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Mirrors > Home > MPE Home > Th. List > Mathboxes > trlcocnv | Structured version Visualization version GIF version |
Description: Swap the arguments of the trace of a composition with converse. (Contributed by NM, 1-Jul-2013.) |
Ref | Expression |
---|---|
trlcocnv.h | ⊢ 𝐻 = (LHyp‘𝐾) |
trlcocnv.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
trlcocnv.r | ⊢ 𝑅 = ((trL‘𝐾)‘𝑊) |
Ref | Expression |
---|---|
trlcocnv | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝑅‘(𝐹 ∘ ◡𝐺)) = (𝑅‘(𝐺 ∘ ◡𝐹))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnvco 5722 | . . . 4 ⊢ ◡(𝐹 ∘ ◡𝐺) = (◡◡𝐺 ∘ ◡𝐹) | |
2 | cocnvcnv1 6084 | . . . 4 ⊢ (◡◡𝐺 ∘ ◡𝐹) = (𝐺 ∘ ◡𝐹) | |
3 | 1, 2 | eqtri 2761 | . . 3 ⊢ ◡(𝐹 ∘ ◡𝐺) = (𝐺 ∘ ◡𝐹) |
4 | 3 | fveq2i 6671 | . 2 ⊢ (𝑅‘◡(𝐹 ∘ ◡𝐺)) = (𝑅‘(𝐺 ∘ ◡𝐹)) |
5 | simp1 1137 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
6 | trlcocnv.h | . . . . . 6 ⊢ 𝐻 = (LHyp‘𝐾) | |
7 | trlcocnv.t | . . . . . 6 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
8 | 6, 7 | ltrncnv 37772 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇) → ◡𝐺 ∈ 𝑇) |
9 | 8 | 3adant2 1132 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → ◡𝐺 ∈ 𝑇) |
10 | 6, 7 | ltrnco 38345 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ◡𝐺 ∈ 𝑇) → (𝐹 ∘ ◡𝐺) ∈ 𝑇) |
11 | 9, 10 | syld3an3 1410 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝐹 ∘ ◡𝐺) ∈ 𝑇) |
12 | trlcocnv.r | . . . 4 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊) | |
13 | 6, 7, 12 | trlcnv 37791 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∘ ◡𝐺) ∈ 𝑇) → (𝑅‘◡(𝐹 ∘ ◡𝐺)) = (𝑅‘(𝐹 ∘ ◡𝐺))) |
14 | 5, 11, 13 | syl2anc 587 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝑅‘◡(𝐹 ∘ ◡𝐺)) = (𝑅‘(𝐹 ∘ ◡𝐺))) |
15 | 4, 14 | syl5reqr 2788 | 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 399 ∧ w3a 1088 = wceq 1542 ∈ wcel 2113 ◡ccnv 5518 ∘ ccom 5523 ‘cfv 6333 HLchlt 36976 LHypclh 37610 LTrncltrn 37727 trLctrl 37784 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-rep 5151 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 ax-riotaBAD 36579 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-op 4520 df-uni 4794 df-iun 4880 df-iin 4881 df-br 5028 df-opab 5090 df-mpt 5108 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-riota 7121 df-ov 7167 df-oprab 7168 df-mpo 7169 df-1st 7707 df-2nd 7708 df-undef 7961 df-map 8432 df-proset 17647 df-poset 17665 df-plt 17677 df-lub 17693 df-glb 17694 df-join 17695 df-meet 17696 df-p0 17758 df-p1 17759 df-lat 17765 df-clat 17827 df-oposet 36802 df-ol 36804 df-oml 36805 df-covers 36892 df-ats 36893 df-atl 36924 df-cvlat 36948 df-hlat 36977 df-llines 37124 df-lplanes 37125 df-lvols 37126 df-lines 37127 df-psubsp 37129 df-pmap 37130 df-padd 37422 df-lhyp 37614 df-laut 37615 df-ldil 37730 df-ltrn 37731 df-trl 37785 |
This theorem is referenced by: cdlemk9bN 38466 cdlemk14 38480 cdlemk21N 38499 cdlemk20 38500 cdlemk22 38519 cdlemkfid1N 38547 |
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