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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 2738 | . . . 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 27099 | . . 3 ⊢ (𝜑 → ∃!𝑚 ∈ 𝑃 𝐵 = ((𝑆‘𝑚)‘𝐴)) |
12 | fveq2 6774 | . . . . . 6 ⊢ (𝑚 = 𝑀 → (𝑆‘𝑚) = (𝑆‘𝑀)) | |
13 | 12 | fveq1d 6776 | . . . . 5 ⊢ (𝑚 = 𝑀 → ((𝑆‘𝑚)‘𝐴) = ((𝑆‘𝑀)‘𝐴)) |
14 | 13 | eqeq2d 2749 | . . . 4 ⊢ (𝑚 = 𝑀 → (𝐵 = ((𝑆‘𝑚)‘𝐴) ↔ 𝐵 = ((𝑆‘𝑀)‘𝐴))) |
15 | 14 | riota2 7258 | . . 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 584 | . 2 ⊢ (𝜑 → (𝐵 = ((𝑆‘𝑀)‘𝐴) ↔ (℩𝑚 ∈ 𝑃 𝐵 = ((𝑆‘𝑚)‘𝐴)) = 𝑀)) |
17 | df-mid 27135 | . . . . 5 ⊢ midG = (𝑔 ∈ V ↦ (𝑎 ∈ (Base‘𝑔), 𝑏 ∈ (Base‘𝑔) ↦ (℩𝑚 ∈ (Base‘𝑔)𝑏 = (((pInvG‘𝑔)‘𝑚)‘𝑎)))) | |
18 | fveq2 6774 | . . . . . . 7 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺)) | |
19 | 18, 2 | eqtr4di 2796 | . . . . . 6 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝑃) |
20 | fveq2 6774 | . . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → (pInvG‘𝑔) = (pInvG‘𝐺)) | |
21 | 20, 7 | eqtr4di 2796 | . . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (pInvG‘𝑔) = 𝑆) |
22 | 21 | fveq1d 6776 | . . . . . . . . 9 ⊢ (𝑔 = 𝐺 → ((pInvG‘𝑔)‘𝑚) = (𝑆‘𝑚)) |
23 | 22 | fveq1d 6776 | . . . . . . . 8 ⊢ (𝑔 = 𝐺 → (((pInvG‘𝑔)‘𝑚)‘𝑎) = ((𝑆‘𝑚)‘𝑎)) |
24 | 23 | eqeq2d 2749 | . . . . . . 7 ⊢ (𝑔 = 𝐺 → (𝑏 = (((pInvG‘𝑔)‘𝑚)‘𝑎) ↔ 𝑏 = ((𝑆‘𝑚)‘𝑎))) |
25 | 19, 24 | riotaeqbidv 7235 | . . . . . 6 ⊢ (𝑔 = 𝐺 → (℩𝑚 ∈ (Base‘𝑔)𝑏 = (((pInvG‘𝑔)‘𝑚)‘𝑎)) = (℩𝑚 ∈ 𝑃 𝑏 = ((𝑆‘𝑚)‘𝑎))) |
26 | 19, 19, 25 | mpoeq123dv 7350 | . . . . 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 3452 | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ V) |
28 | 2 | fvexi 6788 | . . . . . . 7 ⊢ 𝑃 ∈ V |
29 | 28, 28 | mpoex 7920 | . . . . . 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 6898 | . . . 4 ⊢ (𝜑 → (midG‘𝐺) = (𝑎 ∈ 𝑃, 𝑏 ∈ 𝑃 ↦ (℩𝑚 ∈ 𝑃 𝑏 = ((𝑆‘𝑚)‘𝑎)))) |
32 | simprr 770 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑎 = 𝐴 ∧ 𝑏 = 𝐵)) → 𝑏 = 𝐵) | |
33 | simprl 768 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 = 𝐴 ∧ 𝑏 = 𝐵)) → 𝑎 = 𝐴) | |
34 | 33 | fveq2d 6778 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑎 = 𝐴 ∧ 𝑏 = 𝐵)) → ((𝑆‘𝑚)‘𝑎) = ((𝑆‘𝑚)‘𝐴)) |
35 | 32, 34 | eqeq12d 2754 | . . . . 5 ⊢ ((𝜑 ∧ (𝑎 = 𝐴 ∧ 𝑏 = 𝐵)) → (𝑏 = ((𝑆‘𝑚)‘𝑎) ↔ 𝐵 = ((𝑆‘𝑚)‘𝐴))) |
36 | 35 | riotabidv 7234 | . . . 4 ⊢ ((𝜑 ∧ (𝑎 = 𝐴 ∧ 𝑏 = 𝐵)) → (℩𝑚 ∈ 𝑃 𝑏 = ((𝑆‘𝑚)‘𝑎)) = (℩𝑚 ∈ 𝑃 𝐵 = ((𝑆‘𝑚)‘𝐴))) |
37 | riotacl 7250 | . . . . 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 7425 | . . 3 ⊢ (𝜑 → (𝐴(midG‘𝐺)𝐵) = (℩𝑚 ∈ 𝑃 𝐵 = ((𝑆‘𝑚)‘𝐴))) |
40 | 39 | eqeq1d 2740 | . 2 ⊢ (𝜑 → ((𝐴(midG‘𝐺)𝐵) = 𝑀 ↔ (℩𝑚 ∈ 𝑃 𝐵 = ((𝑆‘𝑚)‘𝐴)) = 𝑀)) |
41 | 16, 40 | bitr4d 281 | 1 ⊢ (𝜑 → (𝐵 = ((𝑆‘𝑀)‘𝐴) ↔ (𝐴(midG‘𝐺)𝐵) = 𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ∃!wreu 3066 Vcvv 3432 class class class wbr 5074 ‘cfv 6433 ℩crio 7231 (class class class)co 7275 ∈ cmpo 7277 2c2 12028 Basecbs 16912 distcds 16971 TarskiGcstrkg 26788 DimTarskiG≥cstrkgld 26792 Itvcitv 26794 LineGclng 26795 pInvGcmir 27013 midGcmid 27133 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-oadd 8301 df-er 8498 df-map 8617 df-pm 8618 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-dju 9659 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-3 12037 df-n0 12234 df-xnn0 12306 df-z 12320 df-uz 12583 df-fz 13240 df-fzo 13383 df-hash 14045 df-word 14218 df-concat 14274 df-s1 14301 df-s2 14561 df-s3 14562 df-trkgc 26809 df-trkgb 26810 df-trkgcb 26811 df-trkgld 26813 df-trkg 26814 df-cgrg 26872 df-leg 26944 df-mir 27014 df-rag 27055 df-perpg 27057 df-mid 27135 |
This theorem is referenced by: midbtwn 27140 midcgr 27141 midcom 27143 mirmid 27144 lmieu 27145 lmimid 27155 lmiisolem 27157 hypcgrlem1 27160 hypcgrlem2 27161 hypcgr 27162 trgcopyeulem 27166 |
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