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Theorem opphllem 28809

Description: Lemma 8.24 of [Schwabhauser] p. 66. This is used later for mideulem 28810 and later for opphl 28828. (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 480	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝐺 ∈ TarskiG)
8		eqid 2736	. . . 4 ⊢ (𝑆‘𝑥) = (𝑆‘𝑥)
9		mideu.2	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
10	9	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝐵 ∈ 𝑃)
11		mideulem.3	. . . . 5 ⊢ (𝜑 → 𝑂 ∈ 𝑃)
12	11	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝑂 ∈ 𝑃)
13		mideu.1	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
14	13	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝐴 ∈ 𝑃)
15		opphllem.1	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ 𝑃)
16	15	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝑅 ∈ 𝑃)
17		simprl 770	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝑥 ∈ 𝑃)
18		mideulem.1	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ≠ 𝐵)
19	18	necomd 2987	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 ≠ 𝐴)
20	19	neneqd 2937	. . . . . . . . . . . 12 ⊢ (𝜑 → ¬ 𝐵 = 𝐴)
21	20	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑂 ∈ (𝐵𝐿𝐴)) → ¬ 𝐵 = 𝐴)
22		mideulem.6	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐴𝐿𝐵)(⟂G‘𝐺)(𝐴𝐿𝑂))
23	4, 6, 22	perpln2 28785	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐴𝐿𝑂) ∈ ran 𝐿)
24	1, 3, 4, 6, 13, 11, 23	tglnne 28702	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ≠ 𝑂)
25	24	necomd 2987	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑂 ≠ 𝐴)
26	25	neneqd 2937	. . . . . . . . . . . 12 ⊢ (𝜑 → ¬ 𝑂 = 𝐴)
27	26	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑂 ∈ (𝐵𝐿𝐴)) → ¬ 𝑂 = 𝐴)
28	21, 27	jca 511	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑂 ∈ (𝐵𝐿𝐴)) → (¬ 𝐵 = 𝐴 ∧ ¬ 𝑂 = 𝐴))
29	6	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑂 ∈ (𝐵𝐿𝐴)) → 𝐺 ∈ TarskiG)
30	9	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑂 ∈ (𝐵𝐿𝐴)) → 𝐵 ∈ 𝑃)
31	13	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑂 ∈ (𝐵𝐿𝐴)) → 𝐴 ∈ 𝑃)
32	11	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑂 ∈ (𝐵𝐿𝐴)) → 𝑂 ∈ 𝑃)
33	1, 3, 4, 6, 9, 13, 19	tglinerflx2 28708	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ∈ (𝐵𝐿𝐴))
34	1, 3, 4, 6, 13, 9, 18	tglinecom 28709	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐴𝐿𝐵) = (𝐵𝐿𝐴))
35	34, 22	eqbrtrrd 5122	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐵𝐿𝐴)(⟂G‘𝐺)(𝐴𝐿𝑂))
36	1, 2, 3, 4, 6, 9, 13, 33, 11, 35	perprag 28800	. . . . . . . . . . . . 13 ⊢ (𝜑 → ⟨“𝐵𝐴𝑂”⟩ ∈ (∟G‘𝐺))
37	36	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑂 ∈ (𝐵𝐿𝐴)) → ⟨“𝐵𝐴𝑂”⟩ ∈ (∟G‘𝐺))
38		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑂 ∈ (𝐵𝐿𝐴)) → 𝑂 ∈ (𝐵𝐿𝐴))
39	38	orcd 873	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑂 ∈ (𝐵𝐿𝐴)) → (𝑂 ∈ (𝐵𝐿𝐴) ∨ 𝐵 = 𝐴))
40	1, 2, 3, 4, 5, 29, 30, 31, 32, 37, 39	ragflat3 28780	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑂 ∈ (𝐵𝐿𝐴)) → (𝐵 = 𝐴 ∨ 𝑂 = 𝐴))
41		oran 991	. . . . . . . . . . 11 ⊢ ((𝐵 = 𝐴 ∨ 𝑂 = 𝐴) ↔ ¬ (¬ 𝐵 = 𝐴 ∧ ¬ 𝑂 = 𝐴))
42	40, 41	sylib 218	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑂 ∈ (𝐵𝐿𝐴)) → ¬ (¬ 𝐵 = 𝐴 ∧ ¬ 𝑂 = 𝐴))
43	28, 42	pm2.65da 816	. . . . . . . . 9 ⊢ (𝜑 → ¬ 𝑂 ∈ (𝐵𝐿𝐴))
44	43	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → ¬ 𝑂 ∈ (𝐵𝐿𝐴))
45	34	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → (𝐴𝐿𝐵) = (𝐵𝐿𝐴))
46	44, 45	neleqtrrd 2859	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → ¬ 𝑂 ∈ (𝐴𝐿𝐵))
47	18	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝐴 ≠ 𝐵)
48	47	neneqd 2937	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → ¬ 𝐴 = 𝐵)
49	46, 48	jca 511	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → (¬ 𝑂 ∈ (𝐴𝐿𝐵) ∧ ¬ 𝐴 = 𝐵))
50		pm4.56 990	. . . . . 6 ⊢ ((¬ 𝑂 ∈ (𝐴𝐿𝐵) ∧ ¬ 𝐴 = 𝐵) ↔ ¬ (𝑂 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))
