Theorem lmif 28960

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 28950	. . . . 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 6869	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (LineG‘𝑔) = (LineG‘𝐺))
4		lmif.l	. . . . . . . 8 ⊢ 𝐿 = (LineG‘𝐺)
5	3, 4	eqtr4di 2817	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (LineG‘𝑔) = 𝐿)
6	5	rneqd 5916	. . . . . 6 ⊢ (𝑔 = 𝐺 → ran (LineG‘𝑔) = ran 𝐿)
7		fveq2 6869	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
8		ismid.p	. . . . . . . 8 ⊢ 𝑃 = (Base‘𝐺)
9	7, 8	eqtr4di 2817	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝑃)
10		fveq2 6869	. . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → (midG‘𝑔) = (midG‘𝐺))
11	10	oveqd 7415	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (𝑎(midG‘𝑔)𝑏) = (𝑎(midG‘𝐺)𝑏))
12	11	eleq1d 2849	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → ((𝑎(midG‘𝑔)𝑏) ∈ 𝑑 ↔ (𝑎(midG‘𝐺)𝑏) ∈ 𝑑))
13		eqidd 2765	. . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → 𝑑 = 𝑑)
14		fveq2 6869	. . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → (⟂G‘𝑔) = (⟂G‘𝐺))
15	5	oveqd 7415	. . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → (𝑎(LineG‘𝑔)𝑏) = (𝑎𝐿𝑏))
16	13, 14, 15	breq123d 5116	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (𝑑(⟂G‘𝑔)(𝑎(LineG‘𝑔)𝑏) ↔ 𝑑(⟂G‘𝐺)(𝑎𝐿𝑏)))
17	16	orbi1d 927	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → ((𝑑(⟂G‘𝑔)(𝑎(LineG‘𝑔)𝑏) ∨ 𝑎 = 𝑏) ↔ (𝑑(⟂G‘𝐺)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏)))
18	12, 17	anbi12d 641	. . . . . . . 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 7358	. . . . . . 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 5189	. . . . . 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 5189	. . . . 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 3479	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ V)
24	4	fvexi 6883	. . . . . . 7 ⊢ 𝐿 ∈ V
25		rnexg 7885	. . . . . . 7 ⊢ (𝐿 ∈ V → ran 𝐿 ∈ V)
26		mptexg 7207	. . . . . . 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 7001	. . . 4 ⊢ (𝜑 → (lInvG‘𝐺) = (𝑑 ∈ ran 𝐿 ↦ (𝑎 ∈ 𝑃 ↦ (℩𝑏 ∈ 𝑃 ((𝑎(midG‘𝐺)𝑏) ∈ 𝑑 ∧ (𝑑(⟂G‘𝐺)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏))))))
30		eleq2 2853	. . . . . . . 8 ⊢ (𝑑 = 𝐷 → ((𝑎(midG‘𝐺)𝑏) ∈ 𝑑 ↔ (𝑎(midG‘𝐺)𝑏) ∈ 𝐷))
31		breq1 5105	. . . . . . . . 9 ⊢ (𝑑 = 𝐷 → (𝑑(⟂G‘𝐺)(𝑎𝐿𝑏) ↔ 𝐷(⟂G‘𝐺)(𝑎𝐿𝑏)))
32	31	orbi1d 927	. . . . . . . 8 ⊢ (𝑑 = 𝐷 → ((𝑑(⟂G‘𝐺)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏) ↔ (𝐷(⟂G‘𝐺)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏)))
33	30, 32	anbi12d 641	. . . . . . 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 7357	. . . . . 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 5196	. . . . 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 485	. . . 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 6883	. . . . . 6 ⊢ 𝑃 ∈ V
39	38	mptex 7209	. . . . 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 6985	. . 3 ⊢ (𝜑 → ((lInvG‘𝐺)‘𝐷) = (𝑎 ∈ 𝑃 ↦ (℩𝑏 ∈ 𝑃 ((𝑎(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏)))))
42	1, 41	eqtrid 2811	. 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 484	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑃) → 𝐺 ∈ TarskiG)
46		ismid.1	. . . . 5 ⊢ (𝜑 → 𝐺DimTarskiG≥2)
47	46	adantr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑃) → 𝐺DimTarskiG≥2)
48	37	adantr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑃) → 𝐷 ∈ ran 𝐿)
49		simpr 488	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑃) → 𝑎 ∈ 𝑃)
50	8, 43, 44, 45, 47, 4, 48, 49	lmieu 28959	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑃) → ∃!𝑏 ∈ 𝑃 ((𝑎(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏)))
51		riotacl 7372	. . 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 7099	1 ⊢ (𝜑 → 𝑀:𝑃⟶𝑃)

Syntax hints:   → wi 4   ∧ wa 399   ∨ wo 858   = wceq 1562   ∈ wcel 2144  ∃!wreu 3367  Vcvv 3456   class class class wbr 5102   ↦ cmpt 5183  ran crn 5650  ⟶wf 6519  ‘cfv 6523  ℩crio 7354  (class class class)co 7398  2c2 12274  Basecbs 17247  distcds 17297  TarskiGcstrkg 28598  Dim_TarskiG≥cstrkgld 28602  Itvcitv 28604  LineGclng 28605  ⟂Gcperpg 28870  midGcmid 28947  lInvGclmi 28948

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-rep 5229  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392  ax-un 7720  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5544  df-eprel 5549  df-po 5557  df-so 5558  df-fr 5602  df-we 5604  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-pred 6290  df-ord 6351  df-on 6352  df-lim 6353  df-suc 6354  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-riota 7355  df-ov 7401  df-oprab 7402  df-mpo 7403  df-om 7849  df-1st 7972  df-2nd 7973  df-frecs 8264  df-wrecs 8295  df-recs 8344  df-rdg 8383  df-1o 8439  df-oadd 8443  df-er 8680  df-map 8812  df-pm 8813  df-en 8930  df-dom 8931  df-sdom 8932  df-fin 8933  df-dju 9861  df-card 9899  df-pnf 11220  df-mnf 11221  df-xr 11222  df-ltxr 11223  df-le 11224  df-sub 11418  df-neg 11419  df-nn 12213  df-2 12282  df-3 12283  df-n0 12484  df-xnn0 12557  df-z 12571  df-uz 12842  df-fz 13515  df-fzo 13662  df-hash 14346  df-word 14529  df-concat 14586  df-s1 14612  df-s2 14863  df-s3 14864  df-trkgc 28619  df-trkgb 28620  df-trkgcb 28621  df-trkgld 28623  df-trkg 28624  df-cgrg 28682  df-leg 28754  df-mir 28828  df-rag 28869  df-perpg 28871  df-mid 28949  df-lmi 28950

This theorem is referenced by:  lmicl  28961  lmif1o  28970