Theorem opphllem 28555

Description: Lemma 8.24 of [Schwabhauser] p. 66. This is used later for mideulem 28556 and later for opphl 28574. (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 479	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝐺 ∈ TarskiG)
8		eqid 2725	. . . 4 ⊢ (𝑆‘𝑥) = (𝑆‘𝑥)
9		mideu.2	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
10	9	adantr 479	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝐵 ∈ 𝑃)
11		mideulem.3	. . . . 5 ⊢ (𝜑 → 𝑂 ∈ 𝑃)
12	11	adantr 479	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝑂 ∈ 𝑃)
13		mideu.1	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
14	13	adantr 479	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝐴 ∈ 𝑃)
15		opphllem.1	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ 𝑃)
16	15	adantr 479	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝑅 ∈ 𝑃)
17		simprl 769	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝑥 ∈ 𝑃)
18		mideulem.1	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ≠ 𝐵)
19	18	necomd 2986	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 ≠ 𝐴)
20	19	neneqd 2935	. . . . . . . . . . . 12 ⊢ (𝜑 → ¬ 𝐵 = 𝐴)
21	20	adantr 479	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑂 ∈ (𝐵𝐿𝐴)) → ¬ 𝐵 = 𝐴)
22		mideulem.6	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐴𝐿𝐵)(⟂G‘𝐺)(𝐴𝐿𝑂))
23	4, 6, 22	perpln2 28531	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐴𝐿𝑂) ∈ ran 𝐿)
24	1, 3, 4, 6, 13, 11, 23	tglnne 28448	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ≠ 𝑂)
25	24	necomd 2986	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑂 ≠ 𝐴)
26	25	neneqd 2935	. . . . . . . . . . . 12 ⊢ (𝜑 → ¬ 𝑂 = 𝐴)
27	26	adantr 479	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑂 ∈ (𝐵𝐿𝐴)) → ¬ 𝑂 = 𝐴)
28	21, 27	jca 510	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑂 ∈ (𝐵𝐿𝐴)) → (¬ 𝐵 = 𝐴 ∧ ¬ 𝑂 = 𝐴))
29	6	adantr 479	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑂 ∈ (𝐵𝐿𝐴)) → 𝐺 ∈ TarskiG)
30	9	adantr 479	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑂 ∈ (𝐵𝐿𝐴)) → 𝐵 ∈ 𝑃)
31	13	adantr 479	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑂 ∈ (𝐵𝐿𝐴)) → 𝐴 ∈ 𝑃)
32	11	adantr 479	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑂 ∈ (𝐵𝐿𝐴)) → 𝑂 ∈ 𝑃)
33	1, 3, 4, 6, 9, 13, 19	tglinerflx2 28454	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ∈ (𝐵𝐿𝐴))
34	1, 3, 4, 6, 13, 9, 18	tglinecom 28455	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐴𝐿𝐵) = (𝐵𝐿𝐴))
35	34, 22	eqbrtrrd 5165	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐵𝐿𝐴)(⟂G‘𝐺)(𝐴𝐿𝑂))
36	1, 2, 3, 4, 6, 9, 13, 33, 11, 35	perprag 28546	. . . . . . . . . . . . 13 ⊢ (𝜑 → ⟨“𝐵𝐴𝑂”⟩ ∈ (∟G‘𝐺))
37	36	adantr 479	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑂 ∈ (𝐵𝐿𝐴)) → ⟨“𝐵𝐴𝑂”⟩ ∈ (∟G‘𝐺))
38		simpr 483	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑂 ∈ (𝐵𝐿𝐴)) → 𝑂 ∈ (𝐵𝐿𝐴))
39	38	orcd 871	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑂 ∈ (𝐵𝐿𝐴)) → (𝑂 ∈ (𝐵𝐿𝐴) ∨ 𝐵 = 𝐴))
40	1, 2, 3, 4, 5, 29, 30, 31, 32, 37, 39	ragflat3 28526	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑂 ∈ (𝐵𝐿𝐴)) → (𝐵 = 𝐴 ∨ 𝑂 = 𝐴))
41		oran 987	. . . . . . . . . . 11 ⊢ ((𝐵 = 𝐴 ∨ 𝑂 = 𝐴) ↔ ¬ (¬ 𝐵 = 𝐴 ∧ ¬ 𝑂 = 𝐴))
42	40, 41	sylib 217	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑂 ∈ (𝐵𝐿𝐴)) → ¬ (¬ 𝐵 = 𝐴 ∧ ¬ 𝑂 = 𝐴))
43	28, 42	pm2.65da 815	. . . . . . . . 9 ⊢ (𝜑 → ¬ 𝑂 ∈ (𝐵𝐿𝐴))
44	43	adantr 479	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → ¬ 𝑂 ∈ (𝐵𝐿𝐴))
45	34	adantr 479	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → (𝐴𝐿𝐵) = (𝐵𝐿𝐴))
46	44, 45	neleqtrrd 2848	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → ¬ 𝑂 ∈ (𝐴𝐿𝐵))
47	18	adantr 479	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝐴 ≠ 𝐵)
48	47	neneqd 2935	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → ¬ 𝐴 = 𝐵)
49	46, 48	jca 510	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → (¬ 𝑂 ∈ (𝐴𝐿𝐵) ∧ ¬ 𝐴 = 𝐵))
50		pm4.56 986	. . . . . 6 ⊢ ((¬ 𝑂 ∈ (𝐴𝐿𝐵) ∧ ¬ 𝐴 = 𝐵) ↔ ¬ (𝑂 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))
51	49, 50	sylib 217	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → ¬ (𝑂 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))
