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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 2824 | . . . . 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 2823 | . . . . . 6 ⊢ (pInvG‘𝐺) = (pInvG‘𝐺) | |
10 | 2, 3, 4, 5, 6, 7, 8 | midcl 26565 | . . . . . 6 ⊢ (𝜑 → (𝐴(midG‘𝐺)𝐵) ∈ 𝑃) |
11 | 2, 3, 4, 5, 6, 7, 8, 9, 10 | ismidb 26566 | . . . . 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 6676 | . . 3 ⊢ (𝜑 → (𝑆‘𝐵) = (𝑆‘(((pInvG‘𝐺)‘(𝐴(midG‘𝐺)𝐵))‘𝐴))) |
14 | eqid 2823 | . . . 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 26462 | . . 3 ⊢ (𝜑 → (𝑆‘(((pInvG‘𝐺)‘(𝐴(midG‘𝐺)𝐵))‘𝐴)) = (((pInvG‘𝐺)‘(𝑆‘(𝐴(midG‘𝐺)𝐵)))‘(𝑆‘𝐴))) |
18 | 13, 17 | eqtrd 2858 | . 2 ⊢ (𝜑 → (𝑆‘𝐵) = (((pInvG‘𝐺)‘(𝑆‘(𝐴(midG‘𝐺)𝐵)))‘(𝑆‘𝐴))) |
19 | 2, 3, 4, 14, 9, 5, 15, 16, 7 | mircl 26449 | . . 3 ⊢ (𝜑 → (𝑆‘𝐴) ∈ 𝑃) |
20 | 2, 3, 4, 14, 9, 5, 15, 16, 8 | mircl 26449 | . . 3 ⊢ (𝜑 → (𝑆‘𝐵) ∈ 𝑃) |
21 | 2, 3, 4, 14, 9, 5, 15, 16, 10 | mircl 26449 | . . 3 ⊢ (𝜑 → (𝑆‘(𝐴(midG‘𝐺)𝐵)) ∈ 𝑃) |
22 | 2, 3, 4, 5, 6, 19, 20, 9, 21 | ismidb 26566 | . 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 1537 ∈ wcel 2114 class class class wbr 5068 ‘cfv 6357 (class class class)co 7158 2c2 11695 Basecbs 16485 distcds 16576 TarskiGcstrkg 26218 DimTarskiG≥cstrkgld 26222 Itvcitv 26224 LineGclng 26225 pInvGcmir 26440 midGcmid 26560 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-map 8410 df-pm 8411 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-dju 9332 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-3 11704 df-n0 11901 df-xnn0 11971 df-z 11985 df-uz 12247 df-fz 12896 df-fzo 13037 df-hash 13694 df-word 13865 df-concat 13925 df-s1 13952 df-s2 14212 df-s3 14213 df-trkgc 26236 df-trkgb 26237 df-trkgcb 26238 df-trkgld 26240 df-trkg 26241 df-cgrg 26299 df-leg 26371 df-mir 26441 df-rag 26482 df-perpg 26484 df-mid 26562 |
This theorem is referenced by: lmiisolem 26584 |
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