51	49, 50	sylib 218	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → ¬ (𝑂 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))
52	1, 4, 3, 7, 14, 10, 12, 51	ncolrot2 28637	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → ¬ (𝐵 ∈ (𝑂𝐿𝐴) ∨ 𝑂 = 𝐴))
53		simprrr 781	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝑥 ∈ (𝑅𝐼𝑂))
54	1, 2, 3, 7, 16, 17, 12, 53	tgbtwncom 28562	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝑥 ∈ (𝑂𝐼𝑅))
55		mideulem.4	. . . . . . . 8 ⊢ (𝜑 → 𝑇 ∈ 𝑃)
56	55	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝑇 ∈ 𝑃)
57		mideulem.7	. . . . . . . 8 ⊢ (𝜑 → 𝑇 ∈ (𝐴𝐿𝐵))
58	57	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝑇 ∈ (𝐴𝐿𝐵))
59		simprrl 780	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝑥 ∈ (𝑇𝐼𝐵))
60	1, 3, 4, 7, 56, 14, 10, 17, 58, 59	coltr3 28722	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝑥 ∈ (𝐴𝐿𝐵))
61	43, 34	neleqtrrd 2859	. . . . . . 7 ⊢ (𝜑 → ¬ 𝑂 ∈ (𝐴𝐿𝐵))
62	61	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → ¬ 𝑂 ∈ (𝐴𝐿𝐵))
63		nelne2 3030	. . . . . 6 ⊢ ((𝑥 ∈ (𝐴𝐿𝐵) ∧ ¬ 𝑂 ∈ (𝐴𝐿𝐵)) → 𝑥 ≠ 𝑂)
64	60, 62, 63	syl2anc 584	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝑥 ≠ 𝑂)
65	1, 2, 3, 7, 12, 17, 16, 54, 64	tgbtwnne 28564	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝑂 ≠ 𝑅)
66	1, 2, 3, 4, 5, 6, 9, 13, 11	israg 28771	. . . . . . . 8 ⊢ (𝜑 → (⟨“𝐵𝐴𝑂”⟩ ∈ (∟G‘𝐺) ↔ (𝐵 − 𝑂) = (𝐵 − ((𝑆‘𝐴)‘𝑂))))
67	36, 66	mpbid 232	. . . . . . 7 ⊢ (𝜑 → (𝐵 − 𝑂) = (𝐵 − ((𝑆‘𝐴)‘𝑂)))
68	67	ad3antrrr 730	. . . . . 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 730	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠 ∈ 𝑃) ∧ (𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 − 𝑠) = (𝑥 − 𝑅))) → 𝐺 ∈ TarskiG)
70		eqid 2736	. . . . . . . . 9 ⊢ (𝑆‘𝐴) = (𝑆‘𝐴)
71	1, 2, 3, 4, 5, 7, 14, 70, 12	mircl 28735	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → ((𝑆‘𝐴)‘𝑂) ∈ 𝑃)
72	71	ad2antrr 726	. . . . . . 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 730	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠 ∈ 𝑃) ∧ (𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 − 𝑠) = (𝑥 − 𝑅))) → 𝐴 ∈ 𝑃)
74	11	ad3antrrr 730	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠 ∈ 𝑃) ∧ (𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 − 𝑠) = (𝑥 − 𝑅))) → 𝑂 ∈ 𝑃)
75	15	ad3antrrr 730	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠 ∈ 𝑃) ∧ (𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 − 𝑠) = (𝑥 − 𝑅))) → 𝑅 ∈ 𝑃)
76	9	ad3antrrr 730	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠 ∈ 𝑃) ∧ (𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 − 𝑠) = (𝑥 − 𝑅))) → 𝐵 ∈ 𝑃)
77		simplr 768	. . . . . . 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 28732	. . . . . . 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 2736	. . . . . . . . 9 ⊢ (𝑆‘𝐵) = (𝑆‘𝐵)
80	1, 2, 3, 4, 5, 69, 76, 79, 77	mirbtwn 28732	. . . . . . . 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 484	. . . . . . . . . . 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 726	. . . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . 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 732	. . . . . . . . . . . . . . 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 734	. . . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . 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 732	. . . . . . . . . . . . . . 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 734	. . . . . . . . . . . . . . 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 734	. . . . . . . . . . . . . . 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 732	. . . . . . . . . . . . . . 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 734	. . . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . 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 734	. . . . . . . . . . . . . . 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 734	. . . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠 ∈ 𝑃) ∧ (𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 − 𝑠) = (𝑥 − 𝑅))) → 𝑥 ∈ 𝑃)
