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Mirrors > Home > MPE Home > Th. List > ismidb | Structured version Visualization version GIF version |
Description: Property of the midpoint. (Contributed by Thierry Arnoux, 1-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) |
ismidb.s | ⊢ 𝑆 = (pInvG‘𝐺) |
ismidb.m | ⊢ (𝜑 → 𝑀 ∈ 𝑃) |
Ref | Expression |
---|---|
ismidb | ⊢ (𝜑 → (𝐵 = ((𝑆‘𝑀)‘𝐴) ↔ (𝐴(midG‘𝐺)𝐵) = 𝑀)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ismidb.m | . . 3 ⊢ (𝜑 → 𝑀 ∈ 𝑃) | |
2 | ismid.p | . . . 4 ⊢ 𝑃 = (Base‘𝐺) | |
3 | ismid.d | . . . 4 ⊢ − = (dist‘𝐺) | |
4 | ismid.i | . . . 4 ⊢ 𝐼 = (Itv‘𝐺) | |
5 | eqid 2737 | . . . 4 ⊢ (LineG‘𝐺) = (LineG‘𝐺) | |
6 | ismid.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
7 | ismidb.s | . . . 4 ⊢ 𝑆 = (pInvG‘𝐺) | |
8 | midcl.1 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
9 | midcl.2 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
10 | ismid.1 | . . . 4 ⊢ (𝜑 → 𝐺DimTarskiG≥2) | |
11 | 2, 3, 4, 5, 6, 7, 8, 9, 10 | mideu 26829 | . . 3 ⊢ (𝜑 → ∃!𝑚 ∈ 𝑃 𝐵 = ((𝑆‘𝑚)‘𝐴)) |
12 | fveq2 6717 | . . . . . 6 ⊢ (𝑚 = 𝑀 → (𝑆‘𝑚) = (𝑆‘𝑀)) | |
13 | 12 | fveq1d 6719 | . . . . 5 ⊢ (𝑚 = 𝑀 → ((𝑆‘𝑚)‘𝐴) = ((𝑆‘𝑀)‘𝐴)) |
14 | 13 | eqeq2d 2748 | . . . 4 ⊢ (𝑚 = 𝑀 → (𝐵 = ((𝑆‘𝑚)‘𝐴) ↔ 𝐵 = ((𝑆‘𝑀)‘𝐴))) |
15 | 14 | riota2 7196 | . . 3 ⊢ ((𝑀 ∈ 𝑃 ∧ ∃!𝑚 ∈ 𝑃 𝐵 = ((𝑆‘𝑚)‘𝐴)) → (𝐵 = ((𝑆‘𝑀)‘𝐴) ↔ (℩𝑚 ∈ 𝑃 𝐵 = ((𝑆‘𝑚)‘𝐴)) = 𝑀)) |
16 | 1, 11, 15 | syl2anc 587 | . 2 ⊢ (𝜑 → (𝐵 = ((𝑆‘𝑀)‘𝐴) ↔ (℩𝑚 ∈ 𝑃 𝐵 = ((𝑆‘𝑚)‘𝐴)) = 𝑀)) |
17 | df-mid 26865 | . . . . 5 ⊢ midG = (𝑔 ∈ V ↦ (𝑎 ∈ (Base‘𝑔), 𝑏 ∈ (Base‘𝑔) ↦ (℩𝑚 ∈ (Base‘𝑔)𝑏 = (((pInvG‘𝑔)‘𝑚)‘𝑎)))) | |
18 | fveq2 6717 | . . . . . . 7 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺)) | |
19 | 18, 2 | eqtr4di 2796 | . . . . . 6 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝑃) |
20 | fveq2 6717 | . . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → (pInvG‘𝑔) = (pInvG‘𝐺)) | |
21 | 20, 7 | eqtr4di 2796 | . . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (pInvG‘𝑔) = 𝑆) |
22 | 21 | fveq1d 6719 | . . . . . . . . 9 ⊢ (𝑔 = 𝐺 → ((pInvG‘𝑔)‘𝑚) = (𝑆‘𝑚)) |
23 | 22 | fveq1d 6719 | . . . . . . . 8 ⊢ (𝑔 = 𝐺 → (((pInvG‘𝑔)‘𝑚)‘𝑎) = ((𝑆‘𝑚)‘𝑎)) |
24 | 23 | eqeq2d 2748 | . . . . . . 7 ⊢ (𝑔 = 𝐺 → (𝑏 = (((pInvG‘𝑔)‘𝑚)‘𝑎) ↔ 𝑏 = ((𝑆‘𝑚)‘𝑎))) |
25 | 19, 24 | riotaeqbidv 7173 | . . . . . 6 ⊢ (𝑔 = 𝐺 → (℩𝑚 ∈ (Base‘𝑔)𝑏 = (((pInvG‘𝑔)‘𝑚)‘𝑎)) = (℩𝑚 ∈ 𝑃 𝑏 = ((𝑆‘𝑚)‘𝑎))) |
26 | 19, 19, 25 | mpoeq123dv 7286 | . . . . 5 ⊢ (𝑔 = 𝐺 → (𝑎 ∈ (Base‘𝑔), 𝑏 ∈ (Base‘𝑔) ↦ (℩𝑚 ∈ (Base‘𝑔)𝑏 = (((pInvG‘𝑔)‘𝑚)‘𝑎))) = (𝑎 ∈ 𝑃, 𝑏 ∈ 𝑃 ↦ (℩𝑚 ∈ 𝑃 𝑏 = ((𝑆‘𝑚)‘𝑎)))) |
