Theorem symquadlem 26002

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 25800	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ (𝐵𝐼𝐴))
10	2, 3, 4, 5, 6, 7, 7, 9	btwncolg1 25868	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 ∈ (𝐵𝐿𝐴) ∨ 𝐵 = 𝐴))
11	10	adantr 474	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → (𝐴 ∈ (𝐵𝐿𝐴) ∨ 𝐵 = 𝐴))
12		simpr 479	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐴 = 𝐶)
13	12	oveq2d 6922	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → (𝐵𝐿𝐴) = (𝐵𝐿𝐶))
14	13	eleq2d 2893	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → (𝐴 ∈ (𝐵𝐿𝐴) ↔ 𝐴 ∈ (𝐵𝐿𝐶)))
15	12	eqeq2d 2836	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → (𝐵 = 𝐴 ↔ 𝐵 = 𝐶))
16	14, 15	orbi12d 949	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → ((𝐴 ∈ (𝐵𝐿𝐴) ∨ 𝐵 = 𝐴) ↔ (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶)))
17	11, 16	mpbid 224	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶))
18	1, 17	mtand 852	. . . . . . 7 ⊢ (𝜑 → ¬ 𝐴 = 𝐶)
19	18	neqned 3007	. . . . . 6 ⊢ (𝜑 → 𝐴 ≠ 𝐶)
20	19	ad2antrr 719	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝐴 ≠ 𝐶)
21	20	necomd 3055	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝐶 ≠ 𝐴)
22	21	neneqd 3005	. . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → ¬ 𝐶 = 𝐴)
23		mirval.s	. . . . . 6 ⊢ 𝑆 = (pInvG‘𝐺)
24	5	ad2antrr 719	. . . . . 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 719	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝐶 ∈ 𝑃)
28	7	ad2antrr 719	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝐴 ∈ 𝑃)
29		symquadlem.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
30	29	ad2antrr 719	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝑋 ∈ 𝑃)
31		symquadlem.5	. . . . . . . 8 ⊢ (𝜑 → (𝑋 ∈ (𝐴𝐿𝐶) ∨ 𝐴 = 𝐶))
32	31	ad2antrr 719	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝑋 ∈ (𝐴𝐿𝐶) ∨ 𝐴 = 𝐶))
33	2, 3, 4, 24, 28, 27, 30, 32	colcom 25871	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝑋 ∈ (𝐶𝐿𝐴) ∨ 𝐶 = 𝐴))
34	6	ad2antrr 719	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝐵 ∈ 𝑃)
35		symquadlem.d	. . . . . . . . 9 ⊢ (𝜑 → 𝐷 ∈ 𝑃)
36	35	ad2antrr 719	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝐷 ∈ 𝑃)
37		eqid 2826	. . . . . . . 8 ⊢ (cgrG‘𝐺) = (cgrG‘𝐺)
38		simplr 787	. . . . . . . 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 25873	. . . . . . . . . 10 ⊢ (𝜑 → (𝐷 ∈ (𝑋𝐿𝐵) ∨ 𝑋 = 𝐵))
41	2, 3, 4, 5, 29, 6, 35, 40	colcom 25871	. . . . . . . . 9 ⊢ (𝜑 → (𝐷 ∈ (𝐵𝐿𝑋) ∨ 𝐵 = 𝑋))
42	41	ad2antrr 719	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝐷 ∈ (𝐵𝐿𝑋) ∨ 𝐵 = 𝑋))
43		simpr 479	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩)
44		symquadlem.4	. . . . . . . . 9 ⊢ (𝜑 → (𝐵 − 𝐶) = (𝐷 − 𝐴))
45	44	ad2antrr 719	. . . . . . . 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 25793	. . . . . . . . . 10 ⊢ (𝜑 → (𝐵 − 𝐴) = (𝐷 − 𝐶))
48	47	ad2antrr 719	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝐵 − 𝐴) = (𝐷 − 𝐶))
49	48	eqcomd 2832	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝐷 − 𝐶) = (𝐵 − 𝐴))
50		symquadlem.2	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ≠ 𝐷)
51	50	ad2antrr 719	. . . . . . . 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 25881	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝑋 − 𝐶) = (𝑥 − 𝐴))
53	2, 3, 4, 5, 6, 26, 7, 1	ncolcom 25874	. . . . . . . . . 10 ⊢ (𝜑 → ¬ (𝐴 ∈ (𝐶𝐿𝐵) ∨ 𝐶 = 𝐵))
54	53	ad2antrr 719	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → ¬ (𝐴 ∈ (𝐶𝐿𝐵) ∨ 𝐶 = 𝐵))
55	31	orcomd 904	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 = 𝐶 ∨ 𝑋 ∈ (𝐴𝐿𝐶)))
56	55	ord 897	. . . . . . . . . . 11 ⊢ (𝜑 → (¬ 𝐴 = 𝐶 → 𝑋 ∈ (𝐴𝐿𝐶)))
57	18, 56	mpd 15	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ (𝐴𝐿𝐶))
58	57	ad2antrr 719	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝑋 ∈ (𝐴𝐿𝐶))
59	18	ad2antrr 719	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → ¬ 𝐴 = 𝐶)
60	45	eqcomd 2832	. . . . . . . . . . . . . . . . 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 25881	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝑋 − 𝐴) = (𝑥 − 𝐶))
62	2, 8, 4, 24, 30, 28, 38, 27, 61	tgcgrcomlr 25793	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝐴 − 𝑋) = (𝐶 − 𝑥))
63	2, 8, 4, 24, 27, 28	axtgcgrrflx 25775	. . . . . . . . . . . . . . 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 25829	. . . . . . . . . . . . . 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 25879	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝑥 ∈ (𝐶𝐿𝐴) ∨ 𝐶 = 𝐴))
66	2, 3, 4, 24, 27, 28, 38, 65	colcom 25871	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝑥 ∈ (𝐴𝐿𝐶) ∨ 𝐴 = 𝐶))
67	66	orcomd 904	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝐴 = 𝐶 ∨ 𝑥 ∈ (𝐴𝐿𝐶)))
68	67	ord 897	. . . . . . . . . 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 3005	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ 𝐵 = 𝐷)
71	39	orcomd 904	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐵 = 𝐷 ∨ 𝑋 ∈ (𝐵𝐿𝐷)))
72	71	ord 897	. . . . . . . . . . 11 ⊢ (𝜑 → (¬ 𝐵 = 𝐷 → 𝑋 ∈ (𝐵𝐿𝐷)))
73	70, 72	mpd 15	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ (𝐵𝐿𝐷))
74	73	ad2antrr 719	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝑋 ∈ (𝐵𝐿𝐷))
75	70	ad2antrr 719	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → ¬ 𝐵 = 𝐷)
76	39	ad2antrr 719	. . . . . . . . . . . . . 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 25837	. . . . . . . . . . . . . 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 25879	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝑥 ∈ (𝐷𝐿𝐵) ∨ 𝐷 = 𝐵))
79	2, 3, 4, 24, 36, 34, 38, 78	colcom 25871	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝑥 ∈ (𝐵𝐿𝐷) ∨ 𝐵 = 𝐷))
80	79	orcomd 904	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝐵 = 𝐷 ∨ 𝑥 ∈ (𝐵𝐿𝐷)))
81	80	ord 897	. . . . . . . . . 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 25958	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝑋 = 𝑥)
84	83	oveq1d 6921	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝑋 − 𝐴) = (𝑥 − 𝐴))
85	52, 84	eqtr4d 2865	. . . . . 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 26001	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝐴 = (𝑀‘𝐶) ∨ 𝐶 = 𝐴))
87	86	orcomd 904	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝐶 = 𝐴 ∨ 𝐴 = (𝑀‘𝐶)))
88	87	ord 897	. . 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 25775	. . 3 ⊢ (𝜑 → (𝐵 − 𝐷) = (𝐷 − 𝐵))
91	2, 3, 4, 5, 6, 35, 29, 37, 35, 6, 8, 41, 90	lnext 25880	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩)
92	89, 91	r19.29a 3289	1 ⊢ (𝜑 → 𝐴 = (𝑀‘𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 386   ∨ wo 880   = wceq 1658   ∈ wcel 2166   ≠ wne 3000   class class class wbr 4874  ‘cfv 6124  (class class class)co 6906  ⟨“cs3 13964  Basecbs 16223  distcds 16315  TarskiGcstrkg 25743  Itvcitv 25749  LineGclng 25750  cgrGccgrg 25823  pInvGcmir 25965