102	101	ad2antrr 726	. . . . . . . . . . . . . . 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 785	. . . . . . . . . . . . . . . . 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 495	. . . . . . . . . . . . . . . 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 494	. . . . . . . . . . . . . . 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 495	. . . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . 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 775	. . . . . . . . . . . . . . . 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 494	. . . . . . . . . . . . . . 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 772	. . . . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . 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 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)) → 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 28808	. . . . . . . . . . . . . 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 2742	. . . . . . . . . . . . 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 6838	. . . . . . . . . . . 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 6836	. . . . . . . . . . 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 2771	. . . . . . . . . 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 2736	. . . . . . . . . . 11 ⊢ (𝑆‘𝑚) = (𝑆‘𝑚)
119	1, 2, 3, 4, 5, 69, 118, 77, 75, 101, 110	midexlem 28766	. . . . . . . . . 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 3144	. . . . . . . . 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 7373	. . . . . . . 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 2839	. . . . . . 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 28731	. . . . . . . . . 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 730	. . . . . . . . . 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 2771	. . . . . . . . 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 28554	. . . . . . . 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 7374	. . . . . . . . 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 28731	. . . . . . . . 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 2775	. . . . . . . 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 28559	. . . . . . 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 28536	. . . . . . 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 28536	. . . . . . . 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 726	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠 ∈ 𝑃) ∧ (𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 − 𝑠) = (𝑥 − 𝑅))) → 𝑥 ∈ (𝑅𝐼𝑂))
134		simprl 770	. . . . . . . . . 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 28562	. . . . . . . . 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 28554	. . . . . . . . . 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 2742	. . . . . . . . 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 730	. . . . . . . . . . 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 2987	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝐵 ≠ 𝐴)
140	139	ad2antrr 726	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠 ∈ 𝑃) ∧ (𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 − 𝑠) = (𝑥 − 𝑅))) → 𝐵 ≠ 𝐴)
141	60	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠 ∈ 𝑃) ∧ (𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 − 𝑠) = (𝑥 − 𝑅))) → 𝑥 ∈ (𝐴𝐿𝐵))
