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Theorem lmiisolem 25908
Description: Lemma for lmiiso 25909. (Contributed by Thierry Arnoux, 14-Dec-2019.)
Hypotheses
Ref Expression
ismid.p 𝑃 = (Base‘𝐺)
ismid.d = (dist‘𝐺)
ismid.i 𝐼 = (Itv‘𝐺)
ismid.g (𝜑𝐺 ∈ TarskiG)
ismid.1 (𝜑𝐺DimTarskiG≥2)
lmif.m 𝑀 = ((lInvG‘𝐺)‘𝐷)
lmif.l 𝐿 = (LineG‘𝐺)
lmif.d (𝜑𝐷 ∈ ran 𝐿)
lmiiso.1 (𝜑𝐴𝑃)
lmiiso.2 (𝜑𝐵𝑃)
lmiisolem.s 𝑆 = ((pInvG‘𝐺)‘𝑍)
lmiisolem.z 𝑍 = ((𝐴(midG‘𝐺)(𝑀𝐴))(midG‘𝐺)(𝐵(midG‘𝐺)(𝑀𝐵)))
Assertion
Ref Expression
lmiisolem (𝜑 → ((𝑀𝐴) (𝑀𝐵)) = (𝐴 𝐵))

Proof of Theorem lmiisolem
StepHypRef Expression
1 ismid.p . . . . . . . 8 𝑃 = (Base‘𝐺)
2 ismid.d . . . . . . . 8 = (dist‘𝐺)
3 ismid.i . . . . . . . 8 𝐼 = (Itv‘𝐺)
4 ismid.g . . . . . . . . 9 (𝜑𝐺 ∈ TarskiG)
54adantr 472 . . . . . . . 8 ((𝜑 ∧ (𝑆𝐴) = 𝑍) → 𝐺 ∈ TarskiG)
6 lmiisolem.z . . . . . . . . . 10 𝑍 = ((𝐴(midG‘𝐺)(𝑀𝐴))(midG‘𝐺)(𝐵(midG‘𝐺)(𝑀𝐵)))
7 ismid.1 . . . . . . . . . . 11 (𝜑𝐺DimTarskiG≥2)
8 lmiiso.1 . . . . . . . . . . . 12 (𝜑𝐴𝑃)
9 lmif.m . . . . . . . . . . . . 13 𝑀 = ((lInvG‘𝐺)‘𝐷)
10 lmif.l . . . . . . . . . . . . 13 𝐿 = (LineG‘𝐺)
11 lmif.d . . . . . . . . . . . . 13 (𝜑𝐷 ∈ ran 𝐿)
121, 2, 3, 4, 7, 9, 10, 11, 8lmicl 25898 . . . . . . . . . . . 12 (𝜑 → (𝑀𝐴) ∈ 𝑃)
131, 2, 3, 4, 7, 8, 12midcl 25889 . . . . . . . . . . 11 (𝜑 → (𝐴(midG‘𝐺)(𝑀𝐴)) ∈ 𝑃)
14 lmiiso.2 . . . . . . . . . . . 12 (𝜑𝐵𝑃)
151, 2, 3, 4, 7, 9, 10, 11, 14lmicl 25898 . . . . . . . . . . . 12 (𝜑 → (𝑀𝐵) ∈ 𝑃)
161, 2, 3, 4, 7, 14, 15midcl 25889 . . . . . . . . . . 11 (𝜑 → (𝐵(midG‘𝐺)(𝑀𝐵)) ∈ 𝑃)
171, 2, 3, 4, 7, 13, 16midcl 25889 . . . . . . . . . 10 (𝜑 → ((𝐴(midG‘𝐺)(𝑀𝐴))(midG‘𝐺)(𝐵(midG‘𝐺)(𝑀𝐵))) ∈ 𝑃)
186, 17syl5eqel 2843 . . . . . . . . 9 (𝜑𝑍𝑃)
1918adantr 472 . . . . . . . 8 ((𝜑 ∧ (𝑆𝐴) = 𝑍) → 𝑍𝑃)
20 eqid 2760 . . . . . . . . . 10 (pInvG‘𝐺) = (pInvG‘𝐺)
21 lmiisolem.s . . . . . . . . . 10 𝑆 = ((pInvG‘𝐺)‘𝑍)
221, 2, 3, 10, 20, 4, 18, 21, 8mircl 25776 . . . . . . . . 9 (𝜑 → (𝑆𝐴) ∈ 𝑃)
2322adantr 472 . . . . . . . 8 ((𝜑 ∧ (𝑆𝐴) = 𝑍) → (𝑆𝐴) ∈ 𝑃)
248adantr 472 . . . . . . . 8 ((𝜑 ∧ (𝑆𝐴) = 𝑍) → 𝐴𝑃)
251, 2, 3, 10, 20, 5, 19, 21, 24mircgr 25772 . . . . . . . 8 ((𝜑 ∧ (𝑆𝐴) = 𝑍) → (𝑍 (𝑆𝐴)) = (𝑍 𝐴))
26 simpr 479 . . . . . . . . 9 ((𝜑 ∧ (𝑆𝐴) = 𝑍) → (𝑆𝐴) = 𝑍)
