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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 16995. (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 2818 | . . . . . . . . 9 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
3 | 1, 2 | oppcbas 16976 | . . . . . . . 8 ⊢ (Base‘𝐶) = (Base‘𝑂) |
4 | eqid 2818 | . . . . . . . 8 ⊢ (comp‘𝑂) = (comp‘𝑂) | |
5 | eqid 2818 | . . . . . . . 8 ⊢ (oppCat‘𝑂) = (oppCat‘𝑂) | |
6 | simpr1 1186 | . . . . . . . 8 ⊢ ((⊤ ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶)) | |
7 | simpr2 1187 | . . . . . . . 8 ⊢ ((⊤ ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶)) | |
8 | simpr3 1188 | . . . . . . . 8 ⊢ ((⊤ ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → 𝑧 ∈ (Base‘𝐶)) | |
9 | 3, 4, 5, 6, 7, 8 | oppcco 16975 | . . . . . . 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 2818 | . . . . . . . 8 ⊢ (comp‘𝐶) = (comp‘𝐶) | |
11 | 2, 10, 1, 8, 7, 6 | oppcco 16975 | . . . . . . 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 2854 | . . . . . 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 3180 | . . . . 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 3180 | . . . 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 3189 | . . 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 2818 | . . . 4 ⊢ (comp‘(oppCat‘𝑂)) = (comp‘(oppCat‘𝑂)) | |
17 | eqid 2818 | . . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
18 | eqidd 2819 | . . . 4 ⊢ (⊤ → (Base‘𝐶) = (Base‘𝐶)) | |
19 | 1, 2 | 2oppcbas 16981 | . . . . 5 ⊢ (Base‘𝐶) = (Base‘(oppCat‘𝑂)) |
20 | 19 | a1i 11 | . . . 4 ⊢ (⊤ → (Base‘𝐶) = (Base‘(oppCat‘𝑂))) |
21 | 1 | 2oppchomf 16982 | . . . . 5 ⊢ (Homf ‘𝐶) = (Homf ‘(oppCat‘𝑂)) |
22 | 21 | a1i 11 | . . . 4 ⊢ (⊤ → (Homf ‘𝐶) = (Homf ‘(oppCat‘𝑂))) |
23 | 10, 16, 17, 18, 20, 22 | comfeq 16964 | . . 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 258 | . 2 ⊢ (⊤ → (compf‘𝐶) = (compf‘(oppCat‘𝑂))) |
25 | 24 | mptru 1535 | 1 ⊢ (compf‘𝐶) = (compf‘(oppCat‘𝑂)) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 ∧ w3a 1079 = wceq 1528 ⊤wtru 1529 ∈ wcel 2105 ∀wral 3135 〈cop 4563 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 Hom chom 16564 compcco 16565 Homf chomf 16925 compfccomf 16926 oppCatcoppc 16969 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 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 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-tpos 7881 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-hom 16577 df-cco 16578 df-homf 16929 df-comf 16930 df-oppc 16970 |
This theorem is referenced by: oppcepi 16997 oppchofcl 17498 oppcyon 17507 oyoncl 17508 |
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