Theorem oppgoppcco 49586

Description: The converted opposite monoid has the same composition as that of the opposite category. Example 3.6(2) of [Adamek] p. 25. (Contributed by Zhi Wang, 22-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))
oppgoppcco.o	⊢ (𝜑 → · = (comp‘𝐷))
oppgoppcco.x	⊢ (𝜑 → ∙ = (comp‘𝑂))

Assertion

Ref	Expression
oppgoppcco	⊢ (𝜑 → (⟨𝑋, 𝑋⟩ · 𝑋) = (⟨𝑌, 𝑌⟩ ∙ 𝑌))

Proof of Theorem oppgoppcco

Step	Hyp	Ref	Expression
1		mndtccat.c	. . . . 5 ⊢ (𝜑 → 𝐶 = (MndToCat‘𝑀))
2		mndtccat.m	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ Mnd)
3		oppgoppchom.o	. . . . . . . 8 ⊢ 𝑂 = (oppCat‘𝐶)
4		eqid 2729	. . . . . . . 8 ⊢ (Base‘𝐶) = (Base‘𝐶)
5	3, 4	oppcbas 17624	. . . . . . 7 ⊢ (Base‘𝐶) = (Base‘𝑂)
6	5	eqcomi 2738	. . . . . 6 ⊢ (Base‘𝑂) = (Base‘𝐶)
7	6	a1i 11	. . . . 5 ⊢ (𝜑 → (Base‘𝑂) = (Base‘𝐶))
8		oppgoppchom.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝑂))
9		eqidd 2730	. . . . 5 ⊢ (𝜑 → (comp‘𝐶) = (comp‘𝐶))
10	1, 2, 7, 8, 8, 8, 9	mndtcco 49580	. . . 4 ⊢ (𝜑 → (⟨𝑌, 𝑌⟩(comp‘𝐶)𝑌) = (+g‘𝑀))
11	10	tposeqd 8162	. . 3 ⊢ (𝜑 → tpos (⟨𝑌, 𝑌⟩(comp‘𝐶)𝑌) = tpos (+g‘𝑀))
12		eqid 2729	. . . 4 ⊢ (comp‘𝐶) = (comp‘𝐶)
13	6, 12, 3, 8, 8, 8	oppccofval 17622	. . 3 ⊢ (𝜑 → (⟨𝑌, 𝑌⟩(comp‘𝑂)𝑌) = tpos (⟨𝑌, 𝑌⟩(comp‘𝐶)𝑌))
14		oppgoppchom.d	. . . . 5 ⊢ (𝜑 → 𝐷 = (MndToCat‘(oppg‘𝑀)))
15		eqid 2729	. . . . . . 7 ⊢ (oppg‘𝑀) = (oppg‘𝑀)
16	15	oppgmnd 19233	. . . . . 6 ⊢ (𝑀 ∈ Mnd → (oppg‘𝑀) ∈ Mnd)
17	2, 16	syl 17	. . . . 5 ⊢ (𝜑 → (oppg‘𝑀) ∈ Mnd)
18		eqidd 2730	. . . . 5 ⊢ (𝜑 → (Base‘𝐷) = (Base‘𝐷))
19		oppgoppchom.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐷))
20		oppgoppcco.o	. . . . 5 ⊢ (𝜑 → · = (comp‘𝐷))
21	14, 17, 18, 19, 19, 19, 20	mndtcco 49580	. . . 4 ⊢ (𝜑 → (⟨𝑋, 𝑋⟩ · 𝑋) = (+g‘(oppg‘𝑀)))
22		eqid 2729	. . . . 5 ⊢ (+g‘𝑀) = (+g‘𝑀)
23		eqid 2729	. . . . 5 ⊢ (+g‘(oppg‘𝑀)) = (+g‘(oppg‘𝑀))
24	22, 15, 23	oppgplusfval 19227	. . . 4 ⊢ (+g‘(oppg‘𝑀)) = tpos (+g‘𝑀)
25	21, 24	eqtrdi 2780	. . 3 ⊢ (𝜑 → (⟨𝑋, 𝑋⟩ · 𝑋) = tpos (+g‘𝑀))
26	11, 13, 25	3eqtr4rd 2775	. 2 ⊢ (𝜑 → (⟨𝑋, 𝑋⟩ · 𝑋) = (⟨𝑌, 𝑌⟩(comp‘𝑂)𝑌))
27		oppgoppcco.x	. . 3 ⊢ (𝜑 → ∙ = (comp‘𝑂))
28	27	oveqd 7366	. 2 ⊢ (𝜑 → (⟨𝑌, 𝑌⟩ ∙ 𝑌) = (⟨𝑌, 𝑌⟩(comp‘𝑂)𝑌))
29	26, 28	eqtr4d 2767	1 ⊢ (𝜑 → (⟨𝑋, 𝑋⟩ · 𝑋) = (⟨𝑌, 𝑌⟩ ∙ 𝑌))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  ⟨cop 4583  ‘cfv 6482  (class class class)co 7349  tpos ctpos 8158  Basecbs 17120  +_gcplusg 17161  compcco 17173  oppCatcoppc 17617  Mndcmnd 18608  opp_gcoppg 19224  MndToCatcmndtc 49572

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-tp 4582  df-op 4584  df-ot 4586  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-1st 7924  df-2nd 7925  df-tpos 8159  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-er 8625  df-en 8873  df-dom 8874  df-sdom 8875  df-fin 8876  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-sub 11349  df-neg 11350  df-nn 12129  df-2 12191  df-3 12192  df-4 12193  df-5 12194  df-6 12195  df-7 12196  df-8 12197  df-9 12198  df-n0 12385  df-z 12472  df-dec 12592  df-uz 12736  df-fz 13411  df-struct 17058  df-sets 17075  df-slot 17093  df-ndx 17105  df-base 17121  df-plusg 17174  df-hom 17185  df-cco 17186  df-0g 17345  df-oppc 17618  df-mgm 18514  df-sgrp 18593  df-mnd 18609  df-oppg 19225  df-mndtc 49573

This theorem is referenced by: (None)