Theorem islmib 28721

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 28709	. . . . . . 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 6861	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (LineG‘𝑔) = (LineG‘𝐺))
4		lmif.l	. . . . . . . . . 10 ⊢ 𝐿 = (LineG‘𝐺)
5	3, 4	eqtr4di 2783	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (LineG‘𝑔) = 𝐿)
6	5	rneqd 5905	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → ran (LineG‘𝑔) = ran 𝐿)
7		fveq2 6861	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
8		ismid.p	. . . . . . . . . 10 ⊢ 𝑃 = (Base‘𝐺)
9	7, 8	eqtr4di 2783	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝑃)
10		fveq2 6861	. . . . . . . . . . . . 13 ⊢ (𝑔 = 𝐺 → (midG‘𝑔) = (midG‘𝐺))
11	10	oveqd 7407	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝐺 → (𝑎(midG‘𝑔)𝑏) = (𝑎(midG‘𝐺)𝑏))
12	11	eleq1d 2814	. . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → ((𝑎(midG‘𝑔)𝑏) ∈ 𝑑 ↔ (𝑎(midG‘𝐺)𝑏) ∈ 𝑑))
13		eqidd 2731	. . . . . . . . . . . . 13 ⊢ (𝑔 = 𝐺 → 𝑑 = 𝑑)
14		fveq2 6861	. . . . . . . . . . . . 13 ⊢ (𝑔 = 𝐺 → (⟂G‘𝑔) = (⟂G‘𝐺))
15	5	oveqd 7407	. . . . . . . . . . . . 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 916	. . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → ((𝑑(⟂G‘𝑔)(𝑎(LineG‘𝑔)𝑏) ∨ 𝑎 = 𝑏) ↔ (𝑑(⟂G‘𝐺)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏)))
18	12, 17	anbi12d 632	. . . . . . . . . 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 7350	. . . . . . . . 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 5197	. . . . . . . 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 5197	. . . . . . 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 6875	. . . . . . . . 9 ⊢ 𝐿 ∈ V
25		rnexg 7881	. . . . . . . . 9 ⊢ (𝐿 ∈ V → ran 𝐿 ∈ V)
26		mptexg 7198	. . . . . . . . 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 6994	. . . . . 6 ⊢ (𝜑 → (lInvG‘𝐺) = (𝑑 ∈ ran 𝐿 ↦ (𝑎 ∈ 𝑃 ↦ (℩𝑏 ∈ 𝑃 ((𝑎(midG‘𝐺)𝑏) ∈ 𝑑 ∧ (𝑑(⟂G‘𝐺)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏))))))
30		eleq2 2818	. . . . . . . . . 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 916	. . . . . . . . . 10 ⊢ (𝑑 = 𝐷 → ((𝑑(⟂G‘𝐺)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏) ↔ (𝐷(⟂G‘𝐺)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏)))
33	30, 32	anbi12d 632	. . . . . . . . 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 7349	. . . . . . . 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 5204	. . . . . . 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 481	. . . . . 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 6875	. . . . . . . 8 ⊢ 𝑃 ∈ V
39	38	mptex 7200	. . . . . . 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 6978	. . . . 5 ⊢ (𝜑 → ((lInvG‘𝐺)‘𝐷) = (𝑎 ∈ 𝑃 ↦ (℩𝑏 ∈ 𝑃 ((𝑎(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏)))))
42	1, 41	eqtrid 2777	. . . 4 ⊢ (𝜑 → 𝑀 = (𝑎 ∈ 𝑃 ↦ (℩𝑏 ∈ 𝑃 ((𝑎(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏)))))
43		oveq1 7397	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → (𝑎(midG‘𝐺)𝑏) = (𝐴(midG‘𝐺)𝑏))
44	43	eleq1d 2814	. . . . . . 7 ⊢ (𝑎 = 𝐴 → ((𝑎(midG‘𝐺)𝑏) ∈ 𝐷 ↔ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷))
45		oveq1 7397	. . . . . . . . 9 ⊢ (𝑎 = 𝐴 → (𝑎𝐿𝑏) = (𝐴𝐿𝑏))
