Theorem lmif 26571

Description: Line mirror as a function. (Contributed by Thierry Arnoux, 11-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)
lmif.m	⊢ 𝑀 = ((lInvG‘𝐺)‘𝐷)
lmif.l	⊢ 𝐿 = (LineG‘𝐺)
lmif.d	⊢ (𝜑 → 𝐷 ∈ ran 𝐿)

Assertion

Ref	Expression
lmif	⊢ (𝜑 → 𝑀:𝑃⟶𝑃)

Proof of Theorem lmif

Dummy variables 𝑎 𝑏 𝑔 𝑑 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lmif.m	. . 3 ⊢ 𝑀 = ((lInvG‘𝐺)‘𝐷)
2		df-lmi 26561	. . . . 5 ⊢ lInvG = (𝑔 ∈ V ↦ (𝑑 ∈ ran (LineG‘𝑔) ↦ (𝑎 ∈ (Base‘𝑔) ↦ (℩𝑏 ∈ (Base‘𝑔)((𝑎(midG‘𝑔)𝑏) ∈ 𝑑 ∧ (𝑑(⟂G‘𝑔)(𝑎(LineG‘𝑔)𝑏) ∨ 𝑎 = 𝑏))))))
3		fveq2 6670	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (LineG‘𝑔) = (LineG‘𝐺))
4		lmif.l	. . . . . . . 8 ⊢ 𝐿 = (LineG‘𝐺)
5	3, 4	syl6eqr 2874	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (LineG‘𝑔) = 𝐿)
6	5	rneqd 5808	. . . . . 6 ⊢ (𝑔 = 𝐺 → ran (LineG‘𝑔) = ran 𝐿)
7		fveq2 6670	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
8		ismid.p	. . . . . . . 8 ⊢ 𝑃 = (Base‘𝐺)
9	7, 8	syl6eqr 2874	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝑃)
10		fveq2 6670	. . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → (midG‘𝑔) = (midG‘𝐺))
11	10	oveqd 7173	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (𝑎(midG‘𝑔)𝑏) = (𝑎(midG‘𝐺)𝑏))
12	11	eleq1d 2897	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → ((𝑎(midG‘𝑔)𝑏) ∈ 𝑑 ↔ (𝑎(midG‘𝐺)𝑏) ∈ 𝑑))
13		eqidd 2822	. . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → 𝑑 = 𝑑)
14		fveq2 6670	. . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → (⟂G‘𝑔) = (⟂G‘𝐺))
15	5	oveqd 7173	. . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → (𝑎(LineG‘𝑔)𝑏) = (𝑎𝐿𝑏))
16	13, 14, 15	breq123d 5080	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (𝑑(⟂G‘𝑔)(𝑎(LineG‘𝑔)𝑏) ↔ 𝑑(⟂G‘𝐺)(𝑎𝐿𝑏)))
17	16	orbi1d 913	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → ((𝑑(⟂G‘𝑔)(𝑎(LineG‘𝑔)𝑏) ∨ 𝑎 = 𝑏) ↔ (𝑑(⟂G‘𝐺)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏)))
18	12, 17	anbi12d 632	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (((𝑎(midG‘𝑔)𝑏) ∈ 𝑑 ∧ (𝑑(⟂G‘𝑔)(𝑎(LineG‘𝑔)𝑏) ∨ 𝑎 = 𝑏)) ↔ ((𝑎(midG‘𝐺)𝑏) ∈ 𝑑 ∧ (𝑑(⟂G‘𝐺)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏))))
19	9, 18	riotaeqbidv 7117	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (℩𝑏 ∈ (Base‘𝑔)((𝑎(midG‘𝑔)𝑏) ∈ 𝑑 ∧ (𝑑(⟂G‘𝑔)(𝑎(LineG‘𝑔)𝑏) ∨ 𝑎 = 𝑏))) = (℩𝑏 ∈ 𝑃 ((𝑎(midG‘𝐺)𝑏) ∈ 𝑑 ∧ (𝑑(⟂G‘𝐺)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏))))
20	9, 19	mpteq12dv 5151	. . . . . 6 ⊢ (𝑔 = 𝐺 → (𝑎 ∈ (Base‘𝑔) ↦ (℩𝑏 ∈ (Base‘𝑔)((𝑎(midG‘𝑔)𝑏) ∈ 𝑑 ∧ (𝑑(⟂G‘𝑔)(𝑎(LineG‘𝑔)𝑏) ∨ 𝑎 = 𝑏)))) = (𝑎 ∈ 𝑃 ↦ (℩𝑏 ∈ 𝑃 ((𝑎(midG‘𝐺)𝑏) ∈ 𝑑 ∧ (𝑑(⟂G‘𝐺)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏)))))
21	6, 20	mpteq12dv 5151	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑑 ∈ ran (LineG‘𝑔) ↦ (𝑎 ∈ (Base‘𝑔) ↦ (℩𝑏 ∈ (Base‘𝑔)((𝑎(midG‘𝑔)𝑏) ∈ 𝑑 ∧ (𝑑(⟂G‘𝑔)(𝑎(LineG‘𝑔)𝑏) ∨ 𝑎 = 𝑏))))) = (𝑑 ∈ ran 𝐿 ↦ (𝑎 ∈ 𝑃 ↦ (℩𝑏 ∈ 𝑃 ((𝑎(midG‘𝐺)𝑏) ∈ 𝑑 ∧ (𝑑(⟂G‘𝐺)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏))))))
22		ismid.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
23	22	elexd 3514	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ V)
24	4	fvexi 6684	. . . . . . 7 ⊢ 𝐿 ∈ V
25		rnexg 7614	. . . . . . 7 ⊢ (𝐿 ∈ V → ran 𝐿 ∈ V)
26		mptexg 6984	. . . . . . 7 ⊢ (ran 𝐿 ∈ V → (𝑑 ∈ ran 𝐿 ↦ (𝑎 ∈ 𝑃 ↦ (℩𝑏 ∈ 𝑃 ((𝑎(midG‘𝐺)𝑏) ∈ 𝑑 ∧ (𝑑(⟂G‘𝐺)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏))))) ∈ V)
