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Theorem lmieu 26272

Description: Uniqueness of the line mirror point. Theorem 10.2 of [Schwabhauser] p. 88. (Contributed by Thierry Arnoux, 1-Dec-2019.)

**Hypotheses**
Ref	Expression
ismid.p	⊢ 𝑃 = (Base‘𝐺)
ismid.d	⊢ − = (dist‘𝐺)
ismid.i	⊢ 𝐼 = (Itv‘𝐺)
ismid.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
ismid.1	⊢ (𝜑 → 𝐺Dim_TarskiG≥2)
lmieu.l	⊢ 𝐿 = (LineG‘𝐺)
lmieu.1	⊢ (𝜑 → 𝐷 ∈ ran 𝐿)
lmieu.a	⊢ (𝜑 → 𝐴 ∈ 𝑃)

**Assertion**
Ref	Expression
lmieu	⊢ (𝜑 → ∃!𝑏 ∈ 𝑃 ((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏)))

Distinct variable groups: 𝐺,𝑏 𝑃,𝑏 𝜑,𝑏 𝐴,𝑏 𝐷,𝑏 𝐿,𝑏 Allowed substitution hints: 𝐼(𝑏) − (𝑏)

Proof of Theorem lmieu Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		lmieu.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
2	1	adantr 473	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝐷) → 𝐴 ∈ 𝑃)
3		simpr 477	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) ∧ ¬ 𝐴 = 𝑏) → ¬ 𝐴 = 𝑏)
4		eqidd 2779	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) ∧ ¬ 𝐴 = 𝑏) → (𝐴(midG‘𝐺)𝑏) = (𝐴(midG‘𝐺)𝑏))
5		ismid.p	. . . . . . . . . . . . . . . 16 ⊢ 𝑃 = (Base‘𝐺)
6		ismid.d	. . . . . . . . . . . . . . . 16 ⊢ − = (dist‘𝐺)
7		ismid.i	. . . . . . . . . . . . . . . 16 ⊢ 𝐼 = (Itv‘𝐺)
8		ismid.g	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
9	8	ad4antr 719	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) ∧ ¬ 𝐴 = 𝑏) → 𝐺 ∈ TarskiG)
10		ismid.1	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐺Dim_TarskiG≥2)
11	10	ad4antr 719	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) ∧ ¬ 𝐴 = 𝑏) → 𝐺Dim_TarskiG≥2)
12	2	ad3antrrr 717	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) ∧ ¬ 𝐴 = 𝑏) → 𝐴 ∈ 𝑃)
13		simpllr 763	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) ∧ ¬ 𝐴 = 𝑏) → 𝑏 ∈ 𝑃)
14		eqid 2778	. . . . . . . . . . . . . . . 16 ⊢ (pInvG‘𝐺) = (pInvG‘𝐺)
15	5, 6, 7, 9, 11, 12, 13	midcl 26265	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) ∧ ¬ 𝐴 = 𝑏) → (𝐴(midG‘𝐺)𝑏) ∈ 𝑃)
16	5, 6, 7, 9, 11, 12, 13, 14, 15	ismidb 26266	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) ∧ ¬ 𝐴 = 𝑏) → (𝑏 = (((pInvG‘𝐺)‘(𝐴(midG‘𝐺)𝑏))‘𝐴) ↔ (𝐴(midG‘𝐺)𝑏) = (𝐴(midG‘𝐺)𝑏)))
17	4, 16	mpbird 249	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) ∧ ¬ 𝐴 = 𝑏) → 𝑏 = (((pInvG‘𝐺)‘(𝐴(midG‘𝐺)𝑏))‘𝐴))
18	17	adantr 473	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) ∧ ¬ 𝐴 = 𝑏) ∧ 𝐷 ≠ (𝐴𝐿𝑏)) → 𝑏 = (((pInvG‘𝐺)‘(𝐴(midG‘𝐺)𝑏))‘𝐴))
19		lmieu.l	. . . . . . . . . . . . . . . 16 ⊢ 𝐿 = (LineG‘𝐺)
20	9	adantr 473	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) ∧ ¬ 𝐴 = 𝑏) ∧ 𝐷 ≠ (𝐴𝐿𝑏)) → 𝐺 ∈ TarskiG)
21		lmieu.1	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐷 ∈ ran 𝐿)
22	21	ad4antr 719	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) ∧ ¬ 𝐴 = 𝑏) → 𝐷 ∈ ran 𝐿)
23	22	adantr 473	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) ∧ ¬ 𝐴 = 𝑏) ∧ 𝐷 ≠ (𝐴𝐿𝑏)) → 𝐷 ∈ ran 𝐿)
24	12	adantr 473	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) ∧ ¬ 𝐴 = 𝑏) ∧ 𝐷 ≠ (𝐴𝐿𝑏)) → 𝐴 ∈ 𝑃)
