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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 28463 | . . 3 ⊢ (𝜑 → (𝐻‘𝐴) ∈ 𝑃) |
11 | 8, 10 | eqeltrd 2826 | . 2 ⊢ (𝜑 → 𝐷 ∈ 𝑃) |
12 | motcgr3.e | . . 3 ⊢ (𝜑 → 𝐸 = (𝐻‘𝐵)) | |
13 | 1, 2, 4, 9, 6 | motcl 28463 | . . 3 ⊢ (𝜑 → (𝐻‘𝐵) ∈ 𝑃) |
14 | 12, 13 | eqeltrd 2826 | . 2 ⊢ (𝜑 → 𝐸 ∈ 𝑃) |
15 | motcgr3.f | . . 3 ⊢ (𝜑 → 𝐹 = (𝐻‘𝐶)) | |
16 | 1, 2, 4, 9, 7 | motcl 28463 | . . 3 ⊢ (𝜑 → (𝐻‘𝐶) ∈ 𝑃) |
17 | 15, 16 | eqeltrd 2826 | . 2 ⊢ (𝜑 → 𝐹 ∈ 𝑃) |
18 | 8, 12 | oveq12d 7434 | . . 3 ⊢ (𝜑 → (𝐷 − 𝐸) = ((𝐻‘𝐴) − (𝐻‘𝐵))) |
19 | 1, 2, 4, 5, 6, 9 | motcgr 28460 | . . 3 ⊢ (𝜑 → ((𝐻‘𝐴) − (𝐻‘𝐵)) = (𝐴 − 𝐵)) |
20 | 18, 19 | eqtr2d 2767 | . 2 ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐷 − 𝐸)) |
21 | 12, 15 | oveq12d 7434 | . . 3 ⊢ (𝜑 → (𝐸 − 𝐹) = ((𝐻‘𝐵) − (𝐻‘𝐶))) |
22 | 1, 2, 4, 6, 7, 9 | motcgr 28460 | . . 3 ⊢ (𝜑 → ((𝐻‘𝐵) − (𝐻‘𝐶)) = (𝐵 − 𝐶)) |
23 | 21, 22 | eqtr2d 2767 | . 2 ⊢ (𝜑 → (𝐵 − 𝐶) = (𝐸 − 𝐹)) |
24 | 15, 8 | oveq12d 7434 | . . 3 ⊢ (𝜑 → (𝐹 − 𝐷) = ((𝐻‘𝐶) − (𝐻‘𝐴))) |
25 | 1, 2, 4, 7, 5, 9 | motcgr 28460 | . . 3 ⊢ (𝜑 → ((𝐻‘𝐶) − (𝐻‘𝐴)) = (𝐶 − 𝐴)) |
26 | 24, 25 | eqtr2d 2767 | . 2 ⊢ (𝜑 → (𝐶 − 𝐴) = (𝐹 − 𝐷)) |
27 | 1, 2, 3, 4, 5, 6, 7, 11, 14, 17, 20, 23, 26 | trgcgr 28440 | 1 ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝐸𝐹”〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 class class class wbr 5145 ‘cfv 6546 (class class class)co 7416 〈“cs3 14846 Basecbs 17208 distcds 17270 TarskiGcstrkg 28351 cgrGccgrg 28434 Ismtcismt 28456 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5282 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 ax-cnex 11205 ax-resscn 11206 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-addrcl 11210 ax-mulcl 11211 ax-mulrcl 11212 ax-mulcom 11213 ax-addass 11214 ax-mulass 11215 ax-distr 11216 ax-i2m1 11217 ax-1ne0 11218 ax-1rid 11219 ax-rnegex 11220 ax-rrecex 11221 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 ax-pre-ltadd 11225 ax-pre-mulgt0 11226 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4906 df-int 4947 df-iun 4995 df-br 5146 df-opab 5208 df-mpt 5229 df-tr 5263 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6304 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7995 df-2nd 7996 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-1o 8488 df-er 8726 df-map 8849 df-pm 8850 df-en 8967 df-dom 8968 df-sdom 8969 df-fin 8970 df-card 9975 df-pnf 11291 df-mnf 11292 df-xr 11293 df-ltxr 11294 df-le 11295 df-sub 11487 df-neg 11488 df-nn 12259 df-2 12321 df-3 12322 df-n0 12519 df-z 12605 df-uz 12869 df-fz 13533 df-fzo 13676 df-hash 14343 df-word 14518 df-concat 14574 df-s1 14599 df-s2 14852 df-s3 14853 df-trkgc 28372 df-trkgcb 28374 df-trkg 28377 df-cgrg 28435 df-ismt 28457 |
This theorem is referenced by: motrag 28632 |
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