Theorem islmib 29074

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 29062	. . . . . . 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 6883	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (LineG‘𝑔) = (LineG‘𝐺))
4		lmif.l	. . . . . . . . . 10 ⊢ 𝐿 = (LineG‘𝐺)
5	3, 4	eqtr4di 2816	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (LineG‘𝑔) = 𝐿)
6	5	rneqd 5930	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → ran (LineG‘𝑔) = ran 𝐿)
7		fveq2 6883	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
8		ismid.p	. . . . . . . . . 10 ⊢ 𝑃 = (Base‘𝐺)
9	7, 8	eqtr4di 2816	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝑃)
10		fveq2 6883	. . . . . . . . . . . . 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 2848	. . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → ((𝑎(midG‘𝑔)𝑏) ∈ 𝑑 ↔ (𝑎(midG‘𝐺)𝑏) ∈ 𝑑))
13		eqidd 2764	. . . . . . . . . . . . 13 ⊢ (𝑔 = 𝐺 → 𝑑 = 𝑑)
14		fveq2 6883	. . . . . . . . . . . . 13 ⊢ (𝑔 = 𝐺 → (⟂G‘𝑔) = (⟂G‘𝐺))
15	5	oveqd 7429	. . . . . . . . . . . . 13 ⊢ (𝑔 = 𝐺 → (𝑎(LineG‘𝑔)𝑏) = (𝑎𝐿𝑏))
16	13, 14, 15	breq123d 5124	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝐺 → (𝑑(⟂G‘𝑔)(𝑎(LineG‘𝑔)𝑏) ↔ 𝑑(⟂G‘𝐺)(𝑎𝐿𝑏)))
17	16	orbi1d 929	. . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → ((𝑑(⟂G‘𝑔)(𝑎(LineG‘𝑔)𝑏) ∨ 𝑎 = 𝑏) ↔ (𝑑(⟂G‘𝐺)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏)))
18	12, 17	anbi12d 643	. . . . . . . . . 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 5199	. . . . . . . 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 5199	. . . . . . 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 3478	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ V)
24	4	fvexi 6897	. . . . . . . . 9 ⊢ 𝐿 ∈ V
25		rnexg 7900	. . . . . . . . 9 ⊢ (𝐿 ∈ V → ran 𝐿 ∈ V)
26		mptexg 7221	. . . . . . . . 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 7015	. . . . . 6 ⊢ (𝜑 → (lInvG‘𝐺) = (𝑑 ∈ ran 𝐿 ↦ (𝑎 ∈ 𝑃 ↦ (℩𝑏 ∈ 𝑃 ((𝑎(midG‘𝐺)𝑏) ∈ 𝑑 ∧ (𝑑(⟂G‘𝐺)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏))))))
30		eleq2 2852	. . . . . . . . . 10 ⊢ (𝑑 = 𝐷 → ((𝑎(midG‘𝐺)𝑏) ∈ 𝑑 ↔ (𝑎(midG‘𝐺)𝑏) ∈ 𝐷))
31		breq1 5113	. . . . . . . . . . 11 ⊢ (𝑑 = 𝐷 → (𝑑(⟂G‘𝐺)(𝑎𝐿𝑏) ↔ 𝐷(⟂G‘𝐺)(𝑎𝐿𝑏)))
32	31	orbi1d 929	. . . . . . . . . 10 ⊢ (𝑑 = 𝐷 → ((𝑑(⟂G‘𝐺)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏) ↔ (𝐷(⟂G‘𝐺)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏)))
33	30, 32	anbi12d 643	. . . . . . . . 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 5206	. . . . . . 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 486	. . . . . 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 6897	. . . . . . . 8 ⊢ 𝑃 ∈ V
39	38	mptex 7223	. . . . . . 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 6999	. . . . 5 ⊢ (𝜑 → ((lInvG‘𝐺)‘𝐷) = (𝑎 ∈ 𝑃 ↦ (℩𝑏 ∈ 𝑃 ((𝑎(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏)))))
42	1, 41	eqtrid 2810	. . . 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 2848	. . . . . . 7 ⊢ (𝑎 = 𝐴 → ((𝑎(midG‘𝐺)𝑏) ∈ 𝐷 ↔ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷))
45		oveq1 7419	. . . . . . . . 9 ⊢ (𝑎 = 𝐴 → (𝑎𝐿𝑏) = (𝐴𝐿𝑏))
46	45	breq2d 5122	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → (𝐷(⟂G‘𝐺)(𝑎𝐿𝑏) ↔ 𝐷(⟂G‘𝐺)(𝐴𝐿𝑏)))
47		eqeq1 2767	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → (𝑎 = 𝑏 ↔ 𝐴 = 𝑏))
48	46, 47	orbi12d 931	. . . . . . 7 ⊢ (𝑎 = 𝐴 → ((𝐷(⟂G‘𝐺)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏) ↔ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏)))
49	44, 48	anbi12d 643	. . . . . 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 486	. . . 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 29071	. . . . 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 6999	. . 3 ⊢ (𝜑 → (𝑀‘𝐴) = (℩𝑏 ∈ 𝑃 ((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏))))
60	59	eqeq2d 2774	. 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 2848	. . . . . 6 ⊢ (𝑏 = 𝐵 → ((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ↔ (𝐴(midG‘𝐺)𝐵) ∈ 𝐷))
64		oveq2 7420	. . . . . . . 8 ⊢ (𝑏 = 𝐵 → (𝐴𝐿𝑏) = (𝐴𝐿𝐵))
65	64	breq2d 5122	. . . . . . 7 ⊢ (𝑏 = 𝐵 → (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ↔ 𝐷(⟂G‘𝐺)(𝐴𝐿𝐵)))
66		eqeq2 2775	. . . . . . 7 ⊢ (𝑏 = 𝐵 → (𝐴 = 𝑏 ↔ 𝐴 = 𝐵))
67	65, 66	orbi12d 931	. . . . . 6 ⊢ (𝑏 = 𝐵 → ((𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏) ↔ (𝐷(⟂G‘𝐺)(𝐴𝐿𝐵) ∨ 𝐴 = 𝐵)))
68	63, 67	anbi12d 643	. . . . 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 595	. . 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 2770	. . 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))))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   = wceq 1570   ∈ wcel 2143  ∃!wreu 3367  Vcvv 3455   class class class wbr 5110   ↦ cmpt 5193  ran crn 5664  ‘cfv 6538  ℩crio 7368  (class class class)co 7412  2c2 12296  Basecbs 17270  distcds 17320  TarskiGcstrkg 28674  Dim_TarskiG≥cstrkgld 28678  Itvcitv 28680  LineGclng 28681  ⟂Gcperpg 28953  midGcmid 29059  lInvGclmi 29060

