Theorem islmib 29274

Description: Property of the line mirror. (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)
lmicl.1	⊢ (𝜑 → 𝐴 ∈ 𝑃)
islmib.b	⊢ (𝜑 → 𝐵 ∈ 𝑃)

Assertion

Ref	Expression
islmib	⊢ (𝜑 → (𝐵 = (𝑀‘𝐴) ↔ ((𝐴(midG‘𝐺)𝐵) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))))

Proof of Theorem islmib

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

Step	Hyp	Ref	Expression
1		lmif.m	. . . . 5 ⊢ 𝑀 = ((lInvG‘𝐺)‘𝐷)
2		df-lmi 29262	. . . . . . 7 ⊢ 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 6877	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (LineG‘𝑔) = (LineG‘𝐺))
4		lmif.l	. . . . . . . . . 10 ⊢ 𝐿 = (LineG‘𝐺)
5	3, 4	eqtr4di 2814	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (LineG‘𝑔) = 𝐿)
6	5	rneqd 5920	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → ran (LineG‘𝑔) = ran 𝐿)
7		fveq2 6877	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
8		ismid.p	. . . . . . . . . 10 ⊢ 𝑃 = (Base‘𝐺)
9	7, 8	eqtr4di 2814	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝑃)
10		fveq2 6877	. . . . . . . . . . . . 13 ⊢ (𝑔 = 𝐺 → (midG‘𝑔) = (midG‘𝐺))
11	10	oveqd 7429	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝐺 → (𝑎(midG‘𝑔)𝑏) = (𝑎(midG‘𝐺)𝑏))
12	11	eleq1d 2846	. . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → ((𝑎(midG‘𝑔)𝑏) ∈ 𝑑 ↔ (𝑎(midG‘𝐺)𝑏) ∈ 𝑑))
13		eqidd 2762	. . . . . . . . . . . . 13 ⊢ (𝑔 = 𝐺 → 𝑑 = 𝑑)
14		fveq2 6877	. . . . . . . . . . . . 13 ⊢ (𝑔 = 𝐺 → (⟂G‘𝑔) = (⟂G‘𝐺))
15	5	oveqd 7429	. . . . . . . . . . . . 13 ⊢ (𝑔 = 𝐺 → (𝑎(LineG‘𝑔)𝑏) = (𝑎𝐿𝑏))
16	13, 14, 15	breq123d 5117	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝐺 → (𝑑(⟂G‘𝑔)(𝑎(LineG‘𝑔)𝑏) ↔ 𝑑(⟂G‘𝐺)(𝑎𝐿𝑏)))
17	16	orbi1d 930	. . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → ((𝑑(⟂G‘𝑔)(𝑎(LineG‘𝑔)𝑏) ∨ 𝑎 = 𝑏) ↔ (𝑑(⟂G‘𝐺)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏)))
18	12, 17	anbi12d 644	. . . . . . . . . 10 ⊢ (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 7372	. . . . . . . . 9 ⊢ (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 5192	. . . . . . . 8 ⊢ (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 5192	. . . . . . 7 ⊢ (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	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
23	22	elexd 3474	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ V)
24	4	fvexi 6891	. . . . . . . . 9 ⊢ 𝐿 ∈ V
25		rnexg 7903	. . . . . . . . 9 ⊢ (𝐿 ∈ V → ran 𝐿 ∈ V)
26		mptexg 7219	. . . . . . . . 9 ⊢ (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	. . . . . . . 8 ⊢ (𝑑 ∈ ran 𝐿 ↦ (𝑎 ∈ 𝑃 ↦ (℩𝑏 ∈ 𝑃 ((𝑎(midG‘𝐺)𝑏) ∈ 𝑑 ∧ (𝑑(⟂G‘𝐺)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏))))) ∈ V
28	27	a1i 11	. . . . . . 7 ⊢ (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 7009	. . . . . 6 ⊢ (𝜑 → (lInvG‘𝐺) = (𝑑 ∈ ran 𝐿 ↦ (𝑎 ∈ 𝑃 ↦ (℩𝑏 ∈ 𝑃 ((𝑎(midG‘𝐺)𝑏) ∈ 𝑑 ∧ (𝑑(⟂G‘𝐺)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏))))))
30		eleq2 2850	. . . . . . . . . 10 ⊢ (𝑑 = 𝐷 → ((𝑎(midG‘𝐺)𝑏) ∈ 𝑑 ↔ (𝑎(midG‘𝐺)𝑏) ∈ 𝐷))
31		breq1 5106	. . . . . . . . . . 11 ⊢ (𝑑 = 𝐷 → (𝑑(⟂G‘𝐺)(𝑎𝐿𝑏) ↔ 𝐷(⟂G‘𝐺)(𝑎𝐿𝑏)))
32	31	orbi1d 930	. . . . . . . . . 10 ⊢ (𝑑 = 𝐷 → ((𝑑(⟂G‘𝐺)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏) ↔ (𝐷(⟂G‘𝐺)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏)))
33	30, 32	anbi12d 644	. . . . . . . . 9 ⊢ (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 7371	. . . . . . . 8 ⊢ (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 5199	. . . . . . 7 ⊢ (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 487	. . . . . 6 ⊢ ((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	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ ran 𝐿)
38	8	fvexi 6891	. . . . . . . 8 ⊢ 𝑃 ∈ V
39	38	mptex 7221	. . . . . . 7 ⊢ (𝑎 ∈ 𝑃 ↦ (℩𝑏 ∈ 𝑃 ((𝑎(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏)))) ∈ V
40	39	a1i 11	. . . . . 6 ⊢ (𝜑 → (𝑎 ∈ 𝑃 ↦ (℩𝑏 ∈ 𝑃 ((𝑎(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏)))) ∈ V)