27 | 6 | elexd 3428 | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ V) |
28 | 2 | fvexi 6731 | . . . . . . 7 ⊢ 𝑃 ∈ V |
29 | 28, 28 | mpoex 7850 | . . . . . 6 ⊢ (𝑎 ∈ 𝑃, 𝑏 ∈ 𝑃 ↦ (℩𝑚 ∈ 𝑃 𝑏 = ((𝑆‘𝑚)‘𝑎))) ∈ V |
30 | 29 | a1i 11 | . . . . 5 ⊢ (𝜑 → (𝑎 ∈ 𝑃, 𝑏 ∈ 𝑃 ↦ (℩𝑚 ∈ 𝑃 𝑏 = ((𝑆‘𝑚)‘𝑎))) ∈ V) |
31 | 17, 26, 27, 30 | fvmptd3 6841 | . . . 4 ⊢ (𝜑 → (midG‘𝐺) = (𝑎 ∈ 𝑃, 𝑏 ∈ 𝑃 ↦ (℩𝑚 ∈ 𝑃 𝑏 = ((𝑆‘𝑚)‘𝑎)))) |
32 | simprr 773 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑎 = 𝐴 ∧ 𝑏 = 𝐵)) → 𝑏 = 𝐵) | |
33 | simprl 771 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 = 𝐴 ∧ 𝑏 = 𝐵)) → 𝑎 = 𝐴) | |
34 | 33 | fveq2d 6721 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑎 = 𝐴 ∧ 𝑏 = 𝐵)) → ((𝑆‘𝑚)‘𝑎) = ((𝑆‘𝑚)‘𝐴)) |
35 | 32, 34 | eqeq12d 2753 | . . . . 5 ⊢ ((𝜑 ∧ (𝑎 = 𝐴 ∧ 𝑏 = 𝐵)) → (𝑏 = ((𝑆‘𝑚)‘𝑎) ↔ 𝐵 = ((𝑆‘𝑚)‘𝐴))) |
36 | 35 | riotabidv 7172 | . . . 4 ⊢ ((𝜑 ∧ (𝑎 = 𝐴 ∧ 𝑏 = 𝐵)) → (℩𝑚 ∈ 𝑃 𝑏 = ((𝑆‘𝑚)‘𝑎)) = (℩𝑚 ∈ 𝑃 𝐵 = ((𝑆‘𝑚)‘𝐴))) |
37 | riotacl 7188 | . . . . 5 ⊢ (∃!𝑚 ∈ 𝑃 𝐵 = ((𝑆‘𝑚)‘𝐴) → (℩𝑚 ∈ 𝑃 𝐵 = ((𝑆‘𝑚)‘𝐴)) ∈ 𝑃) | |
38 | 11, 37 | syl 17 | . . . 4 ⊢ (𝜑 → (℩𝑚 ∈ 𝑃 𝐵 = ((𝑆‘𝑚)‘𝐴)) ∈ 𝑃) |
39 | 31, 36, 8, 9, 38 | ovmpod 7361 | . . 3 ⊢ (𝜑 → (𝐴(midG‘𝐺)𝐵) = (℩𝑚 ∈ 𝑃 𝐵 = ((𝑆‘𝑚)‘𝐴))) |
40 | 39 | eqeq1d 2739 | . 2 ⊢ (𝜑 → ((𝐴(midG‘𝐺)𝐵) = 𝑀 ↔ (℩𝑚 ∈ 𝑃 𝐵 = ((𝑆‘𝑚)‘𝐴)) = 𝑀)) |
41 | 16, 40 | bitr4d 285 | 1 ⊢ (𝜑 → (𝐵 = ((𝑆‘𝑀)‘𝐴) ↔ (𝐴(midG‘𝐺)𝐵) = 𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ∃!wreu 3063 Vcvv 3408 class class class wbr 5053 ‘cfv 6380 ℩crio 7169 (class class class)co 7213 ∈ cmpo 7215 2c2 11885 Basecbs 16760 distcds 16811 TarskiGcstrkg 26521 DimTarskiG≥cstrkgld 26525 Itvcitv 26527 LineGclng 26528 pInvGcmir 26743 midGcmid 26863 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-1st 7761 df-2nd 7762 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-oadd 8206 df-er 8391 df-map 8510 df-pm 8511 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-dju 9517 df-card 9555 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-nn 11831 df-2 11893 df-3 11894 df-n0 12091 df-xnn0 12163 df-z 12177 df-uz 12439 df-fz 13096 df-fzo 13239 df-hash 13897 df-word 14070 df-concat 14126 df-s1 14153 df-s2 14413 df-s3 14414 df-trkgc 26539 df-trkgb 26540 df-trkgcb 26541 df-trkgld 26543 df-trkg 26544 df-cgrg 26602 df-leg 26674 df-mir 26744 df-rag 26785 df-perpg 26787 df-mid 26865 |
This theorem is referenced by: midbtwn 26870 midcgr 26871 midcom 26873 mirmid 26874 lmieu 26875 lmimid 26885 lmiisolem 26887 hypcgrlem1 26890 hypcgrlem2 26891 hypcgr 26892 trgcopyeulem 26896 |
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