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Theorem ismidb 25570
Description: Property of the midpoint. (Contributed by Thierry Arnoux, 1-Dec-2019.)
Hypotheses
Ref Expression
ismid.p 𝑃 = (Base‘𝐺)
ismid.d = (dist‘𝐺)
ismid.i 𝐼 = (Itv‘𝐺)
ismid.g (𝜑𝐺 ∈ TarskiG)
ismid.1 (𝜑𝐺DimTarskiG≥2)
midcl.1 (𝜑𝐴𝑃)
midcl.2 (𝜑𝐵𝑃)
ismidb.s 𝑆 = (pInvG‘𝐺)
ismidb.m (𝜑𝑀𝑃)
Assertion
Ref Expression
ismidb (𝜑 → (𝐵 = ((𝑆𝑀)‘𝐴) ↔ (𝐴(midG‘𝐺)𝐵) = 𝑀))

Proof of Theorem ismidb
Dummy variables 𝑚 𝑎 𝑏 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ismidb.m . . 3 (𝜑𝑀𝑃)
2 ismid.p . . . 4 𝑃 = (Base‘𝐺)
3 ismid.d . . . 4 = (dist‘𝐺)
4 ismid.i . . . 4 𝐼 = (Itv‘𝐺)
5 eqid 2621 . . . 4 (LineG‘𝐺) = (LineG‘𝐺)
6 ismid.g . . . 4 (𝜑𝐺 ∈ TarskiG)
7 ismidb.s . . . 4 𝑆 = (pInvG‘𝐺)
8 midcl.1 . . . 4 (𝜑𝐴𝑃)
9 midcl.2 . . . 4 (𝜑𝐵𝑃)
10 ismid.1 . . . 4 (𝜑𝐺DimTarskiG≥2)
112, 3, 4, 5, 6, 7, 8, 9, 10mideu 25530 . . 3 (𝜑 → ∃!𝑚𝑃 𝐵 = ((𝑆𝑚)‘𝐴))
12 fveq2 6148 . . . . . 6 (𝑚 = 𝑀 → (𝑆𝑚) = (𝑆𝑀))
1312fveq1d 6150 . . . . 5 (𝑚 = 𝑀 → ((𝑆𝑚)‘𝐴) = ((𝑆𝑀)‘𝐴))
1413eqeq2d 2631 . . . 4 (𝑚 = 𝑀 → (𝐵 = ((𝑆𝑚)‘𝐴) ↔ 𝐵 = ((𝑆𝑀)‘𝐴)))
1514riota2 6587 . . 3 ((𝑀𝑃 ∧ ∃!𝑚𝑃 𝐵 = ((𝑆𝑚)‘𝐴)) → (𝐵 = ((𝑆𝑀)‘𝐴) ↔ (𝑚𝑃 𝐵 = ((𝑆𝑚)‘𝐴)) = 𝑀))
161, 11, 15syl2anc 692 . 2 (𝜑 → (𝐵 = ((𝑆𝑀)‘𝐴) ↔ (𝑚𝑃 𝐵 = ((𝑆𝑚)‘𝐴)) = 𝑀))
17 df-mid 25566 . . . . . 6 midG = (𝑔 ∈ V ↦ (𝑎 ∈ (Base‘𝑔), 𝑏 ∈ (Base‘𝑔) ↦ (𝑚 ∈ (Base‘𝑔)𝑏 = (((pInvG‘𝑔)‘𝑚)‘𝑎))))
1817a1i 11 . . . . 5 (𝜑 → midG = (𝑔 ∈ V ↦ (𝑎 ∈ (Base‘𝑔), 𝑏 ∈ (Base‘𝑔) ↦ (𝑚 ∈ (Base‘𝑔)𝑏 = (((pInvG‘𝑔)‘𝑚)‘𝑎)))))
19 fveq2 6148 . . . . . . . 8 (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
2019, 2syl6eqr 2673 . . . . . . 7 (𝑔 = 𝐺 → (Base‘𝑔) = 𝑃)
21 fveq2 6148 . . . . . . . . . . . 12 (𝑔 = 𝐺 → (pInvG‘𝑔) = (pInvG‘𝐺))
2221, 7syl6eqr 2673 . . . . . . . . . . 11 (𝑔 = 𝐺 → (pInvG‘𝑔) = 𝑆)
2322fveq1d 6150 . . . . . . . . . 10 (𝑔 = 𝐺 → ((pInvG‘𝑔)‘𝑚) = (𝑆𝑚))
2423fveq1d 6150 . . . . . . . . 9 (𝑔 = 𝐺 → (((pInvG‘𝑔)‘𝑚)‘𝑎) = ((𝑆𝑚)‘𝑎))
2524eqeq2d 2631 . . . . . . . 8 (𝑔 = 𝐺 → (𝑏 = (((pInvG‘𝑔)‘𝑚)‘𝑎) ↔ 𝑏 = ((𝑆𝑚)‘𝑎)))
2620, 25riotaeqbidv 6568 . . . . . . 7 (𝑔 = 𝐺 → (𝑚 ∈ (Base‘𝑔)𝑏 = (((pInvG‘𝑔)‘𝑚)‘𝑎)) = (𝑚𝑃 𝑏 = ((𝑆𝑚)‘𝑎)))
2720, 20, 26mpt2eq123dv 6670 . . . . . 6 (𝑔 = 𝐺 → (𝑎 ∈ (Base‘𝑔), 𝑏 ∈ (Base‘𝑔) ↦ (𝑚 ∈ (Base‘𝑔)𝑏 = (((pInvG‘𝑔)‘𝑚)‘𝑎))) = (𝑎𝑃, 𝑏𝑃 ↦ (𝑚𝑃 𝑏 = ((𝑆𝑚)‘𝑎))))
2827adantl 482 . . . . 5 ((𝜑𝑔 = 𝐺) → (𝑎 ∈ (Base‘𝑔), 𝑏 ∈ (Base‘𝑔) ↦ (𝑚 ∈ (Base‘𝑔)𝑏 = (((pInvG‘𝑔)‘𝑚)‘𝑎))) = (𝑎𝑃, 𝑏𝑃 ↦ (𝑚𝑃 𝑏 = ((𝑆𝑚)‘𝑎))))