25	13	adantr 473	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) ∧ ¬ 𝐴 = 𝑏) ∧ 𝐷 ≠ (𝐴𝐿𝑏)) → 𝑏 ∈ 𝑃)
26	3	neqned 2974	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) ∧ ¬ 𝐴 = 𝑏) → 𝐴 ≠ 𝑏)
27	26	adantr 473	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) ∧ ¬ 𝐴 = 𝑏) ∧ 𝐷 ≠ (𝐴𝐿𝑏)) → 𝐴 ≠ 𝑏)
28	5, 7, 19, 20, 24, 25, 27	tgelrnln 26118	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) ∧ ¬ 𝐴 = 𝑏) ∧ 𝐷 ≠ (𝐴𝐿𝑏)) → (𝐴𝐿𝑏) ∈ ran 𝐿)
29		simpr 477	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) ∧ ¬ 𝐴 = 𝑏) ∧ 𝐷 ≠ (𝐴𝐿𝑏)) → 𝐷 ≠ (𝐴𝐿𝑏))
30		simp-4r 771	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) ∧ ¬ 𝐴 = 𝑏) → 𝐴 ∈ 𝐷)
31	30	adantr 473	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) ∧ ¬ 𝐴 = 𝑏) ∧ 𝐷 ≠ (𝐴𝐿𝑏)) → 𝐴 ∈ 𝐷)
32	5, 7, 19, 20, 24, 25, 27	tglinerflx1 26121	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) ∧ ¬ 𝐴 = 𝑏) ∧ 𝐷 ≠ (𝐴𝐿𝑏)) → 𝐴 ∈ (𝐴𝐿𝑏))
33	31, 32	elind 4059	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) ∧ ¬ 𝐴 = 𝑏) ∧ 𝐷 ≠ (𝐴𝐿𝑏)) → 𝐴 ∈ (𝐷 ∩ (𝐴𝐿𝑏)))
34		simpllr 763	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) ∧ ¬ 𝐴 = 𝑏) ∧ 𝐷 ≠ (𝐴𝐿𝑏)) → (𝐴(midG‘𝐺)𝑏) ∈ 𝐷)
35	5, 6, 7, 9, 11, 12, 13	midbtwn 26267	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) ∧ ¬ 𝐴 = 𝑏) → (𝐴(midG‘𝐺)𝑏) ∈ (𝐴𝐼𝑏))
36	5, 7, 19, 9, 12, 13, 15, 26, 35	btwnlng1 26107	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) ∧ ¬ 𝐴 = 𝑏) → (𝐴(midG‘𝐺)𝑏) ∈ (𝐴𝐿𝑏))
37	36	adantr 473	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) ∧ ¬ 𝐴 = 𝑏) ∧ 𝐷 ≠ (𝐴𝐿𝑏)) → (𝐴(midG‘𝐺)𝑏) ∈ (𝐴𝐿𝑏))
38	34, 37	elind 4059	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) ∧ ¬ 𝐴 = 𝑏) ∧ 𝐷 ≠ (𝐴𝐿𝑏)) → (𝐴(midG‘𝐺)𝑏) ∈ (𝐷 ∩ (𝐴𝐿𝑏)))
39	5, 7, 19, 20, 23, 28, 29, 33, 38	tglineineq 26131	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) ∧ ¬ 𝐴 = 𝑏) ∧ 𝐷 ≠ (𝐴𝐿𝑏)) → 𝐴 = (𝐴(midG‘𝐺)𝑏))
40	39	fveq2d 6503	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) ∧ ¬ 𝐴 = 𝑏) ∧ 𝐷 ≠ (𝐴𝐿𝑏)) → ((pInvG‘𝐺)‘𝐴) = ((pInvG‘𝐺)‘(𝐴(midG‘𝐺)𝑏)))
41	40	fveq1d 6501	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) ∧ ¬ 𝐴 = 𝑏) ∧ 𝐷 ≠ (𝐴𝐿𝑏)) → (((pInvG‘𝐺)‘𝐴)‘𝐴) = (((pInvG‘𝐺)‘(𝐴(midG‘𝐺)𝑏))‘𝐴))
42		eqid 2778	. . . . . . . . . . . . . 14 ⊢ ((pInvG‘𝐺)‘𝐴) = ((pInvG‘𝐺)‘𝐴)
43	5, 6, 7, 19, 14, 20, 24, 42	mircinv 26156	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) ∧ ¬ 𝐴 = 𝑏) ∧ 𝐷 ≠ (𝐴𝐿𝑏)) → (((pInvG‘𝐺)‘𝐴)‘𝐴) = 𝐴)
44	18, 41, 43	3eqtr2rd 2821	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) ∧ ¬ 𝐴 = 𝑏) ∧ 𝐷 ≠ (𝐴𝐿𝑏)) → 𝐴 = 𝑏)
45	3, 44	mtand 803	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) ∧ ¬ 𝐴 = 𝑏) → ¬ 𝐷 ≠ (𝐴𝐿𝑏))
46	8	ad5antr 721	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) ∧ ¬ 𝐴 = 𝑏) ∧ 𝐷(⟂G‘𝐺)(𝐴𝐿𝑏)) → 𝐺 ∈ TarskiG)
47	21	ad5antr 721	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) ∧ ¬ 𝐴 = 𝑏) ∧ 𝐷(⟂G‘𝐺)(𝐴𝐿𝑏)) → 𝐷 ∈ ran 𝐿)
48		nne 2971	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝐷 ≠ (𝐴𝐿𝑏) ↔ 𝐷 = (𝐴𝐿𝑏))
49	45, 48	sylib 210	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) ∧ ¬ 𝐴 = 𝑏) → 𝐷 = (𝐴𝐿𝑏))
50	49	adantr 473	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) ∧ ¬ 𝐴 = 𝑏) ∧ 𝐷(⟂G‘𝐺)(𝐴𝐿𝑏)) → 𝐷 = (𝐴𝐿𝑏))
