Theorem oppgoppcco 49873

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 2735	. . . . . . . 8 ⊢ (Base‘𝐶) = (Base‘𝐶)
5	3, 4	oppcbas 17643	. . . . . . 7 ⊢ (Base‘𝐶) = (Base‘𝑂)
6	5	eqcomi 2744	. . . . . 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 2736	. . . . 5 ⊢ (𝜑 → (comp‘𝐶) = (comp‘𝐶))
10	1, 2, 7, 8, 8, 8, 9	mndtcco 49867	. . . 4 ⊢ (𝜑 → (⟨𝑌, 𝑌⟩(comp‘𝐶)𝑌) = (+g‘𝑀))
11	10	tposeqd 8171	. . 3 ⊢ (𝜑 → tpos (⟨𝑌, 𝑌⟩(comp‘𝐶)𝑌) = tpos (+g‘𝑀))
12		eqid 2735	. . . 4 ⊢ (comp‘𝐶) = (comp‘𝐶)
13	6, 12, 3, 8, 8, 8	oppccofval 17641	. . 3 ⊢ (𝜑 → (⟨𝑌, 𝑌⟩(comp‘𝑂)𝑌) = tpos (⟨𝑌, 𝑌⟩(comp‘𝐶)𝑌))
14		oppgoppchom.d	. . . . 5 ⊢ (𝜑 → 𝐷 = (MndToCat‘(oppg‘𝑀)))
15		eqid 2735	. . . . . . 7 ⊢ (oppg‘𝑀) = (oppg‘𝑀)
16	15	oppgmnd 19285	. . . . . 6 ⊢ (𝑀 ∈ Mnd → (oppg‘𝑀) ∈ Mnd)
17	2, 16	syl 17	. . . . 5 ⊢ (𝜑 → (oppg‘𝑀) ∈ Mnd)
18		eqidd 2736	. . . . 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 49867	. . . 4 ⊢ (𝜑 → (⟨𝑋, 𝑋⟩ · 𝑋) = (+g‘(oppg‘𝑀)))
22		eqid 2735	. . . . 5 ⊢ (+g‘𝑀) = (+g‘𝑀)
23		eqid 2735	. . . . 5 ⊢ (+g‘(oppg‘𝑀)) = (+g‘(oppg‘𝑀))
24	22, 15, 23	oppgplusfval 19279	. . . 4 ⊢ (+g‘(oppg‘𝑀)) = tpos (+g‘𝑀)
25	21, 24	eqtrdi 2786	. . 3 ⊢ (𝜑 → (⟨𝑋, 𝑋⟩ · 𝑋) = tpos (+g‘𝑀))
26	11, 13, 25	3eqtr4rd 2781	. 2 ⊢ (𝜑 → (⟨𝑋, 𝑋⟩ · 𝑋) = (⟨𝑌, 𝑌⟩(comp‘𝑂)𝑌))
27		oppgoppcco.x	. . 3 ⊢ (𝜑 → ∙ = (comp‘𝑂))
28	27	oveqd 7375	. 2 ⊢ (𝜑 → (⟨𝑌, 𝑌⟩ ∙ 𝑌) = (⟨𝑌, 𝑌⟩(comp‘𝑂)𝑌))
29	26, 28	eqtr4d 2773	1 ⊢ (𝜑 → (⟨𝑋, 𝑋⟩ · 𝑋) = (⟨𝑌, 𝑌⟩ ∙ 𝑌))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  ⟨cop 4585  ‘cfv 6491  (class class class)co 7358  tpos ctpos 8167  Basecbs 17138  +_gcplusg 17179  compcco 17191  oppCatcoppc 17636  Mndcmnd 18661  opp_gcoppg 19276  MndToCatcmndtc 49859

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 2183  ax-ext 2707  ax-rep 5223  ax-sep 5240  ax-nul 5250  ax-pow 5309  ax-pr 5376  ax-un 7680  ax-cnex 11084  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104  ax-pre-mulgt0 11105

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 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2932  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3349  df-reu 3350  df-rab 3399  df-v 3441  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4285  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-tp 4584  df-op 4586  df-ot 4588  df-uni 4863  df-iun 4947  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6258  df-ord 6319  df-on 6320  df-lim 6321  df-suc 6322  df-iota 6447  df-fun 6493  df-fn 6494  df-f 6495  df-f1 6496  df-fo 6497  df-f1o 6498  df-fv 6499  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-tpos 8168  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-er 8635  df-en 8886  df-dom 8887  df-sdom 8888  df-fin 8889  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11368  df-neg 11369  df-nn 12148  df-2 12210  df-3 12211  df-4 12212  df-5 12213  df-6 12214  df-7 12215  df-8 12216  df-9 12217  df-n0 12404  df-z 12491  df-dec 12610  df-uz 12754  df-fz 13426  df-struct 17076  df-sets 17093  df-slot 17111  df-ndx 17123  df-base 17139  df-plusg 17192  df-hom 17203  df-cco 17204  df-0g 17363  df-oppc 17637  df-mgm 18567  df-sgrp 18646  df-mnd 18662  df-oppg 19277  df-mndtc 49860

This theorem is referenced by: (None)