trlcocnv - Mathbox for Norm Megill

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Theorem trlcocnv 41968

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 1154	. . 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 41394	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇) → ^◡𝐺 ∈ 𝑇)
5	4	3adant2 1149	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → ^◡𝐺 ∈ 𝑇)
6	2, 3	ltrnco 41967	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ^◡𝐺 ∈ 𝑇) → (𝐹 ∘ ^◡𝐺) ∈ 𝑇)
7	5, 6	syld3an3 1437	. . 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 41413	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∘ ^◡𝐺) ∈ 𝑇) → (𝑅‘^◡(𝐹 ∘ ^◡𝐺)) = (𝑅‘(𝐹 ∘ ^◡𝐺)))
10	1, 7, 9	syl2anc 596	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝑅‘^◡(𝐹 ∘ ^◡𝐺)) = (𝑅‘(𝐹 ∘ ^◡𝐺)))
11		cnvco 5873	. . . 4 ⊢ ^◡(𝐹 ∘ ^◡𝐺) = (^◡^◡𝐺 ∘ ^◡𝐹)
12		cocnvcnv1 6271	. . . 4 ⊢ (^◡^◡𝐺 ∘ ^◡𝐹) = (𝐺 ∘ ^◡𝐹)
13	11, 12	eqtri 2786	. . 3 ⊢ ^◡(𝐹 ∘ ^◡𝐺) = (𝐺 ∘ ^◡𝐹)
14	13	fveq2i 6908	. 2 ⊢ (𝑅‘^◡(𝐹 ∘ ^◡𝐺)) = (𝑅‘(𝐺 ∘ ^◡𝐹))
15	10, 14	eqtr3di 2813	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝑅‘(𝐹 ∘ ^◡𝐺)) = (𝑅‘(𝐺 ∘ ^◡𝐹)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2147 ^◡ccnv 5654 ∘ ccom 5659 ‘cfv 6555 HLchlt 40598 LHypclh 41232 LTrncltrn 41349 trLctrl 41406

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7772 ax-riotaBAD 40201

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-iun 4956 df-iin 4957 df-br 5107 df-opab 5171 df-mpt 5190 df-id 5550 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-iota 6509 df-fun 6557 df-fn 6558 df-f 6559 df-f1 6560 df-fo 6561 df-f1o 6562 df-fv 6563 df-riota 7397 df-ov 7443 df-oprab 7444 df-mpo 7445 df-1st 8022 df-2nd 8023 df-undef 8315 df-map 8874 df-proset 18494 df-poset 18513 df-plt 18528 df-lub 18544 df-glb 18545 df-join 18546 df-meet 18547 df-p0 18623 df-p1 18624 df-lat 18632 df-clat 18699 df-oposet 40424 df-ol 40426 df-oml 40427 df-covers 40514 df-ats 40515 df-atl 40546 df-cvlat 40570 df-hlat 40599 df-llines 40746 df-lplanes 40747 df-lvols 40748 df-lines 40749 df-psubsp 40751 df-pmap 40752 df-padd 41044 df-lhyp 41236 df-laut 41237 df-ldil 41352 df-ltrn 41353 df-trl 41407

This theorem is used by: cdlemk9bN 42088 cdlemk14 42102 cdlemk21N 42121 cdlemk20 42122 cdlemk22 42141 cdlemkfid1N 42169

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