Theorem opphllem 25627

Description: Lemma 8.24 of [Schwabhauser] p. 66. This is used later for mideulem 25628 and later for opphl 25646. (Contributed by Thierry Arnoux, 21-Dec-2019.)

Hypotheses

Ref	Expression
colperpex.p	⊢ 𝑃 = (Base‘𝐺)
colperpex.d	⊢ − = (dist‘𝐺)
colperpex.i	⊢ 𝐼 = (Itv‘𝐺)
colperpex.l	⊢ 𝐿 = (LineG‘𝐺)
colperpex.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
mideu.s	⊢ 𝑆 = (pInvG‘𝐺)
mideu.1	⊢ (𝜑 → 𝐴 ∈ 𝑃)
mideu.2	⊢ (𝜑 → 𝐵 ∈ 𝑃)
mideulem.1	⊢ (𝜑 → 𝐴 ≠ 𝐵)
mideulem.2	⊢ (𝜑 → 𝑄 ∈ 𝑃)
mideulem.3	⊢ (𝜑 → 𝑂 ∈ 𝑃)
mideulem.4	⊢ (𝜑 → 𝑇 ∈ 𝑃)
mideulem.5	⊢ (𝜑 → (𝐴𝐿𝐵)(⟂G‘𝐺)(𝑄𝐿𝐵))
mideulem.6	⊢ (𝜑 → (𝐴𝐿𝐵)(⟂G‘𝐺)(𝐴𝐿𝑂))
mideulem.7	⊢ (𝜑 → 𝑇 ∈ (𝐴𝐿𝐵))
mideulem.8	⊢ (𝜑 → 𝑇 ∈ (𝑄𝐼𝑂))
opphllem.1	⊢ (𝜑 → 𝑅 ∈ 𝑃)
opphllem.2	⊢ (𝜑 → 𝑅 ∈ (𝐵𝐼𝑄))
opphllem.3	⊢ (𝜑 → (𝐴 − 𝑂) = (𝐵 − 𝑅))

Assertion

Ref	Expression
opphllem	⊢ (𝜑 → ∃𝑥 ∈ 𝑃 (𝐵 = ((𝑆‘𝑥)‘𝐴) ∧ 𝑂 = ((𝑆‘𝑥)‘𝑅)))

Distinct variable groups:   𝑥, −   𝑥,𝐴   𝑥,𝐵   𝑥,𝐼   𝑥,𝑂   𝑥,𝑃   𝑥,𝑄   𝑥,𝑅   𝑥,𝑇   𝜑,𝑥

Allowed substitution hints:   𝑆(𝑥)   𝐺(𝑥)   𝐿(𝑥)

Proof of Theorem opphllem

Dummy variables 𝑚 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		colperpex.p	. . . 4 ⊢ 𝑃 = (Base‘𝐺)
2		colperpex.d	. . . 4 ⊢ − = (dist‘𝐺)
3		colperpex.i	. . . 4 ⊢ 𝐼 = (Itv‘𝐺)
4		colperpex.l	. . . 4 ⊢ 𝐿 = (LineG‘𝐺)
5		mideu.s	. . . 4 ⊢ 𝑆 = (pInvG‘𝐺)
6		colperpex.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
7	6	adantr 481	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝐺 ∈ TarskiG)
8		eqid 2622	. . . 4 ⊢ (𝑆‘𝑥) = (𝑆‘𝑥)
9		mideu.2	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
10	9	adantr 481	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝐵 ∈ 𝑃)
11		mideulem.3	. . . . 5 ⊢ (𝜑 → 𝑂 ∈ 𝑃)
12	11	adantr 481	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝑂 ∈ 𝑃)
13		mideu.1	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
14	13	adantr 481	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝐴 ∈ 𝑃)
15		opphllem.1	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ 𝑃)
16	15	adantr 481	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝑅 ∈ 𝑃)
17		simprl 794	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝑥 ∈ 𝑃)
18		mideulem.1	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ≠ 𝐵)
19	18	necomd 2849	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 ≠ 𝐴)
20	19	neneqd 2799	. . . . . . . . . . . 12 ⊢ (𝜑 → ¬ 𝐵 = 𝐴)
21	20	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑂 ∈ (𝐵𝐿𝐴)) → ¬ 𝐵 = 𝐴)
22		mideulem.6	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐴𝐿𝐵)(⟂G‘𝐺)(𝐴𝐿𝑂))
23	4, 6, 22	perpln2 25606	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐴𝐿𝑂) ∈ ran 𝐿)
24	1, 3, 4, 6, 13, 11, 23	tglnne 25523	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ≠ 𝑂)
25	24	necomd 2849	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑂 ≠ 𝐴)
26	25	neneqd 2799	. . . . . . . . . . . 12 ⊢ (𝜑 → ¬ 𝑂 = 𝐴)
27	26	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑂 ∈ (𝐵𝐿𝐴)) → ¬ 𝑂 = 𝐴)
28	21, 27	jca 554	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑂 ∈ (𝐵𝐿𝐴)) → (¬ 𝐵 = 𝐴 ∧ ¬ 𝑂 = 𝐴))
29	6	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑂 ∈ (𝐵𝐿𝐴)) → 𝐺 ∈ TarskiG)
30	9	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑂 ∈ (𝐵𝐿𝐴)) → 𝐵 ∈ 𝑃)
