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Mirrors > Home > MPE Home > Th. List > motcgr3 | Structured version Visualization version GIF version |
Description: Property of a motion: distances are preserved, special case of triangles. (Contributed by Thierry Arnoux, 15-Dec-2019.) |
Ref | Expression |
---|---|
motcgr3.p | ⊢ 𝑃 = (Base‘𝐺) |
motcgr3.m | ⊢ − = (dist‘𝐺) |
motcgr3.r | ⊢ ∼ = (cgrG‘𝐺) |
motcgr3.g | ⊢ (𝜑 → 𝐺 ∈ TarskiG) |
motcgr3.a | ⊢ (𝜑 → 𝐴 ∈ 𝑃) |
motcgr3.b | ⊢ (𝜑 → 𝐵 ∈ 𝑃) |
motcgr3.c | ⊢ (𝜑 → 𝐶 ∈ 𝑃) |
motcgr3.d | ⊢ (𝜑 → 𝐷 = (𝐻‘𝐴)) |
motcgr3.e | ⊢ (𝜑 → 𝐸 = (𝐻‘𝐵)) |
motcgr3.f | ⊢ (𝜑 → 𝐹 = (𝐻‘𝐶)) |
motcgr3.h | ⊢ (𝜑 → 𝐻 ∈ (𝐺Ismt𝐺)) |
Ref | Expression |
---|---|
motcgr3 | ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝐸𝐹”〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | motcgr3.p | . 2 ⊢ 𝑃 = (Base‘𝐺) | |
2 | motcgr3.m | . 2 ⊢ − = (dist‘𝐺) | |
3 | motcgr3.r | . 2 ⊢ ∼ = (cgrG‘𝐺) | |
4 | motcgr3.g | . 2 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
5 | motcgr3.a | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
6 | motcgr3.b | . 2 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
7 | motcgr3.c | . 2 ⊢ (𝜑 → 𝐶 ∈ 𝑃) | |
8 | motcgr3.d | . . 3 ⊢ (𝜑 → 𝐷 = (𝐻‘𝐴)) | |
9 | motcgr3.h | . . . 4 ⊢ (𝜑 → 𝐻 ∈ (𝐺Ismt𝐺)) | |
10 | 1, 2, 4, 9, 5 | motcl 25907 | . . 3 ⊢ (𝜑 → (𝐻‘𝐴) ∈ 𝑃) |
11 | 8, 10 | eqeltrd 2859 | . 2 ⊢ (𝜑 → 𝐷 ∈ 𝑃) |
12 | motcgr3.e | . . 3 ⊢ (𝜑 → 𝐸 = (𝐻‘𝐵)) | |
13 | 1, 2, 4, 9, 6 | motcl 25907 | . . 3 ⊢ (𝜑 → (𝐻‘𝐵) ∈ 𝑃) |
14 | 12, 13 | eqeltrd 2859 | . 2 ⊢ (𝜑 → 𝐸 ∈ 𝑃) |
15 | motcgr3.f | . . 3 ⊢ (𝜑 → 𝐹 = (𝐻‘𝐶)) | |
16 | 1, 2, 4, 9, 7 | motcl 25907 | . . 3 ⊢ (𝜑 → (𝐻‘𝐶) ∈ 𝑃) |
17 | 15, 16 | eqeltrd 2859 | . 2 ⊢ (𝜑 → 𝐹 ∈ 𝑃) |
18 | 8, 12 | oveq12d 6942 | . . 3 ⊢ (𝜑 → (𝐷 − 𝐸) = ((𝐻‘𝐴) − (𝐻‘𝐵))) |
19 | 1, 2, 4, 5, 6, 9 | motcgr 25904 | . . 3 ⊢ (𝜑 → ((𝐻‘𝐴) − (𝐻‘𝐵)) = (𝐴 − 𝐵)) |
20 | 18, 19 | eqtr2d 2815 | . 2 ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐷 − 𝐸)) |
21 | 12, 15 | oveq12d 6942 | . . 3 ⊢ (𝜑 → (𝐸 − 𝐹) = ((𝐻‘𝐵) − (𝐻‘𝐶))) |
22 | 1, 2, 4, 6, 7, 9 | motcgr 25904 | . . 3 ⊢ (𝜑 → ((𝐻‘𝐵) − (𝐻‘𝐶)) = (𝐵 − 𝐶)) |
23 | 21, 22 | eqtr2d 2815 | . 2 ⊢ (𝜑 → (𝐵 − 𝐶) = (𝐸 − 𝐹)) |
24 | 15, 8 | oveq12d 6942 | . . 3 ⊢ (𝜑 → (𝐹 − 𝐷) = ((𝐻‘𝐶) − (𝐻‘𝐴))) |
25 | 1, 2, 4, 7, 5, 9 | motcgr 25904 | . . 3 ⊢ (𝜑 → ((𝐻‘𝐶) − (𝐻‘𝐴)) = (𝐶 − 𝐴)) |
26 | 24, 25 | eqtr2d 2815 | . 2 ⊢ (𝜑 → (𝐶 − 𝐴) = (𝐹 − 𝐷)) |
27 | 1, 2, 3, 4, 5, 6, 7, 11, 14, 17, 20, 23, 26 | trgcgr 25884 | 1 ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝐸𝐹”〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2107 class class class wbr 4888 ‘cfv 6137 (class class class)co 6924 〈“cs3 13999 Basecbs 16266 distcds 16358 TarskiGcstrkg 25798 cgrGccgrg 25878 Ismtcismt 25900 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-int 4713 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-om 7346 df-1st 7447 df-2nd 7448 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-1o 7845 df-oadd 7849 df-er 8028 df-map 8144 df-pm 8145 df-en 8244 df-dom 8245 df-sdom 8246 df-fin 8247 df-card 9100 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-nn 11380 df-2 11443 df-3 11444 df-n0 11648 df-z 11734 df-uz 11998 df-fz 12649 df-fzo 12790 df-hash 13442 df-word 13606 df-concat 13667 df-s1 13692 df-s2 14005 df-s3 14006 df-trkgc 25816 df-trkgcb 25818 df-trkg 25821 df-cgrg 25879 df-ismt 25901 |
This theorem is referenced by: motrag 26076 |
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