Theorem symquadlem 28711

Description: Lemma of the symetrial quadrilateral. The diagonals of quadrilaterals with congruent opposing sides intersect at their middle point. In Euclidean geometry, such quadrilaterals are called parallelograms, as opposing sides are parallel. However, this is not necessarily true in the case of absolute geometry. Lemma 7.21 of [Schwabhauser] p. 52. (Contributed by Thierry Arnoux, 6-Aug-2019.)

Hypotheses

Ref	Expression
mirval.p	⊢ 𝑃 = (Base‘𝐺)
mirval.d	⊢ − = (dist‘𝐺)
mirval.i	⊢ 𝐼 = (Itv‘𝐺)
mirval.l	⊢ 𝐿 = (LineG‘𝐺)
mirval.s	⊢ 𝑆 = (pInvG‘𝐺)
mirval.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
symquadlem.m	⊢ 𝑀 = (𝑆‘𝑋)
symquadlem.a	⊢ (𝜑 → 𝐴 ∈ 𝑃)
symquadlem.b	⊢ (𝜑 → 𝐵 ∈ 𝑃)
symquadlem.c	⊢ (𝜑 → 𝐶 ∈ 𝑃)
symquadlem.d	⊢ (𝜑 → 𝐷 ∈ 𝑃)
symquadlem.x	⊢ (𝜑 → 𝑋 ∈ 𝑃)
symquadlem.1	⊢ (𝜑 → ¬ (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶))
symquadlem.2	⊢ (𝜑 → 𝐵 ≠ 𝐷)
symquadlem.3	⊢ (𝜑 → (𝐴 − 𝐵) = (𝐶 − 𝐷))
symquadlem.4	⊢ (𝜑 → (𝐵 − 𝐶) = (𝐷 − 𝐴))
symquadlem.5	⊢ (𝜑 → (𝑋 ∈ (𝐴𝐿𝐶) ∨ 𝐴 = 𝐶))
symquadlem.6	⊢ (𝜑 → (𝑋 ∈ (𝐵𝐿𝐷) ∨ 𝐵 = 𝐷))

Assertion

Ref	Expression
symquadlem	⊢ (𝜑 → 𝐴 = (𝑀‘𝐶))

