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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 28746 | . . 3 ⊢ (𝜑 → ∃!𝑚 ∈ 𝑃 𝐵 = ((𝑆‘𝑚)‘𝐴)) |
| 12 | fveq2 6906 | . . . . . 6 ⊢ (𝑚 = 𝑀 → (𝑆‘𝑚) = (𝑆‘𝑀)) | |
| 13 | 12 | fveq1d 6908 | . . . . 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 7413 | . . 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 28782 | . . . . 5 ⊢ midG = (𝑔 ∈ V ↦ (𝑎 ∈ (Base‘𝑔), 𝑏 ∈ (Base‘𝑔) ↦ (℩𝑚 ∈ (Base‘𝑔)𝑏 = (((pInvG‘𝑔)‘𝑚)‘𝑎)))) | |
| 18 | fveq2 6906 | . . . . . . 7 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺)) | |
| 19 | 18, 2 | eqtr4di 2795 | . . . . . 6 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝑃) |
| 20 | fveq2 6906 | . . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → (pInvG‘𝑔) = (pInvG‘𝐺)) | |
| 21 | 20, 7 | eqtr4di 2795 | . . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (pInvG‘𝑔) = 𝑆) |
| 22 | 21 | fveq1d 6908 | . . . . . . . . 9 ⊢ (𝑔 = 𝐺 → ((pInvG‘𝑔)‘𝑚) = (𝑆‘𝑚)) |
| 23 | 22 | fveq1d 6908 | . . . . . . . 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 7391 | . . . . . 6 ⊢ (𝑔 = 𝐺 → (℩𝑚 ∈ (Base‘𝑔)𝑏 = (((pInvG‘𝑔)‘𝑚)‘𝑎)) = (℩𝑚 ∈ 𝑃 𝑏 = ((𝑆‘𝑚)‘𝑎))) |
| 26 | 19, 19, 25 | mpoeq123dv 7508 | . . . . 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 3504 | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ V) |
| 28 | 2 | fvexi 6920 | . . . . . . 7 ⊢ 𝑃 ∈ V |
| 29 | 28, 28 | mpoex 8104 | . . . . . 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 7039 | . . . 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 6910 | . . . . . 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 7390 | . . . 4 ⊢ ((𝜑 ∧ (𝑎 = 𝐴 ∧ 𝑏 = 𝐵)) → (℩𝑚 ∈ 𝑃 𝑏 = ((𝑆‘𝑚)‘𝑎)) = (℩𝑚 ∈ 𝑃 𝐵 = ((𝑆‘𝑚)‘𝐴))) |
| 37 | riotacl 7405 | . . . . 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 7585 | . . 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 282 | 1 ⊢ (𝜑 → (𝐵 = ((𝑆‘𝑀)‘𝐴) ↔ (𝐴(midG‘𝐺)𝐵) = 𝑀)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∃!wreu 3378 Vcvv 3480 class class class wbr 5143 ‘cfv 6561 ℩crio 7387 (class class class)co 7431 ∈ cmpo 7433 2c2 12321 Basecbs 17247 distcds 17306 TarskiGcstrkg 28435 DimTarskiG≥cstrkgld 28439 Itvcitv 28441 LineGclng 28442 pInvGcmir 28660 midGcmid 28780 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-oadd 8510 df-er 8745 df-map 8868 df-pm 8869 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-dju 9941 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-xnn0 12600 df-z 12614 df-uz 12879 df-fz 13548 df-fzo 13695 df-hash 14370 df-word 14553 df-concat 14609 df-s1 14634 df-s2 14887 df-s3 14888 df-trkgc 28456 df-trkgb 28457 df-trkgcb 28458 df-trkgld 28460 df-trkg 28461 df-cgrg 28519 df-leg 28591 df-mir 28661 df-rag 28702 df-perpg 28704 df-mid 28782 |
| This theorem is referenced by: midbtwn 28787 midcgr 28788 midcom 28790 mirmid 28791 lmieu 28792 lmimid 28802 lmiisolem 28804 hypcgrlem1 28807 hypcgrlem2 28808 hypcgr 28809 trgcopyeulem 28813 |
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