2726eqcomd 2766 . . . . . . . 8 ((𝜑 ∧ (𝑆𝐴) = 𝑍) → 𝑍 = (𝑆𝐴))
281, 2, 3, 5, 19, 23, 19, 24, 25, 27tgcgreq 25597 . . . . . . 7 ((𝜑 ∧ (𝑆𝐴) = 𝑍) → 𝑍 = 𝐴)
29 simpr 479 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝐴(midG‘𝐺)(𝑀𝐴)) = (𝐵(midG‘𝐺)(𝑀𝐵))) → (𝐴(midG‘𝐺)(𝑀𝐴)) = (𝐵(midG‘𝐺)(𝑀𝐵)))
3029oveq2d 6830 . . . . . . . . . . . 12 ((𝜑 ∧ (𝐴(midG‘𝐺)(𝑀𝐴)) = (𝐵(midG‘𝐺)(𝑀𝐵))) → ((𝐴(midG‘𝐺)(𝑀𝐴))(midG‘𝐺)(𝐴(midG‘𝐺)(𝑀𝐴))) = ((𝐴(midG‘𝐺)(𝑀𝐴))(midG‘𝐺)(𝐵(midG‘𝐺)(𝑀𝐵))))
3130, 6syl6reqr 2813 . . . . . . . . . . 11 ((𝜑 ∧ (𝐴(midG‘𝐺)(𝑀𝐴)) = (𝐵(midG‘𝐺)(𝑀𝐵))) → 𝑍 = ((𝐴(midG‘𝐺)(𝑀𝐴))(midG‘𝐺)(𝐴(midG‘𝐺)(𝑀𝐴))))
324adantr 472 . . . . . . . . . . . 12 ((𝜑 ∧ (𝐴(midG‘𝐺)(𝑀𝐴)) = (𝐵(midG‘𝐺)(𝑀𝐵))) → 𝐺 ∈ TarskiG)
337adantr 472 . . . . . . . . . . . 12 ((𝜑 ∧ (𝐴(midG‘𝐺)(𝑀𝐴)) = (𝐵(midG‘𝐺)(𝑀𝐵))) → 𝐺DimTarskiG≥2)
3413adantr 472 . . . . . . . . . . . 12 ((𝜑 ∧ (𝐴(midG‘𝐺)(𝑀𝐴)) = (𝐵(midG‘𝐺)(𝑀𝐵))) → (𝐴(midG‘𝐺)(𝑀𝐴)) ∈ 𝑃)
351, 2, 3, 32, 33, 34, 34midid 25893 . . . . . . . . . . 11 ((𝜑 ∧ (𝐴(midG‘𝐺)(𝑀𝐴)) = (𝐵(midG‘𝐺)(𝑀𝐵))) → ((𝐴(midG‘𝐺)(𝑀𝐴))(midG‘𝐺)(𝐴(midG‘𝐺)(𝑀𝐴))) = (𝐴(midG‘𝐺)(𝑀𝐴)))
3631, 35eqtrd 2794 . . . . . . . . . 10 ((𝜑 ∧ (𝐴(midG‘𝐺)(𝑀𝐴)) = (𝐵(midG‘𝐺)(𝑀𝐵))) → 𝑍 = (𝐴(midG‘𝐺)(𝑀𝐴)))
37 eqidd 2761 . . . . . . . . . . . . 13 (𝜑 → (𝑀𝐴) = (𝑀𝐴))
381, 2, 3, 4, 7, 9, 10, 11, 8, 12islmib 25899 . . . . . . . . . . . . 13 (𝜑 → ((𝑀𝐴) = (𝑀𝐴) ↔ ((𝐴(midG‘𝐺)(𝑀𝐴)) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿(𝑀𝐴)) ∨ 𝐴 = (𝑀𝐴)))))
3937, 38mpbid 222 . . . . . . . . . . . 12 (𝜑 → ((𝐴(midG‘𝐺)(𝑀𝐴)) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿(𝑀𝐴)) ∨ 𝐴 = (𝑀𝐴))))
4039simpld 477 . . . . . . . . . . 11 (𝜑 → (𝐴(midG‘𝐺)(𝑀𝐴)) ∈ 𝐷)
4140adantr 472 . . . . . . . . . 10 ((𝜑 ∧ (𝐴(midG‘𝐺)(𝑀𝐴)) = (𝐵(midG‘𝐺)(𝑀𝐵))) → (𝐴(midG‘𝐺)(𝑀𝐴)) ∈ 𝐷)
4236, 41eqeltrd 2839 . . . . . . . . 9 ((𝜑 ∧ (𝐴(midG‘𝐺)(𝑀𝐴)) = (𝐵(midG‘𝐺)(𝑀𝐵))) → 𝑍𝐷)
434adantr 472 . . . . . . . . . . 11 ((𝜑 ∧ (𝐴(midG‘𝐺)(𝑀𝐴)) ≠ (𝐵(midG‘𝐺)(𝑀𝐵))) → 𝐺 ∈ TarskiG)
4413adantr 472 . . . . . . . . . . 11 ((𝜑 ∧ (𝐴(midG‘𝐺)(𝑀𝐴)) ≠ (𝐵(midG‘𝐺)(𝑀𝐵))) → (𝐴(midG‘𝐺)(𝑀𝐴)) ∈ 𝑃)
4516adantr 472 . . . . . . . . . . 11 ((𝜑 ∧ (𝐴(midG‘𝐺)(𝑀𝐴)) ≠ (𝐵(midG‘𝐺)(𝑀𝐵))) → (𝐵(midG‘𝐺)(𝑀𝐵)) ∈ 𝑃)
4618adantr 472 . . . . . . . . . . 11 ((𝜑 ∧ (𝐴(midG‘𝐺)(𝑀𝐴)) ≠ (𝐵(midG‘𝐺)(𝑀𝐵))) → 𝑍𝑃)
