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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 17617. (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 2736 | . . . . . . . . 9 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
3 | 1, 2 | oppcbas 17596 | . . . . . . . 8 ⊢ (Base‘𝐶) = (Base‘𝑂) |
4 | eqid 2736 | . . . . . . . 8 ⊢ (comp‘𝑂) = (comp‘𝑂) | |
5 | eqid 2736 | . . . . . . . 8 ⊢ (oppCat‘𝑂) = (oppCat‘𝑂) | |
6 | simpr1 1194 | . . . . . . . 8 ⊢ ((⊤ ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶)) | |
7 | simpr2 1195 | . . . . . . . 8 ⊢ ((⊤ ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶)) | |
8 | simpr3 1196 | . . . . . . . 8 ⊢ ((⊤ ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → 𝑧 ∈ (Base‘𝐶)) | |
9 | 3, 4, 5, 6, 7, 8 | oppcco 17595 | . . . . . . 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 2736 | . . . . . . . 8 ⊢ (comp‘𝐶) = (comp‘𝐶) | |
11 | 2, 10, 1, 8, 7, 6 | oppcco 17595 | . . . . . . 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 2777 | . . . . . 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 3146 | . . . . 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 3146 | . . . 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 3199 | . . 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 2736 | . . . 4 ⊢ (comp‘(oppCat‘𝑂)) = (comp‘(oppCat‘𝑂)) | |
17 | eqid 2736 | . . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
18 | eqidd 2737 | . . . 4 ⊢ (⊤ → (Base‘𝐶) = (Base‘𝐶)) | |
19 | 1, 2 | 2oppcbas 17602 | . . . . 5 ⊢ (Base‘𝐶) = (Base‘(oppCat‘𝑂)) |
20 | 19 | a1i 11 | . . . 4 ⊢ (⊤ → (Base‘𝐶) = (Base‘(oppCat‘𝑂))) |
21 | 1 | 2oppchomf 17603 | . . . . 5 ⊢ (Homf ‘𝐶) = (Homf ‘(oppCat‘𝑂)) |
22 | 21 | a1i 11 | . . . 4 ⊢ (⊤ → (Homf ‘𝐶) = (Homf ‘(oppCat‘𝑂))) |
23 | 10, 16, 17, 18, 20, 22 | comfeq 17583 | . . 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 256 | . 2 ⊢ (⊤ → (compf‘𝐶) = (compf‘(oppCat‘𝑂))) |
25 | 24 | mptru 1548 | 1 ⊢ (compf‘𝐶) = (compf‘(oppCat‘𝑂)) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 ∧ w3a 1087 = wceq 1541 ⊤wtru 1542 ∈ wcel 2106 ∀wral 3063 〈cop 4591 ‘cfv 6494 (class class class)co 7354 Basecbs 17080 Hom chom 17141 compcco 17142 Homf chomf 17543 compfccomf 17544 oppCatcoppc 17588 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7669 ax-cnex 11104 ax-resscn 11105 ax-1cn 11106 ax-icn 11107 ax-addcl 11108 ax-addrcl 11109 ax-mulcl 11110 ax-mulrcl 11111 ax-mulcom 11112 ax-addass 11113 ax-mulass 11114 ax-distr 11115 ax-i2m1 11116 ax-1ne0 11117 ax-1rid 11118 ax-rnegex 11119 ax-rrecex 11120 ax-cnre 11121 ax-pre-lttri 11122 ax-pre-lttrn 11123 ax-pre-ltadd 11124 ax-pre-mulgt0 11125 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5530 df-eprel 5536 df-po 5544 df-so 5545 df-fr 5587 df-we 5589 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6252 df-ord 6319 df-on 6320 df-lim 6321 df-suc 6322 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-riota 7310 df-ov 7357 df-oprab 7358 df-mpo 7359 df-om 7800 df-1st 7918 df-2nd 7919 df-tpos 8154 df-frecs 8209 df-wrecs 8240 df-recs 8314 df-rdg 8353 df-er 8645 df-en 8881 df-dom 8882 df-sdom 8883 df-pnf 11188 df-mnf 11189 df-xr 11190 df-ltxr 11191 df-le 11192 df-sub 11384 df-neg 11385 df-nn 12151 df-2 12213 df-3 12214 df-4 12215 df-5 12216 df-6 12217 df-7 12218 df-8 12219 df-9 12220 df-n0 12411 df-z 12497 df-dec 12616 df-sets 17033 df-slot 17051 df-ndx 17063 df-base 17081 df-hom 17154 df-cco 17155 df-homf 17547 df-comf 17548 df-oppc 17589 |
This theorem is referenced by: oppcepi 17619 oppchofcl 18146 oppcyon 18155 oyoncl 18156 |
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