31	13	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑂 ∈ (𝐵𝐿𝐴)) → 𝐴 ∈ 𝑃)
32	11	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑂 ∈ (𝐵𝐿𝐴)) → 𝑂 ∈ 𝑃)
33	1, 3, 4, 6, 9, 13, 19	tglinerflx2 25529	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ∈ (𝐵𝐿𝐴))
34	1, 3, 4, 6, 13, 9, 18	tglinecom 25530	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐴𝐿𝐵) = (𝐵𝐿𝐴))
35	34, 22	eqbrtrrd 4677	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐵𝐿𝐴)(⟂G‘𝐺)(𝐴𝐿𝑂))
36	1, 2, 3, 4, 6, 9, 13, 33, 11, 35	perprag 25618	. . . . . . . . . . . . 13 ⊢ (𝜑 → ⟨“𝐵𝐴𝑂”⟩ ∈ (∟G‘𝐺))
37	36	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑂 ∈ (𝐵𝐿𝐴)) → ⟨“𝐵𝐴𝑂”⟩ ∈ (∟G‘𝐺))
38		simpr 477	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑂 ∈ (𝐵𝐿𝐴)) → 𝑂 ∈ (𝐵𝐿𝐴))
39	38	orcd 407	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑂 ∈ (𝐵𝐿𝐴)) → (𝑂 ∈ (𝐵𝐿𝐴) ∨ 𝐵 = 𝐴))
40	1, 2, 3, 4, 5, 29, 30, 31, 32, 37, 39	ragflat3 25601	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑂 ∈ (𝐵𝐿𝐴)) → (𝐵 = 𝐴 ∨ 𝑂 = 𝐴))
41		oran 517	. . . . . . . . . . 11 ⊢ ((𝐵 = 𝐴 ∨ 𝑂 = 𝐴) ↔ ¬ (¬ 𝐵 = 𝐴 ∧ ¬ 𝑂 = 𝐴))
42	40, 41	sylib 208	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑂 ∈ (𝐵𝐿𝐴)) → ¬ (¬ 𝐵 = 𝐴 ∧ ¬ 𝑂 = 𝐴))
43	28, 42	pm2.65da 600	. . . . . . . . 9 ⊢ (𝜑 → ¬ 𝑂 ∈ (𝐵𝐿𝐴))
44	43	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → ¬ 𝑂 ∈ (𝐵𝐿𝐴))
45	34	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → (𝐴𝐿𝐵) = (𝐵𝐿𝐴))
46	44, 45	neleqtrrd 2723	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → ¬ 𝑂 ∈ (𝐴𝐿𝐵))
47	18	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝐴 ≠ 𝐵)
48	47	neneqd 2799	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → ¬ 𝐴 = 𝐵)
49	46, 48	jca 554	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → (¬ 𝑂 ∈ (𝐴𝐿𝐵) ∧ ¬ 𝐴 = 𝐵))
50		pm4.56 516	. . . . . 6 ⊢ ((¬ 𝑂 ∈ (𝐴𝐿𝐵) ∧ ¬ 𝐴 = 𝐵) ↔ ¬ (𝑂 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))
51	49, 50	sylib 208	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → ¬ (𝑂 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))
52	1, 4, 3, 7, 14, 10, 12, 51	ncolrot2 25458	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → ¬ (𝐵 ∈ (𝑂𝐿𝐴) ∨ 𝑂 = 𝐴))
53		simprrr 805	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝑥 ∈ (𝑅𝐼𝑂))
54	1, 2, 3, 7, 16, 17, 12, 53	tgbtwncom 25383	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝑥 ∈ (𝑂𝐼𝑅))
55		mideulem.4	. . . . . . . 8 ⊢ (𝜑 → 𝑇 ∈ 𝑃)
56	55	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝑇 ∈ 𝑃)
57		mideulem.7	. . . . . . . 8 ⊢ (𝜑 → 𝑇 ∈ (𝐴𝐿𝐵))
58	57	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝑇 ∈ (𝐴𝐿𝐵))
59		simprrl 804	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝑥 ∈ (𝑇𝐼𝐵))
60	1, 3, 4, 7, 56, 14, 10, 17, 58, 59	coltr3 25543	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝑥 ∈ (𝐴𝐿𝐵))
61	43, 34	neleqtrrd 2723	. . . . . . 7 ⊢ (𝜑 → ¬ 𝑂 ∈ (𝐴𝐿𝐵))
62	61	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → ¬ 𝑂 ∈ (𝐴𝐿𝐵))
63		nelne2 2891	. . . . . 6 ⊢ ((𝑥 ∈ (𝐴𝐿𝐵) ∧ ¬ 𝑂 ∈ (𝐴𝐿𝐵)) → 𝑥 ≠ 𝑂)
64	60, 62, 63	syl2anc 693	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝑥 ≠ 𝑂)