47 simpr 479 . . . . . . . . . . 11 ((𝜑 ∧ (𝐴(midG‘𝐺)(𝑀𝐴)) ≠ (𝐵(midG‘𝐺)(𝑀𝐵))) → (𝐴(midG‘𝐺)(𝑀𝐴)) ≠ (𝐵(midG‘𝐺)(𝑀𝐵)))
481, 2, 3, 4, 7, 13, 16midbtwn 25891 . . . . . . . . . . . . 13 (𝜑 → ((𝐴(midG‘𝐺)(𝑀𝐴))(midG‘𝐺)(𝐵(midG‘𝐺)(𝑀𝐵))) ∈ ((𝐴(midG‘𝐺)(𝑀𝐴))𝐼(𝐵(midG‘𝐺)(𝑀𝐵))))
496, 48syl5eqel 2843 . . . . . . . . . . . 12 (𝜑𝑍 ∈ ((𝐴(midG‘𝐺)(𝑀𝐴))𝐼(𝐵(midG‘𝐺)(𝑀𝐵))))
5049adantr 472 . . . . . . . . . . 11 ((𝜑 ∧ (𝐴(midG‘𝐺)(𝑀𝐴)) ≠ (𝐵(midG‘𝐺)(𝑀𝐵))) → 𝑍 ∈ ((𝐴(midG‘𝐺)(𝑀𝐴))𝐼(𝐵(midG‘𝐺)(𝑀𝐵))))
511, 3, 10, 43, 44, 45, 46, 47, 50btwnlng1 25734 . . . . . . . . . 10 ((𝜑 ∧ (𝐴(midG‘𝐺)(𝑀𝐴)) ≠ (𝐵(midG‘𝐺)(𝑀𝐵))) → 𝑍 ∈ ((𝐴(midG‘𝐺)(𝑀𝐴))𝐿(𝐵(midG‘𝐺)(𝑀𝐵))))
5211adantr 472 . . . . . . . . . . 11 ((𝜑 ∧ (𝐴(midG‘𝐺)(𝑀𝐴)) ≠ (𝐵(midG‘𝐺)(𝑀𝐵))) → 𝐷 ∈ ran 𝐿)
5340adantr 472 . . . . . . . . . . 11 ((𝜑 ∧ (𝐴(midG‘𝐺)(𝑀𝐴)) ≠ (𝐵(midG‘𝐺)(𝑀𝐵))) → (𝐴(midG‘𝐺)(𝑀𝐴)) ∈ 𝐷)
54 eqidd 2761 . . . . . . . . . . . . . 14 (𝜑 → (𝑀𝐵) = (𝑀𝐵))
551, 2, 3, 4, 7, 9, 10, 11, 14, 15islmib 25899 . . . . . . . . . . . . . 14 (𝜑 → ((𝑀𝐵) = (𝑀𝐵) ↔ ((𝐵(midG‘𝐺)(𝑀𝐵)) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐵𝐿(𝑀𝐵)) ∨ 𝐵 = (𝑀𝐵)))))
5654, 55mpbid 222 . . . . . . . . . . . . 13 (𝜑 → ((𝐵(midG‘𝐺)(𝑀𝐵)) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐵𝐿(𝑀𝐵)) ∨ 𝐵 = (𝑀𝐵))))
5756simpld 477 . . . . . . . . . . . 12 (𝜑 → (𝐵(midG‘𝐺)(𝑀𝐵)) ∈ 𝐷)
5857adantr 472 . . . . . . . . . . 11 ((𝜑 ∧ (𝐴(midG‘𝐺)(𝑀𝐴)) ≠ (𝐵(midG‘𝐺)(𝑀𝐵))) → (𝐵(midG‘𝐺)(𝑀𝐵)) ∈ 𝐷)
591, 3, 10, 43, 44, 45, 47, 47, 52, 53, 58tglinethru 25751 . . . . . . . . . 10 ((𝜑 ∧ (𝐴(midG‘𝐺)(𝑀𝐴)) ≠ (𝐵(midG‘𝐺)(𝑀𝐵))) → 𝐷 = ((𝐴(midG‘𝐺)(𝑀𝐴))𝐿(𝐵(midG‘𝐺)(𝑀𝐵))))
6051, 59eleqtrrd 2842 . . . . . . . . 9 ((𝜑 ∧ (𝐴(midG‘𝐺)(𝑀𝐴)) ≠ (𝐵(midG‘𝐺)(𝑀𝐵))) → 𝑍𝐷)
6142, 60pm2.61dane 3019 . . . . . . . 8 (𝜑𝑍𝐷)
6261adantr 472 . . . . . . 7 ((𝜑 ∧ (𝑆𝐴) = 𝑍) → 𝑍𝐷)
6328, 62eqeltrrd 2840 . . . . . 6 ((𝜑 ∧ (𝑆𝐴) = 𝑍) → 𝐴𝐷)
641, 2, 3, 4, 7, 9, 10, 11, 8lmiinv 25904 . . . . . . 7 (𝜑 → ((𝑀𝐴) = 𝐴𝐴𝐷))
6564biimpar 503 . . . . . 6 ((𝜑𝐴𝐷) → (𝑀𝐴) = 𝐴)
6663, 65syldan 488 . . . . 5 ((𝜑 ∧ (𝑆𝐴) = 𝑍) → (𝑀𝐴) = 𝐴)
6766, 28eqtr4d 2797 . . . 4 ((𝜑 ∧ (𝑆𝐴) = 𝑍) → (𝑀𝐴) = 𝑍)
6867oveq1d 6829 . . 3 ((𝜑 ∧ (𝑆𝐴) = 𝑍) → ((𝑀𝐴) (𝑀𝐵)) = (𝑍 (𝑀𝐵)))
69 eqidd 2761 . . . . . . . . 9 ((𝜑𝐵 = (𝑀𝐵)) → 𝑍 = 𝑍)
