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 40047

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 1133	. . 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 39473	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇) → ^◡𝐺 ∈ 𝑇)
5	4	3adant2 1128	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → ^◡𝐺 ∈ 𝑇)
6	2, 3	ltrnco 40046	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ^◡𝐺 ∈ 𝑇) → (𝐹 ∘ ^◡𝐺) ∈ 𝑇)
7	5, 6	syld3an3 1406	. . 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 39492	. . 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 5875	. . . 4 ⊢ ^◡(𝐹 ∘ ^◡𝐺) = (^◡^◡𝐺 ∘ ^◡𝐹)
12		cocnvcnv1 6246	. . . 4 ⊢ (^◡^◡𝐺 ∘ ^◡𝐹) = (𝐺 ∘ ^◡𝐹)
13	11, 12	eqtri 2752	. . 3 ⊢ ^◡(𝐹 ∘ ^◡𝐺) = (𝐺 ∘ ^◡𝐹)
14	13	fveq2i 6884	. 2 ⊢ (𝑅‘^◡(𝐹 ∘ ^◡𝐺)) = (𝑅‘(𝐺 ∘ ^◡𝐹))
15	10, 14	eqtr3di 2779	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 1084 = wceq 1533 ∈ wcel 2098 ^◡ccnv 5665 ∘ ccom 5670 ‘cfv 6533 HLchlt 38676 LHypclh 39311 LTrncltrn 39428 trLctrl 39485

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-riotaBAD 38279

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-iun 4989 df-iin 4990 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-1st 7968 df-2nd 7969 df-undef 8253 df-map 8817 df-proset 18247 df-poset 18265 df-plt 18282 df-lub 18298 df-glb 18299 df-join 18300 df-meet 18301 df-p0 18377 df-p1 18378 df-lat 18384 df-clat 18451 df-oposet 38502 df-ol 38504 df-oml 38505 df-covers 38592 df-ats 38593 df-atl 38624 df-cvlat 38648 df-hlat 38677 df-llines 38825 df-lplanes 38826 df-lvols 38827 df-lines 38828 df-psubsp 38830 df-pmap 38831 df-padd 39123 df-lhyp 39315 df-laut 39316 df-ldil 39431 df-ltrn 39432 df-trl 39486

This theorem is referenced by: cdlemk9bN 40167 cdlemk14 40181 cdlemk21N 40200 cdlemk20 40201 cdlemk22 40220 cdlemkfid1N 40248

W3C validator