52	1, 4, 3, 7, 14, 10, 12, 51	ncolrot2 28383	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → ¬ (𝐵 ∈ (𝑂𝐿𝐴) ∨ 𝑂 = 𝐴))
53		simprrr 780	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝑥 ∈ (𝑅𝐼𝑂))
54	1, 2, 3, 7, 16, 17, 12, 53	tgbtwncom 28308	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝑥 ∈ (𝑂𝐼𝑅))
55		mideulem.4	. . . . . . . 8 ⊢ (𝜑 → 𝑇 ∈ 𝑃)
56	55	adantr 479	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝑇 ∈ 𝑃)
57		mideulem.7	. . . . . . . 8 ⊢ (𝜑 → 𝑇 ∈ (𝐴𝐿𝐵))
58	57	adantr 479	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝑇 ∈ (𝐴𝐿𝐵))
59		simprrl 779	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝑥 ∈ (𝑇𝐼𝐵))
60	1, 3, 4, 7, 56, 14, 10, 17, 58, 59	coltr3 28468	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝑥 ∈ (𝐴𝐿𝐵))
61	43, 34	neleqtrrd 2848	. . . . . . 7 ⊢ (𝜑 → ¬ 𝑂 ∈ (𝐴𝐿𝐵))
62	61	adantr 479	. . . . . 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 582	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝑥 ≠ 𝑂)
65	1, 2, 3, 7, 12, 17, 16, 54, 64	tgbtwnne 28310	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝑂 ≠ 𝑅)
66	1, 2, 3, 4, 5, 6, 9, 13, 11	israg 28517	. . . . . . . 8 ⊢ (𝜑 → (⟨“𝐵𝐴𝑂”⟩ ∈ (∟G‘𝐺) ↔ (𝐵 − 𝑂) = (𝐵 − ((𝑆‘𝐴)‘𝑂))))
67	36, 66	mpbid 231	. . . . . . 7 ⊢ (𝜑 → (𝐵 − 𝑂) = (𝐵 − ((𝑆‘𝐴)‘𝑂)))
68	67	ad3antrrr 728	. . . . . 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 728	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠 ∈ 𝑃) ∧ (𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 − 𝑠) = (𝑥 − 𝑅))) → 𝐺 ∈ TarskiG)
70		eqid 2725	. . . . . . . . 9 ⊢ (𝑆‘𝐴) = (𝑆‘𝐴)
71	1, 2, 3, 4, 5, 7, 14, 70, 12	mircl 28481	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → ((𝑆‘𝐴)‘𝑂) ∈ 𝑃)
72	71	ad2antrr 724	. . . . . . 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 728	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠 ∈ 𝑃) ∧ (𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 − 𝑠) = (𝑥 − 𝑅))) → 𝐴 ∈ 𝑃)
74	11	ad3antrrr 728	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠 ∈ 𝑃) ∧ (𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 − 𝑠) = (𝑥 − 𝑅))) → 𝑂 ∈ 𝑃)
75	15	ad3antrrr 728	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠 ∈ 𝑃) ∧ (𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 − 𝑠) = (𝑥 − 𝑅))) → 𝑅 ∈ 𝑃)
76	9	ad3antrrr 728	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠 ∈ 𝑃) ∧ (𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 − 𝑠) = (𝑥 − 𝑅))) → 𝐵 ∈ 𝑃)
77		simplr 767	. . . . . . 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 28478	. . . . . . 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 2725	. . . . . . . . 9 ⊢ (𝑆‘𝐵) = (𝑆‘𝐵)
80	1, 2, 3, 4, 5, 69, 76, 79, 77	mirbtwn 28478	. . . . . . . 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 483	. . . . . . . . . . 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 724	. . . . . . . . . . . . . . 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 724	. . . . . . . . . . . . . . 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 724	. . . . . . . . . . . . . . 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 730	. . . . . . . . . . . . . . 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 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)) → 𝑄 ∈ 𝑃)
88	74	ad2antrr 724	. . . . . . . . . . . . . . 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 730	. . . . . . . . . . . . . . 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 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)) → (𝐴𝐿𝐵)(⟂G‘𝐺)(𝑄𝐿𝐵))
92	22	ad5antr 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)) → (𝐴𝐿𝐵)(⟂G‘𝐺)(𝐴𝐿𝑂))
93	58	ad4antr 730	. . . . . . . . . . . . . . 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 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 ∈ (𝑄𝐼𝑂))
96	75	ad2antrr 724	. . . . . . . . . . . . . . 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 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)) → 𝑅 ∈ (𝐵𝐼𝑄))
99		opphllem.3	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐴 − 𝑂) = (𝐵 − 𝑅))
100	99	ad5antr 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 − 𝑂) = (𝐵 − 𝑅))
101	17	ad2antrr 724	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠 ∈ 𝑃) ∧ (𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 − 𝑠) = (𝑥 − 𝑅))) → 𝑥 ∈ 𝑃)
