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Theorem lmieu 29104

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 485	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝐷) → 𝐴 ∈ 𝑃)
3		simpr 489	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) ∧ ¬ 𝐴 = 𝑏) → ¬ 𝐴 = 𝑏)
4		eqidd 2763	. . . . . . . . . . . . . . 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 744	. . . . . . . . . . . . . . . 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 744	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) ∧ ¬ 𝐴 = 𝑏) → 𝐺Dim_TarskiG≥2)
12	2	ad3antrrr 742	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) ∧ ¬ 𝐴 = 𝑏) → 𝐴 ∈ 𝑃)
13		simpllr 787	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) ∧ ¬ 𝐴 = 𝑏) → 𝑏 ∈ 𝑃)
14		eqid 2762	. . . . . . . . . . . . . . . 16 ⊢ (pInvG‘𝐺) = (pInvG‘𝐺)
15	5, 6, 7, 9, 11, 12, 13	midcl 29097	. . . . . . . . . . . . . . . 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 29098	. . . . . . . . . . . . . . 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 260	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) ∧ ¬ 𝐴 = 𝑏) → 𝑏 = (((pInvG‘𝐺)‘(𝐴(midG‘𝐺)𝑏))‘𝐴))
18	17	adantr 485	. . . . . . . . . . . . 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 485	. . . . . . . . . . . . . . . 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 744	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) ∧ ¬ 𝐴 = 𝑏) → 𝐷 ∈ ran 𝐿)
23	22	adantr 485	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) ∧ ¬ 𝐴 = 𝑏) ∧ 𝐷 ≠ (𝐴𝐿𝑏)) → 𝐷 ∈ ran 𝐿)
24	12	adantr 485	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) ∧ ¬ 𝐴 = 𝑏) ∧ 𝐷 ≠ (𝐴𝐿𝑏)) → 𝐴 ∈ 𝑃)
25	13	adantr 485	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) ∧ ¬ 𝐴 = 𝑏) ∧ 𝐷 ≠ (𝐴𝐿𝑏)) → 𝑏 ∈ 𝑃)
26	3	neqned 2964	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) ∧ ¬ 𝐴 = 𝑏) → 𝐴 ≠ 𝑏)
27	26	adantr 485	. . . . . . . . . . . . . . . . 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 28914	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) ∧ ¬ 𝐴 = 𝑏) ∧ 𝐷 ≠ (𝐴𝐿𝑏)) → (𝐴𝐿𝑏) ∈ ran 𝐿)
29		simpr 489	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) ∧ ¬ 𝐴 = 𝑏) ∧ 𝐷 ≠ (𝐴𝐿𝑏)) → 𝐷 ≠ (𝐴𝐿𝑏))
30		simp-4r 795	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) ∧ ¬ 𝐴 = 𝑏) → 𝐴 ∈ 𝐷)
31	30	adantr 485	. . . . . . . . . . . . . . . . 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 28917	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) ∧ ¬ 𝐴 = 𝑏) ∧ 𝐷 ≠ (𝐴𝐿𝑏)) → 𝐴 ∈ (𝐴𝐿𝑏))
33	31, 32	elind 4152	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) ∧ ¬ 𝐴 = 𝑏) ∧ 𝐷 ≠ (𝐴𝐿𝑏)) → 𝐴 ∈ (𝐷 ∩ (𝐴𝐿𝑏)))
34		simpllr 787	. . . . . . . . . . . . . . . . 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 29099	. . . . . . . . . . . . . . . . . . 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 28903	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) ∧ ¬ 𝐴 = 𝑏) → (𝐴(midG‘𝐺)𝑏) ∈ (𝐴𝐿𝑏))
37	36	adantr 485	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) ∧ ¬ 𝐴 = 𝑏) ∧ 𝐷 ≠ (𝐴𝐿𝑏)) → (𝐴(midG‘𝐺)𝑏) ∈ (𝐴𝐿𝑏))
38	34, 37	elind 4152	. . . . . . . . . . . . . . . 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 28927	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) ∧ ¬ 𝐴 = 𝑏) ∧ 𝐷 ≠ (𝐴𝐿𝑏)) → 𝐴 = (𝐴(midG‘𝐺)𝑏))
40	39	fveq2d 6885	. . . . . . . . . . . . . 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 6883	. . . . . . . . . . . . 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 2762	. . . . . . . . . . . . . 14 ⊢ ((pInvG‘𝐺)‘𝐴) = ((pInvG‘𝐺)‘𝐴)
43	5, 6, 7, 19, 14, 20, 24, 42	mircinv 28956	. . . . . . . . . . . . 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 2804	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) ∧ ¬ 𝐴 = 𝑏) ∧ 𝐷 ≠ (𝐴𝐿𝑏)) → 𝐴 = 𝑏)
45	3, 44	mtand 827	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) ∧ ¬ 𝐴 = 𝑏) → ¬ 𝐷 ≠ (𝐴𝐿𝑏))
46	8	ad5antr 746	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) ∧ ¬ 𝐴 = 𝑏) ∧ 𝐷(⟂G‘𝐺)(𝐴𝐿𝑏)) → 𝐺 ∈ TarskiG)
47	21	ad5antr 746	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) ∧ ¬ 𝐴 = 𝑏) ∧ 𝐷(⟂G‘𝐺)(𝐴𝐿𝑏)) → 𝐷 ∈ ran 𝐿)
48		nne 2961	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝐷 ≠ (𝐴𝐿𝑏) ↔ 𝐷 = (𝐴𝐿𝑏))
49	45, 48	sylib 221	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) ∧ ¬ 𝐴 = 𝑏) → 𝐷 = (𝐴𝐿𝑏))