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-rep 4995  ax-sep 5006  ax-nul 5014  ax-pow 5066  ax-pr 5128  ax-un 7210  ax-cnex 10309  ax-resscn 10310  ax-1cn 10311  ax-icn 10312  ax-addcl 10313  ax-addrcl 10314  ax-mulcl 10315  ax-mulrcl 10316  ax-mulcom 10317  ax-addass 10318  ax-mulass 10319  ax-distr 10320  ax-i2m1 10321  ax-1ne0 10322  ax-1rid 10323  ax-rnegex 10324  ax-rrecex 10325  ax-cnre 10326  ax-pre-lttri 10327  ax-pre-lttrn 10328  ax-pre-ltadd 10329  ax-pre-mulgt0 10330

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ne 3001  df-nel 3104  df-ral 3123  df-rex 3124  df-reu 3125  df-rmo 3126  df-rab 3127  df-v 3417  df-sbc 3664  df-csb 3759  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-pss 3815  df-nul 4146  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4660  df-int 4699  df-iun 4743  df-br 4875  df-opab 4937  df-mpt 4954  df-tr 4977  df-id 5251  df-eprel 5256  df-po 5264  df-so 5265  df-fr 5302  df-we 5304  df-xp 5349  df-rel 5350  df-cnv 5351  df-co 5352  df-dm 5353  df-rn 5354  df-res 5355  df-ima 5356  df-pred 5921  df-ord 5967  df-on 5968  df-lim 5969  df-suc 5970  df-iota 6087  df-fun 6126  df-fn 6127  df-f 6128  df-f1 6129  df-fo 6130  df-f1o 6131  df-fv 6132  df-riota 6867  df-ov 6909  df-oprab 6910  df-mpt2 6911  df-om 7328  df-1st 7429  df-2nd 7430  df-wrecs 7673  df-recs 7735  df-rdg 7773  df-1o 7827  df-oadd 7831  df-er 8010  df-pm 8126  df-en 8224  df-dom 8225  df-sdom 8226  df-fin 8227  df-card 9079  df-cda 9306  df-pnf 10394  df-mnf 10395  df-xr 10396  df-ltxr 10397  df-le 10398  df-sub 10588  df-neg 10589  df-nn 11352  df-2 11415  df-3 11416  df-n0 11620  df-xnn0 11692  df-z 11706  df-uz 11970  df-fz 12621  df-fzo 12762  df-hash 13412  df-word 13576  df-concat 13632  df-s1 13657  df-s2 13970  df-s3 13971  df-trkgc 25761  df-trkgb 25762  df-trkgcb 25763  df-trkg 25766  df-cgrg 25824  df-mir 25966

This theorem is referenced by:  opphllem  26045