41	29, 36, 37, 40	fvmptd 6993	. . . . 5 ⊢ (𝜑 → ((lInvG‘𝐺)‘𝐷) = (𝑎 ∈ 𝑃 ↦ (℩𝑏 ∈ 𝑃 ((𝑎(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏)))))
42	1, 41	eqtrid 2808	. . . 4 ⊢ (𝜑 → 𝑀 = (𝑎 ∈ 𝑃 ↦ (℩𝑏 ∈ 𝑃 ((𝑎(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏)))))
43		oveq1 7419	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → (𝑎(midG‘𝐺)𝑏) = (𝐴(midG‘𝐺)𝑏))
44	43	eleq1d 2846	. . . . . . 7 ⊢ (𝑎 = 𝐴 → ((𝑎(midG‘𝐺)𝑏) ∈ 𝐷 ↔ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷))
45		oveq1 7419	. . . . . . . . 9 ⊢ (𝑎 = 𝐴 → (𝑎𝐿𝑏) = (𝐴𝐿𝑏))
46	45	breq2d 5115	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → (𝐷(⟂G‘𝐺)(𝑎𝐿𝑏) ↔ 𝐷(⟂G‘𝐺)(𝐴𝐿𝑏)))
47		eqeq1 2765	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → (𝑎 = 𝑏 ↔ 𝐴 = 𝑏))
48	46, 47	orbi12d 932	. . . . . . 7 ⊢ (𝑎 = 𝐴 → ((𝐷(⟂G‘𝐺)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏) ↔ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏)))
49	44, 48	anbi12d 644	. . . . . 6 ⊢ (𝑎 = 𝐴 → (((𝑎(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏)) ↔ ((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏))))
50	49	riotabidv 7371	. . . . 5 ⊢ (𝑎 = 𝐴 → (℩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))))
51	50	adantl 487	. . . 4 ⊢ ((𝜑 ∧ 𝑎 = 𝐴) → (℩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))))
52		lmicl.1	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
53		ismid.d	. . . . . 6 ⊢ − = (dist‘𝐺)
54		ismid.i	. . . . . 6 ⊢ 𝐼 = (Itv‘𝐺)
55		ismid.1	. . . . . 6 ⊢ (𝜑 → 𝐺DimTarskiG≥2)
56	8, 53, 54, 22, 55, 4, 37, 52	lmieu 29271	. . . . 5 ⊢ (𝜑 → ∃!𝑏 ∈ 𝑃 ((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏)))
57		riotacl 7386	. . . . 5 ⊢ (∃!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)
58	56, 57	syl 18	. . . 4 ⊢ (𝜑 → (℩𝑏 ∈ 𝑃 ((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏))) ∈ 𝑃)
59	42, 51, 52, 58	fvmptd 6993	. . 3 ⊢ (𝜑 → (𝑀‘𝐴) = (℩𝑏 ∈ 𝑃 ((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏))))
60	59	eqeq2d 2772	. 2 ⊢ (𝜑 → (𝐵 = (𝑀‘𝐴) ↔ 𝐵 = (℩𝑏 ∈ 𝑃 ((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏)))))
61		islmib.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
62		oveq2 7420	. . . . . . 7 ⊢ (𝑏 = 𝐵 → (𝐴(midG‘𝐺)𝑏) = (𝐴(midG‘𝐺)𝐵))
63	62	eleq1d 2846	. . . . . 6 ⊢ (𝑏 = 𝐵 → ((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ↔ (𝐴(midG‘𝐺)𝐵) ∈ 𝐷))
64		oveq2 7420	. . . . . . . 8 ⊢ (𝑏 = 𝐵 → (𝐴𝐿𝑏) = (𝐴𝐿𝐵))
65	64	breq2d 5115	. . . . . . 7 ⊢ (𝑏 = 𝐵 → (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ↔ 𝐷(⟂G‘𝐺)(𝐴𝐿𝐵)))
66		eqeq2 2773	. . . . . . 7 ⊢ (𝑏 = 𝐵 → (𝐴 = 𝑏 ↔ 𝐴 = 𝐵))
67	65, 66	orbi12d 932	. . . . . 6 ⊢ (𝑏 = 𝐵 → ((𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏) ↔ (𝐷(⟂G‘𝐺)(𝐴𝐿𝐵) ∨ 𝐴 = 𝐵)))
68	63, 67	anbi12d 644	. . . . 5 ⊢ (𝑏 = 𝐵 → (((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏)) ↔ ((𝐴(midG‘𝐺)𝐵) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))))
69	68	riota2 7394	. . . 4 ⊢ ((𝐵 ∈ 𝑃 ∧ ∃!𝑏 ∈ 𝑃 ((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏))) → (((𝐴(midG‘𝐺)𝐵) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝐵) ∨ 𝐴 = 𝐵)) ↔ (℩𝑏 ∈ 𝑃 ((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏))) = 𝐵))
70	61, 56, 69	syl2anc 596	. . 3 ⊢ (𝜑 → (((𝐴(midG‘𝐺)𝐵) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝐵) ∨ 𝐴 = 𝐵)) ↔ (℩𝑏 ∈ 𝑃 ((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏))) = 𝐵))
71		eqcom 2768	. . 3 ⊢ (𝐵 = (℩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))) = 𝐵)
72	70, 71	bitr4di 292	. 2 ⊢ (𝜑 → (((𝐴(midG‘𝐺)𝐵) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝐵) ∨ 𝐴 = 𝐵)) ↔ 𝐵 = (℩𝑏 ∈ 𝑃 ((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏)))))
73	60, 72	bitr4d 285	1 ⊢ (𝜑 → (𝐵 = (𝑀‘𝐴) ↔ ((𝐴(midG‘𝐺)𝐵) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2145  ∃!wreu 3364  Vcvv 3451   class class class wbr 5103   ↦ cmpt 5186  ran crn 5652  ‘cfv 6531  ℩crio 7368  (class class class)co 7412  2c2 12378  Basecbs 17367  distcds 17417  TarskiGcstrkg 28871  Dim_TarskiG≥cstrkgld 28875  Itvcitv 28877  LineGclng 28878  ⟂Gcperpg 29152  midGcmid 29259  lInvGclmi 29260