704adantr 472 . . . . . . . . . 10 ((𝜑𝐵 = (𝑀𝐵)) → 𝐺 ∈ TarskiG)
7114adantr 472 . . . . . . . . . 10 ((𝜑𝐵 = (𝑀𝐵)) → 𝐵𝑃)
7216adantr 472 . . . . . . . . . 10 ((𝜑𝐵 = (𝑀𝐵)) → (𝐵(midG‘𝐺)(𝑀𝐵)) ∈ 𝑃)
731, 2, 3, 4, 7, 14, 15midbtwn 25891 . . . . . . . . . . . 12 (𝜑 → (𝐵(midG‘𝐺)(𝑀𝐵)) ∈ (𝐵𝐼(𝑀𝐵)))
7473adantr 472 . . . . . . . . . . 11 ((𝜑𝐵 = (𝑀𝐵)) → (𝐵(midG‘𝐺)(𝑀𝐵)) ∈ (𝐵𝐼(𝑀𝐵)))
75 simpr 479 . . . . . . . . . . . 12 ((𝜑𝐵 = (𝑀𝐵)) → 𝐵 = (𝑀𝐵))
7675oveq2d 6830 . . . . . . . . . . 11 ((𝜑𝐵 = (𝑀𝐵)) → (𝐵𝐼𝐵) = (𝐵𝐼(𝑀𝐵)))
7774, 76eleqtrrd 2842 . . . . . . . . . 10 ((𝜑𝐵 = (𝑀𝐵)) → (𝐵(midG‘𝐺)(𝑀𝐵)) ∈ (𝐵𝐼𝐵))
781, 2, 3, 70, 71, 72, 77axtgbtwnid 25585 . . . . . . . . 9 ((𝜑𝐵 = (𝑀𝐵)) → 𝐵 = (𝐵(midG‘𝐺)(𝑀𝐵)))
79 eqidd 2761 . . . . . . . . 9 ((𝜑𝐵 = (𝑀𝐵)) → 𝐵 = 𝐵)
8069, 78, 79s3eqd 13829 . . . . . . . 8 ((𝜑𝐵 = (𝑀𝐵)) → ⟨“𝑍𝐵𝐵”⟩ = ⟨“𝑍(𝐵(midG‘𝐺)(𝑀𝐵))𝐵”⟩)
811, 2, 3, 10, 20, 4, 18, 14, 14ragtrivb 25817 . . . . . . . . 9 (𝜑 → ⟨“𝑍𝐵𝐵”⟩ ∈ (∟G‘𝐺))
8281adantr 472 . . . . . . . 8 ((𝜑𝐵 = (𝑀𝐵)) → ⟨“𝑍𝐵𝐵”⟩ ∈ (∟G‘𝐺))
8380, 82eqeltrrd 2840 . . . . . . 7 ((𝜑𝐵 = (𝑀𝐵)) → ⟨“𝑍(𝐵(midG‘𝐺)(𝑀𝐵))𝐵”⟩ ∈ (∟G‘𝐺))
844adantr 472 . . . . . . . 8 ((𝜑𝐵 ≠ (𝑀𝐵)) → 𝐺 ∈ TarskiG)
8561adantr 472 . . . . . . . 8 ((𝜑𝐵 ≠ (𝑀𝐵)) → 𝑍𝐷)
8657adantr 472 . . . . . . . 8 ((𝜑𝐵 ≠ (𝑀𝐵)) → (𝐵(midG‘𝐺)(𝑀𝐵)) ∈ 𝐷)
8714adantr 472 . . . . . . . 8 ((𝜑𝐵 ≠ (𝑀𝐵)) → 𝐵𝑃)
88 df-ne 2933 . . . . . . . . . 10 (𝐵 ≠ (𝑀𝐵) ↔ ¬ 𝐵 = (𝑀𝐵))
8956simprd 482 . . . . . . . . . . . 12 (𝜑 → (𝐷(⟂G‘𝐺)(𝐵𝐿(𝑀𝐵)) ∨ 𝐵 = (𝑀𝐵)))
9089orcomd 402 . . . . . . . . . . 11 (𝜑 → (𝐵 = (𝑀𝐵) ∨ 𝐷(⟂G‘𝐺)(𝐵𝐿(𝑀𝐵))))
9190orcanai 990 . . . . . . . . . 10 ((𝜑 ∧ ¬ 𝐵 = (𝑀𝐵)) → 𝐷(⟂G‘𝐺)(𝐵𝐿(𝑀𝐵)))
9288, 91sylan2b 493 . . . . . . . . 9 ((𝜑𝐵 ≠ (𝑀𝐵)) → 𝐷(⟂G‘𝐺)(𝐵𝐿(𝑀𝐵)))
9315adantr 472 . . . . . . . . . . 11 ((𝜑𝐵 ≠ (𝑀𝐵)) → (𝑀𝐵) ∈ 𝑃)
94 simpr 479 . . . . . . . . . . 11 ((𝜑𝐵 ≠ (𝑀𝐵)) → 𝐵 ≠ (𝑀𝐵))
9516adantr 472 . . . . . . . . . . 11 ((𝜑𝐵 ≠ (𝑀𝐵)) → (𝐵(midG‘𝐺)(𝑀𝐵)) ∈ 𝑃)
964adantr 472 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝐵(midG‘𝐺)(𝑀𝐵)) = 𝐵) → 𝐺 ∈ TarskiG)
9714adantr 472 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝐵(midG‘𝐺)(𝑀𝐵)) = 𝐵) → 𝐵𝑃)
9815adantr 472 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝐵(midG‘𝐺)(𝑀𝐵)) = 𝐵) → (𝑀𝐵) ∈ 𝑃)
