Theorem symquadlem 28668

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 28466	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ (𝐵𝐼𝐴))
10	2, 3, 4, 5, 6, 7, 7, 9	btwncolg1 28534	. . . . . . . . . 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 7362	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → (𝐵𝐿𝐴) = (𝐵𝐿𝐶))
14	13	eleq2d 2817	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → (𝐴 ∈ (𝐵𝐿𝐴) ↔ 𝐴 ∈ (𝐵𝐿𝐶)))
15	12	eqeq2d 2742	. . . . . . . . . 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 815	. . . . . . 7 ⊢ (𝜑 → ¬ 𝐴 = 𝐶)
19	18	neqned 2935	. . . . . 6 ⊢ (𝜑 → 𝐴 ≠ 𝐶)
20	19	ad2antrr 726	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝐴 ≠ 𝐶)
21	20	necomd 2983	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝐶 ≠ 𝐴)
22	21	neneqd 2933	. . 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 28537	. . . . . 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 2731	. . . . . . . 8 ⊢ (cgrG‘𝐺) = (cgrG‘𝐺)
38		simplr 768	. . . . . . . 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 28539	. . . . . . . . . 10 ⊢ (𝜑 → (𝐷 ∈ (𝑋𝐿𝐵) ∨ 𝑋 = 𝐵))
41	2, 3, 4, 5, 29, 6, 35, 40	colcom 28537	. . . . . . . . 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 28459	. . . . . . . . . 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 2737	. . . . . . . 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 28547	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝑋 − 𝐶) = (𝑥 − 𝐴))
53	2, 3, 4, 5, 6, 26, 7, 1	ncolcom 28540	. . . . . . . . . 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 2737	. . . . . . . . . . . . . . . . 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 28547	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝑋 − 𝐴) = (𝑥 − 𝐶))
62	2, 8, 4, 24, 30, 28, 38, 27, 61	tgcgrcomlr 28459	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝐴 − 𝑋) = (𝐶 − 𝑥))
63	2, 8, 4, 24, 27, 28	axtgcgrrflx 28441	. . . . . . . . . . . . . . 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 28495	. . . . . . . . . . . . . 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 28545	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝑥 ∈ (𝐶𝐿𝐴) ∨ 𝐶 = 𝐴))
66	2, 3, 4, 24, 27, 28, 38, 65	colcom 28537	. . . . . . . . . . . 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 2933	. . . . . . . . . . 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 28503	. . . . . . . . . . . . . 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 28545	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝑥 ∈ (𝐷𝐿𝐵) ∨ 𝐷 = 𝐵))
79	2, 3, 4, 24, 36, 34, 38, 78	colcom 28537	. . . . . . . . . . . 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 28624	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝑋 = 𝑥)
84	83	oveq1d 7361	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝑋 − 𝐴) = (𝑥 − 𝐴))
85	52, 84	eqtr4d 2769	. . . . . 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 28667	. . . . 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 28441	. . 3 ⊢ (𝜑 → (𝐵 − 𝐷) = (𝐷 − 𝐵))
91	2, 3, 4, 5, 6, 35, 29, 37, 35, 6, 8, 41, 90	lnext 28546	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩)
92	89, 91	r19.29a 3140	1 ⊢ (𝜑 → 𝐴 = (𝑀‘𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 847   = wceq 1541   ∈ wcel 2111   ≠ wne 2928   class class class wbr 5091  ‘cfv 6481  (class class class)co 7346  ⟨“cs3 14749  Basecbs 17120  distcds 17170  TarskiGcstrkg 28406  Itvcitv 28412  LineGclng 28413  cgrGccgrg 28489  pInvGcmir 28631

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668  ax-cnex 11062  ax-resscn 11063  ax-1cn 11064  ax-icn 11065  ax-addcl 11066  ax-addrcl 11067  ax-mulcl 11068  ax-mulrcl 11069  ax-mulcom 11070  ax-addass 11071  ax-mulass 11072  ax-distr 11073  ax-i2m1 11074  ax-1ne0 11075  ax-1rid 11076  ax-rnegex 11077  ax-rrecex 11078  ax-cnre 11079  ax-pre-lttri 11080  ax-pre-lttrn 11081  ax-pre-ltadd 11082  ax-pre-mulgt0 11083

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-tp 4581  df-op 4583  df-uni 4860  df-int 4898  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-tr 5199  df-id 5511  df-eprel 5516  df-po 5524  df-so 5525  df-fr 5569  df-we 5571  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-1st 7921  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-1o 8385  df-oadd 8389  df-er 8622  df-pm 8753  df-en 8870  df-dom 8871  df-sdom 8872  df-fin 8873  df-dju 9794  df-card 9832  df-pnf 11148  df-mnf 11149  df-xr 11150  df-ltxr 11151  df-le 11152  df-sub 11346  df-neg 11347  df-nn 12126  df-2 12188  df-3 12189  df-n0 12382  df-xnn0 12455  df-z 12469  df-uz 12733  df-fz 13408  df-fzo 13555  df-hash 14238  df-word 14421  df-concat 14478  df-s1 14504  df-s2 14755  df-s3 14756  df-trkgc 28427  df-trkgb 28428  df-trkgcb 28429  df-trkg 28432  df-cgrg 28490  df-mir 28632

This theorem is referenced by:  opphllem  28714