46	45	breq2d 5122	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → (𝐷(⟂G‘𝐺)(𝑎𝐿𝑏) ↔ 𝐷(⟂G‘𝐺)(𝐴𝐿𝑏)))
47		eqeq1 2734	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → (𝑎 = 𝑏 ↔ 𝐴 = 𝑏))
48	46, 47	orbi12d 918	. . . . . . 7 ⊢ (𝑎 = 𝐴 → ((𝐷(⟂G‘𝐺)(𝑎𝐿𝑏) ∨ 𝑎 = 𝑏) ↔ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏)))
49	44, 48	anbi12d 632	. . . . . 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 7349	. . . . 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 481	. . . 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 28718	. . . . 5 ⊢ (𝜑 → ∃!𝑏 ∈ 𝑃 ((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏)))
57		riotacl 7364	. . . . 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 17	. . . 4 ⊢ (𝜑 → (℩𝑏 ∈ 𝑃 ((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏))) ∈ 𝑃)
59	42, 51, 52, 58	fvmptd 6978	. . 3 ⊢ (𝜑 → (𝑀‘𝐴) = (℩𝑏 ∈ 𝑃 ((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏))))
60	59	eqeq2d 2741	. 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 7398	. . . . . . 7 ⊢ (𝑏 = 𝐵 → (𝐴(midG‘𝐺)𝑏) = (𝐴(midG‘𝐺)𝐵))
63	62	eleq1d 2814	. . . . . 6 ⊢ (𝑏 = 𝐵 → ((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ↔ (𝐴(midG‘𝐺)𝐵) ∈ 𝐷))
64		oveq2 7398	. . . . . . . 8 ⊢ (𝑏 = 𝐵 → (𝐴𝐿𝑏) = (𝐴𝐿𝐵))
65	64	breq2d 5122	. . . . . . 7 ⊢ (𝑏 = 𝐵 → (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ↔ 𝐷(⟂G‘𝐺)(𝐴𝐿𝐵)))
66		eqeq2 2742	. . . . . . 7 ⊢ (𝑏 = 𝐵 → (𝐴 = 𝑏 ↔ 𝐴 = 𝐵))
67	65, 66	orbi12d 918	. . . . . 6 ⊢ (𝑏 = 𝐵 → ((𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏) ↔ (𝐷(⟂G‘𝐺)(𝐴𝐿𝐵) ∨ 𝐴 = 𝐵)))
68	63, 67	anbi12d 632	. . . . 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 7372	. . . 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 584	. . 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 2737	. . 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 289	. 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 282	1 ⊢ (𝜑 → (𝐵 = (𝑀‘𝐴) ↔ ((𝐴(midG‘𝐺)𝐵) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1540   ∈ wcel 2109  ∃!wreu 3354  Vcvv 3450   class class class wbr 5110   ↦ cmpt 5191  ran crn 5642  ‘cfv 6514  ℩crio 7346  (class class class)co 7390  2c2 12248  Basecbs 17186  distcds 17236  TarskiGcstrkg 28361  Dim_TarskiG≥cstrkgld 28365  Itvcitv 28367  LineGclng 28368  ⟂Gcperpg 28629  midGcmid 28706  lInvGclmi 28707

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  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 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-tp 4597  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-oadd 8441  df-er 8674  df-map 8804  df-pm 8805  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-dju 9861  df-card 9899  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-nn 12194  df-2 12256  df-3 12257  df-n0 12450  df-xnn0 12523  df-z 12537  df-uz 12801  df-fz 13476  df-fzo 13623  df-hash 14303  df-word 14486  df-concat 14543  df-s1 14568  df-s2 14821  df-s3 14822  df-trkgc 28382  df-trkgb 28383  df-trkgcb 28384  df-trkgld 28386  df-trkg 28387  df-cgrg 28445  df-leg 28517  df-mir 28587  df-rag 28628  df-perpg 28630  df-mid 28708  df-lmi 28709

This theorem is referenced by:  lmicom  28722  lmiinv  28726  lmimid  28728  lmiisolem  28730  hypcgrlem1  28733  hypcgrlem2  28734  lmiopp  28736  trgcopyeulem  28739