65	1, 2, 3, 7, 12, 17, 16, 54, 64	tgbtwnne 25385	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝑂 ≠ 𝑅)
66	1, 2, 3, 4, 5, 6, 9, 13, 11	israg 25592	. . . . . . . 8 ⊢ (𝜑 → (⟨“𝐵𝐴𝑂”⟩ ∈ (∟G‘𝐺) ↔ (𝐵 − 𝑂) = (𝐵 − ((𝑆‘𝐴)‘𝑂))))
67	36, 66	mpbid 222	. . . . . . 7 ⊢ (𝜑 → (𝐵 − 𝑂) = (𝐵 − ((𝑆‘𝐴)‘𝑂)))
68	67	ad3antrrr 766	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠 ∈ 𝑃) ∧ (𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 − 𝑠) = (𝑥 − 𝑅))) → (𝐵 − 𝑂) = (𝐵 − ((𝑆‘𝐴)‘𝑂)))
69	6	ad3antrrr 766	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠 ∈ 𝑃) ∧ (𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 − 𝑠) = (𝑥 − 𝑅))) → 𝐺 ∈ TarskiG)
70		eqid 2622	. . . . . . . . 9 ⊢ (𝑆‘𝐴) = (𝑆‘𝐴)
71	1, 2, 3, 4, 5, 7, 14, 70, 12	mircl 25556	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → ((𝑆‘𝐴)‘𝑂) ∈ 𝑃)
72	71	ad2antrr 762	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠 ∈ 𝑃) ∧ (𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 − 𝑠) = (𝑥 − 𝑅))) → ((𝑆‘𝐴)‘𝑂) ∈ 𝑃)
73	13	ad3antrrr 766	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠 ∈ 𝑃) ∧ (𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 − 𝑠) = (𝑥 − 𝑅))) → 𝐴 ∈ 𝑃)
74	11	ad3antrrr 766	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠 ∈ 𝑃) ∧ (𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 − 𝑠) = (𝑥 − 𝑅))) → 𝑂 ∈ 𝑃)
75	15	ad3antrrr 766	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠 ∈ 𝑃) ∧ (𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 − 𝑠) = (𝑥 − 𝑅))) → 𝑅 ∈ 𝑃)
76	9	ad3antrrr 766	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠 ∈ 𝑃) ∧ (𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 − 𝑠) = (𝑥 − 𝑅))) → 𝐵 ∈ 𝑃)
77		simplr 792	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠 ∈ 𝑃) ∧ (𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 − 𝑠) = (𝑥 − 𝑅))) → 𝑠 ∈ 𝑃)
78	1, 2, 3, 4, 5, 69, 73, 70, 74	mirbtwn 25553	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠 ∈ 𝑃) ∧ (𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 − 𝑠) = (𝑥 − 𝑅))) → 𝐴 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑂))
79		eqid 2622	. . . . . . . . 9 ⊢ (𝑆‘𝐵) = (𝑆‘𝐵)
80	1, 2, 3, 4, 5, 69, 76, 79, 77	mirbtwn 25553	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠 ∈ 𝑃) ∧ (𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 − 𝑠) = (𝑥 − 𝑅))) → 𝐵 ∈ (((𝑆‘𝐵)‘𝑠)𝐼𝑠))
81		simpr 477	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠 ∈ 𝑃) ∧ (𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 − 𝑠) = (𝑥 − 𝑅))) ∧ 𝑚 ∈ 𝑃) ∧ 𝑅 = ((𝑆‘𝑚)‘𝑠)) → 𝑅 = ((𝑆‘𝑚)‘𝑠))
82	69	ad2antrr 762	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠 ∈ 𝑃) ∧ (𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 − 𝑠) = (𝑥 − 𝑅))) ∧ 𝑚 ∈ 𝑃) ∧ 𝑅 = ((𝑆‘𝑚)‘𝑠)) → 𝐺 ∈ TarskiG)
83	73	ad2antrr 762	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠 ∈ 𝑃) ∧ (𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 − 𝑠) = (𝑥 − 𝑅))) ∧ 𝑚 ∈ 𝑃) ∧ 𝑅 = ((𝑆‘𝑚)‘𝑠)) → 𝐴 ∈ 𝑃)
84	76	ad2antrr 762	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠 ∈ 𝑃) ∧ (𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 − 𝑠) = (𝑥 − 𝑅))) ∧ 𝑚 ∈ 𝑃) ∧ 𝑅 = ((𝑆‘𝑚)‘𝑠)) → 𝐵 ∈ 𝑃)
85	47	ad4antr 768	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠 ∈ 𝑃) ∧ (𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 − 𝑠) = (𝑥 − 𝑅))) ∧ 𝑚 ∈ 𝑃) ∧ 𝑅 = ((𝑆‘𝑚)‘𝑠)) → 𝐴 ≠ 𝐵)
