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Mirrors > Home > MPE Home > Th. List > mirmid | Structured version Visualization version GIF version |
Description: Point inversion preserves midpoints. (Contributed by Thierry Arnoux, 12-Dec-2019.) |
Ref | Expression |
---|---|
ismid.p | ⊢ 𝑃 = (Base‘𝐺) |
ismid.d | ⊢ − = (dist‘𝐺) |
ismid.i | ⊢ 𝐼 = (Itv‘𝐺) |
ismid.g | ⊢ (𝜑 → 𝐺 ∈ TarskiG) |
ismid.1 | ⊢ (𝜑 → 𝐺DimTarskiG≥2) |
midcl.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑃) |
midcl.2 | ⊢ (𝜑 → 𝐵 ∈ 𝑃) |
mirmid.s | ⊢ 𝑆 = ((pInvG‘𝐺)‘𝑀) |
mirmid.x | ⊢ (𝜑 → 𝑀 ∈ 𝑃) |
Ref | Expression |
---|---|
mirmid | ⊢ (𝜑 → ((𝑆‘𝐴)(midG‘𝐺)(𝑆‘𝐵)) = (𝑆‘(𝐴(midG‘𝐺)𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqidd 2822 | . . . . 5 ⊢ (𝜑 → (𝐴(midG‘𝐺)𝐵) = (𝐴(midG‘𝐺)𝐵)) | |
2 | ismid.p | . . . . . 6 ⊢ 𝑃 = (Base‘𝐺) | |
3 | ismid.d | . . . . . 6 ⊢ − = (dist‘𝐺) | |
4 | ismid.i | . . . . . 6 ⊢ 𝐼 = (Itv‘𝐺) | |
5 | ismid.g | . . . . . 6 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
6 | ismid.1 | . . . . . 6 ⊢ (𝜑 → 𝐺DimTarskiG≥2) | |
7 | midcl.1 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
8 | midcl.2 | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
9 | eqid 2821 | . . . . . 6 ⊢ (pInvG‘𝐺) = (pInvG‘𝐺) | |
10 | 2, 3, 4, 5, 6, 7, 8 | midcl 26557 | . . . . . 6 ⊢ (𝜑 → (𝐴(midG‘𝐺)𝐵) ∈ 𝑃) |
11 | 2, 3, 4, 5, 6, 7, 8, 9, 10 | ismidb 26558 | . . . . 5 ⊢ (𝜑 → (𝐵 = (((pInvG‘𝐺)‘(𝐴(midG‘𝐺)𝐵))‘𝐴) ↔ (𝐴(midG‘𝐺)𝐵) = (𝐴(midG‘𝐺)𝐵))) |
12 | 1, 11 | mpbird 259 | . . . 4 ⊢ (𝜑 → 𝐵 = (((pInvG‘𝐺)‘(𝐴(midG‘𝐺)𝐵))‘𝐴)) |
13 | 12 | fveq2d 6668 | . . 3 ⊢ (𝜑 → (𝑆‘𝐵) = (𝑆‘(((pInvG‘𝐺)‘(𝐴(midG‘𝐺)𝐵))‘𝐴))) |
14 | eqid 2821 | . . . 4 ⊢ (LineG‘𝐺) = (LineG‘𝐺) | |
15 | mirmid.x | . . . 4 ⊢ (𝜑 → 𝑀 ∈ 𝑃) | |
16 | mirmid.s | . . . 4 ⊢ 𝑆 = ((pInvG‘𝐺)‘𝑀) | |
17 | 2, 3, 4, 14, 9, 5, 15, 16, 7, 10 | mirmir2 26454 | . . 3 ⊢ (𝜑 → (𝑆‘(((pInvG‘𝐺)‘(𝐴(midG‘𝐺)𝐵))‘𝐴)) = (((pInvG‘𝐺)‘(𝑆‘(𝐴(midG‘𝐺)𝐵)))‘(𝑆‘𝐴))) |
18 | 13, 17 | eqtrd 2856 | . 2 ⊢ (𝜑 → (𝑆‘𝐵) = (((pInvG‘𝐺)‘(𝑆‘(𝐴(midG‘𝐺)𝐵)))‘(𝑆‘𝐴))) |
19 | 2, 3, 4, 14, 9, 5, 15, 16, 7 | mircl 26441 | . . 3 ⊢ (𝜑 → (𝑆‘𝐴) ∈ 𝑃) |
20 | 2, 3, 4, 14, 9, 5, 15, 16, 8 | mircl 26441 | . . 3 ⊢ (𝜑 → (𝑆‘𝐵) ∈ 𝑃) |
21 | 2, 3, 4, 14, 9, 5, 15, 16, 10 | mircl 26441 | . . 3 ⊢ (𝜑 → (𝑆‘(𝐴(midG‘𝐺)𝐵)) ∈ 𝑃) |
22 | 2, 3, 4, 5, 6, 19, 20, 9, 21 | ismidb 26558 | . 2 ⊢ (𝜑 → ((𝑆‘𝐵) = (((pInvG‘𝐺)‘(𝑆‘(𝐴(midG‘𝐺)𝐵)))‘(𝑆‘𝐴)) ↔ ((𝑆‘𝐴)(midG‘𝐺)(𝑆‘𝐵)) = (𝑆‘(𝐴(midG‘𝐺)𝐵)))) |
23 | 18, 22 | mpbid 234 | 1 ⊢ (𝜑 → ((𝑆‘𝐴)(midG‘𝐺)(𝑆‘𝐵)) = (𝑆‘(𝐴(midG‘𝐺)𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 class class class wbr 5058 ‘cfv 6349 (class class class)co 7150 2c2 11686 Basecbs 16477 distcds 16568 TarskiGcstrkg 26210 DimTarskiG≥cstrkgld 26214 Itvcitv 26216 LineGclng 26217 pInvGcmir 26432 midGcmid 26552 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-map 8402 df-pm 8403 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-dju 9324 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-n0 11892 df-xnn0 11962 df-z 11976 df-uz 12238 df-fz 12887 df-fzo 13028 df-hash 13685 df-word 13856 df-concat 13917 df-s1 13944 df-s2 14204 df-s3 14205 df-trkgc 26228 df-trkgb 26229 df-trkgcb 26230 df-trkgld 26232 df-trkg 26233 df-cgrg 26291 df-leg 26363 df-mir 26433 df-rag 26474 df-perpg 26476 df-mid 26554 |
This theorem is referenced by: lmiisolem 26576 |
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