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Mirrors > Home > MPE Home > Th. List > 2oppccomf | Structured version Visualization version GIF version |
Description: The double opposite category has the same composition as the original category. Intended for use with property lemmas such as monpropd 16855. (Contributed by Mario Carneiro, 3-Jan-2017.) |
Ref | Expression |
---|---|
oppcbas.1 | ⊢ 𝑂 = (oppCat‘𝐶) |
Ref | Expression |
---|---|
2oppccomf | ⊢ (compf‘𝐶) = (compf‘(oppCat‘𝑂)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oppcbas.1 | . . . . . . . . 9 ⊢ 𝑂 = (oppCat‘𝐶) | |
2 | eqid 2772 | . . . . . . . . 9 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
3 | 1, 2 | oppcbas 16836 | . . . . . . . 8 ⊢ (Base‘𝐶) = (Base‘𝑂) |
4 | eqid 2772 | . . . . . . . 8 ⊢ (comp‘𝑂) = (comp‘𝑂) | |
5 | eqid 2772 | . . . . . . . 8 ⊢ (oppCat‘𝑂) = (oppCat‘𝑂) | |
6 | simpr1 1174 | . . . . . . . 8 ⊢ ((⊤ ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶)) | |
7 | simpr2 1175 | . . . . . . . 8 ⊢ ((⊤ ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶)) | |
8 | simpr3 1176 | . . . . . . . 8 ⊢ ((⊤ ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → 𝑧 ∈ (Base‘𝐶)) | |
9 | 3, 4, 5, 6, 7, 8 | oppcco 16835 | . . . . . . 7 ⊢ ((⊤ ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → (𝑔(〈𝑥, 𝑦〉(comp‘(oppCat‘𝑂))𝑧)𝑓) = (𝑓(〈𝑧, 𝑦〉(comp‘𝑂)𝑥)𝑔)) |
10 | eqid 2772 | . . . . . . . 8 ⊢ (comp‘𝐶) = (comp‘𝐶) | |
11 | 2, 10, 1, 8, 7, 6 | oppcco 16835 | . . . . . . 7 ⊢ ((⊤ ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → (𝑓(〈𝑧, 𝑦〉(comp‘𝑂)𝑥)𝑔) = (𝑔(〈𝑥, 𝑦〉(comp‘𝐶)𝑧)𝑓)) |
12 | 9, 11 | eqtr2d 2809 | . . . . . 6 ⊢ ((⊤ ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → (𝑔(〈𝑥, 𝑦〉(comp‘𝐶)𝑧)𝑓) = (𝑔(〈𝑥, 𝑦〉(comp‘(oppCat‘𝑂))𝑧)𝑓)) |
13 | 12 | ralrimivw 3127 | . . . . 5 ⊢ ((⊤ ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → ∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)(𝑔(〈𝑥, 𝑦〉(comp‘𝐶)𝑧)𝑓) = (𝑔(〈𝑥, 𝑦〉(comp‘(oppCat‘𝑂))𝑧)𝑓)) |
14 | 13 | ralrimivw 3127 | . . . 4 ⊢ ((⊤ ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)(𝑔(〈𝑥, 𝑦〉(comp‘𝐶)𝑧)𝑓) = (𝑔(〈𝑥, 𝑦〉(comp‘(oppCat‘𝑂))𝑧)𝑓)) |
15 | 14 | ralrimivvva 3136 | . . 3 ⊢ (⊤ → ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)(𝑔(〈𝑥, 𝑦〉(comp‘𝐶)𝑧)𝑓) = (𝑔(〈𝑥, 𝑦〉(comp‘(oppCat‘𝑂))𝑧)𝑓)) |
16 | eqid 2772 | . . . 4 ⊢ (comp‘(oppCat‘𝑂)) = (comp‘(oppCat‘𝑂)) | |
17 | eqid 2772 | . . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
18 | eqidd 2773 | . . . 4 ⊢ (⊤ → (Base‘𝐶) = (Base‘𝐶)) | |
19 | 1, 2 | 2oppcbas 16841 | . . . . 5 ⊢ (Base‘𝐶) = (Base‘(oppCat‘𝑂)) |
20 | 19 | a1i 11 | . . . 4 ⊢ (⊤ → (Base‘𝐶) = (Base‘(oppCat‘𝑂))) |
21 | 1 | 2oppchomf 16842 | . . . . 5 ⊢ (Homf ‘𝐶) = (Homf ‘(oppCat‘𝑂)) |
22 | 21 | a1i 11 | . . . 4 ⊢ (⊤ → (Homf ‘𝐶) = (Homf ‘(oppCat‘𝑂))) |
23 | 10, 16, 17, 18, 20, 22 | comfeq 16824 | . . 3 ⊢ (⊤ → ((compf‘𝐶) = (compf‘(oppCat‘𝑂)) ↔ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)(𝑔(〈𝑥, 𝑦〉(comp‘𝐶)𝑧)𝑓) = (𝑔(〈𝑥, 𝑦〉(comp‘(oppCat‘𝑂))𝑧)𝑓))) |
24 | 15, 23 | mpbird 249 | . 2 ⊢ (⊤ → (compf‘𝐶) = (compf‘(oppCat‘𝑂))) |
25 | 24 | mptru 1514 | 1 ⊢ (compf‘𝐶) = (compf‘(oppCat‘𝑂)) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 387 ∧ w3a 1068 = wceq 1507 ⊤wtru 1508 ∈ wcel 2048 ∀wral 3082 〈cop 4441 ‘cfv 6182 (class class class)co 6970 Basecbs 16329 Hom chom 16422 compcco 16423 Homf chomf 16785 compfccomf 16786 oppCatcoppc 16829 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-cnex 10383 ax-resscn 10384 ax-1cn 10385 ax-icn 10386 ax-addcl 10387 ax-addrcl 10388 ax-mulcl 10389 ax-mulrcl 10390 ax-mulcom 10391 ax-addass 10392 ax-mulass 10393 ax-distr 10394 ax-i2m1 10395 ax-1ne0 10396 ax-1rid 10397 ax-rnegex 10398 ax-rrecex 10399 ax-cnre 10400 ax-pre-lttri 10401 ax-pre-lttrn 10402 ax-pre-ltadd 10403 ax-pre-mulgt0 10404 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-pss 3841 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5305 df-eprel 5310 df-po 5319 df-so 5320 df-fr 5359 df-we 5361 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-om 7391 df-1st 7494 df-2nd 7495 df-tpos 7688 df-wrecs 7743 df-recs 7805 df-rdg 7843 df-er 8081 df-en 8299 df-dom 8300 df-sdom 8301 df-pnf 10468 df-mnf 10469 df-xr 10470 df-ltxr 10471 df-le 10472 df-sub 10664 df-neg 10665 df-nn 11432 df-2 11496 df-3 11497 df-4 11498 df-5 11499 df-6 11500 df-7 11501 df-8 11502 df-9 11503 df-n0 11701 df-z 11787 df-dec 11905 df-ndx 16332 df-slot 16333 df-base 16335 df-sets 16336 df-hom 16435 df-cco 16436 df-homf 16789 df-comf 16790 df-oppc 16830 |
This theorem is referenced by: oppcepi 16857 oppchofcl 17358 oppcyon 17367 oyoncl 17368 |
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