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Theorem lmieu 28608

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 479	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝐷) → 𝐴 ∈ 𝑃)
3		simpr 483	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) ∧ ¬ 𝐴 = 𝑏) → ¬ 𝐴 = 𝑏)
4		eqidd 2729	. . . . . . . . . . . . . . 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 730	. . . . . . . . . . . . . . . 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 730	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) ∧ ¬ 𝐴 = 𝑏) → 𝐺Dim_TarskiG≥2)
12	2	ad3antrrr 728	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) ∧ ¬ 𝐴 = 𝑏) → 𝐴 ∈ 𝑃)
13		simpllr 774	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) ∧ ¬ 𝐴 = 𝑏) → 𝑏 ∈ 𝑃)
14		eqid 2728	. . . . . . . . . . . . . . . 16 ⊢ (pInvG‘𝐺) = (pInvG‘𝐺)
15	5, 6, 7, 9, 11, 12, 13	midcl 28601	. . . . . . . . . . . . . . . 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 28602	. . . . . . . . . . . . . . 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 256	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) ∧ ¬ 𝐴 = 𝑏) → 𝑏 = (((pInvG‘𝐺)‘(𝐴(midG‘𝐺)𝑏))‘𝐴))
18	17	adantr 479	. . . . . . . . . . . . 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 479	. . . . . . . . . . . . . . . 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 730	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) ∧ ¬ 𝐴 = 𝑏) → 𝐷 ∈ ran 𝐿)
23	22	adantr 479	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) ∧ ¬ 𝐴 = 𝑏) ∧ 𝐷 ≠ (𝐴𝐿𝑏)) → 𝐷 ∈ ran 𝐿)
24	12	adantr 479	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) ∧ ¬ 𝐴 = 𝑏) ∧ 𝐷 ≠ (𝐴𝐿𝑏)) → 𝐴 ∈ 𝑃)
25	13	adantr 479	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) ∧ ¬ 𝐴 = 𝑏) ∧ 𝐷 ≠ (𝐴𝐿𝑏)) → 𝑏 ∈ 𝑃)
26	3	neqned 2944	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) ∧ ¬ 𝐴 = 𝑏) → 𝐴 ≠ 𝑏)
27	26	adantr 479	. . . . . . . . . . . . . . . . 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 28454	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) ∧ ¬ 𝐴 = 𝑏) ∧ 𝐷 ≠ (𝐴𝐿𝑏)) → (𝐴𝐿𝑏) ∈ ran 𝐿)
29		simpr 483	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) ∧ ¬ 𝐴 = 𝑏) ∧ 𝐷 ≠ (𝐴𝐿𝑏)) → 𝐷 ≠ (𝐴𝐿𝑏))
30		simp-4r 782	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) ∧ ¬ 𝐴 = 𝑏) → 𝐴 ∈ 𝐷)
31	30	adantr 479	. . . . . . . . . . . . . . . . 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 28457	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) ∧ ¬ 𝐴 = 𝑏) ∧ 𝐷 ≠ (𝐴𝐿𝑏)) → 𝐴 ∈ (𝐴𝐿𝑏))
33	31, 32	elind 4196	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) ∧ ¬ 𝐴 = 𝑏) ∧ 𝐷 ≠ (𝐴𝐿𝑏)) → 𝐴 ∈ (𝐷 ∩ (𝐴𝐿𝑏)))
34		simpllr 774	. . . . . . . . . . . . . . . . 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 28603	. . . . . . . . . . . . . . . . . . 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 28443	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) ∧ ¬ 𝐴 = 𝑏) → (𝐴(midG‘𝐺)𝑏) ∈ (𝐴𝐿𝑏))
37	36	adantr 479	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) ∧ ¬ 𝐴 = 𝑏) ∧ 𝐷 ≠ (𝐴𝐿𝑏)) → (𝐴(midG‘𝐺)𝑏) ∈ (𝐴𝐿𝑏))
38	34, 37	elind 4196	. . . . . . . . . . . . . . . 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 28467	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) ∧ ¬ 𝐴 = 𝑏) ∧ 𝐷 ≠ (𝐴𝐿𝑏)) → 𝐴 = (𝐴(midG‘𝐺)𝑏))
40	39	fveq2d 6906	. . . . . . . . . . . . . 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 6904	. . . . . . . . . . . . 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 2728	. . . . . . . . . . . . . 14 ⊢ ((pInvG‘𝐺)‘𝐴) = ((pInvG‘𝐺)‘𝐴)
43	5, 6, 7, 19, 14, 20, 24, 42	mircinv 28492	. . . . . . . . . . . . 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 2775	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) ∧ ¬ 𝐴 = 𝑏) ∧ 𝐷 ≠ (𝐴𝐿𝑏)) → 𝐴 = 𝑏)
45	3, 44	mtand 814	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) ∧ ¬ 𝐴 = 𝑏) → ¬ 𝐷 ≠ (𝐴𝐿𝑏))
46	8	ad5antr 732	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) ∧ ¬ 𝐴 = 𝑏) ∧ 𝐷(⟂G‘𝐺)(𝐴𝐿𝑏)) → 𝐺 ∈ TarskiG)