27	24, 25, 26	mp2b 10	. . . . . 6 ⊢ (𝑑 ∈ ran 𝐿 ↦ (𝑎 ∈ 𝑃 ↦ (℩𝑏 ∈ 𝑃 ((𝑎(midG‘𝐺)𝑏) ∈ 𝑑 ∧ (𝑑(⟂G‘𝐺)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏))))) ∈ V
28	27	a1i 11	. . . . 5 ⊢ (𝜑 → (𝑑 ∈ ran 𝐿 ↦ (𝑎 ∈ 𝑃 ↦ (℩𝑏 ∈ 𝑃 ((𝑎(midG‘𝐺)𝑏) ∈ 𝑑 ∧ (𝑑(⟂G‘𝐺)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏))))) ∈ V)
29	2, 21, 23, 28	fvmptd3 6791	. . . 4 ⊢ (𝜑 → (lInvG‘𝐺) = (𝑑 ∈ ran 𝐿 ↦ (𝑎 ∈ 𝑃 ↦ (℩𝑏 ∈ 𝑃 ((𝑎(midG‘𝐺)𝑏) ∈ 𝑑 ∧ (𝑑(⟂G‘𝐺)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏))))))
30		eleq2 2901	. . . . . . . 8 ⊢ (𝑑 = 𝐷 → ((𝑎(midG‘𝐺)𝑏) ∈ 𝑑 ↔ (𝑎(midG‘𝐺)𝑏) ∈ 𝐷))
31		breq1 5069	. . . . . . . . 9 ⊢ (𝑑 = 𝐷 → (𝑑(⟂G‘𝐺)(𝑎𝐿𝑏) ↔ 𝐷(⟂G‘𝐺)(𝑎𝐿𝑏)))
32	31	orbi1d 913	. . . . . . . 8 ⊢ (𝑑 = 𝐷 → ((𝑑(⟂G‘𝐺)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏) ↔ (𝐷(⟂G‘𝐺)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏)))
33	30, 32	anbi12d 632	. . . . . . 7 ⊢ (𝑑 = 𝐷 → (((𝑎(midG‘𝐺)𝑏) ∈ 𝑑 ∧ (𝑑(⟂G‘𝐺)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏)) ↔ ((𝑎(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏))))
34	33	riotabidv 7116	. . . . . 6 ⊢ (𝑑 = 𝐷 → (℩𝑏 ∈ 𝑃 ((𝑎(midG‘𝐺)𝑏) ∈ 𝑑 ∧ (𝑑(⟂G‘𝐺)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏))) = (℩𝑏 ∈ 𝑃 ((𝑎(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏))))
35	34	mpteq2dv 5162	. . . . 5 ⊢ (𝑑 = 𝐷 → (𝑎 ∈ 𝑃 ↦ (℩𝑏 ∈ 𝑃 ((𝑎(midG‘𝐺)𝑏) ∈ 𝑑 ∧ (𝑑(⟂G‘𝐺)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏)))) = (𝑎 ∈ 𝑃 ↦ (℩𝑏 ∈ 𝑃 ((𝑎(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏)))))
36	35	adantl 484	. . . 4 ⊢ ((𝜑 ∧ 𝑑 = 𝐷) → (𝑎 ∈ 𝑃 ↦ (℩𝑏 ∈ 𝑃 ((𝑎(midG‘𝐺)𝑏) ∈ 𝑑 ∧ (𝑑(⟂G‘𝐺)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏)))) = (𝑎 ∈ 𝑃 ↦ (℩𝑏 ∈ 𝑃 ((𝑎(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏)))))
37		lmif.d	. . . 4 ⊢ (𝜑 → 𝐷 ∈ ran 𝐿)
38	8	fvexi 6684	. . . . . 6 ⊢ 𝑃 ∈ V
39	38	mptex 6986	. . . . 5 ⊢ (𝑎 ∈ 𝑃 ↦ (℩𝑏 ∈ 𝑃 ((𝑎(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏)))) ∈ V
40	39	a1i 11	. . . 4 ⊢ (𝜑 → (𝑎 ∈ 𝑃 ↦ (℩𝑏 ∈ 𝑃 ((𝑎(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏)))) ∈ V)
41	29, 36, 37, 40	fvmptd 6775	. . 3 ⊢ (𝜑 → ((lInvG‘𝐺)‘𝐷) = (𝑎 ∈ 𝑃 ↦ (℩𝑏 ∈ 𝑃 ((𝑎(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏)))))
42	1, 41	syl5eq 2868	. 2 ⊢ (𝜑 → 𝑀 = (𝑎 ∈ 𝑃 ↦ (℩𝑏 ∈ 𝑃 ((𝑎(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏)))))
43		ismid.d	. . . 4 ⊢ − = (dist‘𝐺)
44		ismid.i	. . . 4 ⊢ 𝐼 = (Itv‘𝐺)
45	22	adantr 483	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑃) → 𝐺 ∈ TarskiG)
46		ismid.1	. . . . 5 ⊢ (𝜑 → 𝐺DimTarskiG≥2)
47	46	adantr 483	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑃) → 𝐺DimTarskiG≥2)
48	37	adantr 483	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑃) → 𝐷 ∈ ran 𝐿)
49		simpr 487	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑃) → 𝑎 ∈ 𝑃)
50	8, 43, 44, 45, 47, 4, 48, 49	lmieu 26570	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑃) → ∃!𝑏 ∈ 𝑃 ((𝑎(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏)))
51		riotacl 7131	. . 3 ⊢ (∃!𝑏 ∈ 𝑃 ((𝑎(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏)) → (℩𝑏 ∈ 𝑃 ((𝑎(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏))) ∈ 𝑃)
52	50, 51	syl 17	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑃) → (℩𝑏 ∈ 𝑃 ((𝑎(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏))) ∈ 𝑃)
53	42, 52	fmpt3d 6880	1 ⊢ (𝜑 → 𝑀:𝑃⟶𝑃)