997adantr 472 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝐵(midG‘𝐺)(𝑀𝐵)) = 𝐵) → 𝐺DimTarskiG≥2)
100 simpr 479 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝐵(midG‘𝐺)(𝑀𝐵)) = 𝐵) → (𝐵(midG‘𝐺)(𝑀𝐵)) = 𝐵)
1011, 2, 3, 96, 99, 97, 98, 100midcgr 25892 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝐵(midG‘𝐺)(𝑀𝐵)) = 𝐵) → (𝐵 𝐵) = (𝐵 (𝑀𝐵)))
102101eqcomd 2766 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝐵(midG‘𝐺)(𝑀𝐵)) = 𝐵) → (𝐵 (𝑀𝐵)) = (𝐵 𝐵))
1031, 2, 3, 96, 97, 98, 97, 102axtgcgrid 25582 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝐵(midG‘𝐺)(𝑀𝐵)) = 𝐵) → 𝐵 = (𝑀𝐵))
104103ex 449 . . . . . . . . . . . . 13 (𝜑 → ((𝐵(midG‘𝐺)(𝑀𝐵)) = 𝐵𝐵 = (𝑀𝐵)))
105104necon3d 2953 . . . . . . . . . . . 12 (𝜑 → (𝐵 ≠ (𝑀𝐵) → (𝐵(midG‘𝐺)(𝑀𝐵)) ≠ 𝐵))
106105imp 444 . . . . . . . . . . 11 ((𝜑𝐵 ≠ (𝑀𝐵)) → (𝐵(midG‘𝐺)(𝑀𝐵)) ≠ 𝐵)
10773adantr 472 . . . . . . . . . . . 12 ((𝜑𝐵 ≠ (𝑀𝐵)) → (𝐵(midG‘𝐺)(𝑀𝐵)) ∈ (𝐵𝐼(𝑀𝐵)))
1081, 3, 10, 84, 87, 93, 95, 94, 107btwnlng1 25734 . . . . . . . . . . 11 ((𝜑𝐵 ≠ (𝑀𝐵)) → (𝐵(midG‘𝐺)(𝑀𝐵)) ∈ (𝐵𝐿(𝑀𝐵)))
1091, 3, 10, 84, 87, 93, 94, 95, 106, 108tglineelsb2 25747 . . . . . . . . . 10 ((𝜑𝐵 ≠ (𝑀𝐵)) → (𝐵𝐿(𝑀𝐵)) = (𝐵𝐿(𝐵(midG‘𝐺)(𝑀𝐵))))
1101, 3, 10, 84, 95, 87, 106tglinecom 25750 . . . . . . . . . 10 ((𝜑𝐵 ≠ (𝑀𝐵)) → ((𝐵(midG‘𝐺)(𝑀𝐵))𝐿𝐵) = (𝐵𝐿(𝐵(midG‘𝐺)(𝑀𝐵))))
111109, 110eqtr4d 2797 . . . . . . . . 9 ((𝜑𝐵 ≠ (𝑀𝐵)) → (𝐵𝐿(𝑀𝐵)) = ((𝐵(midG‘𝐺)(𝑀𝐵))𝐿𝐵))
11292, 111breqtrd 4830 . . . . . . . 8 ((𝜑𝐵 ≠ (𝑀𝐵)) → 𝐷(⟂G‘𝐺)((𝐵(midG‘𝐺)(𝑀𝐵))𝐿𝐵))
1131, 2, 3, 10, 84, 85, 86, 87, 112perpdrag 25840 . . . . . . 7 ((𝜑𝐵 ≠ (𝑀𝐵)) → ⟨“𝑍(𝐵(midG‘𝐺)(𝑀𝐵))𝐵”⟩ ∈ (∟G‘𝐺))
11483, 113pm2.61dane 3019 . . . . . 6 (𝜑 → ⟨“𝑍(𝐵(midG‘𝐺)(𝑀𝐵))𝐵”⟩ ∈ (∟G‘𝐺))
1151, 2, 3, 10, 20, 4, 18, 16, 14israg 25812 . . . . . 6 (𝜑 → (⟨“𝑍(𝐵(midG‘𝐺)(𝑀𝐵))𝐵”⟩ ∈ (∟G‘𝐺) ↔ (𝑍 𝐵) = (𝑍 (((pInvG‘𝐺)‘(𝐵(midG‘𝐺)(𝑀𝐵)))‘𝐵))))
116114, 115mpbid 222 . . . . 5 (𝜑 → (𝑍 𝐵) = (𝑍 (((pInvG‘𝐺)‘(𝐵(midG‘𝐺)(𝑀𝐵)))‘𝐵)))
117 eqidd 2761 . . . . . . 7 (𝜑 → (𝐵(midG‘𝐺)(𝑀𝐵)) = (𝐵(midG‘𝐺)(𝑀𝐵)))
1181, 2, 3, 4, 7, 14, 15, 20, 16ismidb 25890 . . . . . . 7 (𝜑 → ((𝑀𝐵) = (((pInvG‘𝐺)‘(𝐵(midG‘𝐺)(𝑀𝐵)))‘𝐵) ↔ (𝐵(midG‘𝐺)(𝑀𝐵)) = (𝐵(midG‘𝐺)(𝑀𝐵))))
119117, 118mpbird 247 . . . . . 6 (𝜑 → (𝑀𝐵) = (((pInvG‘𝐺)‘(𝐵(midG‘𝐺)(𝑀𝐵)))‘𝐵))
120119oveq2d 6830 . . . . 5 (𝜑 → (𝑍 (𝑀𝐵)) = (𝑍 (((pInvG‘𝐺)‘(𝐵(midG‘𝐺)(𝑀𝐵)))‘𝐵)))
