Theorem oppgoppchom 50065

Description: The converted opposite monoid has the same hom-set as that of the opposite category. Example 3.6(2) of [Adamek] p. 25. (Contributed by Zhi Wang, 21-Sep-2025.)

Hypotheses

Ref	Expression
mndtccat.c	⊢ (𝜑 → 𝐶 = (MndToCat‘𝑀))
mndtccat.m	⊢ (𝜑 → 𝑀 ∈ Mnd)
oppgoppchom.d	⊢ (𝜑 → 𝐷 = (MndToCat‘(oppg‘𝑀)))
oppgoppchom.o	⊢ 𝑂 = (oppCat‘𝐶)
oppgoppchom.x	⊢ (𝜑 → 𝑋 ∈ (Base‘𝐷))
oppgoppchom.y	⊢ (𝜑 → 𝑌 ∈ (Base‘𝑂))
oppgoppchom.h	⊢ (𝜑 → 𝐻 = (Hom ‘𝐷))
oppgoppchom.j	⊢ (𝜑 → 𝐽 = (Hom ‘𝑂))

Assertion

Ref	Expression
oppgoppchom	⊢ (𝜑 → (𝑋𝐻𝑋) = (𝑌𝐽𝑌))

Proof of Theorem oppgoppchom

Step	Hyp	Ref	Expression
1		eqid 2736	. . . . . 6 ⊢ (oppg‘𝑀) = (oppg‘𝑀)
2		eqid 2736	. . . . . 6 ⊢ (Base‘𝑀) = (Base‘𝑀)
3	1, 2	oppgbas 19326	. . . . 5 ⊢ (Base‘𝑀) = (Base‘(oppg‘𝑀))
4	3	a1i 11	. . . 4 ⊢ (𝜑 → (Base‘𝑀) = (Base‘(oppg‘𝑀)))
5		mndtccat.c	. . . . 5 ⊢ (𝜑 → 𝐶 = (MndToCat‘𝑀))
6		mndtccat.m	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ Mnd)
7		oppgoppchom.o	. . . . . . . 8 ⊢ 𝑂 = (oppCat‘𝐶)
8		eqid 2736	. . . . . . . 8 ⊢ (Base‘𝐶) = (Base‘𝐶)
9	7, 8	oppcbas 17684	. . . . . . 7 ⊢ (Base‘𝐶) = (Base‘𝑂)
10	9	eqcomi 2745	. . . . . 6 ⊢ (Base‘𝑂) = (Base‘𝐶)
11	10	a1i 11	. . . . 5 ⊢ (𝜑 → (Base‘𝑂) = (Base‘𝐶))
12		oppgoppchom.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝑂))
13		eqidd 2737	. . . . 5 ⊢ (𝜑 → (Hom ‘𝐶) = (Hom ‘𝐶))
14	5, 6, 11, 12, 12, 13	mndtchom 50059	. . . 4 ⊢ (𝜑 → (𝑌(Hom ‘𝐶)𝑌) = (Base‘𝑀))
15		oppgoppchom.d	. . . . 5 ⊢ (𝜑 → 𝐷 = (MndToCat‘(oppg‘𝑀)))
16	1	oppgmnd 19329	. . . . . 6 ⊢ (𝑀 ∈ Mnd → (oppg‘𝑀) ∈ Mnd)
17	6, 16	syl 17	. . . . 5 ⊢ (𝜑 → (oppg‘𝑀) ∈ Mnd)
18		eqidd 2737	. . . . 5 ⊢ (𝜑 → (Base‘𝐷) = (Base‘𝐷))
19		oppgoppchom.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐷))
20		oppgoppchom.h	. . . . 5 ⊢ (𝜑 → 𝐻 = (Hom ‘𝐷))
21	15, 17, 18, 19, 19, 20	mndtchom 50059	. . . 4 ⊢ (𝜑 → (𝑋𝐻𝑋) = (Base‘(oppg‘𝑀)))
22	4, 14, 21	3eqtr4rd 2782	. . 3 ⊢ (𝜑 → (𝑋𝐻𝑋) = (𝑌(Hom ‘𝐶)𝑌))
23		eqid 2736	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
24	23, 7	oppchom 17681	. . 3 ⊢ (𝑌(Hom ‘𝑂)𝑌) = (𝑌(Hom ‘𝐶)𝑌)
25	22, 24	eqtr4di 2789	. 2 ⊢ (𝜑 → (𝑋𝐻𝑋) = (𝑌(Hom ‘𝑂)𝑌))
26		oppgoppchom.j	. . 3 ⊢ (𝜑 → 𝐽 = (Hom ‘𝑂))
27	26	oveqd 7384	. 2 ⊢ (𝜑 → (𝑌𝐽𝑌) = (𝑌(Hom ‘𝑂)𝑌))
28	25, 27	eqtr4d 2774	1 ⊢ (𝜑 → (𝑋𝐻𝑋) = (𝑌𝐽𝑌))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  ‘cfv 6498  (class class class)co 7367  Basecbs 17179  Hom chom 17231  oppCatcoppc 17677  Mndcmnd 18702  opp_gcoppg 19320  MndToCatcmndtc 50052

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 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  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 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 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-ot 4576  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-tpos 8176  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-er 8643  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-nn 12175  df-2 12244  df-3 12245  df-4 12246  df-5 12247  df-6 12248  df-7 12249  df-8 12250  df-9 12251  df-n0 12438  df-z 12525  df-dec 12645  df-uz 12789  df-fz 13462  df-struct 17117  df-sets 17134  df-slot 17152  df-ndx 17164  df-base 17180  df-plusg 17233  df-hom 17244  df-cco 17245  df-0g 17404  df-oppc 17678  df-mgm 18608  df-sgrp 18687  df-mnd 18703  df-oppg 19321  df-mndtc 50053

This theorem is referenced by: (None)