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Mirrors > Home > MPE Home > Th. List > Mathboxes > oppgoppcco | Structured version Visualization version GIF version |
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.) |
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)) |
Ref | Expression |
---|---|
oppgoppcco | ⊢ (𝜑 → (〈𝑋, 𝑋〉 · 𝑋) = (〈𝑌, 𝑌〉 ∙ 𝑌)) |
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 2734 | . . . . . . . 8 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
5 | 3, 4 | oppcbas 17763 | . . . . . . 7 ⊢ (Base‘𝐶) = (Base‘𝑂) |
6 | 5 | eqcomi 2743 | . . . . . 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 2735 | . . . . 5 ⊢ (𝜑 → (comp‘𝐶) = (comp‘𝐶)) | |
10 | 1, 2, 7, 8, 8, 8, 9 | mndtcco 48893 | . . . 4 ⊢ (𝜑 → (〈𝑌, 𝑌〉(comp‘𝐶)𝑌) = (+g‘𝑀)) |
11 | 10 | tposeqd 8252 | . . 3 ⊢ (𝜑 → tpos (〈𝑌, 𝑌〉(comp‘𝐶)𝑌) = tpos (+g‘𝑀)) |
12 | eqid 2734 | . . . 4 ⊢ (comp‘𝐶) = (comp‘𝐶) | |
13 | 6, 12, 3, 8, 8, 8 | oppccofval 17761 | . . 3 ⊢ (𝜑 → (〈𝑌, 𝑌〉(comp‘𝑂)𝑌) = tpos (〈𝑌, 𝑌〉(comp‘𝐶)𝑌)) |
14 | oppgoppchom.d | . . . . 5 ⊢ (𝜑 → 𝐷 = (MndToCat‘(oppg‘𝑀))) | |
15 | eqid 2734 | . . . . . . 7 ⊢ (oppg‘𝑀) = (oppg‘𝑀) | |
16 | 15 | oppgmnd 19387 | . . . . . 6 ⊢ (𝑀 ∈ Mnd → (oppg‘𝑀) ∈ Mnd) |
17 | 2, 16 | syl 17 | . . . . 5 ⊢ (𝜑 → (oppg‘𝑀) ∈ Mnd) |
18 | eqidd 2735 | . . . . 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 48893 | . . . 4 ⊢ (𝜑 → (〈𝑋, 𝑋〉 · 𝑋) = (+g‘(oppg‘𝑀))) |
22 | eqid 2734 | . . . . 5 ⊢ (+g‘𝑀) = (+g‘𝑀) | |
23 | eqid 2734 | . . . . 5 ⊢ (+g‘(oppg‘𝑀)) = (+g‘(oppg‘𝑀)) | |
24 | 22, 15, 23 | oppgplusfval 19378 | . . . 4 ⊢ (+g‘(oppg‘𝑀)) = tpos (+g‘𝑀) |
25 | 21, 24 | eqtrdi 2790 | . . 3 ⊢ (𝜑 → (〈𝑋, 𝑋〉 · 𝑋) = tpos (+g‘𝑀)) |
26 | 11, 13, 25 | 3eqtr4rd 2785 | . 2 ⊢ (𝜑 → (〈𝑋, 𝑋〉 · 𝑋) = (〈𝑌, 𝑌〉(comp‘𝑂)𝑌)) |
27 | oppgoppcco.x | . . 3 ⊢ (𝜑 → ∙ = (comp‘𝑂)) | |
28 | 27 | oveqd 7447 | . 2 ⊢ (𝜑 → (〈𝑌, 𝑌〉 ∙ 𝑌) = (〈𝑌, 𝑌〉(comp‘𝑂)𝑌)) |
29 | 26, 28 | eqtr4d 2777 | 1 ⊢ (𝜑 → (〈𝑋, 𝑋〉 · 𝑋) = (〈𝑌, 𝑌〉 ∙ 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2105 〈cop 4636 ‘cfv 6562 (class class class)co 7430 tpos ctpos 8248 Basecbs 17244 +gcplusg 17297 compcco 17309 oppCatcoppc 17755 Mndcmnd 18759 oppgcoppg 19375 MndToCatcmndtc 48885 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-ot 4639 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-tpos 8249 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-z 12611 df-dec 12731 df-uz 12876 df-fz 13544 df-struct 17180 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-plusg 17310 df-hom 17321 df-cco 17322 df-0g 17487 df-oppc 17756 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-oppg 19376 df-mndtc 48886 |
This theorem is referenced by: (None) |
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