86		mideulem.2	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑄 ∈ 𝑃)
87	86	ad5antr 770	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠 ∈ 𝑃) ∧ (𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 − 𝑠) = (𝑥 − 𝑅))) ∧ 𝑚 ∈ 𝑃) ∧ 𝑅 = ((𝑆‘𝑚)‘𝑠)) → 𝑄 ∈ 𝑃)
88	74	ad2antrr 762	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠 ∈ 𝑃) ∧ (𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 − 𝑠) = (𝑥 − 𝑅))) ∧ 𝑚 ∈ 𝑃) ∧ 𝑅 = ((𝑆‘𝑚)‘𝑠)) → 𝑂 ∈ 𝑃)
89	56	ad4antr 768	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠 ∈ 𝑃) ∧ (𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 − 𝑠) = (𝑥 − 𝑅))) ∧ 𝑚 ∈ 𝑃) ∧ 𝑅 = ((𝑆‘𝑚)‘𝑠)) → 𝑇 ∈ 𝑃)
90		mideulem.5	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐴𝐿𝐵)(⟂G‘𝐺)(𝑄𝐿𝐵))
91	90	ad5antr 770	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠 ∈ 𝑃) ∧ (𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 − 𝑠) = (𝑥 − 𝑅))) ∧ 𝑚 ∈ 𝑃) ∧ 𝑅 = ((𝑆‘𝑚)‘𝑠)) → (𝐴𝐿𝐵)(⟂G‘𝐺)(𝑄𝐿𝐵))
92	22	ad5antr 770	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠 ∈ 𝑃) ∧ (𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 − 𝑠) = (𝑥 − 𝑅))) ∧ 𝑚 ∈ 𝑃) ∧ 𝑅 = ((𝑆‘𝑚)‘𝑠)) → (𝐴𝐿𝐵)(⟂G‘𝐺)(𝐴𝐿𝑂))
93	58	ad4antr 768	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠 ∈ 𝑃) ∧ (𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 − 𝑠) = (𝑥 − 𝑅))) ∧ 𝑚 ∈ 𝑃) ∧ 𝑅 = ((𝑆‘𝑚)‘𝑠)) → 𝑇 ∈ (𝐴𝐿𝐵))
94		mideulem.8	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑇 ∈ (𝑄𝐼𝑂))
95	94	ad5antr 770	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠 ∈ 𝑃) ∧ (𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 − 𝑠) = (𝑥 − 𝑅))) ∧ 𝑚 ∈ 𝑃) ∧ 𝑅 = ((𝑆‘𝑚)‘𝑠)) → 𝑇 ∈ (𝑄𝐼𝑂))
96	75	ad2antrr 762	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠 ∈ 𝑃) ∧ (𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 − 𝑠) = (𝑥 − 𝑅))) ∧ 𝑚 ∈ 𝑃) ∧ 𝑅 = ((𝑆‘𝑚)‘𝑠)) → 𝑅 ∈ 𝑃)
97		opphllem.2	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑅 ∈ (𝐵𝐼𝑄))
98	97	ad5antr 770	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠 ∈ 𝑃) ∧ (𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 − 𝑠) = (𝑥 − 𝑅))) ∧ 𝑚 ∈ 𝑃) ∧ 𝑅 = ((𝑆‘𝑚)‘𝑠)) → 𝑅 ∈ (𝐵𝐼𝑄))
99		opphllem.3	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐴 − 𝑂) = (𝐵 − 𝑅))
100	99	ad5antr 770	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠 ∈ 𝑃) ∧ (𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 − 𝑠) = (𝑥 − 𝑅))) ∧ 𝑚 ∈ 𝑃) ∧ 𝑅 = ((𝑆‘𝑚)‘𝑠)) → (𝐴 − 𝑂) = (𝐵 − 𝑅))
101	17	ad2antrr 762	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠 ∈ 𝑃) ∧ (𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 − 𝑠) = (𝑥 − 𝑅))) → 𝑥 ∈ 𝑃)
102	101	ad2antrr 762	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠 ∈ 𝑃) ∧ (𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 − 𝑠) = (𝑥 − 𝑅))) ∧ 𝑚 ∈ 𝑃) ∧ 𝑅 = ((𝑆‘𝑚)‘𝑠)) → 𝑥 ∈ 𝑃)
103		simp-5r 809	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠 ∈ 𝑃) ∧ (𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 − 𝑠) = (𝑥 − 𝑅))) ∧ 𝑚 ∈ 𝑃) ∧ 𝑅 = ((𝑆‘𝑚)‘𝑠)) → (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂))))
