Theorem symquadlem 28623

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 28421	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ (𝐵𝐼𝐴))
10	2, 3, 4, 5, 6, 7, 7, 9	btwncolg1 28489	. . . . . . . . . 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 7406	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → (𝐵𝐿𝐴) = (𝐵𝐿𝐶))
14	13	eleq2d 2815	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → (𝐴 ∈ (𝐵𝐿𝐴) ↔ 𝐴 ∈ (𝐵𝐿𝐶)))
15	12	eqeq2d 2741	. . . . . . . . . 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 2933	. . . . . 6 ⊢ (𝜑 → 𝐴 ≠ 𝐶)
20	19	ad2antrr 726	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝐴 ≠ 𝐶)
21	20	necomd 2981	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝐶 ≠ 𝐴)
22	21	neneqd 2931	. . 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 28492	. . . . . 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 2730	. . . . . . . 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 28494	. . . . . . . . . 10 ⊢ (𝜑 → (𝐷 ∈ (𝑋𝐿𝐵) ∨ 𝑋 = 𝐵))
41	2, 3, 4, 5, 29, 6, 35, 40	colcom 28492	. . . . . . . . 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 28414	. . . . . . . . . 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 2736	. . . . . . . 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 28502	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝑋 − 𝐶) = (𝑥 − 𝐴))
53	2, 3, 4, 5, 6, 26, 7, 1	ncolcom 28495	. . . . . . . . . 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 2736	. . . . . . . . . . . . . . . . 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 28502	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝑋 − 𝐴) = (𝑥 − 𝐶))
62	2, 8, 4, 24, 30, 28, 38, 27, 61	tgcgrcomlr 28414	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝐴 − 𝑋) = (𝐶 − 𝑥))
63	2, 8, 4, 24, 27, 28	axtgcgrrflx 28396	. . . . . . . . . . . . . . 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 28450	. . . . . . . . . . . . . 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 28500	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝑥 ∈ (𝐶𝐿𝐴) ∨ 𝐶 = 𝐴))
66	2, 3, 4, 24, 27, 28, 38, 65	colcom 28492	. . . . . . . . . . . 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 2931	. . . . . . . . . . 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 28458	. . . . . . . . . . . . . 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 28500	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝑥 ∈ (𝐷𝐿𝐵) ∨ 𝐷 = 𝐵))
79	2, 3, 4, 24, 36, 34, 38, 78	colcom 28492	. . . . . . . . . . . 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 28579	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝑋 = 𝑥)
84	83	oveq1d 7405	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝑋 − 𝐴) = (𝑥 − 𝐴))
85	52, 84	eqtr4d 2768	. . . . . 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 28622	. . . . 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 28396	. . 3 ⊢ (𝜑 → (𝐵 − 𝐷) = (𝐷 − 𝐵))
91	2, 3, 4, 5, 6, 35, 29, 37, 35, 6, 8, 41, 90	lnext 28501	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩)
92	89, 91	r19.29a 3142	1 ⊢ (𝜑 → 𝐴 = (𝑀‘𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 847   = wceq 1540   ∈ wcel 2109   ≠ wne 2926   class class class wbr 5110  ‘cfv 6514  (class class class)co 7390  ⟨“cs3 14815  Basecbs 17186  distcds 17236  TarskiGcstrkg 28361  Itvcitv 28367  LineGclng 28368  cgrGccgrg 28444  pInvGcmir 28586

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-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-trkg 28387  df-cgrg 28445  df-mir 28587

This theorem is referenced by:  opphllem  28669