51	50, 47	eqeltrrd 2867	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) ∧ ¬ 𝐴 = 𝑏) ∧ 𝐷(⟂G‘𝐺)(𝐴𝐿𝑏)) → (𝐴𝐿𝑏) ∈ ran 𝐿)
52		simpr 477	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) ∧ ¬ 𝐴 = 𝑏) ∧ 𝐷(⟂G‘𝐺)(𝐴𝐿𝑏)) → 𝐷(⟂G‘𝐺)(𝐴𝐿𝑏))
53	5, 6, 7, 19, 46, 47, 51, 52	perpneq 26202	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) ∧ ¬ 𝐴 = 𝑏) ∧ 𝐷(⟂G‘𝐺)(𝐴𝐿𝑏)) → 𝐷 ≠ (𝐴𝐿𝑏))
54	45, 53	mtand 803	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) ∧ ¬ 𝐴 = 𝑏) → ¬ 𝐷(⟂G‘𝐺)(𝐴𝐿𝑏))
55	54	ex 405	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) → (¬ 𝐴 = 𝑏 → ¬ 𝐷(⟂G‘𝐺)(𝐴𝐿𝑏)))
56	55	con4d 115	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) → (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) → 𝐴 = 𝑏))
57		idd 24	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) → (𝐴 = 𝑏 → 𝐴 = 𝑏))
58	56, 57	jaod 845	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) → ((𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏) → 𝐴 = 𝑏))
59	58	impr 447	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ ((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏))) → 𝐴 = 𝑏)
60	59	eqcomd 2784	. . . . 5 ⊢ ((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ ((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏))) → 𝑏 = 𝐴)
61		simpr 477	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ 𝑏 = 𝐴) → 𝑏 = 𝐴)
62	61	oveq2d 6992	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ 𝑏 = 𝐴) → (𝐴(midG‘𝐺)𝑏) = (𝐴(midG‘𝐺)𝐴))
63	5, 6, 7, 8, 10, 1, 1	midid 26269	. . . . . . . . 9 ⊢ (𝜑 → (𝐴(midG‘𝐺)𝐴) = 𝐴)
64	63	ad3antrrr 717	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ 𝑏 = 𝐴) → (𝐴(midG‘𝐺)𝐴) = 𝐴)
65	62, 64	eqtrd 2814	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ 𝑏 = 𝐴) → (𝐴(midG‘𝐺)𝑏) = 𝐴)
66		simpllr 763	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ 𝑏 = 𝐴) → 𝐴 ∈ 𝐷)
67	65, 66	eqeltrd 2866	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ 𝑏 = 𝐴) → (𝐴(midG‘𝐺)𝑏) ∈ 𝐷)
68	61	eqcomd 2784	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ 𝑏 = 𝐴) → 𝐴 = 𝑏)
69	68	olcd 860	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ 𝑏 = 𝐴) → (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏))
70	67, 69	jca 504	. . . . 5 ⊢ ((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ 𝑏 = 𝐴) → ((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏)))
71	60, 70	impbida 788	. . . 4 ⊢ (((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) → (((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏)) ↔ 𝑏 = 𝐴))
72	71	ralrimiva 3132	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝐷) → ∀𝑏 ∈ 𝑃 (((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏)) ↔ 𝑏 = 𝐴))
73		reu6i 3631	. . 3 ⊢ ((𝐴 ∈ 𝑃 ∧ ∀𝑏 ∈ 𝑃 (((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏)) ↔ 𝑏 = 𝐴)) → ∃!𝑏 ∈ 𝑃 ((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏)))
74	2, 72, 73	syl2anc 576	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝐷) → ∃!𝑏 ∈ 𝑃 ((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏)))
75	8	adantr 473	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) → 𝐺 ∈ TarskiG)
76	75	ad2antrr 713	. . . . 5 ⊢ ((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) → 𝐺 ∈ TarskiG)