104	103	simprd 479	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠 ∈ 𝑃) ∧ (𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 − 𝑠) = (𝑥 − 𝑅))) ∧ 𝑚 ∈ 𝑃) ∧ 𝑅 = ((𝑆‘𝑚)‘𝑠)) → (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))
105	104	simpld 475	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠 ∈ 𝑃) ∧ (𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 − 𝑠) = (𝑥 − 𝑅))) ∧ 𝑚 ∈ 𝑃) ∧ 𝑅 = ((𝑆‘𝑚)‘𝑠)) → 𝑥 ∈ (𝑇𝐼𝐵))
106	104	simprd 479	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠 ∈ 𝑃) ∧ (𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 − 𝑠) = (𝑥 − 𝑅))) ∧ 𝑚 ∈ 𝑃) ∧ 𝑅 = ((𝑆‘𝑚)‘𝑠)) → 𝑥 ∈ (𝑅𝐼𝑂))
107	77	ad2antrr 762	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠 ∈ 𝑃) ∧ (𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 − 𝑠) = (𝑥 − 𝑅))) ∧ 𝑚 ∈ 𝑃) ∧ 𝑅 = ((𝑆‘𝑚)‘𝑠)) → 𝑠 ∈ 𝑃)
108		simpllr 799	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠 ∈ 𝑃) ∧ (𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 − 𝑠) = (𝑥 − 𝑅))) ∧ 𝑚 ∈ 𝑃) ∧ 𝑅 = ((𝑆‘𝑚)‘𝑠)) → (𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 − 𝑠) = (𝑥 − 𝑅)))
109	108	simpld 475	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠 ∈ 𝑃) ∧ (𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 − 𝑠) = (𝑥 − 𝑅))) ∧ 𝑚 ∈ 𝑃) ∧ 𝑅 = ((𝑆‘𝑚)‘𝑠)) → 𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠))
110		simprr 796	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠 ∈ 𝑃) ∧ (𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 − 𝑠) = (𝑥 − 𝑅))) → (𝑥 − 𝑠) = (𝑥 − 𝑅))
111	110	ad2antrr 762	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠 ∈ 𝑃) ∧ (𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 − 𝑠) = (𝑥 − 𝑅))) ∧ 𝑚 ∈ 𝑃) ∧ 𝑅 = ((𝑆‘𝑚)‘𝑠)) → (𝑥 − 𝑠) = (𝑥 − 𝑅))
112		simplr 792	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠 ∈ 𝑃) ∧ (𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 − 𝑠) = (𝑥 − 𝑅))) ∧ 𝑚 ∈ 𝑃) ∧ 𝑅 = ((𝑆‘𝑚)‘𝑠)) → 𝑚 ∈ 𝑃)
113	1, 2, 3, 4, 82, 5, 83, 84, 85, 87, 88, 89, 91, 92, 93, 95, 96, 98, 100, 102, 105, 106, 107, 109, 111, 112, 81	mideulem2 25626	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠 ∈ 𝑃) ∧ (𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 − 𝑠) = (𝑥 − 𝑅))) ∧ 𝑚 ∈ 𝑃) ∧ 𝑅 = ((𝑆‘𝑚)‘𝑠)) → 𝐵 = 𝑚)
114	113	eqcomd 2628	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠 ∈ 𝑃) ∧ (𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 − 𝑠) = (𝑥 − 𝑅))) ∧ 𝑚 ∈ 𝑃) ∧ 𝑅 = ((𝑆‘𝑚)‘𝑠)) → 𝑚 = 𝐵)
115	114	fveq2d 6195	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠 ∈ 𝑃) ∧ (𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 − 𝑠) = (𝑥 − 𝑅))) ∧ 𝑚 ∈ 𝑃) ∧ 𝑅 = ((𝑆‘𝑚)‘𝑠)) → (𝑆‘𝑚) = (𝑆‘𝐵))
116	115	fveq1d 6193	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠 ∈ 𝑃) ∧ (𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 − 𝑠) = (𝑥 − 𝑅))) ∧ 𝑚 ∈ 𝑃) ∧ 𝑅 = ((𝑆‘𝑚)‘𝑠)) → ((𝑆‘𝑚)‘𝑠) = ((𝑆‘𝐵)‘𝑠))
117	81, 116	eqtrd 2656	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠 ∈ 𝑃) ∧ (𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 − 𝑠) = (𝑥 − 𝑅))) ∧ 𝑚 ∈ 𝑃) ∧ 𝑅 = ((𝑆‘𝑚)‘𝑠)) → 𝑅 = ((𝑆‘𝐵)‘𝑠))
118		eqid 2622	. . . . . . . . . . 11 ⊢ (𝑆‘𝑚) = (𝑆‘𝑚)
119	1, 2, 3, 4, 5, 69, 118, 77, 75, 101, 110	midexlem 25587	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠 ∈ 𝑃) ∧ (𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 − 𝑠) = (𝑥 − 𝑅))) → ∃𝑚 ∈ 𝑃 𝑅 = ((𝑆‘𝑚)‘𝑠))
