![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > mirrag | Structured version Visualization version GIF version |
Description: Right angle is conserved by point inversion. (Contributed by Thierry Arnoux, 3-Nov-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) |
ragmir.1 | ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉 ∈ (∟G‘𝐺)) |
mirrag.m | ⊢ 𝑀 = (𝑆‘𝐷) |
mirrag.d | ⊢ (𝜑 → 𝐷 ∈ 𝑃) |
Ref | Expression |
---|---|
mirrag | ⊢ (𝜑 → 〈“(𝑀‘𝐴)(𝑀‘𝐵)(𝑀‘𝐶)”〉 ∈ (∟G‘𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | israg.p | . . . 4 ⊢ 𝑃 = (Base‘𝐺) | |
2 | israg.d | . . . 4 ⊢ − = (dist‘𝐺) | |
3 | israg.i | . . . 4 ⊢ 𝐼 = (Itv‘𝐺) | |
4 | israg.l | . . . 4 ⊢ 𝐿 = (LineG‘𝐺) | |
5 | israg.s | . . . 4 ⊢ 𝑆 = (pInvG‘𝐺) | |
6 | israg.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
7 | mirrag.d | . . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑃) | |
8 | mirrag.m | . . . 4 ⊢ 𝑀 = (𝑆‘𝐷) | |
9 | israg.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
10 | israg.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑃) | |
11 | israg.b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
12 | eqid 2798 | . . . . 5 ⊢ (𝑆‘𝐵) = (𝑆‘𝐵) | |
13 | 1, 2, 3, 4, 5, 6, 11, 12, 10 | mircl 26455 | . . . 4 ⊢ (𝜑 → ((𝑆‘𝐵)‘𝐶) ∈ 𝑃) |
14 | ragmir.1 | . . . . 5 ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉 ∈ (∟G‘𝐺)) | |
15 | 1, 2, 3, 4, 5, 6, 9, 11, 10 | israg 26491 | . . . . 5 ⊢ (𝜑 → (〈“𝐴𝐵𝐶”〉 ∈ (∟G‘𝐺) ↔ (𝐴 − 𝐶) = (𝐴 − ((𝑆‘𝐵)‘𝐶)))) |
16 | 14, 15 | mpbid 235 | . . . 4 ⊢ (𝜑 → (𝐴 − 𝐶) = (𝐴 − ((𝑆‘𝐵)‘𝐶))) |
17 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 9, 13, 16 | mircgrs 26467 | . . 3 ⊢ (𝜑 → ((𝑀‘𝐴) − (𝑀‘𝐶)) = ((𝑀‘𝐴) − (𝑀‘((𝑆‘𝐵)‘𝐶)))) |
18 | 1, 2, 3, 4, 5, 6, 7, 8, 10, 11 | mirmir2 26468 | . . . 4 ⊢ (𝜑 → (𝑀‘((𝑆‘𝐵)‘𝐶)) = ((𝑆‘(𝑀‘𝐵))‘(𝑀‘𝐶))) |
19 | 18 | oveq2d 7151 | . . 3 ⊢ (𝜑 → ((𝑀‘𝐴) − (𝑀‘((𝑆‘𝐵)‘𝐶))) = ((𝑀‘𝐴) − ((𝑆‘(𝑀‘𝐵))‘(𝑀‘𝐶)))) |
20 | 17, 19 | eqtrd 2833 | . 2 ⊢ (𝜑 → ((𝑀‘𝐴) − (𝑀‘𝐶)) = ((𝑀‘𝐴) − ((𝑆‘(𝑀‘𝐵))‘(𝑀‘𝐶)))) |
21 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | mircl 26455 | . . 3 ⊢ (𝜑 → (𝑀‘𝐴) ∈ 𝑃) |
22 | 1, 2, 3, 4, 5, 6, 7, 8, 11 | mircl 26455 | . . 3 ⊢ (𝜑 → (𝑀‘𝐵) ∈ 𝑃) |
23 | 1, 2, 3, 4, 5, 6, 7, 8, 10 | mircl 26455 | . . 3 ⊢ (𝜑 → (𝑀‘𝐶) ∈ 𝑃) |
24 | 1, 2, 3, 4, 5, 6, 21, 22, 23 | israg 26491 | . 2 ⊢ (𝜑 → (〈“(𝑀‘𝐴)(𝑀‘𝐵)(𝑀‘𝐶)”〉 ∈ (∟G‘𝐺) ↔ ((𝑀‘𝐴) − (𝑀‘𝐶)) = ((𝑀‘𝐴) − ((𝑆‘(𝑀‘𝐵))‘(𝑀‘𝐶))))) |
25 | 20, 24 | mpbird 260 | 1 ⊢ (𝜑 → 〈“(𝑀‘𝐴)(𝑀‘𝐵)(𝑀‘𝐶)”〉 ∈ (∟G‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 ‘cfv 6324 (class class class)co 7135 〈“cs3 14195 Basecbs 16475 distcds 16566 TarskiGcstrkg 26224 Itvcitv 26230 LineGclng 26231 pInvGcmir 26446 ∟Gcrag 26487 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 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-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-pm 8392 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-dju 9314 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-n0 11886 df-xnn0 11956 df-z 11970 df-uz 12232 df-fz 12886 df-fzo 13029 df-hash 13687 df-word 13858 df-concat 13914 df-s1 13941 df-s2 14201 df-s3 14202 df-trkgc 26242 df-trkgb 26243 df-trkgcb 26244 df-trkg 26247 df-cgrg 26305 df-mir 26447 df-rag 26488 |
This theorem is referenced by: colperpexlem1 26524 hypcgrlem2 26594 hypcgr 26595 |
Copyright terms: Public domain | W3C validator |