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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 2823 | . . . 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 26526 | . . 3 ⊢ (𝜑 → ∃!𝑚 ∈ 𝑃 𝐵 = ((𝑆‘𝑚)‘𝐴)) |
12 | fveq2 6672 | . . . . . 6 ⊢ (𝑚 = 𝑀 → (𝑆‘𝑚) = (𝑆‘𝑀)) | |
13 | 12 | fveq1d 6674 | . . . . 5 ⊢ (𝑚 = 𝑀 → ((𝑆‘𝑚)‘𝐴) = ((𝑆‘𝑀)‘𝐴)) |
14 | 13 | eqeq2d 2834 | . . . 4 ⊢ (𝑚 = 𝑀 → (𝐵 = ((𝑆‘𝑚)‘𝐴) ↔ 𝐵 = ((𝑆‘𝑀)‘𝐴))) |
15 | 14 | riota2 7141 | . . 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 586 | . 2 ⊢ (𝜑 → (𝐵 = ((𝑆‘𝑀)‘𝐴) ↔ (℩𝑚 ∈ 𝑃 𝐵 = ((𝑆‘𝑚)‘𝐴)) = 𝑀)) |
17 | df-mid 26562 | . . . . 5 ⊢ midG = (𝑔 ∈ V ↦ (𝑎 ∈ (Base‘𝑔), 𝑏 ∈ (Base‘𝑔) ↦ (℩𝑚 ∈ (Base‘𝑔)𝑏 = (((pInvG‘𝑔)‘𝑚)‘𝑎)))) | |
18 | fveq2 6672 | . . . . . . 7 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺)) | |
19 | 18, 2 | syl6eqr 2876 | . . . . . 6 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝑃) |
20 | fveq2 6672 | . . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → (pInvG‘𝑔) = (pInvG‘𝐺)) | |
21 | 20, 7 | syl6eqr 2876 | . . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (pInvG‘𝑔) = 𝑆) |
22 | 21 | fveq1d 6674 | . . . . . . . . 9 ⊢ (𝑔 = 𝐺 → ((pInvG‘𝑔)‘𝑚) = (𝑆‘𝑚)) |
23 | 22 | fveq1d 6674 | . . . . . . . 8 ⊢ (𝑔 = 𝐺 → (((pInvG‘𝑔)‘𝑚)‘𝑎) = ((𝑆‘𝑚)‘𝑎)) |
24 | 23 | eqeq2d 2834 | . . . . . . 7 ⊢ (𝑔 = 𝐺 → (𝑏 = (((pInvG‘𝑔)‘𝑚)‘𝑎) ↔ 𝑏 = ((𝑆‘𝑚)‘𝑎))) |
25 | 19, 24 | riotaeqbidv 7119 | . . . . . 6 ⊢ (𝑔 = 𝐺 → (℩𝑚 ∈ (Base‘𝑔)𝑏 = (((pInvG‘𝑔)‘𝑚)‘𝑎)) = (℩𝑚 ∈ 𝑃 𝑏 = ((𝑆‘𝑚)‘𝑎))) |
26 | 19, 19, 25 | mpoeq123dv 7231 | . . . . 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 3516 | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ V) |
28 | 2 | fvexi 6686 | . . . . . . 7 ⊢ 𝑃 ∈ V |
29 | 28, 28 | mpoex 7779 | . . . . . 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 6793 | . . . 4 ⊢ (𝜑 → (midG‘𝐺) = (𝑎 ∈ 𝑃, 𝑏 ∈ 𝑃 ↦ (℩𝑚 ∈ 𝑃 𝑏 = ((𝑆‘𝑚)‘𝑎)))) |
32 | simprr 771 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑎 = 𝐴 ∧ 𝑏 = 𝐵)) → 𝑏 = 𝐵) | |
33 | simprl 769 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 = 𝐴 ∧ 𝑏 = 𝐵)) → 𝑎 = 𝐴) | |
34 | 33 | fveq2d 6676 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑎 = 𝐴 ∧ 𝑏 = 𝐵)) → ((𝑆‘𝑚)‘𝑎) = ((𝑆‘𝑚)‘𝐴)) |
35 | 32, 34 | eqeq12d 2839 | . . . . 5 ⊢ ((𝜑 ∧ (𝑎 = 𝐴 ∧ 𝑏 = 𝐵)) → (𝑏 = ((𝑆‘𝑚)‘𝑎) ↔ 𝐵 = ((𝑆‘𝑚)‘𝐴))) |
36 | 35 | riotabidv 7118 | . . . 4 ⊢ ((𝜑 ∧ (𝑎 = 𝐴 ∧ 𝑏 = 𝐵)) → (℩𝑚 ∈ 𝑃 𝑏 = ((𝑆‘𝑚)‘𝑎)) = (℩𝑚 ∈ 𝑃 𝐵 = ((𝑆‘𝑚)‘𝐴))) |
37 | riotacl 7133 | . . . . 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 7304 | . . 3 ⊢ (𝜑 → (𝐴(midG‘𝐺)𝐵) = (℩𝑚 ∈ 𝑃 𝐵 = ((𝑆‘𝑚)‘𝐴))) |
40 | 39 | eqeq1d 2825 | . 2 ⊢ (𝜑 → ((𝐴(midG‘𝐺)𝐵) = 𝑀 ↔ (℩𝑚 ∈ 𝑃 𝐵 = ((𝑆‘𝑚)‘𝐴)) = 𝑀)) |
41 | 16, 40 | bitr4d 284 | 1 ⊢ (𝜑 → (𝐵 = ((𝑆‘𝑀)‘𝐴) ↔ (𝐴(midG‘𝐺)𝐵) = 𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∃!wreu 3142 Vcvv 3496 class class class wbr 5068 ‘cfv 6357 ℩crio 7115 (class class class)co 7158 ∈ cmpo 7160 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: midbtwn 26567 midcgr 26568 midcom 26570 mirmid 26571 lmieu 26572 lmimid 26582 lmiisolem 26584 hypcgrlem1 26587 hypcgrlem2 26588 hypcgr 26589 trgcopyeulem 26593 |
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