102	101	ad2antrr 724	. . . . . . . . . . . . . . 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 784	. . . . . . . . . . . . . . . . 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 494	. . . . . . . . . . . . . . . 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 493	. . . . . . . . . . . . . . 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 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 ∈ (𝑅𝐼𝑂))
107	77	ad2antrr 724	. . . . . . . . . . . . . . 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 774	. . . . . . . . . . . . . . . 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 493	. . . . . . . . . . . . . . 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 771	. . . . . . . . . . . . . . . 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 724	. . . . . . . . . . . . . . 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 767	. . . . . . . . . . . . . . 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 28554	. . . . . . . . . . . . . 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 2731	. . . . . . . . . . . . 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 6894	. . . . . . . . . . . 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 6892	. . . . . . . . . . 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 2765	. . . . . . . . . 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 2725	. . . . . . . . . . 11 ⊢ (𝑆‘𝑚) = (𝑆‘𝑚)
119	1, 2, 3, 4, 5, 69, 118, 77, 75, 101, 110	midexlem 28512	. . . . . . . . . 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 3152	. . . . . . . . 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 7429	. . . . . . . 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 2828	. . . . . . 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 28477	. . . . . . . . . 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 728	. . . . . . . . . 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 2765	. . . . . . . . 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 28300	. . . . . . . 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 7430	. . . . . . . . 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 28477	. . . . . . . . 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 2769	. . . . . . . 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 28305	. . . . . . 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 28282	. . . . . . 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 28282	. . . . . . . 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 724	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠 ∈ 𝑃) ∧ (𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 − 𝑠) = (𝑥 − 𝑅))) → 𝑥 ∈ (𝑅𝐼𝑂))
134		simprl 769	. . . . . . . . . 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 28308	. . . . . . . . 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 28300	. . . . . . . . . 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 2731	. . . . . . . . 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 728	. . . . . . . . . . 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 2986	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝐵 ≠ 𝐴)
140	139	ad2antrr 724	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠 ∈ 𝑃) ∧ (𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 − 𝑠) = (𝑥 − 𝑅))) → 𝐵 ≠ 𝐴)
141	60	ad2antrr 724	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠 ∈ 𝑃) ∧ (𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 − 𝑠) = (𝑥 − 𝑅))) → 𝑥 ∈ (𝐴𝐿𝐵))
142	141	orcd 871	. . . . . . . . . . . . 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 28378	. . . . . . . . . . . 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 28379	. . . . . . . . . . 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 28519	. . . . . . . . . 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 28517	. . . . . . . . . 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 231	. . . . . . . . 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 28305	. . . . . . . 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 2765	. . . . . . 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 28328	. . . . . 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 2768	. . . . 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 28284	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → ∃𝑠 ∈ 𝑃 (𝑥 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 − 𝑠) = (𝑥 − 𝑅)))