121116, 120eqtr4d 2797 . . . 4 (𝜑 → (𝑍 𝐵) = (𝑍 (𝑀𝐵)))
122121adantr 472 . . 3 ((𝜑 ∧ (𝑆𝐴) = 𝑍) → (𝑍 𝐵) = (𝑍 (𝑀𝐵)))
12328oveq1d 6829 . . 3 ((𝜑 ∧ (𝑆𝐴) = 𝑍) → (𝑍 𝐵) = (𝐴 𝐵))
12468, 122, 1233eqtr2d 2800 . 2 ((𝜑 ∧ (𝑆𝐴) = 𝑍) → ((𝑀𝐴) (𝑀𝐵)) = (𝐴 𝐵))
1254adantr 472 . . . 4 ((𝜑 ∧ (𝑆𝐴) ≠ 𝑍) → 𝐺 ∈ TarskiG)
12622adantr 472 . . . 4 ((𝜑 ∧ (𝑆𝐴) ≠ 𝑍) → (𝑆𝐴) ∈ 𝑃)
12718adantr 472 . . . 4 ((𝜑 ∧ (𝑆𝐴) ≠ 𝑍) → 𝑍𝑃)
1288adantr 472 . . . 4 ((𝜑 ∧ (𝑆𝐴) ≠ 𝑍) → 𝐴𝑃)
1291, 2, 3, 10, 20, 4, 18, 21, 12mircl 25776 . . . . 5 (𝜑 → (𝑆‘(𝑀𝐴)) ∈ 𝑃)
130129adantr 472 . . . 4 ((𝜑 ∧ (𝑆𝐴) ≠ 𝑍) → (𝑆‘(𝑀𝐴)) ∈ 𝑃)
13112adantr 472 . . . 4 ((𝜑 ∧ (𝑆𝐴) ≠ 𝑍) → (𝑀𝐴) ∈ 𝑃)
13214adantr 472 . . . 4 ((𝜑 ∧ (𝑆𝐴) ≠ 𝑍) → 𝐵𝑃)
13315adantr 472 . . . 4 ((𝜑 ∧ (𝑆𝐴) ≠ 𝑍) → (𝑀𝐵) ∈ 𝑃)
134 simpr 479 . . . 4 ((𝜑 ∧ (𝑆𝐴) ≠ 𝑍) → (𝑆𝐴) ≠ 𝑍)
1351, 2, 3, 10, 20, 125, 127, 21, 128mirbtwn 25773 . . . 4 ((𝜑 ∧ (𝑆𝐴) ≠ 𝑍) → 𝑍 ∈ ((𝑆𝐴)𝐼𝐴))
1361, 2, 3, 10, 20, 125, 127, 21, 131mirbtwn 25773 . . . 4 ((𝜑 ∧ (𝑆𝐴) ≠ 𝑍) → 𝑍 ∈ ((𝑆‘(𝑀𝐴))𝐼(𝑀𝐴)))
137 eqidd 2761 . . . . . . . . . . . 12 ((𝜑𝐴 = (𝑀𝐴)) → 𝑍 = 𝑍)
1384adantr 472 . . . . . . . . . . . . 13 ((𝜑𝐴 = (𝑀𝐴)) → 𝐺 ∈ TarskiG)
1398adantr 472 . . . . . . . . . . . . 13 ((𝜑𝐴 = (𝑀𝐴)) → 𝐴𝑃)
14013adantr 472 . . . . . . . . . . . . 13 ((𝜑𝐴 = (𝑀𝐴)) → (𝐴(midG‘𝐺)(𝑀𝐴)) ∈ 𝑃)
1411, 2, 3, 4, 7, 8, 12midbtwn 25891 . . . . . . . . . . . . . . 15 (𝜑 → (𝐴(midG‘𝐺)(𝑀𝐴)) ∈ (𝐴𝐼(𝑀𝐴)))
142141adantr 472 . . . . . . . . . . . . . 14 ((𝜑𝐴 = (𝑀𝐴)) → (𝐴(midG‘𝐺)(𝑀𝐴)) ∈ (𝐴𝐼(𝑀𝐴)))
143 simpr 479 . . . . . . . . . . . . . . 15 ((𝜑𝐴 = (𝑀𝐴)) → 𝐴 = (𝑀𝐴))
144143oveq2d 6830 . . . . . . . . . . . . . 14 ((𝜑𝐴 = (𝑀𝐴)) → (𝐴𝐼𝐴) = (𝐴𝐼(𝑀𝐴)))
145142, 144eleqtrrd 2842 . . . . . . . . . . . . 13 ((𝜑𝐴 = (𝑀𝐴)) → (𝐴(midG‘𝐺)(𝑀𝐴)) ∈ (𝐴𝐼𝐴))
1461, 2, 3, 138, 139, 140, 145axtgbtwnid 25585 . . . . . . . . . . . 12 ((𝜑𝐴 = (𝑀𝐴)) → 𝐴 = (𝐴(midG‘𝐺)(𝑀𝐴)))
147 eqidd 2761 . . . . . . . . . . . 12 ((𝜑𝐴 = (𝑀𝐴)) → 𝐴 = 𝐴)
148137, 146, 147s3eqd 13829 . . . . . . . . . . 11 ((𝜑𝐴 = (𝑀𝐴)) → ⟨“𝑍𝐴𝐴”⟩ = ⟨“𝑍(𝐴(midG‘𝐺)(𝑀𝐴))𝐴”⟩)
1491, 2, 3, 10, 20, 4, 18, 8, 8ragtrivb 25817 . . . . . . . . . . . 12 (𝜑 → ⟨“𝑍𝐴𝐴”⟩ ∈ (∟G‘𝐺))
150149adantr 472 . . . . . . . . . . 11 ((𝜑𝐴 = (𝑀𝐴)) → ⟨“𝑍𝐴𝐴”⟩ ∈ (∟G‘𝐺))