50	49	adantr 485	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) ∧ ¬ 𝐴 = 𝑏) ∧ 𝐷(⟂G‘𝐺)(𝐴𝐿𝑏)) → 𝐷 = (𝐴𝐿𝑏))
51	50, 47	eqeltrrd 2863	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) ∧ ¬ 𝐴 = 𝑏) ∧ 𝐷(⟂G‘𝐺)(𝐴𝐿𝑏)) → (𝐴𝐿𝑏) ∈ ran 𝐿)
52		simpr 489	. . . . . . . . . . . 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 29005	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) ∧ ¬ 𝐴 = 𝑏) ∧ 𝐷(⟂G‘𝐺)(𝐴𝐿𝑏)) → 𝐷 ≠ (𝐴𝐿𝑏))
54	45, 53	mtand 827	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) ∧ ¬ 𝐴 = 𝑏) → ¬ 𝐷(⟂G‘𝐺)(𝐴𝐿𝑏))
55	54	ex 417	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) → (¬ 𝐴 = 𝑏 → ¬ 𝐷(⟂G‘𝐺)(𝐴𝐿𝑏)))
56	55	con4d 116	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) → (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) → 𝐴 = 𝑏))
57		idd 25	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) → (𝐴 = 𝑏 → 𝐴 = 𝑏))
58	56, 57	jaod 872	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) → ((𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏) → 𝐴 = 𝑏))
59	58	impr 459	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ ((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏))) → 𝐴 = 𝑏)
60	59	eqcomd 2768	. . . . 5 ⊢ ((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ ((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏))) → 𝑏 = 𝐴)
61		simpr 489	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ 𝑏 = 𝐴) → 𝑏 = 𝐴)
62	61	oveq2d 7428	. . . . . . . 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 29101	. . . . . . . . 9 ⊢ (𝜑 → (𝐴(midG‘𝐺)𝐴) = 𝐴)
64	63	ad3antrrr 742	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ 𝑏 = 𝐴) → (𝐴(midG‘𝐺)𝐴) = 𝐴)
65	62, 64	eqtrd 2797	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ 𝑏 = 𝐴) → (𝐴(midG‘𝐺)𝑏) = 𝐴)
66		simpllr 787	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ 𝑏 = 𝐴) → 𝐴 ∈ 𝐷)
67	65, 66	eqeltrd 2862	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ 𝑏 = 𝐴) → (𝐴(midG‘𝐺)𝑏) ∈ 𝐷)
68	61	eqcomd 2768	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ 𝑏 = 𝐴) → 𝐴 = 𝑏)
69	68	olcd 887	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ 𝑏 = 𝐴) → (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏))
70	67, 69	jca 520	. . . . 5 ⊢ ((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ 𝑏 = 𝐴) → ((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏)))
71	60, 70	impbida 812	. . . 4 ⊢ (((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) → (((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏)) ↔ 𝑏 = 𝐴))
72	71	ralrimiva 3156	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝐷) → ∀𝑏 ∈ 𝑃 (((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏)) ↔ 𝑏 = 𝐴))
73		reu6i 3690	. . 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 595	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝐷) → ∃!𝑏 ∈ 𝑃 ((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏)))
75	8	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) → 𝐺 ∈ TarskiG)
76	75	ad2antrr 738	. . . . 5 ⊢ ((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) → 𝐺 ∈ TarskiG)
77	21	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) → 𝐷 ∈ ran 𝐿)
78	77	ad2antrr 738	. . . . . 6 ⊢ ((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) → 𝐷 ∈ ran 𝐿)
79		simplr 780	. . . . . 6 ⊢ ((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) → 𝑥 ∈ 𝐷)
80	5, 19, 7, 76, 78, 79	tglnpt 28829	. . . . 5 ⊢ ((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) → 𝑥 ∈ 𝑃)
81		eqid 2762	. . . . 5 ⊢ ((pInvG‘𝐺)‘𝑥) = ((pInvG‘𝐺)‘𝑥)
82	1	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) → 𝐴 ∈ 𝑃)
83	82	ad2antrr 738	. . . . 5 ⊢ ((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) → 𝐴 ∈ 𝑃)
84	5, 6, 7, 19, 14, 76, 80, 81, 83	mircl 28949	. . . 4 ⊢ ((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) → (((pInvG‘𝐺)‘𝑥)‘𝐴) ∈ 𝑃)
85		oveq2 7420	. . . . . . . . . 10 ⊢ (𝑥 = (𝐴(midG‘𝐺)𝑏) → (𝐴𝐿𝑥) = (𝐴𝐿(𝐴(midG‘𝐺)𝑏)))
86	85	breq1d 5118	. . . . . . . . 9 ⊢ (𝑥 = (𝐴(midG‘𝐺)𝑏) → ((𝐴𝐿𝑥)(⟂G‘𝐺)𝐷 ↔ (𝐴𝐿(𝐴(midG‘𝐺)𝑏))(⟂G‘𝐺)𝐷))
87		simprl 782	. . . . . . . . 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 489	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) → ¬ 𝐴 ∈ 𝐷)
89	5, 6, 7, 19, 75, 77, 82, 88	foot 29013	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) → ∃!𝑥 ∈ 𝐷 (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷)