77	21	adantr 473	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) → 𝐷 ∈ ran 𝐿)
78	77	ad2antrr 713	. . . . . 6 ⊢ ((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) → 𝐷 ∈ ran 𝐿)
79		simplr 756	. . . . . 6 ⊢ ((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) → 𝑥 ∈ 𝐷)
80	5, 19, 7, 76, 78, 79	tglnpt 26037	. . . . 5 ⊢ ((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) → 𝑥 ∈ 𝑃)
81		eqid 2778	. . . . 5 ⊢ ((pInvG‘𝐺)‘𝑥) = ((pInvG‘𝐺)‘𝑥)
82	1	adantr 473	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) → 𝐴 ∈ 𝑃)
83	82	ad2antrr 713	. . . . 5 ⊢ ((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) → 𝐴 ∈ 𝑃)
84	5, 6, 7, 19, 14, 76, 80, 81, 83	mircl 26149	. . . 4 ⊢ ((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) → (((pInvG‘𝐺)‘𝑥)‘𝐴) ∈ 𝑃)
85		oveq2 6984	. . . . . . . . . 10 ⊢ (𝑥 = (𝐴(midG‘𝐺)𝑏) → (𝐴𝐿𝑥) = (𝐴𝐿(𝐴(midG‘𝐺)𝑏)))
86	85	breq1d 4939	. . . . . . . . 9 ⊢ (𝑥 = (𝐴(midG‘𝐺)𝑏) → ((𝐴𝐿𝑥)(⟂G‘𝐺)𝐷 ↔ (𝐴𝐿(𝐴(midG‘𝐺)𝑏))(⟂G‘𝐺)𝐷))
87		simprl 758	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) ∧ 𝑏 ∈ 𝑃) ∧ ((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏))) → (𝐴(midG‘𝐺)𝑏) ∈ 𝐷)
88		simpr 477	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) → ¬ 𝐴 ∈ 𝐷)
89	5, 6, 7, 19, 75, 77, 82, 88	foot 26210	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) → ∃!𝑥 ∈ 𝐷 (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷)
90		reurmo 3373	. . . . . . . . . . 11 ⊢ (∃!𝑥 ∈ 𝐷 (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷 → ∃*𝑥 ∈ 𝐷 (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷)
91	89, 90	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) → ∃*𝑥 ∈ 𝐷 (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷)
92	91	ad4antr 719	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) ∧ 𝑏 ∈ 𝑃) ∧ ((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏))) → ∃*𝑥 ∈ 𝐷 (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷)
93	79	ad2antrr 713	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) ∧ 𝑏 ∈ 𝑃) ∧ ((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏))) → 𝑥 ∈ 𝐷)
94		simpllr 763	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) ∧ 𝑏 ∈ 𝑃) ∧ ((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏))) → (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷)
95	76	ad2antrr 713	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) ∧ 𝑏 ∈ 𝑃) ∧ ((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏))) → 𝐺 ∈ TarskiG)
96	83	ad2antrr 713	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) ∧ 𝑏 ∈ 𝑃) ∧ ((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏))) → 𝐴 ∈ 𝑃)
97		simplr 756	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) ∧ 𝑏 ∈ 𝑃) ∧ ((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏))) → 𝑏 ∈ 𝑃)
98	10	ad5antr 721	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) ∧ 𝑏 ∈ 𝑃) ∧ ((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏))) → 𝐺Dim_TarskiG≥2)
99	5, 6, 7, 95, 98, 96, 97	midcl 26265	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) ∧ 𝑏 ∈ 𝑃) ∧ ((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏))) → (𝐴(midG‘𝐺)𝑏) ∈ 𝑃)
