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Mirrors > Home > MPE Home > Th. List > motrag | Structured version Visualization version GIF version |
Description: Right angles are preserved by motions. (Contributed by Thierry Arnoux, 16-Dec-2019.) |
Ref | Expression |
---|---|
israg.p | ⊢ 𝑃 = (Base‘𝐺) |
israg.d | ⊢ − = (dist‘𝐺) |
israg.i | ⊢ 𝐼 = (Itv‘𝐺) |
israg.l | ⊢ 𝐿 = (LineG‘𝐺) |
israg.s | ⊢ 𝑆 = (pInvG‘𝐺) |
israg.g | ⊢ (𝜑 → 𝐺 ∈ TarskiG) |
israg.a | ⊢ (𝜑 → 𝐴 ∈ 𝑃) |
israg.b | ⊢ (𝜑 → 𝐵 ∈ 𝑃) |
israg.c | ⊢ (𝜑 → 𝐶 ∈ 𝑃) |
motrag.f | ⊢ (𝜑 → 𝐹 ∈ (𝐺Ismt𝐺)) |
motrag.1 | ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉 ∈ (∟G‘𝐺)) |
Ref | Expression |
---|---|
motrag | ⊢ (𝜑 → 〈“(𝐹‘𝐴)(𝐹‘𝐵)(𝐹‘𝐶)”〉 ∈ (∟G‘𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | israg.p | . 2 ⊢ 𝑃 = (Base‘𝐺) | |
2 | israg.d | . 2 ⊢ − = (dist‘𝐺) | |
3 | israg.i | . 2 ⊢ 𝐼 = (Itv‘𝐺) | |
4 | israg.l | . 2 ⊢ 𝐿 = (LineG‘𝐺) | |
5 | israg.s | . 2 ⊢ 𝑆 = (pInvG‘𝐺) | |
6 | israg.g | . 2 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
7 | israg.a | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
8 | israg.b | . 2 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
9 | israg.c | . 2 ⊢ (𝜑 → 𝐶 ∈ 𝑃) | |
10 | eqid 2819 | . 2 ⊢ (cgrG‘𝐺) = (cgrG‘𝐺) | |
11 | motrag.f | . . 3 ⊢ (𝜑 → 𝐹 ∈ (𝐺Ismt𝐺)) | |
12 | 1, 2, 6, 11, 7 | motcl 26317 | . 2 ⊢ (𝜑 → (𝐹‘𝐴) ∈ 𝑃) |
13 | 1, 2, 6, 11, 8 | motcl 26317 | . 2 ⊢ (𝜑 → (𝐹‘𝐵) ∈ 𝑃) |
14 | 1, 2, 6, 11, 9 | motcl 26317 | . 2 ⊢ (𝜑 → (𝐹‘𝐶) ∈ 𝑃) |
15 | motrag.1 | . 2 ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉 ∈ (∟G‘𝐺)) | |
16 | eqidd 2820 | . . 3 ⊢ (𝜑 → (𝐹‘𝐴) = (𝐹‘𝐴)) | |
17 | eqidd 2820 | . . 3 ⊢ (𝜑 → (𝐹‘𝐵) = (𝐹‘𝐵)) | |
18 | eqidd 2820 | . . 3 ⊢ (𝜑 → (𝐹‘𝐶) = (𝐹‘𝐶)) | |
19 | 1, 2, 10, 6, 7, 8, 9, 16, 17, 18, 11 | motcgr3 26323 | . 2 ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“(𝐹‘𝐴)(𝐹‘𝐵)(𝐹‘𝐶)”〉) |
20 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 19 | ragcgr 26485 | 1 ⊢ (𝜑 → 〈“(𝐹‘𝐴)(𝐹‘𝐵)(𝐹‘𝐶)”〉 ∈ (∟G‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1531 ∈ wcel 2108 ‘cfv 6348 (class class class)co 7148 〈“cs3 14196 Basecbs 16475 distcds 16566 TarskiGcstrkg 26208 Itvcitv 26214 LineGclng 26215 cgrGccgrg 26288 Ismtcismt 26310 pInvGcmir 26430 ∟Gcrag 26471 |
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 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-cnex 10585 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rmo 3144 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-om 7573 df-1st 7681 df-2nd 7682 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-1o 8094 df-oadd 8098 df-er 8281 df-map 8400 df-pm 8401 df-en 8502 df-dom 8503 df-sdom 8504 df-fin 8505 df-card 9360 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-nn 11631 df-2 11692 df-3 11693 df-n0 11890 df-z 11974 df-uz 12236 df-fz 12885 df-fzo 13026 df-hash 13683 df-word 13854 df-concat 13915 df-s1 13942 df-s2 14202 df-s3 14203 df-trkgc 26226 df-trkgb 26227 df-trkgcb 26228 df-trkg 26231 df-cgrg 26289 df-ismt 26311 df-mir 26431 df-rag 26472 |
This theorem is referenced by: hypcgrlem2 26578 |
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