trlcocnv - Mathbox for Norm Megill

	Mathbox for Norm Megill		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > trlcocnv		Structured version Visualization version GIF version

Theorem trlcocnv 40722

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 40148	. . . . 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 40721	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ^◡𝐺 ∈ 𝑇) → (𝐹 ∘ ^◡𝐺) ∈ 𝑇)
7	5, 6	syld3an3 1411	. . 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 40167	. . 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 5896	. . . 4 ⊢ ^◡(𝐹 ∘ ^◡𝐺) = (^◡^◡𝐺 ∘ ^◡𝐹)
12		cocnvcnv1 6277	. . . 4 ⊢ (^◡^◡𝐺 ∘ ^◡𝐹) = (𝐺 ∘ ^◡𝐹)
13	11, 12	eqtri 2765	. . 3 ⊢ ^◡(𝐹 ∘ ^◡𝐺) = (𝐺 ∘ ^◡𝐹)
14	13	fveq2i 6909	. 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 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ^◡ccnv 5684 ∘ ccom 5689 ‘cfv 6561 HLchlt 39351 LHypclh 39986 LTrncltrn 40103 trLctrl 40160

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-riotaBAD 38954

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-undef 8298 df-map 8868 df-proset 18340 df-poset 18359 df-plt 18375 df-lub 18391 df-glb 18392 df-join 18393 df-meet 18394 df-p0 18470 df-p1 18471 df-lat 18477 df-clat 18544 df-oposet 39177 df-ol 39179 df-oml 39180 df-covers 39267 df-ats 39268 df-atl 39299 df-cvlat 39323 df-hlat 39352 df-llines 39500 df-lplanes 39501 df-lvols 39502 df-lines 39503 df-psubsp 39505 df-pmap 39506 df-padd 39798 df-lhyp 39990 df-laut 39991 df-ldil 40106 df-ltrn 40107 df-trl 40161

This theorem is referenced by: cdlemk9bN 40842 cdlemk14 40856 cdlemk21N 40875 cdlemk20 40876 cdlemk22 40895 cdlemkfid1N 40923

W3C validator