Theorem ismidb 28751

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	⊢ (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)

Assertion

Ref	Expression
ismidb	⊢ (𝜑 → (𝐵 = ((𝑆‘𝑀)‘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.

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 2731	. . . 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 28711	. . 3 ⊢ (𝜑 → ∃!𝑚 ∈ 𝑃 𝐵 = ((𝑆‘𝑚)‘𝐴))
12		fveq2 6817	. . . . . 6 ⊢ (𝑚 = 𝑀 → (𝑆‘𝑚) = (𝑆‘𝑀))
13	12	fveq1d 6819	. . . . 5 ⊢ (𝑚 = 𝑀 → ((𝑆‘𝑚)‘𝐴) = ((𝑆‘𝑀)‘𝐴))
14	13	eqeq2d 2742	. . . 4 ⊢ (𝑚 = 𝑀 → (𝐵 = ((𝑆‘𝑚)‘𝐴) ↔ 𝐵 = ((𝑆‘𝑀)‘𝐴)))
15	14	riota2 7323	. . 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 28747	. . . . 5 ⊢ midG = (𝑔 ∈ V ↦ (𝑎 ∈ (Base‘𝑔), 𝑏 ∈ (Base‘𝑔) ↦ (℩𝑚 ∈ (Base‘𝑔)𝑏 = (((pInvG‘𝑔)‘𝑚)‘𝑎))))
18		fveq2 6817	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
19	18, 2	eqtr4di 2784	. . . . . 6 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝑃)
20		fveq2 6817	. . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → (pInvG‘𝑔) = (pInvG‘𝐺))
21	20, 7	eqtr4di 2784	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (pInvG‘𝑔) = 𝑆)
22	21	fveq1d 6819	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → ((pInvG‘𝑔)‘𝑚) = (𝑆‘𝑚))
23	22	fveq1d 6819	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (((pInvG‘𝑔)‘𝑚)‘𝑎) = ((𝑆‘𝑚)‘𝑎))
24	23	eqeq2d 2742	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (𝑏 = (((pInvG‘𝑔)‘𝑚)‘𝑎) ↔ 𝑏 = ((𝑆‘𝑚)‘𝑎)))
25	19, 24	riotaeqbidv 7301	. . . . . 6 ⊢ (𝑔 = 𝐺 → (℩𝑚 ∈ (Base‘𝑔)𝑏 = (((pInvG‘𝑔)‘𝑚)‘𝑎)) = (℩𝑚 ∈ 𝑃 𝑏 = ((𝑆‘𝑚)‘𝑎)))
26	19, 19, 25	mpoeq123dv 7416	. . . . 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 3460	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ V)
28	2	fvexi 6831	. . . . . . 7 ⊢ 𝑃 ∈ V
29	28, 28	mpoex 8006	. . . . . 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 6947	. . . 4 ⊢ (𝜑 → (midG‘𝐺) = (𝑎 ∈ 𝑃, 𝑏 ∈ 𝑃 ↦ (℩𝑚 ∈ 𝑃 𝑏 = ((𝑆‘𝑚)‘𝑎))))
32		simprr 772	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 = 𝐴 ∧ 𝑏 = 𝐵)) → 𝑏 = 𝐵)
33		simprl 770	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 = 𝐴 ∧ 𝑏 = 𝐵)) → 𝑎 = 𝐴)
34	33	fveq2d 6821	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 = 𝐴 ∧ 𝑏 = 𝐵)) → ((𝑆‘𝑚)‘𝑎) = ((𝑆‘𝑚)‘𝐴))
35	32, 34	eqeq12d 2747	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 = 𝐴 ∧ 𝑏 = 𝐵)) → (𝑏 = ((𝑆‘𝑚)‘𝑎) ↔ 𝐵 = ((𝑆‘𝑚)‘𝐴)))
36	35	riotabidv 7300	. . . 4 ⊢ ((𝜑 ∧ (𝑎 = 𝐴 ∧ 𝑏 = 𝐵)) → (℩𝑚 ∈ 𝑃 𝑏 = ((𝑆‘𝑚)‘𝑎)) = (℩𝑚 ∈ 𝑃 𝐵 = ((𝑆‘𝑚)‘𝐴)))
37		riotacl 7315	. . . . 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 7493	. . 3 ⊢ (𝜑 → (𝐴(midG‘𝐺)𝐵) = (℩𝑚 ∈ 𝑃 𝐵 = ((𝑆‘𝑚)‘𝐴)))
40	39	eqeq1d 2733	. 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))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∃!wreu 3344  Vcvv 3436   class class class wbr 5086  ‘cfv 6476  ℩crio 7297  (class class class)co 7341   ∈ cmpo 7343  2c2 12175  Basecbs 17115  distcds 17165  TarskiGcstrkg 28400  Dim_TarskiG≥cstrkgld 28404  Itvcitv 28406  LineGclng 28407  pInvGcmir 28625  midGcmid 28745

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 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5212  ax-sep 5229  ax-nul 5239  ax-pow 5298  ax-pr 5365  ax-un 7663  ax-cnex 11057  ax-resscn 11058  ax-1cn 11059  ax-icn 11060  ax-addcl 11061  ax-addrcl 11062  ax-mulcl 11063  ax-mulrcl 11064  ax-mulcom 11065  ax-addass 11066  ax-mulass 11067  ax-distr 11068  ax-i2m1 11069  ax-1ne0 11070  ax-1rid 11071  ax-rnegex 11072  ax-rrecex 11073  ax-cnre 11074  ax-pre-lttri 11075  ax-pre-lttrn 11076  ax-pre-ltadd 11077  ax-pre-mulgt0 11078

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-tp 4576  df-op 4578  df-uni 4855  df-int 4893  df-iun 4938  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5506  df-eprel 5511  df-po 5519  df-so 5520  df-fr 5564  df-we 5566  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-pred 6243  df-ord 6304  df-on 6305  df-lim 6306  df-suc 6307  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-riota 7298  df-ov 7344  df-oprab 7345  df-mpo 7346  df-om 7792  df-1st 7916  df-2nd 7917  df-frecs 8206  df-wrecs 8237  df-recs 8286  df-rdg 8324  df-1o 8380  df-oadd 8384  df-er 8617  df-map 8747  df-pm 8748  df-en 8865  df-dom 8866  df-sdom 8867  df-fin 8868  df-dju 9789  df-card 9827  df-pnf 11143  df-mnf 11144  df-xr 11145  df-ltxr 11146  df-le 11147  df-sub 11341  df-neg 11342  df-nn 12121  df-2 12183  df-3 12184  df-n0 12377  df-xnn0 12450  df-z 12464  df-uz 12728  df-fz 13403  df-fzo 13550  df-hash 14233  df-word 14416  df-concat 14473  df-s1 14499  df-s2 14750  df-s3 14751  df-trkgc 28421  df-trkgb 28422  df-trkgcb 28423  df-trkgld 28425  df-trkg 28426  df-cgrg 28484  df-leg 28556  df-mir 28626  df-rag 28667  df-perpg 28669  df-mid 28747

This theorem is referenced by:  midbtwn  28752  midcgr  28753  midcom  28755  mirmid  28756  lmieu  28757  lmimid  28767  lmiisolem  28769  hypcgrlem1  28772  hypcgrlem2  28773  hypcgr  28774  trgcopyeulem  28778