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740  ax-cnex 11237  ax-resscn 11238  ax-1cn 11239  ax-icn 11240  ax-addcl 11241  ax-addrcl 11242  ax-mulcl 11243  ax-mulrcl 11244  ax-mulcom 11245  ax-addass 11246  ax-mulass 11247  ax-distr 11248  ax-i2m1 11249  ax-1ne0 11250  ax-1rid 11251  ax-rnegex 11252  ax-rrecex 11253  ax-cnre 11254  ax-pre-lttri 11255  ax-pre-lttrn 11256  ax-pre-ltadd 11257  ax-pre-mulgt0 11258

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  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 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6297  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7867  df-1st 7990  df-2nd 7991  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-1o 8460  df-oadd 8464  df-er 8701  df-map 8833  df-pm 8834  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-dju 9963  df-card 10001  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11524  df-neg 11525  df-nn 12317  df-2 12386  df-3 12387  df-n0 12588  df-xnn0 12661  df-z 12675  df-uz 12947  df-fz 13621  df-fzo 13769  df-hash 14455  df-word 14639  df-concat 14696  df-s1 14723  df-s2 14979  df-s3 14980  df-trkgc 28892  df-trkgb 28893  df-trkgcb 28894  df-trkgld 28896  df-trkg 28897  df-cgrg 28956  df-leg 29028  df-mir 29107  df-rag 29151  df-perpg 29153  df-mid 29261  df-lmi 29262

This theorem is used by:  lmicom  29275  lmiinv  29279  lmimid  29281  lmiisolem  29283  hypcgrlem1  29287  hypcgrlem2  29288  lmiopp  29290  trgcopyeulem  29294