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Mirrors > Home > MPE Home > Th. List > Mathboxes > trlcocnv | Structured version Visualization version GIF version |
Description: Swap the arguments of the trace of a composition with converse. (Contributed by NM, 1-Jul-2013.) |
Ref | Expression |
---|---|
trlcocnv.h | ⊢ 𝐻 = (LHyp‘𝐾) |
trlcocnv.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
trlcocnv.r | ⊢ 𝑅 = ((trL‘𝐾)‘𝑊) |
Ref | Expression |
---|---|
trlcocnv | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝑅‘(𝐹 ∘ ◡𝐺)) = (𝑅‘(𝐺 ∘ ◡𝐹))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnvco 5751 | . . . 4 ⊢ ◡(𝐹 ∘ ◡𝐺) = (◡◡𝐺 ∘ ◡𝐹) | |
2 | cocnvcnv1 6105 | . . . 4 ⊢ (◡◡𝐺 ∘ ◡𝐹) = (𝐺 ∘ ◡𝐹) | |
3 | 1, 2 | eqtri 2844 | . . 3 ⊢ ◡(𝐹 ∘ ◡𝐺) = (𝐺 ∘ ◡𝐹) |
4 | 3 | fveq2i 6668 | . 2 ⊢ (𝑅‘◡(𝐹 ∘ ◡𝐺)) = (𝑅‘(𝐺 ∘ ◡𝐹)) |
5 | simp1 1132 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
6 | trlcocnv.h | . . . . . 6 ⊢ 𝐻 = (LHyp‘𝐾) | |
7 | trlcocnv.t | . . . . . 6 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
8 | 6, 7 | ltrncnv 37276 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇) → ◡𝐺 ∈ 𝑇) |
9 | 8 | 3adant2 1127 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → ◡𝐺 ∈ 𝑇) |
10 | 6, 7 | ltrnco 37849 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ◡𝐺 ∈ 𝑇) → (𝐹 ∘ ◡𝐺) ∈ 𝑇) |
11 | 9, 10 | syld3an3 1405 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝐹 ∘ ◡𝐺) ∈ 𝑇) |
12 | trlcocnv.r | . . . 4 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊) | |
13 | 6, 7, 12 | trlcnv 37295 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∘ ◡𝐺) ∈ 𝑇) → (𝑅‘◡(𝐹 ∘ ◡𝐺)) = (𝑅‘(𝐹 ∘ ◡𝐺))) |
14 | 5, 11, 13 | syl2anc 586 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝑅‘◡(𝐹 ∘ ◡𝐺)) = (𝑅‘(𝐹 ∘ ◡𝐺))) |
15 | 4, 14 | syl5reqr 2871 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝑅‘(𝐹 ∘ ◡𝐺)) = (𝑅‘(𝐺 ∘ ◡𝐹))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ◡ccnv 5549 ∘ ccom 5554 ‘cfv 6350 HLchlt 36480 LHypclh 37114 LTrncltrn 37231 trLctrl 37288 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-riotaBAD 36083 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-iun 4914 df-iin 4915 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-1st 7683 df-2nd 7684 df-undef 7933 df-map 8402 df-proset 17532 df-poset 17550 df-plt 17562 df-lub 17578 df-glb 17579 df-join 17580 df-meet 17581 df-p0 17643 df-p1 17644 df-lat 17650 df-clat 17712 df-oposet 36306 df-ol 36308 df-oml 36309 df-covers 36396 df-ats 36397 df-atl 36428 df-cvlat 36452 df-hlat 36481 df-llines 36628 df-lplanes 36629 df-lvols 36630 df-lines 36631 df-psubsp 36633 df-pmap 36634 df-padd 36926 df-lhyp 37118 df-laut 37119 df-ldil 37234 df-ltrn 37235 df-trl 37289 |
This theorem is referenced by: cdlemk9bN 37970 cdlemk14 37984 cdlemk21N 38003 cdlemk20 38004 cdlemk22 38023 cdlemkfid1N 38051 |
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