Proof of Theorem symquadlem

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		symquadlem.1	. . . . . . . 8 ⊢ (𝜑 → ¬ (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶))
2		mirval.p	. . . . . . . . . . 11 ⊢ 𝑃 = (Base‘𝐺)
3		mirval.l	. . . . . . . . . . 11 ⊢ 𝐿 = (LineG‘𝐺)
4		mirval.i	. . . . . . . . . . 11 ⊢ 𝐼 = (Itv‘𝐺)
5		mirval.g	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
6		symquadlem.b	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
7		symquadlem.a	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
8		mirval.d	. . . . . . . . . . . 12 ⊢ − = (dist‘𝐺)
9	2, 8, 4, 5, 6, 7	tgbtwntriv2 28509	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ (𝐵𝐼𝐴))
10	2, 3, 4, 5, 6, 7, 7, 9	btwncolg1 28577	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 ∈ (𝐵𝐿𝐴) ∨ 𝐵 = 𝐴))
11	10	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → (𝐴 ∈ (𝐵𝐿𝐴) ∨ 𝐵 = 𝐴))
12		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐴 = 𝐶)
13	12	oveq2d 7446	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → (𝐵𝐿𝐴) = (𝐵𝐿𝐶))
14	13	eleq2d 2824	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → (𝐴 ∈ (𝐵𝐿𝐴) ↔ 𝐴 ∈ (𝐵𝐿𝐶)))
15	12	eqeq2d 2745	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → (𝐵 = 𝐴 ↔ 𝐵 = 𝐶))
16	14, 15	orbi12d 918	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → ((𝐴 ∈ (𝐵𝐿𝐴) ∨ 𝐵 = 𝐴) ↔ (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶)))
17	11, 16	mpbid 232	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶))
18	1, 17	mtand 816	. . . . . . 7 ⊢ (𝜑 → ¬ 𝐴 = 𝐶)
19	18	neqned 2944	. . . . . 6 ⊢ (𝜑 → 𝐴 ≠ 𝐶)
20	19	ad2antrr 726	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝐴 ≠ 𝐶)
21	20	necomd 2993	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝐶 ≠ 𝐴)
22	21	neneqd 2942	. . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → ¬ 𝐶 = 𝐴)
23		mirval.s	. . . . . 6 ⊢ 𝑆 = (pInvG‘𝐺)
24	5	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝐺 ∈ TarskiG)
25		symquadlem.m	. . . . . 6 ⊢ 𝑀 = (𝑆‘𝑋)
26		symquadlem.c	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
27	26	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝐶 ∈ 𝑃)
28	7	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝐴 ∈ 𝑃)
29		symquadlem.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
30	29	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝑋 ∈ 𝑃)
31		symquadlem.5	. . . . . . . 8 ⊢ (𝜑 → (𝑋 ∈ (𝐴𝐿𝐶) ∨ 𝐴 = 𝐶))
32	31	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝑋 ∈ (𝐴𝐿𝐶) ∨ 𝐴 = 𝐶))
33	2, 3, 4, 24, 28, 27, 30, 32	colcom 28580	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝑋 ∈ (𝐶𝐿𝐴) ∨ 𝐶 = 𝐴))
34	6	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝐵 ∈ 𝑃)
35		symquadlem.d	. . . . . . . . 9 ⊢ (𝜑 → 𝐷 ∈ 𝑃)
36	35	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝐷 ∈ 𝑃)
37		eqid 2734	. . . . . . . 8 ⊢ (cgrG‘𝐺) = (cgrG‘𝐺)
38		simplr 769	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝑥 ∈ 𝑃)
39		symquadlem.6	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑋 ∈ (𝐵𝐿𝐷) ∨ 𝐵 = 𝐷))
40	2, 3, 4, 5, 6, 35, 29, 39	colrot2 28582	. . . . . . . . . 10 ⊢ (𝜑 → (𝐷 ∈ (𝑋𝐿𝐵) ∨ 𝑋 = 𝐵))
41	2, 3, 4, 5, 29, 6, 35, 40	colcom 28580	. . . . . . . . 9 ⊢ (𝜑 → (𝐷 ∈ (𝐵𝐿𝑋) ∨ 𝐵 = 𝑋))
42	41	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝐷 ∈ (𝐵𝐿𝑋) ∨ 𝐵 = 𝑋))
43		simpr 484	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩)
44		symquadlem.4	. . . . . . . . 9 ⊢ (𝜑 → (𝐵 − 𝐶) = (𝐷 − 𝐴))
45	44	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝐵 − 𝐶) = (𝐷 − 𝐴))
46		symquadlem.3	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐶 − 𝐷))
47	2, 8, 4, 5, 7, 6, 26, 35, 46	tgcgrcomlr 28502	. . . . . . . . . 10 ⊢ (𝜑 → (𝐵 − 𝐴) = (𝐷 − 𝐶))
48	47	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝐵 − 𝐴) = (𝐷 − 𝐶))
49	48	eqcomd 2740	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝐷 − 𝐶) = (𝐵 − 𝐴))
50		symquadlem.2	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ≠ 𝐷)
51	50	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝐵 ≠ 𝐷)
52	2, 3, 4, 24, 34, 36, 30, 37, 36, 34, 8, 27, 38, 28, 42, 43, 45, 49, 51	tgfscgr 28590	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝑋 − 𝐶) = (𝑥 − 𝐴))
53	2, 3, 4, 5, 6, 26, 7, 1	ncolcom 28583	. . . . . . . . . 10 ⊢ (𝜑 → ¬ (𝐴 ∈ (𝐶𝐿𝐵) ∨ 𝐶 = 𝐵))
54	53	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → ¬ (𝐴 ∈ (𝐶𝐿𝐵) ∨ 𝐶 = 𝐵))
55	31	orcomd 871	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 = 𝐶 ∨ 𝑋 ∈ (𝐴𝐿𝐶)))
56	55	ord 864	. . . . . . . . . . 11 ⊢ (𝜑 → (¬ 𝐴 = 𝐶 → 𝑋 ∈ (𝐴𝐿𝐶)))
