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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 17009. (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 2823 | . . . . . . . . 9 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
3 | 1, 2 | oppcbas 16990 | . . . . . . . 8 ⊢ (Base‘𝐶) = (Base‘𝑂) |
4 | eqid 2823 | . . . . . . . 8 ⊢ (comp‘𝑂) = (comp‘𝑂) | |
5 | eqid 2823 | . . . . . . . 8 ⊢ (oppCat‘𝑂) = (oppCat‘𝑂) | |
6 | simpr1 1190 | . . . . . . . 8 ⊢ ((⊤ ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶)) | |
7 | simpr2 1191 | . . . . . . . 8 ⊢ ((⊤ ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶)) | |
8 | simpr3 1192 | . . . . . . . 8 ⊢ ((⊤ ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → 𝑧 ∈ (Base‘𝐶)) | |
9 | 3, 4, 5, 6, 7, 8 | oppcco 16989 | . . . . . . 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 2823 | . . . . . . . 8 ⊢ (comp‘𝐶) = (comp‘𝐶) | |
11 | 2, 10, 1, 8, 7, 6 | oppcco 16989 | . . . . . . 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 2859 | . . . . . 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 3185 | . . . . 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 3185 | . . . 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 3194 | . . 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 2823 | . . . 4 ⊢ (comp‘(oppCat‘𝑂)) = (comp‘(oppCat‘𝑂)) | |
17 | eqid 2823 | . . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
18 | eqidd 2824 | . . . 4 ⊢ (⊤ → (Base‘𝐶) = (Base‘𝐶)) | |
19 | 1, 2 | 2oppcbas 16995 | . . . . 5 ⊢ (Base‘𝐶) = (Base‘(oppCat‘𝑂)) |
20 | 19 | a1i 11 | . . . 4 ⊢ (⊤ → (Base‘𝐶) = (Base‘(oppCat‘𝑂))) |
21 | 1 | 2oppchomf 16996 | . . . . 5 ⊢ (Homf ‘𝐶) = (Homf ‘(oppCat‘𝑂)) |
22 | 21 | a1i 11 | . . . 4 ⊢ (⊤ → (Homf ‘𝐶) = (Homf ‘(oppCat‘𝑂))) |
23 | 10, 16, 17, 18, 20, 22 | comfeq 16978 | . . 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 259 | . 2 ⊢ (⊤ → (compf‘𝐶) = (compf‘(oppCat‘𝑂))) |
25 | 24 | mptru 1544 | 1 ⊢ (compf‘𝐶) = (compf‘(oppCat‘𝑂)) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 ∧ w3a 1083 = wceq 1537 ⊤wtru 1538 ∈ wcel 2114 ∀wral 3140 〈cop 4575 ‘cfv 6357 (class class class)co 7158 Basecbs 16485 Hom chom 16578 compcco 16579 Homf chomf 16939 compfccomf 16940 oppCatcoppc 16983 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-tpos 7894 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-hom 16591 df-cco 16592 df-homf 16943 df-comf 16944 df-oppc 16984 |
This theorem is referenced by: oppcepi 17011 oppchofcl 17512 oppcyon 17521 oyoncl 17522 |
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