100	5, 6, 7, 95, 98, 96, 97	midbtwn 26267	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) ∧ 𝑏 ∈ 𝑃) ∧ ((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏))) → (𝐴(midG‘𝐺)𝑏) ∈ (𝐴𝐼𝑏))
101	88	ad4antr 719	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) ∧ 𝑏 ∈ 𝑃) ∧ ((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏))) → ¬ 𝐴 ∈ 𝐷)
102		nelne2 3066	. . . . . . . . . . . . 13 ⊢ (((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ ¬ 𝐴 ∈ 𝐷) → (𝐴(midG‘𝐺)𝑏) ≠ 𝐴)
103	87, 101, 102	syl2anc 576	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) ∧ 𝑏 ∈ 𝑃) ∧ ((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏))) → (𝐴(midG‘𝐺)𝑏) ≠ 𝐴)
104	5, 6, 7, 95, 96, 99, 97, 100, 103	tgbtwnne 25978	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) ∧ 𝑏 ∈ 𝑃) ∧ ((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏))) → 𝐴 ≠ 𝑏)
105	5, 7, 19, 95, 96, 97, 99, 104, 100	btwnlng1 26107	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) ∧ 𝑏 ∈ 𝑃) ∧ ((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏))) → (𝐴(midG‘𝐺)𝑏) ∈ (𝐴𝐿𝑏))
106	5, 7, 19, 95, 96, 97, 104, 99, 103, 105	tglineelsb2 26120	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) ∧ 𝑏 ∈ 𝑃) ∧ ((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏))) → (𝐴𝐿𝑏) = (𝐴𝐿(𝐴(midG‘𝐺)𝑏)))
107	78	ad2antrr 713	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) ∧ 𝑏 ∈ 𝑃) ∧ ((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏))) → 𝐷 ∈ ran 𝐿)
108	5, 7, 19, 95, 96, 97, 104	tgelrnln 26118	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) ∧ 𝑏 ∈ 𝑃) ∧ ((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏))) → (𝐴𝐿𝑏) ∈ ran 𝐿)
109	104	neneqd 2972	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) ∧ 𝑏 ∈ 𝑃) ∧ ((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏))) → ¬ 𝐴 = 𝑏)
110		simprr 760	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) ∧ 𝑏 ∈ 𝑃) ∧ ((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏))) → (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏))
111	110	orcomd 857	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) ∧ 𝑏 ∈ 𝑃) ∧ ((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏))) → (𝐴 = 𝑏 ∨ 𝐷(⟂G‘𝐺)(𝐴𝐿𝑏)))
112	111	ord 850	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) ∧ 𝑏 ∈ 𝑃) ∧ ((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏))) → (¬ 𝐴 = 𝑏 → 𝐷(⟂G‘𝐺)(𝐴𝐿𝑏)))
113	109, 112	mpd 15	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) ∧ 𝑏 ∈ 𝑃) ∧ ((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏))) → 𝐷(⟂G‘𝐺)(𝐴𝐿𝑏))
114	5, 6, 7, 19, 95, 107, 108, 113	perpcom 26201	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) ∧ 𝑏 ∈ 𝑃) ∧ ((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏))) → (𝐴𝐿𝑏)(⟂G‘𝐺)𝐷)
115	106, 114	eqbrtrrd 4953	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) ∧ 𝑏 ∈ 𝑃) ∧ ((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏))) → (𝐴𝐿(𝐴(midG‘𝐺)𝑏))(⟂G‘𝐺)𝐷)
116	86, 87, 92, 93, 94, 115	rmoi2 3780	. . . . . . . 8 ⊢ ((((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) ∧ 𝑏 ∈ 𝑃) ∧ ((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏))) → 𝑥 = (𝐴(midG‘𝐺)𝑏))