47	21	ad5antr 732	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) ∧ ¬ 𝐴 = 𝑏) ∧ 𝐷(⟂G‘𝐺)(𝐴𝐿𝑏)) → 𝐷 ∈ ran 𝐿)
48		nne 2941	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝐷 ≠ (𝐴𝐿𝑏) ↔ 𝐷 = (𝐴𝐿𝑏))
49	45, 48	sylib 217	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) ∧ ¬ 𝐴 = 𝑏) → 𝐷 = (𝐴𝐿𝑏))
50	49	adantr 479	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) ∧ ¬ 𝐴 = 𝑏) ∧ 𝐷(⟂G‘𝐺)(𝐴𝐿𝑏)) → 𝐷 = (𝐴𝐿𝑏))
51	50, 47	eqeltrrd 2830	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) ∧ ¬ 𝐴 = 𝑏) ∧ 𝐷(⟂G‘𝐺)(𝐴𝐿𝑏)) → (𝐴𝐿𝑏) ∈ ran 𝐿)
52		simpr 483	. . . . . . . . . . . 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 28538	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) ∧ ¬ 𝐴 = 𝑏) ∧ 𝐷(⟂G‘𝐺)(𝐴𝐿𝑏)) → 𝐷 ≠ (𝐴𝐿𝑏))
54	45, 53	mtand 814	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) ∧ ¬ 𝐴 = 𝑏) → ¬ 𝐷(⟂G‘𝐺)(𝐴𝐿𝑏))
55	54	ex 411	. . . . . . . . 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 857	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ (𝐴(midG‘𝐺)𝑏) ∈ 𝐷) → ((𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏) → 𝐴 = 𝑏))
59	58	impr 453	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ ((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏))) → 𝐴 = 𝑏)
60	59	eqcomd 2734	. . . . 5 ⊢ ((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ ((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏))) → 𝑏 = 𝐴)
61		simpr 483	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ 𝑏 = 𝐴) → 𝑏 = 𝐴)
62	61	oveq2d 7442	. . . . . . . 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 28605	. . . . . . . . 9 ⊢ (𝜑 → (𝐴(midG‘𝐺)𝐴) = 𝐴)
64	63	ad3antrrr 728	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ 𝑏 = 𝐴) → (𝐴(midG‘𝐺)𝐴) = 𝐴)
65	62, 64	eqtrd 2768	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ 𝑏 = 𝐴) → (𝐴(midG‘𝐺)𝑏) = 𝐴)
66		simpllr 774	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ 𝑏 = 𝐴) → 𝐴 ∈ 𝐷)
67	65, 66	eqeltrd 2829	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ 𝑏 = 𝐴) → (𝐴(midG‘𝐺)𝑏) ∈ 𝐷)
68	61	eqcomd 2734	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ 𝑏 = 𝐴) → 𝐴 = 𝑏)
69	68	olcd 872	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ 𝑏 = 𝐴) → (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏))
70	67, 69	jca 510	. . . . 5 ⊢ ((((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) ∧ 𝑏 = 𝐴) → ((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏)))
71	60, 70	impbida 799	. . . 4 ⊢ (((𝜑 ∧ 𝐴 ∈ 𝐷) ∧ 𝑏 ∈ 𝑃) → (((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏)) ↔ 𝑏 = 𝐴))
72	71	ralrimiva 3143	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝐷) → ∀𝑏 ∈ 𝑃 (((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏)) ↔ 𝑏 = 𝐴))
73		reu6i 3725	. . 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 582	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝐷) → ∃!𝑏 ∈ 𝑃 ((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏)))
75	8	adantr 479	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) → 𝐺 ∈ TarskiG)
76	75	ad2antrr 724	. . . . 5 ⊢ ((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) → 𝐺 ∈ TarskiG)
77	21	adantr 479	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) → 𝐷 ∈ ran 𝐿)
78	77	ad2antrr 724	. . . . . 6 ⊢ ((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) → 𝐷 ∈ ran 𝐿)
79		simplr 767	. . . . . 6 ⊢ ((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) → 𝑥 ∈ 𝐷)
80	5, 19, 7, 76, 78, 79	tglnpt 28373	. . . . 5 ⊢ ((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) → 𝑥 ∈ 𝑃)
81		eqid 2728	. . . . 5 ⊢ ((pInvG‘𝐺)‘𝑥) = ((pInvG‘𝐺)‘𝑥)
82	1	adantr 479	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) → 𝐴 ∈ 𝑃)
83	82	ad2antrr 724	. . . . 5 ⊢ ((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) → 𝐴 ∈ 𝑃)
84	5, 6, 7, 19, 14, 76, 80, 81, 83	mircl 28485	. . . 4 ⊢ ((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) → (((pInvG‘𝐺)‘𝑥)‘𝐴) ∈ 𝑃)