57	18, 56	mpd 15	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ (𝐴𝐿𝐶))
58	57	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝑋 ∈ (𝐴𝐿𝐶))
59	18	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → ¬ 𝐴 = 𝐶)
60	45	eqcomd 2740	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝐷 − 𝐴) = (𝐵 − 𝐶))
61	2, 3, 4, 24, 34, 36, 30, 37, 36, 34, 8, 28, 38, 27, 42, 43, 48, 60, 51	tgfscgr 28590	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝑋 − 𝐴) = (𝑥 − 𝐶))
62	2, 8, 4, 24, 30, 28, 38, 27, 61	tgcgrcomlr 28502	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝐴 − 𝑋) = (𝐶 − 𝑥))
63	2, 8, 4, 24, 27, 28	axtgcgrrflx 28484	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝐶 − 𝐴) = (𝐴 − 𝐶))
64	2, 8, 37, 24, 28, 30, 27, 27, 38, 28, 62, 52, 63	trgcgr 28538	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → ⟨“𝐴𝑋𝐶”⟩(cgrG‘𝐺)⟨“𝐶𝑥𝐴”⟩)
65	2, 3, 4, 24, 28, 30, 27, 37, 27, 38, 28, 32, 64	lnxfr 28588	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝑥 ∈ (𝐶𝐿𝐴) ∨ 𝐶 = 𝐴))
66	2, 3, 4, 24, 27, 28, 38, 65	colcom 28580	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝑥 ∈ (𝐴𝐿𝐶) ∨ 𝐴 = 𝐶))
67	66	orcomd 871	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝐴 = 𝐶 ∨ 𝑥 ∈ (𝐴𝐿𝐶)))
68	67	ord 864	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (¬ 𝐴 = 𝐶 → 𝑥 ∈ (𝐴𝐿𝐶)))
69	59, 68	mpd 15	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝑥 ∈ (𝐴𝐿𝐶))
70	50	neneqd 2942	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ 𝐵 = 𝐷)
71	39	orcomd 871	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐵 = 𝐷 ∨ 𝑋 ∈ (𝐵𝐿𝐷)))
72	71	ord 864	. . . . . . . . . . 11 ⊢ (𝜑 → (¬ 𝐵 = 𝐷 → 𝑋 ∈ (𝐵𝐿𝐷)))
73	70, 72	mpd 15	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ (𝐵𝐿𝐷))
74	73	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝑋 ∈ (𝐵𝐿𝐷))
75	70	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → ¬ 𝐵 = 𝐷)
76	39	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝑋 ∈ (𝐵𝐿𝐷) ∨ 𝐵 = 𝐷))
77	2, 8, 4, 37, 24, 34, 36, 30, 36, 34, 38, 43	cgr3swap23 28546	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → ⟨“𝐵𝑋𝐷”⟩(cgrG‘𝐺)⟨“𝐷𝑥𝐵”⟩)
78	2, 3, 4, 24, 34, 30, 36, 37, 36, 38, 34, 76, 77	lnxfr 28588	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝑥 ∈ (𝐷𝐿𝐵) ∨ 𝐷 = 𝐵))
79	2, 3, 4, 24, 36, 34, 38, 78	colcom 28580	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝑥 ∈ (𝐵𝐿𝐷) ∨ 𝐵 = 𝐷))
80	79	orcomd 871	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝐵 = 𝐷 ∨ 𝑥 ∈ (𝐵𝐿𝐷)))
81	80	ord 864	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (¬ 𝐵 = 𝐷 → 𝑥 ∈ (𝐵𝐿𝐷)))
82	75, 81	mpd 15	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝑥 ∈ (𝐵𝐿𝐷))
83	2, 4, 3, 24, 28, 27, 34, 36, 54, 58, 69, 74, 82	tglineinteq 28667	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝑋 = 𝑥)
84	83	oveq1d 7445	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝑋 − 𝐴) = (𝑥 − 𝐴))
85	52, 84	eqtr4d 2777	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝑋 − 𝐶) = (𝑋 − 𝐴))
86	2, 8, 4, 3, 23, 24, 25, 27, 28, 30, 33, 85	colmid 28710	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝐴 = (𝑀‘𝐶) ∨ 𝐶 = 𝐴))
87	86	orcomd 871	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝐶 = 𝐴 ∨ 𝐴 = (𝑀‘𝐶)))
88	87	ord 864	. . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (¬ 𝐶 = 𝐴 → 𝐴 = (𝑀‘𝐶)))
89	22, 88	mpd 15	. 2 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝐴 = (𝑀‘𝐶))
90	2, 8, 4, 5, 6, 35	axtgcgrrflx 28484	. . 3 ⊢ (𝜑 → (𝐵 − 𝐷) = (𝐷 − 𝐵))
91	2, 3, 4, 5, 6, 35, 29, 37, 35, 6, 8, 41, 90	lnext 28589	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩)
92	89, 91	r19.29a 3159	1 ⊢ (𝜑 → 𝐴 = (𝑀‘𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 847   = wceq 1536   ∈ wcel 2105   ≠ wne 2937   class class class wbr 5147  ‘cfv 6562  (class class class)co 7430  ⟨“cs3 14877  Basecbs 17244  distcds 17306  TarskiGcstrkg 28449  Itvcitv 28455  LineGclng 28456  cgrGccgrg 28532  pInvGcmir 28674

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-oadd 8508  df-er 8743  df-pm 8867  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-dju 9938  df-card 9976  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-nn 12264  df-2 12326  df-3 12327  df-n0 12524  df-xnn0 12597  df-z 12611  df-uz 12876  df-fz 13544  df-fzo 13691  df-hash 14366  df-word 14549  df-concat 14605  df-s1 14630  df-s2 14883  df-s3 14884  df-trkgc 28470  df-trkgb 28471  df-trkgcb 28472  df-trkg 28475  df-cgrg 28533  df-mir 28675

This theorem is referenced by:  opphllem  28757