120	117, 119	r19.29a 3078	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠 ∈ 𝑃) ∧ (𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 − 𝑠) = (𝑥 − 𝑅))) → 𝑅 = ((𝑆‘𝐵)‘𝑠))
121	120	oveq1d 6665	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠 ∈ 𝑃) ∧ (𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 − 𝑠) = (𝑥 − 𝑅))) → (𝑅𝐼𝑠) = (((𝑆‘𝐵)‘𝑠)𝐼𝑠))
122	80, 121	eleqtrrd 2704	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠 ∈ 𝑃) ∧ (𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 − 𝑠) = (𝑥 − 𝑅))) → 𝐵 ∈ (𝑅𝐼𝑠))
123	1, 2, 3, 4, 5, 69, 73, 70, 74	mircgr 25552	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠 ∈ 𝑃) ∧ (𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 − 𝑠) = (𝑥 − 𝑅))) → (𝐴 − ((𝑆‘𝐴)‘𝑂)) = (𝐴 − 𝑂))
124	99	ad3antrrr 766	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠 ∈ 𝑃) ∧ (𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 − 𝑠) = (𝑥 − 𝑅))) → (𝐴 − 𝑂) = (𝐵 − 𝑅))
125	123, 124	eqtrd 2656	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠 ∈ 𝑃) ∧ (𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 − 𝑠) = (𝑥 − 𝑅))) → (𝐴 − ((𝑆‘𝐴)‘𝑂)) = (𝐵 − 𝑅))
126	1, 2, 3, 69, 73, 72, 76, 75, 125	tgcgrcomlr 25375	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠 ∈ 𝑃) ∧ (𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 − 𝑠) = (𝑥 − 𝑅))) → (((𝑆‘𝐴)‘𝑂) − 𝐴) = (𝑅 − 𝐵))
127	120	oveq2d 6666	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠 ∈ 𝑃) ∧ (𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 − 𝑠) = (𝑥 − 𝑅))) → (𝐵 − 𝑅) = (𝐵 − ((𝑆‘𝐵)‘𝑠)))
128	1, 2, 3, 4, 5, 69, 76, 79, 77	mircgr 25552	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠 ∈ 𝑃) ∧ (𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 − 𝑠) = (𝑥 − 𝑅))) → (𝐵 − ((𝑆‘𝐵)‘𝑠)) = (𝐵 − 𝑠))
129	124, 127, 128	3eqtrd 2660	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠 ∈ 𝑃) ∧ (𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 − 𝑠) = (𝑥 − 𝑅))) → (𝐴 − 𝑂) = (𝐵 − 𝑠))
130	1, 2, 3, 69, 72, 73, 74, 75, 76, 77, 78, 122, 126, 129	tgcgrextend 25380	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠 ∈ 𝑃) ∧ (𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 − 𝑠) = (𝑥 − 𝑅))) → (((𝑆‘𝐴)‘𝑂) − 𝑂) = (𝑅 − 𝑠))
131	1, 2, 3, 69, 72, 75	axtgcgrrflx 25361	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠 ∈ 𝑃) ∧ (𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 − 𝑠) = (𝑥 − 𝑅))) → (((𝑆‘𝐴)‘𝑂) − 𝑅) = (𝑅 − ((𝑆‘𝐴)‘𝑂)))
132	1, 2, 3, 69, 74, 75	axtgcgrrflx 25361	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠 ∈ 𝑃) ∧ (𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 − 𝑠) = (𝑥 − 𝑅))) → (𝑂 − 𝑅) = (𝑅 − 𝑂))
133	53	ad2antrr 762	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠 ∈ 𝑃) ∧ (𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 − 𝑠) = (𝑥 − 𝑅))) → 𝑥 ∈ (𝑅𝐼𝑂))
134		simprl 794	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠 ∈ 𝑃) ∧ (𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 − 𝑠) = (𝑥 − 𝑅))) → 𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠))
135	1, 2, 3, 69, 72, 101, 77, 134	tgbtwncom 25383	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠 ∈ 𝑃) ∧ (𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 − 𝑠) = (𝑥 − 𝑅))) → 𝑥 ∈ (𝑠𝐼((𝑆‘𝐴)‘𝑂)))