117	116	eqcomd 2784	. . . . . . 7 ⊢ ((((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) ∧ 𝑏 ∈ 𝑃) ∧ ((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏))) → (𝐴(midG‘𝐺)𝑏) = 𝑥)
118	80	ad2antrr 713	. . . . . . . 8 ⊢ ((((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) ∧ 𝑏 ∈ 𝑃) ∧ ((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏))) → 𝑥 ∈ 𝑃)
119	5, 6, 7, 95, 98, 96, 97, 14, 118	ismidb 26266	. . . . . . 7 ⊢ ((((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) ∧ 𝑏 ∈ 𝑃) ∧ ((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏))) → (𝑏 = (((pInvG‘𝐺)‘𝑥)‘𝐴) ↔ (𝐴(midG‘𝐺)𝑏) = 𝑥))
120	117, 119	mpbird 249	. . . . . 6 ⊢ ((((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) ∧ 𝑏 ∈ 𝑃) ∧ ((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏))) → 𝑏 = (((pInvG‘𝐺)‘𝑥)‘𝐴))
121		simpr 477	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) ∧ 𝑏 ∈ 𝑃) ∧ 𝑏 = (((pInvG‘𝐺)‘𝑥)‘𝐴)) → 𝑏 = (((pInvG‘𝐺)‘𝑥)‘𝐴))
122	76	ad2antrr 713	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) ∧ 𝑏 ∈ 𝑃) ∧ 𝑏 = (((pInvG‘𝐺)‘𝑥)‘𝐴)) → 𝐺 ∈ TarskiG)
123	10	ad5antr 721	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) ∧ 𝑏 ∈ 𝑃) ∧ 𝑏 = (((pInvG‘𝐺)‘𝑥)‘𝐴)) → 𝐺Dim_TarskiG≥2)
124	83	ad2antrr 713	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) ∧ 𝑏 ∈ 𝑃) ∧ 𝑏 = (((pInvG‘𝐺)‘𝑥)‘𝐴)) → 𝐴 ∈ 𝑃)
125		simplr 756	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) ∧ 𝑏 ∈ 𝑃) ∧ 𝑏 = (((pInvG‘𝐺)‘𝑥)‘𝐴)) → 𝑏 ∈ 𝑃)
126	80	ad2antrr 713	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) ∧ 𝑏 ∈ 𝑃) ∧ 𝑏 = (((pInvG‘𝐺)‘𝑥)‘𝐴)) → 𝑥 ∈ 𝑃)
127	5, 6, 7, 122, 123, 124, 125, 14, 126	ismidb 26266	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) ∧ 𝑏 ∈ 𝑃) ∧ 𝑏 = (((pInvG‘𝐺)‘𝑥)‘𝐴)) → (𝑏 = (((pInvG‘𝐺)‘𝑥)‘𝐴) ↔ (𝐴(midG‘𝐺)𝑏) = 𝑥))
128	121, 127	mpbid 224	. . . . . . . 8 ⊢ ((((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) ∧ 𝑏 ∈ 𝑃) ∧ 𝑏 = (((pInvG‘𝐺)‘𝑥)‘𝐴)) → (𝐴(midG‘𝐺)𝑏) = 𝑥)
129	79	ad2antrr 713	. . . . . . . 8 ⊢ ((((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) ∧ 𝑏 ∈ 𝑃) ∧ 𝑏 = (((pInvG‘𝐺)‘𝑥)‘𝐴)) → 𝑥 ∈ 𝐷)
130	128, 129	eqeltrd 2866	. . . . . . 7 ⊢ ((((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) ∧ 𝑏 ∈ 𝑃) ∧ 𝑏 = (((pInvG‘𝐺)‘𝑥)‘𝐴)) → (𝐴(midG‘𝐺)𝑏) ∈ 𝐷)
131	122	adantr 473	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) ∧ 𝑏 ∈ 𝑃) ∧ 𝑏 = (((pInvG‘𝐺)‘𝑥)‘𝐴)) ∧ 𝐴 ≠ 𝑏) → 𝐺 ∈ TarskiG)
132		simp-4r 771	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) ∧ 𝑏 ∈ 𝑃) ∧ 𝑏 = (((pInvG‘𝐺)‘𝑥)‘𝐴)) ∧ 𝐴 ≠ 𝑏) → (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷)
133	19, 131, 132	perpln1 26198	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) ∧ 𝑏 ∈ 𝑃) ∧ 𝑏 = (((pInvG‘𝐺)‘𝑥)‘𝐴)) ∧ 𝐴 ≠ 𝑏) → (𝐴𝐿𝑥) ∈ ran 𝐿)
134	78	ad3antrrr 717	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) ∧ 𝑏 ∈ 𝑃) ∧ 𝑏 = (((pInvG‘𝐺)‘𝑥)‘𝐴)) ∧ 𝐴 ≠ 𝑏) → 𝐷 ∈ ran 𝐿)
135	5, 6, 7, 19, 131, 133, 134, 132	perpcom 26201	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) ∧ 𝑏 ∈ 𝑃) ∧ 𝑏 = (((pInvG‘𝐺)‘𝑥)‘𝐴)) ∧ 𝐴 ≠ 𝑏) → 𝐷(⟂G‘𝐺)(𝐴𝐿𝑥))