85		oveq2 7434	. . . . . . . . . 10 ⊢ (𝑥 = (𝐴(midG‘𝐺)𝑏) → (𝐴𝐿𝑥) = (𝐴𝐿(𝐴(midG‘𝐺)𝑏)))
86	85	breq1d 5162	. . . . . . . . 9 ⊢ (𝑥 = (𝐴(midG‘𝐺)𝑏) → ((𝐴𝐿𝑥)(⟂G‘𝐺)𝐷 ↔ (𝐴𝐿(𝐴(midG‘𝐺)𝑏))(⟂G‘𝐺)𝐷))
87		simprl 769	. . . . . . . . 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 483	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) → ¬ 𝐴 ∈ 𝐷)
89	5, 6, 7, 19, 75, 77, 82, 88	foot 28546	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) → ∃!𝑥 ∈ 𝐷 (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷)
90		reurmo 3377	. . . . . . . . . . 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 730	. . . . . . . . 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 724	. . . . . . . . 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 774	. . . . . . . . 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 724	. . . . . . . . . . 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 724	. . . . . . . . . . 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 767	. . . . . . . . . . 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 732	. . . . . . . . . . . . 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 28601	. . . . . . . . . . . 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 28603	. . . . . . . . . . . 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 730	. . . . . . . . . . . . 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 3037	. . . . . . . . . . . . 13 ⊢ (((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ ¬ 𝐴 ∈ 𝐷) → (𝐴(midG‘𝐺)𝑏) ≠ 𝐴)
103	87, 101, 102	syl2anc 582	. . . . . . . . . . . 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 28314	. . . . . . . . . . 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 28443	. . . . . . . . . . 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 28456	. . . . . . . . . 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 724	. . . . . . . . . . 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 28454	. . . . . . . . . . 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 2942	. . . . . . . . . . . 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 771	. . . . . . . . . . . . . 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 869	. . . . . . . . . . . . 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 862	. . . . . . . . . . . 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 28537	. . . . . . . . . 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 5176	. . . . . . . . 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 3888	. . . . . . . 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 2734	. . . . . . 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 724	. . . . . . . 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 28602	. . . . . . 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 256	. . . . . 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 483	. . . . . . . . 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 724	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) ∧ 𝑏 ∈ 𝑃) ∧ 𝑏 = (((pInvG‘𝐺)‘𝑥)‘𝐴)) → 𝐺 ∈ TarskiG)
123	10	ad5antr 732	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) ∧ 𝑏 ∈ 𝑃) ∧ 𝑏 = (((pInvG‘𝐺)‘𝑥)‘𝐴)) → 𝐺Dim_TarskiG≥2)
124	83	ad2antrr 724	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) ∧ 𝑏 ∈ 𝑃) ∧ 𝑏 = (((pInvG‘𝐺)‘𝑥)‘𝐴)) → 𝐴 ∈ 𝑃)
125		simplr 767	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) ∧ 𝑏 ∈ 𝑃) ∧ 𝑏 = (((pInvG‘𝐺)‘𝑥)‘𝐴)) → 𝑏 ∈ 𝑃)
126	80	ad2antrr 724	. . . . . . . . . 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 28602	. . . . . . . . 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 231	. . . . . . . 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 724	. . . . . . . 8 ⊢ ((((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) ∧ 𝑏 ∈ 𝑃) ∧ 𝑏 = (((pInvG‘𝐺)‘𝑥)‘𝐴)) → 𝑥 ∈ 𝐷)