136	1, 2, 3, 69, 101, 77, 101, 75, 110	tgcgrcomlr 25375	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠 ∈ 𝑃) ∧ (𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 − 𝑠) = (𝑥 − 𝑅))) → (𝑠 − 𝑥) = (𝑅 − 𝑥))
137	136	eqcomd 2628	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠 ∈ 𝑃) ∧ (𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 − 𝑠) = (𝑥 − 𝑅))) → (𝑅 − 𝑥) = (𝑠 − 𝑥))
138	36	ad3antrrr 766	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠 ∈ 𝑃) ∧ (𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 − 𝑠) = (𝑥 − 𝑅))) → ⟨“𝐵𝐴𝑂”⟩ ∈ (∟G‘𝐺))
139	47	necomd 2849	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝐵 ≠ 𝐴)
140	139	ad2antrr 762	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠 ∈ 𝑃) ∧ (𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 − 𝑠) = (𝑥 − 𝑅))) → 𝐵 ≠ 𝐴)
141	60	ad2antrr 762	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠 ∈ 𝑃) ∧ (𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 − 𝑠) = (𝑥 − 𝑅))) → 𝑥 ∈ (𝐴𝐿𝐵))
142	141	orcd 407	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠 ∈ 𝑃) ∧ (𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 − 𝑠) = (𝑥 − 𝑅))) → (𝑥 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))
143	1, 4, 3, 69, 73, 76, 101, 142	colcom 25453	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠 ∈ 𝑃) ∧ (𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 − 𝑠) = (𝑥 − 𝑅))) → (𝑥 ∈ (𝐵𝐿𝐴) ∨ 𝐵 = 𝐴))
144	1, 4, 3, 69, 76, 73, 101, 143	colrot1 25454	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠 ∈ 𝑃) ∧ (𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 − 𝑠) = (𝑥 − 𝑅))) → (𝐵 ∈ (𝐴𝐿𝑥) ∨ 𝐴 = 𝑥))
145	1, 2, 3, 4, 5, 69, 76, 73, 74, 101, 138, 140, 144	ragcol 25594	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠 ∈ 𝑃) ∧ (𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 − 𝑠) = (𝑥 − 𝑅))) → ⟨“𝑥𝐴𝑂”⟩ ∈ (∟G‘𝐺))
146	1, 2, 3, 4, 5, 69, 101, 73, 74	israg 25592	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠 ∈ 𝑃) ∧ (𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 − 𝑠) = (𝑥 − 𝑅))) → (⟨“𝑥𝐴𝑂”⟩ ∈ (∟G‘𝐺) ↔ (𝑥 − 𝑂) = (𝑥 − ((𝑆‘𝐴)‘𝑂))))
147	145, 146	mpbid 222	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠 ∈ 𝑃) ∧ (𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 − 𝑠) = (𝑥 − 𝑅))) → (𝑥 − 𝑂) = (𝑥 − ((𝑆‘𝐴)‘𝑂)))
148	1, 2, 3, 69, 75, 101, 74, 77, 101, 72, 133, 135, 137, 147	tgcgrextend 25380	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠 ∈ 𝑃) ∧ (𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 − 𝑠) = (𝑥 − 𝑅))) → (𝑅 − 𝑂) = (𝑠 − ((𝑆‘𝐴)‘𝑂)))
149	132, 148	eqtrd 2656	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠 ∈ 𝑃) ∧ (𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 − 𝑠) = (𝑥 − 𝑅))) → (𝑂 − 𝑅) = (𝑠 − ((𝑆‘𝐴)‘𝑂)))
150	1, 2, 3, 69, 72, 73, 74, 75, 75, 76, 77, 72, 78, 122, 130, 129, 131, 149	tgifscgr 25403	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠 ∈ 𝑃) ∧ (𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 − 𝑠) = (𝑥 − 𝑅))) → (𝐴 − 𝑅) = (𝐵 − ((𝑆‘𝐴)‘𝑂)))
151	68, 150	eqtr4d 2659	. . . . 5 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠 ∈ 𝑃) ∧ (𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 − 𝑠) = (𝑥 − 𝑅))) → (𝐵 − 𝑂) = (𝐴 − 𝑅))
