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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 1135 | . . 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 39321 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇) → ◡𝐺 ∈ 𝑇) |
5 | 4 | 3adant2 1130 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → ◡𝐺 ∈ 𝑇) |
6 | 2, 3 | ltrnco 39894 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ◡𝐺 ∈ 𝑇) → (𝐹 ∘ ◡𝐺) ∈ 𝑇) |
7 | 5, 6 | syld3an3 1408 | . . 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 39340 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∘ ◡𝐺) ∈ 𝑇) → (𝑅‘◡(𝐹 ∘ ◡𝐺)) = (𝑅‘(𝐹 ∘ ◡𝐺))) |
10 | 1, 7, 9 | syl2anc 583 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝑅‘◡(𝐹 ∘ ◡𝐺)) = (𝑅‘(𝐹 ∘ ◡𝐺))) |
11 | cnvco 5886 | . . . 4 ⊢ ◡(𝐹 ∘ ◡𝐺) = (◡◡𝐺 ∘ ◡𝐹) | |
12 | cocnvcnv1 6257 | . . . 4 ⊢ (◡◡𝐺 ∘ ◡𝐹) = (𝐺 ∘ ◡𝐹) | |
13 | 11, 12 | eqtri 2759 | . . 3 ⊢ ◡(𝐹 ∘ ◡𝐺) = (𝐺 ∘ ◡𝐹) |
14 | 13 | fveq2i 6895 | . 2 ⊢ (𝑅‘◡(𝐹 ∘ ◡𝐺)) = (𝑅‘(𝐺 ∘ ◡𝐹)) |
15 | 10, 14 | eqtr3di 2786 | 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 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 ◡ccnv 5676 ∘ ccom 5681 ‘cfv 6544 HLchlt 38524 LHypclh 39159 LTrncltrn 39276 trLctrl 39333 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7728 ax-riotaBAD 38127 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-1st 7978 df-2nd 7979 df-undef 8261 df-map 8825 df-proset 18253 df-poset 18271 df-plt 18288 df-lub 18304 df-glb 18305 df-join 18306 df-meet 18307 df-p0 18383 df-p1 18384 df-lat 18390 df-clat 18457 df-oposet 38350 df-ol 38352 df-oml 38353 df-covers 38440 df-ats 38441 df-atl 38472 df-cvlat 38496 df-hlat 38525 df-llines 38673 df-lplanes 38674 df-lvols 38675 df-lines 38676 df-psubsp 38678 df-pmap 38679 df-padd 38971 df-lhyp 39163 df-laut 39164 df-ldil 39279 df-ltrn 39280 df-trl 39334 |
This theorem is referenced by: cdlemk9bN 40015 cdlemk14 40029 cdlemk21N 40048 cdlemk20 40049 cdlemk22 40068 cdlemkfid1N 40096 |
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