130	128, 129	eqeltrd 2829	. . . . . . 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 479	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) ∧ 𝑏 ∈ 𝑃) ∧ 𝑏 = (((pInvG‘𝐺)‘𝑥)‘𝐴)) ∧ 𝐴 ≠ 𝑏) → 𝐺 ∈ TarskiG)
132		simp-4r 782	. . . . . . . . . . . . 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 28534	. . . . . . . . . . . 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 728	. . . . . . . . . . . 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 28537	. . . . . . . . . . 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 479	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) ∧ 𝑏 ∈ 𝑃) ∧ 𝑏 = (((pInvG‘𝐺)‘𝑥)‘𝐴)) ∧ 𝐴 ≠ 𝑏) → 𝐴 ∈ 𝑃)
137	126	adantr 479	. . . . . . . . . . . 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 28452	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) ∧ 𝑏 ∈ 𝑃) ∧ 𝑏 = (((pInvG‘𝐺)‘𝑥)‘𝐴)) ∧ 𝐴 ≠ 𝑏) → 𝐴 ≠ 𝑥)
139		simpllr 774	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) ∧ 𝑏 ∈ 𝑃) ∧ 𝑏 = (((pInvG‘𝐺)‘𝑥)‘𝐴)) ∧ 𝐴 ≠ 𝑏) → 𝑏 ∈ 𝑃)
140		simpr 483	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷) ∧ 𝑏 ∈ 𝑃) ∧ 𝑏 = (((pInvG‘𝐺)‘𝑥)‘𝐴)) ∧ 𝐴 ≠ 𝑏) → 𝐴 ≠ 𝑏)
141	140	necomd 2993	. . . . . . . . . . . 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 28482	. . . . . . . . . . . . . . 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 767	. . . . . . . . . . . . . . . 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 7441	. . . . . . . . . . . . . . 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 2832	. . . . . . . . . . . . . 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 28443	. . . . . . . . . . . . 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 28447	. . . . . . . . . . . 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 28456	. . . . . . . . . . 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 5178	. . . . . . . . . 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 411	. . . . . . . . 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 2955	. . . . . . . 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 861	. . . . . . 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 510	. . . . . 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 799	. . . . 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 3143	. . . 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 3725	. . . 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 582	. . 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 28545	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) → ∃𝑥 ∈ 𝐷 (𝐴𝐿𝑥)(⟂G‘𝐺)𝐷)
159	157, 158	r19.29a 3159	. 2 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) → ∃!𝑏 ∈ 𝑃 ((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏)))
160	74, 159	pm2.61dan 811	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 205 ∧ wa 394 ∨ wo 845 = wceq 1533 ∈ wcel 2098 ≠ wne 2937 ∀wral 3058 ∃!wreu 3372 ∃*wrmo 3373 class class class wbr 5152 ran crn 5683 ‘cfv 6553 (class class class)co 7426 2c2 12305 Basecbs 17187 distcds 17249 TarskiGcstrkg 28251 Dim_TarskiG≥cstrkgld 28255 Itvcitv 28257 LineGclng 28258 pInvGcmir 28476 ⟂Gcperpg 28519 midGcmid 28596

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 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223

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 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-oadd 8497 df-er 8731 df-map 8853 df-pm 8854 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-dju 9932 df-card 9970 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-2 12313 df-3 12314 df-n0 12511 df-xnn0 12583 df-z 12597 df-uz 12861 df-fz 13525 df-fzo 13668 df-hash 14330 df-word 14505 df-concat 14561 df-s1 14586 df-s2 14839 df-s3 14840 df-trkgc 28272 df-trkgb 28273 df-trkgcb 28274 df-trkgld 28276 df-trkg 28277 df-cgrg 28335 df-leg 28407 df-mir 28477 df-rag 28518 df-perpg 28520 df-mid 28598

This theorem is referenced by: lmif 28609 islmib 28611

W3C validator