136	124	adantr 473	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) ∧ 𝑏 ∈ 𝑃) ∧ 𝑏 = (((pInvG‘𝐺)‘𝑥)‘𝐴)) ∧ 𝐴 ≠ 𝑏) → 𝐴 ∈ 𝑃)
137	126	adantr 473	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) ∧ 𝑏 ∈ 𝑃) ∧ 𝑏 = (((pInvG‘𝐺)‘𝑥)‘𝐴)) ∧ 𝐴 ≠ 𝑏) → 𝑥 ∈ 𝑃)
138	5, 7, 19, 131, 136, 137, 133	tglnne 26116	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) ∧ 𝑏 ∈ 𝑃) ∧ 𝑏 = (((pInvG‘𝐺)‘𝑥)‘𝐴)) ∧ 𝐴 ≠ 𝑏) → 𝐴 ≠ 𝑥)
139		simpllr 763	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) ∧ 𝑏 ∈ 𝑃) ∧ 𝑏 = (((pInvG‘𝐺)‘𝑥)‘𝐴)) ∧ 𝐴 ≠ 𝑏) → 𝑏 ∈ 𝑃)
140		simpr 477	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) ∧ 𝑏 ∈ 𝑃) ∧ 𝑏 = (((pInvG‘𝐺)‘𝑥)‘𝐴)) ∧ 𝐴 ≠ 𝑏) → 𝐴 ≠ 𝑏)
141	140	necomd 3022	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) ∧ 𝑏 ∈ 𝑃) ∧ 𝑏 = (((pInvG‘𝐺)‘𝑥)‘𝐴)) ∧ 𝐴 ≠ 𝑏) → 𝑏 ≠ 𝐴)
142	5, 6, 7, 19, 14, 131, 137, 81, 136	mirbtwn 26146	. . . . . . . . . . . . . . 15 ⊢ (((((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) ∧ 𝑏 ∈ 𝑃) ∧ 𝑏 = (((pInvG‘𝐺)‘𝑥)‘𝐴)) ∧ 𝐴 ≠ 𝑏) → 𝑥 ∈ ((((pInvG‘𝐺)‘𝑥)‘𝐴)𝐼𝐴))
143		simplr 756	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) ∧ 𝑏 ∈ 𝑃) ∧ 𝑏 = (((pInvG‘𝐺)‘𝑥)‘𝐴)) ∧ 𝐴 ≠ 𝑏) → 𝑏 = (((pInvG‘𝐺)‘𝑥)‘𝐴))
144	143	oveq1d 6991	. . . . . . . . . . . . . . 15 ⊢ (((((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) ∧ 𝑏 ∈ 𝑃) ∧ 𝑏 = (((pInvG‘𝐺)‘𝑥)‘𝐴)) ∧ 𝐴 ≠ 𝑏) → (𝑏𝐼𝐴) = ((((pInvG‘𝐺)‘𝑥)‘𝐴)𝐼𝐴))
145	142, 144	eleqtrrd 2869	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) ∧ 𝑏 ∈ 𝑃) ∧ 𝑏 = (((pInvG‘𝐺)‘𝑥)‘𝐴)) ∧ 𝐴 ≠ 𝑏) → 𝑥 ∈ (𝑏𝐼𝐴))
146	5, 7, 19, 131, 139, 136, 137, 141, 145	btwnlng1 26107	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) ∧ 𝑏 ∈ 𝑃) ∧ 𝑏 = (((pInvG‘𝐺)‘𝑥)‘𝐴)) ∧ 𝐴 ≠ 𝑏) → 𝑥 ∈ (𝑏𝐿𝐴))
147	5, 7, 19, 131, 136, 137, 139, 138, 146, 141	lnrot1 26111	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) ∧ 𝑏 ∈ 𝑃) ∧ 𝑏 = (((pInvG‘𝐺)‘𝑥)‘𝐴)) ∧ 𝐴 ≠ 𝑏) → 𝑏 ∈ (𝐴𝐿𝑥))
148	5, 7, 19, 131, 136, 137, 138, 139, 141, 147	tglineelsb2 26120	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) ∧ 𝑏 ∈ 𝑃) ∧ 𝑏 = (((pInvG‘𝐺)‘𝑥)‘𝐴)) ∧ 𝐴 ≠ 𝑏) → (𝐴𝐿𝑥) = (𝐴𝐿𝑏))
149	135, 148	breqtrd 4955	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) ∧ 𝑏 ∈ 𝑃) ∧ 𝑏 = (((pInvG‘𝐺)‘𝑥)‘𝐴)) ∧ 𝐴 ≠ 𝑏) → 𝐷(⟂G‘𝐺)(𝐴𝐿𝑏))
150	149	ex 405	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) ∧ 𝑏 ∈ 𝑃) ∧ 𝑏 = (((pInvG‘𝐺)‘𝑥)‘𝐴)) → (𝐴 ≠ 𝑏 → 𝐷(⟂G‘𝐺)(𝐴𝐿𝑏)))
151	150	necon1bd 2985	. . . . . . . 8 ⊢ ((((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) ∧ 𝑏 ∈ 𝑃) ∧ 𝑏 = (((pInvG‘𝐺)‘𝑥)‘𝐴)) → (¬ 𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) → 𝐴 = 𝑏))
152	151	orrd 849	. . . . . . 7 ⊢ ((((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) ∧ 𝑏 ∈ 𝑃) ∧ 𝑏 = (((pInvG‘𝐺)‘𝑥)‘𝐴)) → (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏))
153	130, 152	jca 504	. . . . . 6 ⊢ ((((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) ∧ 𝑏 ∈ 𝑃) ∧ 𝑏 = (((pInvG‘𝐺)‘𝑥)‘𝐴)) → ((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏)))
154	120, 153	impbida 788	. . . . 5 ⊢ (((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) ∧ 𝑏 ∈ 𝑃) → (((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏)) ↔ 𝑏 = (((pInvG‘𝐺)‘𝑥)‘𝐴)))