151148, 150eqeltrrd 2840 . . . . . . . . . 10 ((𝜑𝐴 = (𝑀𝐴)) → ⟨“𝑍(𝐴(midG‘𝐺)(𝑀𝐴))𝐴”⟩ ∈ (∟G‘𝐺))
1524adantr 472 . . . . . . . . . . 11 ((𝜑𝐴 ≠ (𝑀𝐴)) → 𝐺 ∈ TarskiG)
15361adantr 472 . . . . . . . . . . 11 ((𝜑𝐴 ≠ (𝑀𝐴)) → 𝑍𝐷)
15440adantr 472 . . . . . . . . . . 11 ((𝜑𝐴 ≠ (𝑀𝐴)) → (𝐴(midG‘𝐺)(𝑀𝐴)) ∈ 𝐷)
1558adantr 472 . . . . . . . . . . 11 ((𝜑𝐴 ≠ (𝑀𝐴)) → 𝐴𝑃)
156 df-ne 2933 . . . . . . . . . . . . 13 (𝐴 ≠ (𝑀𝐴) ↔ ¬ 𝐴 = (𝑀𝐴))
15739simprd 482 . . . . . . . . . . . . . . 15 (𝜑 → (𝐷(⟂G‘𝐺)(𝐴𝐿(𝑀𝐴)) ∨ 𝐴 = (𝑀𝐴)))
158157orcomd 402 . . . . . . . . . . . . . 14 (𝜑 → (𝐴 = (𝑀𝐴) ∨ 𝐷(⟂G‘𝐺)(𝐴𝐿(𝑀𝐴))))
159158orcanai 990 . . . . . . . . . . . . 13 ((𝜑 ∧ ¬ 𝐴 = (𝑀𝐴)) → 𝐷(⟂G‘𝐺)(𝐴𝐿(𝑀𝐴)))
160156, 159sylan2b 493 . . . . . . . . . . . 12 ((𝜑𝐴 ≠ (𝑀𝐴)) → 𝐷(⟂G‘𝐺)(𝐴𝐿(𝑀𝐴)))
16112adantr 472 . . . . . . . . . . . . . 14 ((𝜑𝐴 ≠ (𝑀𝐴)) → (𝑀𝐴) ∈ 𝑃)
162 simpr 479 . . . . . . . . . . . . . 14 ((𝜑𝐴 ≠ (𝑀𝐴)) → 𝐴 ≠ (𝑀𝐴))
16313adantr 472 . . . . . . . . . . . . . 14 ((𝜑𝐴 ≠ (𝑀𝐴)) → (𝐴(midG‘𝐺)(𝑀𝐴)) ∈ 𝑃)
1644adantr 472 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝐴(midG‘𝐺)(𝑀𝐴)) = 𝐴) → 𝐺 ∈ TarskiG)
1658adantr 472 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝐴(midG‘𝐺)(𝑀𝐴)) = 𝐴) → 𝐴𝑃)
16612adantr 472 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝐴(midG‘𝐺)(𝑀𝐴)) = 𝐴) → (𝑀𝐴) ∈ 𝑃)
1677adantr 472 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝐴(midG‘𝐺)(𝑀𝐴)) = 𝐴) → 𝐺DimTarskiG≥2)
168 simpr 479 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝐴(midG‘𝐺)(𝑀𝐴)) = 𝐴) → (𝐴(midG‘𝐺)(𝑀𝐴)) = 𝐴)
1691, 2, 3, 164, 167, 165, 166, 168midcgr 25892 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝐴(midG‘𝐺)(𝑀𝐴)) = 𝐴) → (𝐴 𝐴) = (𝐴 (𝑀𝐴)))
170169eqcomd 2766 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝐴(midG‘𝐺)(𝑀𝐴)) = 𝐴) → (𝐴 (𝑀𝐴)) = (𝐴 𝐴))
1711, 2, 3, 164, 165, 166, 165, 170axtgcgrid 25582 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝐴(midG‘𝐺)(𝑀𝐴)) = 𝐴) → 𝐴 = (𝑀𝐴))
172171ex 449 . . . . . . . . . . . . . . . 16 (𝜑 → ((𝐴(midG‘𝐺)(𝑀𝐴)) = 𝐴𝐴 = (𝑀𝐴)))
173172necon3d 2953 . . . . . . . . . . . . . . 15 (𝜑 → (𝐴 ≠ (𝑀𝐴) → (𝐴(midG‘𝐺)(𝑀𝐴)) ≠ 𝐴))
174173imp 444 . . . . . . . . . . . . . 14 ((𝜑𝐴 ≠ (𝑀𝐴)) → (𝐴(midG‘𝐺)(𝑀𝐴)) ≠ 𝐴)
175141adantr 472 . . . . . . . . . . . . . . 15 ((𝜑𝐴 ≠ (𝑀𝐴)) → (𝐴(midG‘𝐺)(𝑀𝐴)) ∈ (𝐴𝐼(𝑀𝐴)))