152	1, 2, 3, 7, 71, 17, 17, 16	axtgsegcon 25363	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → ∃𝑠 ∈ 𝑃 (𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 − 𝑠) = (𝑥 − 𝑅)))
153	151, 152	r19.29a 3078	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → (𝐵 − 𝑂) = (𝐴 − 𝑅))
154	99	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → (𝐴 − 𝑂) = (𝐵 − 𝑅))
155	1, 2, 3, 7, 14, 12, 10, 16, 154	tgcgrcomlr 25375	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → (𝑂 − 𝐴) = (𝑅 − 𝐵))
156	143, 152	r19.29a 3078	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → (𝑥 ∈ (𝐵𝐿𝐴) ∨ 𝐵 = 𝐴))
157	1, 4, 3, 7, 12, 16, 17, 54	btwncolg1 25450	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → (𝑥 ∈ (𝑂𝐿𝑅) ∨ 𝑂 = 𝑅))
158	1, 2, 3, 4, 5, 7, 8, 10, 12, 14, 16, 17, 52, 65, 153, 155, 156, 157	symquadlem 25584	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝐵 = ((𝑆‘𝑥)‘𝐴))
159	1, 2, 3, 4, 5, 7, 17, 8, 14	mirbtwn 25553	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝑥 ∈ (((𝑆‘𝑥)‘𝐴)𝐼𝐴))
160	158	oveq1d 6665	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → (𝐵𝐼𝐴) = (((𝑆‘𝑥)‘𝐴)𝐼𝐴))
161	159, 160	eleqtrrd 2704	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝑥 ∈ (𝐵𝐼𝐴))
162	1, 2, 3, 7, 10, 17, 14, 161	tgbtwncom 25383	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝑥 ∈ (𝐴𝐼𝐵))
163	1, 2, 3, 7, 14, 10	axtgcgrrflx 25361	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → (𝐴 − 𝐵) = (𝐵 − 𝐴))
164	158	oveq2d 6666	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → (𝑥 − 𝐵) = (𝑥 − ((𝑆‘𝑥)‘𝐴)))
165	1, 2, 3, 4, 5, 7, 17, 8, 14	mircgr 25552	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → (𝑥 − ((𝑆‘𝑥)‘𝐴)) = (𝑥 − 𝐴))
166	164, 165	eqtrd 2656	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → (𝑥 − 𝐵) = (𝑥 − 𝐴))
167	1, 2, 3, 7, 14, 17, 10, 12, 10, 17, 14, 16, 162, 161, 163, 166, 154, 153	tgifscgr 25403	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → (𝑥 − 𝑂) = (𝑥 − 𝑅))
168	1, 2, 3, 4, 5, 7, 17, 8, 16, 12, 167, 54	ismir 25554	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝑂 = ((𝑆‘𝑥)‘𝑅))
169	158, 168	jca 554	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → (𝐵 = ((𝑆‘𝑥)‘𝐴) ∧ 𝑂 = ((𝑆‘𝑥)‘𝑅)))
170	1, 2, 3, 6, 86, 55, 11, 94	tgbtwncom 25383	. . 3 ⊢ (𝜑 → 𝑇 ∈ (𝑂𝐼𝑄))
171	1, 2, 3, 6, 11, 9, 86, 55, 15, 170, 97	axtgpasch 25366	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))
172	169, 171	reximddv 3018	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 (𝐵 = ((𝑆‘𝑥)‘𝐴) ∧ 𝑂 = ((𝑆‘𝑥)‘𝑅)))

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 383   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∃wrex 2913   class class class wbr 4653  ‘cfv 5888  (class class class)co 6650  ⟨“cs3 13587  Basecbs 15857  distcds 15950  TarskiGcstrkg 25329  Itvcitv 25335  LineGclng 25336  pInvGcmir 25547  ∟Gcrag 25588  ⟂Gcperpg 25590

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-concat 13301  df-s1 13302  df-s2 13593  df-s3 13594  df-trkgc 25347  df-trkgb 25348  df-trkgcb 25349  df-trkg 25352  df-cgrg 25406  df-leg 25478  df-mir 25548  df-rag 25589  df-perpg 25591

This theorem is referenced by:  mideulem  25628  opphllem3  25641