Syntax hints:   → wi 4   ∧ wa 398   ∨ wo 843   = wceq 1537   ∈ wcel 2114  ∃!wreu 3140  Vcvv 3494   class class class wbr 5066   ↦ cmpt 5146  ran crn 5556  ⟶wf 6351  ‘cfv 6355  ℩crio 7113  (class class class)co 7156  2c2 11693  Basecbs 16483  distcds 16574  TarskiGcstrkg 26216  Dim_TarskiG≥cstrkgld 26220  Itvcitv 26222  LineGclng 26223  ⟂Gcperpg 26481  midGcmid 26558  lInvGclmi 26559

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 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-cnex 10593  ax-resscn 10594  ax-1cn 10595  ax-icn 10596  ax-addcl 10597  ax-addrcl 10598  ax-mulcl 10599  ax-mulrcl 10600  ax-mulcom 10601  ax-addass 10602  ax-mulass 10603  ax-distr 10604  ax-i2m1 10605  ax-1ne0 10606  ax-1rid 10607  ax-rnegex 10608  ax-rrecex 10609  ax-cnre 10610  ax-pre-lttri 10611  ax-pre-lttrn 10612  ax-pre-ltadd 10613  ax-pre-mulgt0 10614

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 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-int 4877  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7581  df-1st 7689  df-2nd 7690  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-1o 8102  df-oadd 8106  df-er 8289  df-map 8408  df-pm 8409  df-en 8510  df-dom 8511  df-sdom 8512  df-fin 8513  df-dju 9330  df-card 9368  df-pnf 10677  df-mnf 10678  df-xr 10679  df-ltxr 10680  df-le 10681  df-sub 10872  df-neg 10873  df-nn 11639  df-2 11701  df-3 11702  df-n0 11899  df-xnn0 11969  df-z 11983  df-uz 12245  df-fz 12894  df-fzo 13035  df-hash 13692  df-word 13863  df-concat 13923  df-s1 13950  df-s2 14210  df-s3 14211  df-trkgc 26234  df-trkgb 26235  df-trkgcb 26236  df-trkgld 26238  df-trkg 26239  df-cgrg 26297  df-leg 26369  df-mir 26439  df-rag 26480  df-perpg 26482  df-mid 26560  df-lmi 26561

This theorem is referenced by:  lmicl  26572  lmif1o  26581