Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 2818 | . . . 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 26451 | . . 3 ⊢ (𝜑 → ∃!𝑚 ∈ 𝑃 𝐵 = ((𝑆‘𝑚)‘𝐴)) |
12 | fveq2 6663 | . . . . . 6 ⊢ (𝑚 = 𝑀 → (𝑆‘𝑚) = (𝑆‘𝑀)) | |
13 | 12 | fveq1d 6665 | . . . . 5 ⊢ (𝑚 = 𝑀 → ((𝑆‘𝑚)‘𝐴) = ((𝑆‘𝑀)‘𝐴)) |
14 | 13 | eqeq2d 2829 | . . . 4 ⊢ (𝑚 = 𝑀 → (𝐵 = ((𝑆‘𝑚)‘𝐴) ↔ 𝐵 = ((𝑆‘𝑀)‘𝐴))) |
15 | 14 | riota2 7128 | . . 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 26487 | . . . . 5 ⊢ midG = (𝑔 ∈ V ↦ (𝑎 ∈ (Base‘𝑔), 𝑏 ∈ (Base‘𝑔) ↦ (℩𝑚 ∈ (Base‘𝑔)𝑏 = (((pInvG‘𝑔)‘𝑚)‘𝑎)))) | |
18 | fveq2 6663 | . . . . . . 7 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺)) | |
19 | 18, 2 | syl6eqr 2871 | . . . . . 6 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝑃) |
20 | fveq2 6663 | . . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → (pInvG‘𝑔) = (pInvG‘𝐺)) | |
21 | 20, 7 | syl6eqr 2871 | . . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (pInvG‘𝑔) = 𝑆) |
22 | 21 | fveq1d 6665 | . . . . . . . . 9 ⊢ (𝑔 = 𝐺 → ((pInvG‘𝑔)‘𝑚) = (𝑆‘𝑚)) |
23 | 22 | fveq1d 6665 | . . . . . . . 8 ⊢ (𝑔 = 𝐺 → (((pInvG‘𝑔)‘𝑚)‘𝑎) = ((𝑆‘𝑚)‘𝑎)) |
24 | 23 | eqeq2d 2829 | . . . . . . 7 ⊢ (𝑔 = 𝐺 → (𝑏 = (((pInvG‘𝑔)‘𝑚)‘𝑎) ↔ 𝑏 = ((𝑆‘𝑚)‘𝑎))) |
25 | 19, 24 | riotaeqbidv 7106 | . . . . . 6 ⊢ (𝑔 = 𝐺 → (℩𝑚 ∈ (Base‘𝑔)𝑏 = (((pInvG‘𝑔)‘𝑚)‘𝑎)) = (℩𝑚 ∈ 𝑃 𝑏 = ((𝑆‘𝑚)‘𝑎))) |
26 | 19, 19, 25 | mpoeq123dv 7218 | . . . . 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 3512 | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ V) |
28 | 2 | fvexi 6677 | . . . . . . 7 ⊢ 𝑃 ∈ V |
29 | 28, 28 | mpoex 7766 | . . . . . 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 6783 | . . . 4 ⊢ (𝜑 → (midG‘𝐺) = (𝑎 ∈ 𝑃, 𝑏 ∈ 𝑃 ↦ (℩𝑚 ∈ 𝑃 𝑏 = ((𝑆‘𝑚)‘𝑎)))) |
32 | simprr 769 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑎 = 𝐴 ∧ 𝑏 = 𝐵)) → 𝑏 = 𝐵) | |
33 | simprl 767 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 = 𝐴 ∧ 𝑏 = 𝐵)) → 𝑎 = 𝐴) | |
34 | 33 | fveq2d 6667 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑎 = 𝐴 ∧ 𝑏 = 𝐵)) → ((𝑆‘𝑚)‘𝑎) = ((𝑆‘𝑚)‘𝐴)) |
35 | 32, 34 | eqeq12d 2834 | . . . . 5 ⊢ ((𝜑 ∧ (𝑎 = 𝐴 ∧ 𝑏 = 𝐵)) → (𝑏 = ((𝑆‘𝑚)‘𝑎) ↔ 𝐵 = ((𝑆‘𝑚)‘𝐴))) |
36 | 35 | riotabidv 7105 | . . . 4 ⊢ ((𝜑 ∧ (𝑎 = 𝐴 ∧ 𝑏 = 𝐵)) → (℩𝑚 ∈ 𝑃 𝑏 = ((𝑆‘𝑚)‘𝑎)) = (℩𝑚 ∈ 𝑃 𝐵 = ((𝑆‘𝑚)‘𝐴))) |
37 | riotacl 7120 | . . . . 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 7291 | . . 3 ⊢ (𝜑 → (𝐴(midG‘𝐺)𝐵) = (℩𝑚 ∈ 𝑃 𝐵 = ((𝑆‘𝑚)‘𝐴))) |
40 | 39 | eqeq1d 2820 | . 2 ⊢ (𝜑 → ((𝐴(midG‘𝐺)𝐵) = 𝑀 ↔ (℩𝑚 ∈ 𝑃 𝐵 = ((𝑆‘𝑚)‘𝐴)) = 𝑀)) |
41 | 16, 40 | bitr4d 283 | 1 ⊢ (𝜑 → (𝐵 = ((𝑆‘𝑀)‘𝐴) ↔ (𝐴(midG‘𝐺)𝐵) = 𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∃!wreu 3137 Vcvv 3492 class class class wbr 5057 ‘cfv 6348 ℩crio 7102 (class class class)co 7145 ∈ cmpo 7147 2c2 11680 Basecbs 16471 distcds 16562 TarskiGcstrkg 26143 DimTarskiG≥cstrkgld 26147 Itvcitv 26149 LineGclng 26150 pInvGcmir 26365 midGcmid 26485 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-map 8397 df-pm 8398 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-dju 9318 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-3 11689 df-n0 11886 df-xnn0 11956 df-z 11970 df-uz 12232 df-fz 12881 df-fzo 13022 df-hash 13679 df-word 13850 df-concat 13911 df-s1 13938 df-s2 14198 df-s3 14199 df-trkgc 26161 df-trkgb 26162 df-trkgcb 26163 df-trkgld 26165 df-trkg 26166 df-cgrg 26224 df-leg 26296 df-mir 26366 df-rag 26407 df-perpg 26409 df-mid 26487 |
This theorem is referenced by: midbtwn 26492 midcgr 26493 midcom 26495 mirmid 26496 lmieu 26497 lmimid 26507 lmiisolem 26509 hypcgrlem1 26512 hypcgrlem2 26513 hypcgr 26514 trgcopyeulem 26518 |
Copyright terms: Public domain | W3C validator |