90		reurmo 3371	. . . . . . . . . . 11 ⊢ (∃!𝑥 ∈ 𝐷 (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷 → ∃*𝑥 ∈ 𝐷 (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷)
91	89, 90	syl 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) → ∃*𝑥 ∈ 𝐷 (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷)
92	91	ad4antr 744	. . . . . . . . 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 738	. . . . . . . . 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 787	. . . . . . . . 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 738	. . . . . . . . . . 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 738	. . . . . . . . . . 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 780	. . . . . . . . . . 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 746	. . . . . . . . . . . . 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 29097	. . . . . . . . . . . 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 29099	. . . . . . . . . . . 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 744	. . . . . . . . . . . . 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 3055	. . . . . . . . . . . . 13 ⊢ (((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ ¬ 𝐴 ∈ 𝐷) → (𝐴(midG‘𝐺)𝑏) ≠ 𝐴)
103	87, 101, 102	syl2anc 595	. . . . . . . . . . . 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 28770	. . . . . . . . . . 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 28903	. . . . . . . . . . 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 28916	. . . . . . . . . 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 738	. . . . . . . . . . 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 28914	. . . . . . . . . . 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 2962	. . . . . . . . . . . 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 784	. . . . . . . . . . . . . 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 884	. . . . . . . . . . . . 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 877	. . . . . . . . . . . 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 16	. . . . . . . . . . 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 29004	. . . . . . . . . 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 5134	. . . . . . . . 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 3846	. . . . . . . 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 2768	. . . . . . 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 738	. . . . . . . 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 29098	. . . . . . 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 260	. . . . . 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 489	. . . . . . . . 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 738	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) ∧ 𝑏 ∈ 𝑃) ∧ 𝑏 = (((pInvG‘𝐺)‘𝑥)‘𝐴)) → 𝐺 ∈ TarskiG)
123	10	ad5antr 746	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) ∧ 𝑏 ∈ 𝑃) ∧ 𝑏 = (((pInvG‘𝐺)‘𝑥)‘𝐴)) → 𝐺Dim_TarskiG≥2)
124	83	ad2antrr 738	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) ∧ 𝑏 ∈ 𝑃) ∧ 𝑏 = (((pInvG‘𝐺)‘𝑥)‘𝐴)) → 𝐴 ∈ 𝑃)
125		simplr 780	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) ∧ 𝑏 ∈ 𝑃) ∧ 𝑏 = (((pInvG‘𝐺)‘𝑥)‘𝐴)) → 𝑏 ∈ 𝑃)
126	80	ad2antrr 738	. . . . . . . . . 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 29098	. . . . . . . . 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 235	. . . . . . . 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 738	. . . . . . . 8 ⊢ ((((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) ∧ 𝑏 ∈ 𝑃) ∧ 𝑏 = (((pInvG‘𝐺)‘𝑥)‘𝐴)) → 𝑥 ∈ 𝐷)
130	128, 129	eqeltrd 2862	. . . . . . 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 485	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) ∧ 𝑏 ∈ 𝑃) ∧ 𝑏 = (((pInvG‘𝐺)‘𝑥)‘𝐴)) ∧ 𝐴 ≠ 𝑏) → 𝐺 ∈ TarskiG)
132		simp-4r 795	. . . . . . . . . . . . 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 29001	. . . . . . . . . . . 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 742	. . . . . . . . . . . 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 29004	. . . . . . . . . . 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 485	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) ∧ 𝑏 ∈ 𝑃) ∧ 𝑏 = (((pInvG‘𝐺)‘𝑥)‘𝐴)) ∧ 𝐴 ≠ 𝑏) → 𝐴 ∈ 𝑃)