29 elex 3198 . . . . . 6 (𝐺 ∈ TarskiG → 𝐺 ∈ V)
306, 29syl 17 . . . . 5 (𝜑𝐺 ∈ V)
31 fvex 6158 . . . . . . . 8 (Base‘𝐺) ∈ V
322, 31eqeltri 2694 . . . . . . 7 𝑃 ∈ V
3332, 32mpt2ex 7192 . . . . . 6 (𝑎𝑃, 𝑏𝑃 ↦ (𝑚𝑃 𝑏 = ((𝑆𝑚)‘𝑎))) ∈ V
3433a1i 11 . . . . 5 (𝜑 → (𝑎𝑃, 𝑏𝑃 ↦ (𝑚𝑃 𝑏 = ((𝑆𝑚)‘𝑎))) ∈ V)
3518, 28, 30, 34fvmptd 6245 . . . 4 (𝜑 → (midG‘𝐺) = (𝑎𝑃, 𝑏𝑃 ↦ (𝑚𝑃 𝑏 = ((𝑆𝑚)‘𝑎))))
36 simprr 795 . . . . . 6 ((𝜑 ∧ (𝑎 = 𝐴𝑏 = 𝐵)) → 𝑏 = 𝐵)
37 simprl 793 . . . . . . 7 ((𝜑 ∧ (𝑎 = 𝐴𝑏 = 𝐵)) → 𝑎 = 𝐴)
3837fveq2d 6152 . . . . . 6 ((𝜑 ∧ (𝑎 = 𝐴𝑏 = 𝐵)) → ((𝑆𝑚)‘𝑎) = ((𝑆𝑚)‘𝐴))
3936, 38eqeq12d 2636 . . . . 5 ((𝜑 ∧ (𝑎 = 𝐴𝑏 = 𝐵)) → (𝑏 = ((𝑆𝑚)‘𝑎) ↔ 𝐵 = ((𝑆𝑚)‘𝐴)))
4039riotabidv 6567 . . . 4 ((𝜑 ∧ (𝑎 = 𝐴𝑏 = 𝐵)) → (𝑚𝑃 𝑏 = ((𝑆𝑚)‘𝑎)) = (𝑚𝑃 𝐵 = ((𝑆𝑚)‘𝐴)))
41 riotacl 6579 . . . . 5 (∃!𝑚𝑃 𝐵 = ((𝑆𝑚)‘𝐴) → (𝑚𝑃 𝐵 = ((𝑆𝑚)‘𝐴)) ∈ 𝑃)
4211, 41syl 17 . . . 4 (𝜑 → (𝑚𝑃 𝐵 = ((𝑆𝑚)‘𝐴)) ∈ 𝑃)
4335, 40, 8, 9, 42ovmpt2d 6741 . . 3 (𝜑 → (𝐴(midG‘𝐺)𝐵) = (𝑚𝑃 𝐵 = ((𝑆𝑚)‘𝐴)))
4443eqeq1d 2623 . 2 (𝜑 → ((𝐴(midG‘𝐺)𝐵) = 𝑀 ↔ (𝑚𝑃 𝐵 = ((𝑆𝑚)‘𝐴)) = 𝑀))
4516, 44bitr4d 271 1 (𝜑 → (𝐵 = ((𝑆𝑀)‘𝐴) ↔ (𝐴(midG‘𝐺)𝐵) = 𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  ∃!wreu 2909  Vcvv 3186   class class class wbr 4613  cmpt 4673  cfv 5847  crio 6564  (class class class)co 6604  cmpt2 6606  2c2 11014  Basecbs 15781  distcds 15871  TarskiGcstrkg 25229  DimTarskiGcstrkgld 25233  Itvcitv 25235  LineGclng 25236  pInvGcmir 25447  midGcmid 25564
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oadd 7509  df-er 7687  df-map 7804  df-pm 7805  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-card 8709  df-cda 8934  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-2 11023  df-3 11024  df-n0 11237  df-xnn0 11308  df-z 11322  df-uz 11632  df-fz 12269  df-fzo 12407  df-hash 13058  df-word 13238  df-concat 13240  df-s1 13241  df-s2 13530  df-s3 13531  df-trkgc 25247  df-trkgb 25248  df-trkgcb 25249  df-trkgld 25251  df-trkg 25252  df-cgrg 25306  df-leg 25378  df-mir 25448  df-rag 25489  df-perpg 25491  df-mid 25566
This theorem is referenced by:  midbtwn  25571  midcgr  25572  midcom  25574  mirmid  25575  lmieu  25576  lmimid  25586  lmiisolem  25588  hypcgrlem1  25591  hypcgrlem2  25592  hypcgr  25593  trgcopyeulem  25597
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