18656  df-mnd 18672  df-bj-end 37569

This theorem is referenced by: (None)