153	151, 152	r19.29a 3152	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → (𝐵 − 𝑂) = (𝐴 − 𝑅))
154	99	adantr 479	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → (𝐴 − 𝑂) = (𝐵 − 𝑅))
155	1, 2, 3, 7, 14, 12, 10, 16, 154	tgcgrcomlr 28300	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → (𝑂 − 𝐴) = (𝑅 − 𝐵))
156	143, 152	r19.29a 3152	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → (𝑥 ∈ (𝐵𝐿𝐴) ∨ 𝐵 = 𝐴))
157	1, 4, 3, 7, 12, 16, 17, 54	btwncolg1 28375	. . . 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 28509	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝐵 = ((𝑆‘𝑥)‘𝐴))
159	1, 2, 3, 4, 5, 7, 17, 8, 14	mirbtwn 28478	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝑥 ∈ (((𝑆‘𝑥)‘𝐴)𝐼𝐴))
160	158	oveq1d 7429	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → (𝐵𝐼𝐴) = (((𝑆‘𝑥)‘𝐴)𝐼𝐴))
161	159, 160	eleqtrrd 2828	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝑥 ∈ (𝐵𝐼𝐴))
162	1, 2, 3, 7, 10, 17, 14, 161	tgbtwncom 28308	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝑥 ∈ (𝐴𝐼𝐵))
163	1, 2, 3, 7, 14, 10	axtgcgrrflx 28282	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → (𝐴 − 𝐵) = (𝐵 − 𝐴))
164	158	oveq2d 7430	. . . . . 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 28477	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → (𝑥 − ((𝑆‘𝑥)‘𝐴)) = (𝑥 − 𝐴))
166	164, 165	eqtrd 2765	. . . . 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 28328	. . . 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 28479	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝑂 = ((𝑆‘𝑥)‘𝑅))
169	158, 168	jca 510	. 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 28308	. . 3 ⊢ (𝜑 → 𝑇 ∈ (𝑂𝐼𝑄))
171	1, 2, 3, 6, 11, 9, 86, 55, 15, 170, 97	axtgpasch 28287	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))
172	169, 171	reximddv 3161	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 (𝐵 = ((𝑆‘𝑥)‘𝐴) ∧ 𝑂 = ((𝑆‘𝑥)‘𝑅)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 394   ∨ wo 845   = wceq 1533   ∈ wcel 2098   ≠ wne 2930  ∃wrex 3060   class class class wbr 5141  ‘cfv 6541  (class class class)co 7414  ⟨“cs3 14823  Basecbs 17177  distcds 17239  TarskiGcstrkg 28247  Itvcitv 28253  LineGclng 28254  pInvGcmir 28472  ∟Gcrag 28513  ⟂Gcperpg 28515

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5357  ax-pr 5421  ax-un 7736  ax-cnex 11192  ax-resscn 11193  ax-1cn 11194  ax-icn 11195  ax-addcl 11196  ax-addrcl 11197  ax-mulcl 11198  ax-mulrcl 11199  ax-mulcom 11200  ax-addass 11201  ax-mulass 11202  ax-distr 11203  ax-i2m1 11204  ax-1ne0 11205  ax-1rid 11206  ax-rnegex 11207  ax-rrecex 11208  ax-cnre 11209  ax-pre-lttri 11210  ax-pre-lttrn 11211  ax-pre-ltadd 11212  ax-pre-mulgt0 11213

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-pss 3958  df-nul 4317  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-tp 4627  df-op 4629  df-uni 4902  df-int 4943  df-iun 4991  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5568  df-eprel 5574  df-po 5582  df-so 5583  df-fr 5625  df-we 5627  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-rn 5681  df-res 5682  df-ima 5683  df-pred 6298  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7867  df-1st 7989  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8388  df-rdg 8427  df-1o 8483  df-oadd 8487  df-er 8721  df-map 8843  df-pm 8844  df-en 8961  df-dom 8962  df-sdom 8963  df-fin 8964  df-dju 9922  df-card 9960  df-pnf 11278  df-mnf 11279  df-xr 11280  df-ltxr 11281  df-le 11282  df-sub 11474  df-neg 11475  df-nn 12241  df-2 12303  df-3 12304  df-n0 12501  df-xnn0 12573  df-z 12587  df-uz 12851  df-fz 13515  df-fzo 13658  df-hash 14320  df-word 14495  df-concat 14551  df-s1 14576  df-s2 14829  df-s3 14830  df-trkgc 28268  df-trkgb 28269  df-trkgcb 28270  df-trkg 28273  df-cgrg 28331  df-leg 28403  df-mir 28473  df-rag 28514  df-perpg 28516

This theorem is referenced by:  mideulem  28556  opphllem3  28569