142	141	orcd 873	. . . . . . . . . . . . 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 28632	. . . . . . . . . . . 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 28633	. . . . . . . . . . 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 28773	. . . . . . . . . 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 28771	. . . . . . . . . 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 232	. . . . . . . . 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 28559	. . . . . . . 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 2771	. . . . . . 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 28582	. . . . . 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 2774	. . . . 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 28538	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → ∃𝑠 ∈ 𝑃 (𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 − 𝑠) = (𝑥 − 𝑅)))
153	151, 152	r19.29a 3144	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → (𝐵 − 𝑂) = (𝐴 − 𝑅))
154	99	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → (𝐴 − 𝑂) = (𝐵 − 𝑅))
155	1, 2, 3, 7, 14, 12, 10, 16, 154	tgcgrcomlr 28554	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → (𝑂 − 𝐴) = (𝑅 − 𝐵))
156	143, 152	r19.29a 3144	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → (𝑥 ∈ (𝐵𝐿𝐴) ∨ 𝐵 = 𝐴))
157	1, 4, 3, 7, 12, 16, 17, 54	btwncolg1 28629	. . . 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 28763	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝐵 = ((𝑆‘𝑥)‘𝐴))
159	1, 2, 3, 4, 5, 7, 17, 8, 14	mirbtwn 28732	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝑥 ∈ (((𝑆‘𝑥)‘𝐴)𝐼𝐴))
160	158	oveq1d 7373	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → (𝐵𝐼𝐴) = (((𝑆‘𝑥)‘𝐴)𝐼𝐴))
161	159, 160	eleqtrrd 2839	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝑥 ∈ (𝐵𝐼𝐴))
162	1, 2, 3, 7, 10, 17, 14, 161	tgbtwncom 28562	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝑥 ∈ (𝐴𝐼𝐵))
163	1, 2, 3, 7, 14, 10	axtgcgrrflx 28536	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → (𝐴 − 𝐵) = (𝐵 − 𝐴))
164	158	oveq2d 7374	. . . . . 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 28731	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → (𝑥 − ((𝑆‘𝑥)‘𝐴)) = (𝑥 − 𝐴))
166	164, 165	eqtrd 2771	. . . . 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 28582	. . . 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 28733	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝑂 = ((𝑆‘𝑥)‘𝑅))
169	158, 168	jca 511	. 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 28562	. . 3 ⊢ (𝜑 → 𝑇 ∈ (𝑂𝐼𝑄))
171	1, 2, 3, 6, 11, 9, 86, 55, 15, 170, 97	axtgpasch 28541	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))
172	169, 171	reximddv 3152	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 (𝐵 = ((𝑆‘𝑥)‘𝐴) ∧ 𝑂 = ((𝑆‘𝑥)‘𝑅)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∨ wo 847 = wceq 1541 ∈ wcel 2113 ≠ wne 2932 ∃wrex 3060 class class class wbr 5098 ‘cfv 6492 (class class class)co 7358 ⟨“cs3 14767 Basecbs 17138 distcds 17188 TarskiGcstrkg 28501 Itvcitv 28507 LineGclng 28508 pInvGcmir 28726 ∟Gcrag 28767 ⟂Gcperpg 28769

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 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105

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 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-tp 4585 df-op 4587 df-uni 4864 df-int 4903 df-iun 4948 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7809 df-1st 7933 df-2nd 7934 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-oadd 8401 df-er 8635 df-map 8767 df-pm 8768 df-en 8886 df-dom 8887 df-sdom 8888 df-fin 8889 df-dju 9815 df-card 9853 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-nn 12148 df-2 12210 df-3 12211 df-n0 12404 df-xnn0 12477 df-z 12491 df-uz 12754 df-fz 13426 df-fzo 13573 df-hash 14256 df-word 14439 df-concat 14496 df-s1 14522 df-s2 14773 df-s3 14774 df-trkgc 28522 df-trkgb 28523 df-trkgcb 28524 df-trkg 28527 df-cgrg 28585 df-leg 28657 df-mir 28727 df-rag 28768 df-perpg 28770

This theorem is referenced by: mideulem 28810 opphllem3 28823

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