Theorem endmndlem 49597

Description: A diagonal hom-set in a category equipped with the restriction of the composition has a structure of monoid. See also df-mndtc 50160 for converting a monoid to a category. Lemma for bj-endmnd 37771. (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 1145	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑋) ∧ 𝑔 ∈ (𝑋𝐻𝑋)) → 𝐶 ∈ Cat)
8		endmndlem.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
9	8	3ad2ant1 1145	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑋) ∧ 𝑔 ∈ (𝑋𝐻𝑋)) → 𝑋 ∈ 𝐵)
10		simp3 1150	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑋) ∧ 𝑔 ∈ (𝑋𝐻𝑋)) → 𝑔 ∈ (𝑋𝐻𝑋))
11		simp2 1149	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑋) ∧ 𝑔 ∈ (𝑋𝐻𝑋)) → 𝑓 ∈ (𝑋𝐻𝑋))
12	3, 4, 5, 7, 9, 9, 9, 10, 11	catcocl 17708	. 2 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑋) ∧ 𝑔 ∈ (𝑋𝐻𝑋)) → (𝑓(⟨𝑋, 𝑋⟩ · 𝑋)𝑔) ∈ (𝑋𝐻𝑋))
13	6	adantr 484	. . 3 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑋) ∧ 𝑔 ∈ (𝑋𝐻𝑋) ∧ 𝑘 ∈ (𝑋𝐻𝑋))) → 𝐶 ∈ Cat)
14	8	adantr 484	. . 3 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑋) ∧ 𝑔 ∈ (𝑋𝐻𝑋) ∧ 𝑘 ∈ (𝑋𝐻𝑋))) → 𝑋 ∈ 𝐵)
15		simpr3 1209	. . 3 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑋) ∧ 𝑔 ∈ (𝑋𝐻𝑋) ∧ 𝑘 ∈ (𝑋𝐻𝑋))) → 𝑘 ∈ (𝑋𝐻𝑋))
16		simpr2 1208	. . 3 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑋) ∧ 𝑔 ∈ (𝑋𝐻𝑋) ∧ 𝑘 ∈ (𝑋𝐻𝑋))) → 𝑔 ∈ (𝑋𝐻𝑋))
17		simpr1 1207	. . 3 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑋) ∧ 𝑔 ∈ (𝑋𝐻𝑋) ∧ 𝑘 ∈ (𝑋𝐻𝑋))) → 𝑓 ∈ (𝑋𝐻𝑋))
18	3, 4, 5, 13, 14, 14, 14, 15, 16, 14, 17	catass 17709	. 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 2761	. . 3 ⊢ (Id‘𝐶) = (Id‘𝐶)
20	3, 4, 19, 6, 8	catidcl 17705	. 2 ⊢ (𝜑 → ((Id‘𝐶)‘𝑋) ∈ (𝑋𝐻𝑋))
21	6	adantr 484	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑋)) → 𝐶 ∈ Cat)
22	8	adantr 484	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑋)) → 𝑋 ∈ 𝐵)
23		simpr 488	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑋)) → 𝑓 ∈ (𝑋𝐻𝑋))
24	3, 4, 19, 21, 22, 5, 22, 23	catlid 17706	. 2 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑋)) → (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑋⟩ · 𝑋)𝑓) = 𝑓)
25	3, 4, 19, 21, 22, 5, 22, 23	catrid 17707	. 2 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑋)) → (𝑓(⟨𝑋, 𝑋⟩ · 𝑋)((Id‘𝐶)‘𝑋)) = 𝑓)
26	1, 2, 12, 18, 20, 24, 25	ismndd 18781	1 ⊢ (𝜑 → 𝑀 ∈ Mnd)

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141  ⟨cop 4585  ‘cfv 6516  (class class class)co 7391  Basecbs 17236  +_gcplusg 17277  Hom chom 17288  compcco 17289  Catccat 17687  Idccid 17688  Mndcmnd 18759

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5224  ax-sep 5243  ax-nul 5253  ax-pr 5387

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-iota 6472  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-riota 7348  df-ov 7394  df-cat 17691  df-cid 17692  df-mgm 18665  df-sgrp 18744  df-mnd 18760

This theorem is referenced by: (None)