155	154	ralrimiva 3132	. . . 4 ⊢ ((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) → ∀𝑏 ∈ 𝑃 (((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏)) ↔ 𝑏 = (((pInvG‘𝐺)‘𝑥)‘𝐴)))
156		reu6i 3631	. . . 4 ⊢ (((((pInvG‘𝐺)‘𝑥)‘𝐴) ∈ 𝑃 ∧ ∀𝑏 ∈ 𝑃 (((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏)) ↔ 𝑏 = (((pInvG‘𝐺)‘𝑥)‘𝐴))) → ∃!𝑏 ∈ 𝑃 ((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏)))
157	84, 155, 156	syl2anc 576	. . 3 ⊢ ((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) → ∃!𝑏 ∈ 𝑃 ((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏)))
158	5, 6, 7, 19, 75, 77, 82, 88	footex 26209	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) → ∃𝑥 ∈ 𝐷 (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷)
159	157, 158	r19.29a 3234	. 2 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) → ∃!𝑏 ∈ 𝑃 ((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏)))
160	74, 159	pm2.61dan 800	1 ⊢ (𝜑 → ∃!𝑏 ∈ 𝑃 ((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ wa 387 ∨ wo 833 = wceq 1507 ∈ wcel 2050 ≠ wne 2967 ∀wral 3088 ∃!wreu 3090 ∃*wrmo 3091 class class class wbr 4929 ran crn 5408 ‘cfv 6188 (class class class)co 6976 2c2 11495 Basecbs 16339 distcds 16430 TarskiGcstrkg 25918 Dim_TarskiG≥cstrkgld 25922 Itvcitv 25924 LineGclng 25925 pInvGcmir 26140 ⟂Gcperpg 26183 midGcmid 26260

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-rep 5049 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 ax-cnex 10391 ax-resscn 10392 ax-1cn 10393 ax-icn 10394 ax-addcl 10395 ax-addrcl 10396 ax-mulcl 10397 ax-mulrcl 10398 ax-mulcom 10399 ax-addass 10400 ax-mulass 10401 ax-distr 10402 ax-i2m1 10403 ax-1ne0 10404 ax-1rid 10405 ax-rnegex 10406 ax-rrecex 10407 ax-cnre 10408 ax-pre-lttri 10409 ax-pre-lttrn 10410 ax-pre-ltadd 10411 ax-pre-mulgt0 10412

This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rmo 3096 df-rab 3097 df-v 3417 df-sbc 3682 df-csb 3787 df-dif 3832 df-un 3834 df-in 3836 df-ss 3843 df-pss 3845 df-nul 4179 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-tp 4446 df-op 4448 df-uni 4713 df-int 4750 df-iun 4794 df-br 4930 df-opab 4992 df-mpt 5009 df-tr 5031 df-id 5312 df-eprel 5317 df-po 5326 df-so 5327 df-fr 5366 df-we 5368 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-pred 5986 df-ord 6032 df-on 6033 df-lim 6034 df-suc 6035 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-riota 6937 df-ov 6979 df-oprab 6980 df-mpo 6981 df-om 7397 df-1st 7501 df-2nd 7502 df-wrecs 7750 df-recs 7812 df-rdg 7850 df-1o 7905 df-oadd 7909 df-er 8089 df-map 8208 df-pm 8209 df-en 8307 df-dom 8308 df-sdom 8309 df-fin 8310 df-dju 9124 df-card 9162 df-pnf 10476 df-mnf 10477 df-xr 10478 df-ltxr 10479 df-le 10480 df-sub 10672 df-neg 10673 df-nn 11440 df-2 11503 df-3 11504 df-n0 11708 df-xnn0 11780 df-z 11794 df-uz 12059 df-fz 12709 df-fzo 12850 df-hash 13506 df-word 13673 df-concat 13734 df-s1 13759 df-s2 14072 df-s3 14073 df-trkgc 25936 df-trkgb 25937 df-trkgcb 25938 df-trkgld 25940 df-trkg 25941 df-cgrg 25999 df-leg 26071 df-mir 26141 df-rag 26182 df-perpg 26184 df-mid 26262

This theorem is referenced by: lmif 26273 islmib 26275

W3C validator