137	126	adantr 485	. . . . . . . . . . . 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 28912	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) ∧ 𝑏 ∈ 𝑃) ∧ 𝑏 = (((pInvG‘𝐺)‘𝑥)‘𝐴)) ∧ 𝐴 ≠ 𝑏) → 𝐴 ≠ 𝑥)
139		simpllr 787	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) ∧ 𝑏 ∈ 𝑃) ∧ 𝑏 = (((pInvG‘𝐺)‘𝑥)‘𝐴)) ∧ 𝐴 ≠ 𝑏) → 𝑏 ∈ 𝑃)
140		simpr 489	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) ∧ 𝑏 ∈ 𝑃) ∧ 𝑏 = (((pInvG‘𝐺)‘𝑥)‘𝐴)) ∧ 𝐴 ≠ 𝑏) → 𝐴 ≠ 𝑏)
141	140	necomd 3012	. . . . . . . . . . . 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 28946	. . . . . . . . . . . . . . 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 780	. . . . . . . . . . . . . . . 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 7427	. . . . . . . . . . . . . . 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 2865	. . . . . . . . . . . . . 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 28903	. . . . . . . . . . . . 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 28907	. . . . . . . . . . . 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 28916	. . . . . . . . . . 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 5136	. . . . . . . . . 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 417	. . . . . . . . 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 2975	. . . . . . . 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 876	. . . . . . 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 520	. . . . . 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 812	. . . . 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 3156	. . . 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 3690	. . . 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 595	. . 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 29012	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) → ∃𝑥 ∈ 𝐷 (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷)
159	157, 158	r19.29a 3172	. 2 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) → ∃!𝑏 ∈ 𝑃 ((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏)))
160	74, 159	pm2.61dan 824	1 ⊢ (𝜑 → ∃!𝑏 ∈ 𝑃 ((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 ∨ wo 860 = wceq 1569 ∈ wcel 2142 ≠ wne 2957 ∀wral 3078 ∃!wreu 3366 ∃*wrmo 3367 class class class wbr 5108 ran crn 5661 ‘cfv 6536 (class class class)co 7412 2c2 12301 Basecbs 17275 distcds 17325 TarskiGcstrkg 28707 Dim_TarskiG≥cstrkgld 28711 Itvcitv 28713 LineGclng 28714 pInvGcmir 28940 ⟂Gcperpg 28986 midGcmid 29092

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1824 ax-4 1838 ax-5 1939 ax-6 1996 ax-7 2037 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5237 ax-sep 5256 ax-nul 5268 ax-pow 5335 ax-pr 5403 ax-un 7734 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1103 df-3an 1104 df-tru 1572 df-fal 1582 df-ex 1809 df-nf 1813 df-sb 2096 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3368 df-reu 3369 df-rab 3416 df-v 3456 df-sbc 3744 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4487 df-pw 4563 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4595 df-uni 4872 df-int 4912 df-iun 4957 df-br 5109 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5555 df-eprel 5560 df-po 5568 df-so 5569 df-fr 5613 df-we 5615 df-xp 5666 df-rel 5667 df-cnv 5668 df-co 5669 df-dm 5670 df-rn 5671 df-res 5672 df-ima 5673 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7861 df-1st 7984 df-2nd 7985 df-frecs 8276 df-wrecs 8307 df-recs 8356 df-rdg 8395 df-1o 8451 df-oadd 8455 df-er 8692 df-map 8824 df-pm 8825 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-dju 9894 df-card 9932 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11449 df-neg 11450 df-nn 12240 df-2 12309 df-3 12310 df-n0 12511 df-xnn0 12584 df-z 12598 df-uz 12869 df-fz 13542 df-fzo 13690 df-hash 14374 df-word 14558 df-concat 14615 df-s1 14641 df-s2 14892 df-s3 14893 df-trkgc 28728 df-trkgb 28729 df-trkgcb 28730 df-trkgld 28732 df-trkg 28733 df-cgrg 28791 df-leg 28863 df-mir 28941 df-rag 28985 df-perpg 28987 df-mid 29094

This theorem is used by: lmif 29105 islmib 29107

W3C validator