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734  ax-cnex 11157  ax-resscn 11158  ax-1cn 11159  ax-icn 11160  ax-addcl 11161  ax-addrcl 11162  ax-mulcl 11163  ax-mulrcl 11164  ax-mulcom 11165  ax-addass 11166  ax-mulass 11167  ax-distr 11168  ax-i2m1 11169  ax-1ne0 11170  ax-1rid 11171  ax-rnegex 11172  ax-rrecex 11173  ax-cnre 11174  ax-pre-lttri 11175  ax-pre-lttrn 11176  ax-pre-ltadd 11177  ax-pre-mulgt0 11178

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-tp 4595  df-op 4597  df-uni 4874  df-int 4914  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7864  df-1st 7987  df-2nd 7988  df-frecs 8279  df-wrecs 8310  df-recs 8359  df-rdg 8398  df-1o 8454  df-oadd 8458  df-er 8695  df-map 8827  df-pm 8828  df-en 8945  df-dom 8946  df-sdom 8947  df-fin 8948  df-dju 9888  df-card 9926  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11444  df-neg 11445  df-nn 12235  df-2 12304  df-3 12305  df-n0 12506  df-xnn0 12579  df-z 12593  df-uz 12864  df-fz 13537  df-fzo 13685  df-hash 14369  df-word 14553  df-concat 14610  df-s1 14636  df-s2 14887  df-s3 14888  df-trkgc 28695  df-trkgb 28696  df-trkgcb 28697  df-trkgld 28699  df-trkg 28700  df-cgrg 28758  df-leg 28830  df-mir 28908  df-rag 28952  df-perpg 28954  df-mid 29061  df-lmi 29062

This theorem is referenced by:  lmicom  29075  lmiinv  29079  lmimid  29081  lmiisolem  29083  hypcgrlem1  29087  hypcgrlem2  29088  lmiopp  29090  trgcopyeulem  29094