trlcocnv - Mathbox for Norm Megill

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Theorem trlcocnv 40702

Description: Swap the arguments of the trace of a composition with converse. (Contributed by NM, 1-Jul-2013.)

**Hypotheses**
Ref	Expression
trlcocnv.h	⊢ 𝐻 = (LHyp‘𝐾)
trlcocnv.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
trlcocnv.r	⊢ 𝑅 = ((trL‘𝐾)‘𝑊)

**Assertion**
Ref	Expression
trlcocnv	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝑅‘(𝐹 ∘ ^◡𝐺)) = (𝑅‘(𝐺 ∘ ^◡𝐹)))

**Proof of Theorem trlcocnv**
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 40128	. . . . 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 40701	. . . 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 40147	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∘ ^◡𝐺) ∈ 𝑇) → (𝑅‘^◡(𝐹 ∘ ^◡𝐺)) = (𝑅‘(𝐹 ∘ ^◡𝐺)))
10	1, 7, 9	syl2anc 584	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝑅‘^◡(𝐹 ∘ ^◡𝐺)) = (𝑅‘(𝐹 ∘ ^◡𝐺)))
11		cnvco 5898	. . . 4 ⊢ ^◡(𝐹 ∘ ^◡𝐺) = (^◡^◡𝐺 ∘ ^◡𝐹)
12		cocnvcnv1 6278	. . . 4 ⊢ (^◡^◡𝐺 ∘ ^◡𝐹) = (𝐺 ∘ ^◡𝐹)
13	11, 12	eqtri 2762	. . 3 ⊢ ^◡(𝐹 ∘ ^◡𝐺) = (𝐺 ∘ ^◡𝐹)
14	13	fveq2i 6909	. 2 ⊢ (𝑅‘^◡(𝐹 ∘ ^◡𝐺)) = (𝑅‘(𝐺 ∘ ^◡𝐹))
15	10, 14	eqtr3di 2789	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 1536 ∈ wcel 2105 ^◡ccnv 5687 ∘ ccom 5692 ‘cfv 6562 HLchlt 39331 LHypclh 39966 LTrncltrn 40083 trLctrl 40140

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-riotaBAD 38934

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-iin 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-1st 8012 df-2nd 8013 df-undef 8296 df-map 8866 df-proset 18351 df-poset 18370 df-plt 18387 df-lub 18403 df-glb 18404 df-join 18405 df-meet 18406 df-p0 18482 df-p1 18483 df-lat 18489 df-clat 18556 df-oposet 39157 df-ol 39159 df-oml 39160 df-covers 39247 df-ats 39248 df-atl 39279 df-cvlat 39303 df-hlat 39332 df-llines 39480 df-lplanes 39481 df-lvols 39482 df-lines 39483 df-psubsp 39485 df-pmap 39486 df-padd 39778 df-lhyp 39970 df-laut 39971 df-ldil 40086 df-ltrn 40087 df-trl 40141

This theorem is referenced by: cdlemk9bN 40822 cdlemk14 40836 cdlemk21N 40855 cdlemk20 40856 cdlemk22 40875 cdlemkfid1N 40903

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