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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 17239. (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 2737 | . . . . . . . . 9 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
3 | 1, 2 | oppcbas 17219 | . . . . . . . 8 ⊢ (Base‘𝐶) = (Base‘𝑂) |
4 | eqid 2737 | . . . . . . . 8 ⊢ (comp‘𝑂) = (comp‘𝑂) | |
5 | eqid 2737 | . . . . . . . 8 ⊢ (oppCat‘𝑂) = (oppCat‘𝑂) | |
6 | simpr1 1196 | . . . . . . . 8 ⊢ ((⊤ ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶)) | |
7 | simpr2 1197 | . . . . . . . 8 ⊢ ((⊤ ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶)) | |
8 | simpr3 1198 | . . . . . . . 8 ⊢ ((⊤ ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → 𝑧 ∈ (Base‘𝐶)) | |
9 | 3, 4, 5, 6, 7, 8 | oppcco 17218 | . . . . . . 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 2737 | . . . . . . . 8 ⊢ (comp‘𝐶) = (comp‘𝐶) | |
11 | 2, 10, 1, 8, 7, 6 | oppcco 17218 | . . . . . . 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 2778 | . . . . . 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 3103 | . . . . 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 3103 | . . . 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 3110 | . . 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 2737 | . . . 4 ⊢ (comp‘(oppCat‘𝑂)) = (comp‘(oppCat‘𝑂)) | |
17 | eqid 2737 | . . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
18 | eqidd 2738 | . . . 4 ⊢ (⊤ → (Base‘𝐶) = (Base‘𝐶)) | |
19 | 1, 2 | 2oppcbas 17224 | . . . . 5 ⊢ (Base‘𝐶) = (Base‘(oppCat‘𝑂)) |
20 | 19 | a1i 11 | . . . 4 ⊢ (⊤ → (Base‘𝐶) = (Base‘(oppCat‘𝑂))) |
21 | 1 | 2oppchomf 17225 | . . . . 5 ⊢ (Homf ‘𝐶) = (Homf ‘(oppCat‘𝑂)) |
22 | 21 | a1i 11 | . . . 4 ⊢ (⊤ → (Homf ‘𝐶) = (Homf ‘(oppCat‘𝑂))) |
23 | 10, 16, 17, 18, 20, 22 | comfeq 17206 | . . 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 260 | . 2 ⊢ (⊤ → (compf‘𝐶) = (compf‘(oppCat‘𝑂))) |
25 | 24 | mptru 1550 | 1 ⊢ (compf‘𝐶) = (compf‘(oppCat‘𝑂)) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 ∧ w3a 1089 = wceq 1543 ⊤wtru 1544 ∈ wcel 2110 ∀wral 3058 〈cop 4544 ‘cfv 6377 (class class class)co 7210 Basecbs 16757 Hom chom 16810 compcco 16811 Homf chomf 17166 compfccomf 17167 oppCatcoppc 17211 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5176 ax-sep 5189 ax-nul 5196 ax-pow 5255 ax-pr 5319 ax-un 7520 ax-cnex 10782 ax-resscn 10783 ax-1cn 10784 ax-icn 10785 ax-addcl 10786 ax-addrcl 10787 ax-mulcl 10788 ax-mulrcl 10789 ax-mulcom 10790 ax-addass 10791 ax-mulass 10792 ax-distr 10793 ax-i2m1 10794 ax-1ne0 10795 ax-1rid 10796 ax-rnegex 10797 ax-rrecex 10798 ax-cnre 10799 ax-pre-lttri 10800 ax-pre-lttrn 10801 ax-pre-ltadd 10802 ax-pre-mulgt0 10803 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2940 df-nel 3044 df-ral 3063 df-rex 3064 df-reu 3065 df-rab 3067 df-v 3407 df-sbc 3692 df-csb 3809 df-dif 3866 df-un 3868 df-in 3870 df-ss 3880 df-pss 3882 df-nul 4235 df-if 4437 df-pw 4512 df-sn 4539 df-pr 4541 df-tp 4543 df-op 4545 df-uni 4817 df-iun 4903 df-br 5051 df-opab 5113 df-mpt 5133 df-tr 5159 df-id 5452 df-eprel 5457 df-po 5465 df-so 5466 df-fr 5506 df-we 5508 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6157 df-ord 6213 df-on 6214 df-lim 6215 df-suc 6216 df-iota 6335 df-fun 6379 df-fn 6380 df-f 6381 df-f1 6382 df-fo 6383 df-f1o 6384 df-fv 6385 df-riota 7167 df-ov 7213 df-oprab 7214 df-mpo 7215 df-om 7642 df-1st 7758 df-2nd 7759 df-tpos 7965 df-wrecs 8044 df-recs 8105 df-rdg 8143 df-er 8388 df-en 8624 df-dom 8625 df-sdom 8626 df-pnf 10866 df-mnf 10867 df-xr 10868 df-ltxr 10869 df-le 10870 df-sub 11061 df-neg 11062 df-nn 11828 df-2 11890 df-3 11891 df-4 11892 df-5 11893 df-6 11894 df-7 11895 df-8 11896 df-9 11897 df-n0 12088 df-z 12174 df-dec 12291 df-sets 16714 df-slot 16732 df-ndx 16742 df-base 16758 df-hom 16823 df-cco 16824 df-homf 17170 df-comf 17171 df-oppc 17212 |
This theorem is referenced by: oppcepi 17241 oppchofcl 17765 oppcyon 17774 oyoncl 17775 |
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