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Mathbox for Norm Megill |
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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 | simp1 1137 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
2 | trlcocnv.h | . . . . . 6 ⊢ 𝐻 = (LHyp‘𝐾) | |
3 | trlcocnv.t | . . . . . 6 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
4 | 2, 3 | ltrncnv 38638 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇) → ◡𝐺 ∈ 𝑇) |
5 | 4 | 3adant2 1132 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → ◡𝐺 ∈ 𝑇) |
6 | 2, 3 | ltrnco 39211 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ◡𝐺 ∈ 𝑇) → (𝐹 ∘ ◡𝐺) ∈ 𝑇) |
7 | 5, 6 | syld3an3 1410 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝐹 ∘ ◡𝐺) ∈ 𝑇) |
8 | trlcocnv.r | . . . 4 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊) | |
9 | 2, 3, 8 | trlcnv 38657 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∘ ◡𝐺) ∈ 𝑇) → (𝑅‘◡(𝐹 ∘ ◡𝐺)) = (𝑅‘(𝐹 ∘ ◡𝐺))) |
10 | 1, 7, 9 | syl2anc 585 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝑅‘◡(𝐹 ∘ ◡𝐺)) = (𝑅‘(𝐹 ∘ ◡𝐺))) |
11 | cnvco 5846 | . . . 4 ⊢ ◡(𝐹 ∘ ◡𝐺) = (◡◡𝐺 ∘ ◡𝐹) | |
12 | cocnvcnv1 6214 | . . . 4 ⊢ (◡◡𝐺 ∘ ◡𝐹) = (𝐺 ∘ ◡𝐹) | |
13 | 11, 12 | eqtri 2765 | . . 3 ⊢ ◡(𝐹 ∘ ◡𝐺) = (𝐺 ∘ ◡𝐹) |
14 | 13 | fveq2i 6850 | . 2 ⊢ (𝑅‘◡(𝐹 ∘ ◡𝐺)) = (𝑅‘(𝐺 ∘ ◡𝐹)) |
15 | 10, 14 | eqtr3di 2792 | 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 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ◡ccnv 5637 ∘ ccom 5642 ‘cfv 6501 HLchlt 37841 LHypclh 38476 LTrncltrn 38593 trLctrl 38650 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-riotaBAD 37444 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-iin 4962 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-1st 7926 df-2nd 7927 df-undef 8209 df-map 8774 df-proset 18191 df-poset 18209 df-plt 18226 df-lub 18242 df-glb 18243 df-join 18244 df-meet 18245 df-p0 18321 df-p1 18322 df-lat 18328 df-clat 18395 df-oposet 37667 df-ol 37669 df-oml 37670 df-covers 37757 df-ats 37758 df-atl 37789 df-cvlat 37813 df-hlat 37842 df-llines 37990 df-lplanes 37991 df-lvols 37992 df-lines 37993 df-psubsp 37995 df-pmap 37996 df-padd 38288 df-lhyp 38480 df-laut 38481 df-ldil 38596 df-ltrn 38597 df-trl 38651 |
This theorem is referenced by: cdlemk9bN 39332 cdlemk14 39346 cdlemk21N 39365 cdlemk20 39366 cdlemk22 39385 cdlemkfid1N 39413 |
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