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Mirrors > Home > MPE Home > Th. List > Mathboxes > dia2dimlem4 | Structured version Visualization version GIF version |
Description: Lemma for dia2dim 41060. Show that the composition (sum) of translations (vectors) 𝐺 and 𝐷 equals 𝐹. Part of proof of Lemma M in [Crawley] p. 121 line 5. (Contributed by NM, 8-Sep-2014.) |
Ref | Expression |
---|---|
dia2dimlem4.l | ⊢ ≤ = (le‘𝐾) |
dia2dimlem4.a | ⊢ 𝐴 = (Atoms‘𝐾) |
dia2dimlem4.h | ⊢ 𝐻 = (LHyp‘𝐾) |
dia2dimlem4.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
dia2dimlem4.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
dia2dimlem4.p | ⊢ (𝜑 → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) |
dia2dimlem4.f | ⊢ (𝜑 → 𝐹 ∈ 𝑇) |
dia2dimlem4.g | ⊢ (𝜑 → 𝐺 ∈ 𝑇) |
dia2dimlem4.gv | ⊢ (𝜑 → (𝐺‘𝑃) = 𝑄) |
dia2dimlem4.d | ⊢ (𝜑 → 𝐷 ∈ 𝑇) |
dia2dimlem4.dv | ⊢ (𝜑 → (𝐷‘𝑄) = (𝐹‘𝑃)) |
Ref | Expression |
---|---|
dia2dimlem4 | ⊢ (𝜑 → (𝐷 ∘ 𝐺) = 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dia2dimlem4.k | . 2 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
2 | dia2dimlem4.d | . . 3 ⊢ (𝜑 → 𝐷 ∈ 𝑇) | |
3 | dia2dimlem4.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ 𝑇) | |
4 | dia2dimlem4.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
5 | dia2dimlem4.t | . . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
6 | 4, 5 | ltrnco 40702 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐷 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝐷 ∘ 𝐺) ∈ 𝑇) |
7 | 1, 2, 3, 6 | syl3anc 1370 | . 2 ⊢ (𝜑 → (𝐷 ∘ 𝐺) ∈ 𝑇) |
8 | dia2dimlem4.f | . 2 ⊢ (𝜑 → 𝐹 ∈ 𝑇) | |
9 | dia2dimlem4.p | . 2 ⊢ (𝜑 → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) | |
10 | 9 | simpld 494 | . . . 4 ⊢ (𝜑 → 𝑃 ∈ 𝐴) |
11 | dia2dimlem4.l | . . . . 5 ⊢ ≤ = (le‘𝐾) | |
12 | dia2dimlem4.a | . . . . 5 ⊢ 𝐴 = (Atoms‘𝐾) | |
13 | 11, 12, 4, 5 | ltrncoval 40128 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐷 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑃 ∈ 𝐴) → ((𝐷 ∘ 𝐺)‘𝑃) = (𝐷‘(𝐺‘𝑃))) |
14 | 1, 2, 3, 10, 13 | syl121anc 1374 | . . 3 ⊢ (𝜑 → ((𝐷 ∘ 𝐺)‘𝑃) = (𝐷‘(𝐺‘𝑃))) |
15 | dia2dimlem4.gv | . . . 4 ⊢ (𝜑 → (𝐺‘𝑃) = 𝑄) | |
16 | 15 | fveq2d 6911 | . . 3 ⊢ (𝜑 → (𝐷‘(𝐺‘𝑃)) = (𝐷‘𝑄)) |
17 | dia2dimlem4.dv | . . 3 ⊢ (𝜑 → (𝐷‘𝑄) = (𝐹‘𝑃)) | |
18 | 14, 16, 17 | 3eqtrd 2779 | . 2 ⊢ (𝜑 → ((𝐷 ∘ 𝐺)‘𝑃) = (𝐹‘𝑃)) |
19 | 11, 12, 4, 5 | cdlemd 40190 | . 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐷 ∘ 𝐺) ∈ 𝑇 ∧ 𝐹 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ ((𝐷 ∘ 𝐺)‘𝑃) = (𝐹‘𝑃)) → (𝐷 ∘ 𝐺) = 𝐹) |
20 | 1, 7, 8, 9, 18, 19 | syl311anc 1383 | 1 ⊢ (𝜑 → (𝐷 ∘ 𝐺) = 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 class class class wbr 5148 ∘ ccom 5693 ‘cfv 6563 lecple 17305 Atomscatm 39245 HLchlt 39332 LHypclh 39967 LTrncltrn 40084 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-riotaBAD 38935 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-undef 8297 df-map 8867 df-proset 18352 df-poset 18371 df-plt 18388 df-lub 18404 df-glb 18405 df-join 18406 df-meet 18407 df-p0 18483 df-p1 18484 df-lat 18490 df-clat 18557 df-oposet 39158 df-ol 39160 df-oml 39161 df-covers 39248 df-ats 39249 df-atl 39280 df-cvlat 39304 df-hlat 39333 df-llines 39481 df-lplanes 39482 df-lvols 39483 df-lines 39484 df-psubsp 39486 df-pmap 39487 df-padd 39779 df-lhyp 39971 df-laut 39972 df-ldil 40087 df-ltrn 40088 df-trl 40142 |
This theorem is referenced by: dia2dimlem5 41051 |
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