trlcocnv - Mathbox for Norm Megill

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Theorem trlcocnv 41125

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 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 40551	. . . . 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 41124	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ^◡𝐺 ∈ 𝑇) → (𝐹 ∘ ^◡𝐺) ∈ 𝑇)
7	5, 6	syld3an3 1412	. . 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 40570	. . 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 5844	. . . 4 ⊢ ^◡(𝐹 ∘ ^◡𝐺) = (^◡^◡𝐺 ∘ ^◡𝐹)
12		cocnvcnv1 6226	. . . 4 ⊢ (^◡^◡𝐺 ∘ ^◡𝐹) = (𝐺 ∘ ^◡𝐹)
13	11, 12	eqtri 2760	. . 3 ⊢ ^◡(𝐹 ∘ ^◡𝐺) = (𝐺 ∘ ^◡𝐹)
14	13	fveq2i 6847	. 2 ⊢ (𝑅‘^◡(𝐹 ∘ ^◡𝐺)) = (𝑅‘(𝐺 ∘ ^◡𝐹))
15	10, 14	eqtr3di 2787	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 1087 = wceq 1542 ∈ wcel 2114 ^◡ccnv 5633 ∘ ccom 5638 ‘cfv 6502 HLchlt 39755 LHypclh 40389 LTrncltrn 40506 trLctrl 40563

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5245 ax-nul 5255 ax-pow 5314 ax-pr 5381 ax-un 7692 ax-riotaBAD 39358

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-iun 4950 df-iin 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-id 5529 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6458 df-fun 6504 df-fn 6505 df-f 6506 df-f1 6507 df-fo 6508 df-f1o 6509 df-fv 6510 df-riota 7327 df-ov 7373 df-oprab 7374 df-mpo 7375 df-1st 7945 df-2nd 7946 df-undef 8227 df-map 8779 df-proset 18231 df-poset 18250 df-plt 18265 df-lub 18281 df-glb 18282 df-join 18283 df-meet 18284 df-p0 18360 df-p1 18361 df-lat 18369 df-clat 18436 df-oposet 39581 df-ol 39583 df-oml 39584 df-covers 39671 df-ats 39672 df-atl 39703 df-cvlat 39727 df-hlat 39756 df-llines 39903 df-lplanes 39904 df-lvols 39905 df-lines 39906 df-psubsp 39908 df-pmap 39909 df-padd 40201 df-lhyp 40393 df-laut 40394 df-ldil 40509 df-ltrn 40510 df-trl 40564

This theorem is referenced by: cdlemk9bN 41245 cdlemk14 41259 cdlemk21N 41278 cdlemk20 41279 cdlemk22 41298 cdlemkfid1N 41326

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