1761, 3, 10, 152, 155, 161, 163, 162, 175btwnlng1 25734 . . . . . . . . . . . . . 14 ((𝜑𝐴 ≠ (𝑀𝐴)) → (𝐴(midG‘𝐺)(𝑀𝐴)) ∈ (𝐴𝐿(𝑀𝐴)))
1771, 3, 10, 152, 155, 161, 162, 163, 174, 176tglineelsb2 25747 . . . . . . . . . . . . 13 ((𝜑𝐴 ≠ (𝑀𝐴)) → (𝐴𝐿(𝑀𝐴)) = (𝐴𝐿(𝐴(midG‘𝐺)(𝑀𝐴))))
1781, 3, 10, 152, 163, 155, 174tglinecom 25750 . . . . . . . . . . . . 13 ((𝜑𝐴 ≠ (𝑀𝐴)) → ((𝐴(midG‘𝐺)(𝑀𝐴))𝐿𝐴) = (𝐴𝐿(𝐴(midG‘𝐺)(𝑀𝐴))))
179177, 178eqtr4d 2797 . . . . . . . . . . . 12 ((𝜑𝐴 ≠ (𝑀𝐴)) → (𝐴𝐿(𝑀𝐴)) = ((𝐴(midG‘𝐺)(𝑀𝐴))𝐿𝐴))
180160, 179breqtrd 4830 . . . . . . . . . . 11 ((𝜑𝐴 ≠ (𝑀𝐴)) → 𝐷(⟂G‘𝐺)((𝐴(midG‘𝐺)(𝑀𝐴))𝐿𝐴))
1811, 2, 3, 10, 152, 153, 154, 155, 180perpdrag 25840 . . . . . . . . . 10 ((𝜑𝐴 ≠ (𝑀𝐴)) → ⟨“𝑍(𝐴(midG‘𝐺)(𝑀𝐴))𝐴”⟩ ∈ (∟G‘𝐺))
182151, 181pm2.61dane 3019 . . . . . . . . 9 (𝜑 → ⟨“𝑍(𝐴(midG‘𝐺)(𝑀𝐴))𝐴”⟩ ∈ (∟G‘𝐺))
1831, 2, 3, 10, 20, 4, 18, 13, 8israg 25812 . . . . . . . . 9 (𝜑 → (⟨“𝑍(𝐴(midG‘𝐺)(𝑀𝐴))𝐴”⟩ ∈ (∟G‘𝐺) ↔ (𝑍 𝐴) = (𝑍 (((pInvG‘𝐺)‘(𝐴(midG‘𝐺)(𝑀𝐴)))‘𝐴))))
184182, 183mpbid 222 . . . . . . . 8 (𝜑 → (𝑍 𝐴) = (𝑍 (((pInvG‘𝐺)‘(𝐴(midG‘𝐺)(𝑀𝐴)))‘𝐴)))
185 eqidd 2761 . . . . . . . . . 10 (𝜑 → (𝐴(midG‘𝐺)(𝑀𝐴)) = (𝐴(midG‘𝐺)(𝑀𝐴)))
1861, 2, 3, 4, 7, 8, 12, 20, 13ismidb 25890 . . . . . . . . . 10 (𝜑 → ((𝑀𝐴) = (((pInvG‘𝐺)‘(𝐴(midG‘𝐺)(𝑀𝐴)))‘𝐴) ↔ (𝐴(midG‘𝐺)(𝑀𝐴)) = (𝐴(midG‘𝐺)(𝑀𝐴))))
187185, 186mpbird 247 . . . . . . . . 9 (𝜑 → (𝑀𝐴) = (((pInvG‘𝐺)‘(𝐴(midG‘𝐺)(𝑀𝐴)))‘𝐴))
188187oveq2d 6830 . . . . . . . 8 (𝜑 → (𝑍 (𝑀𝐴)) = (𝑍 (((pInvG‘𝐺)‘(𝐴(midG‘𝐺)(𝑀𝐴)))‘𝐴)))
189184, 188eqtr4d 2797 . . . . . . 7 (𝜑 → (𝑍 𝐴) = (𝑍 (𝑀𝐴)))
1901, 2, 3, 10, 20, 4, 18, 21, 8mircgr 25772 . . . . . . 7 (𝜑 → (𝑍 (𝑆𝐴)) = (𝑍 𝐴))
1911, 2, 3, 10, 20, 4, 18, 21, 12mircgr 25772 . . . . . . 7 (𝜑 → (𝑍 (𝑆‘(𝑀𝐴))) = (𝑍 (𝑀𝐴)))
192189, 190, 1913eqtr4d 2804 . . . . . 6 (𝜑 → (𝑍 (𝑆𝐴)) = (𝑍 (𝑆‘(𝑀𝐴))))
193192adantr 472 . . . . 5 ((𝜑 ∧ (𝑆𝐴) ≠ 𝑍) → (𝑍 (𝑆𝐴)) = (𝑍 (𝑆‘(𝑀𝐴))))
1941, 2, 3, 125, 127, 126, 127, 130, 193tgcgrcomlr 25595 . . . 4 ((𝜑 ∧ (𝑆𝐴) ≠ 𝑍) → ((𝑆𝐴) 𝑍) = ((𝑆‘(𝑀𝐴)) 𝑍))
195189adantr 472 . . . 4 ((𝜑 ∧ (𝑆𝐴) ≠ 𝑍) → (𝑍 𝐴) = (𝑍 (𝑀𝐴)))
19621fveq1i 6354 . . . . . . . . . 10 (𝑆‘(𝐴(midG‘𝐺)(𝑀𝐴))) = (((pInvG‘𝐺)‘𝑍)‘(𝐴(midG‘𝐺)(𝑀𝐴)))
1971, 2, 3, 4, 7, 8, 12, 21, 18mirmid 25895 . . . . . . . . . 10 (𝜑 → ((𝑆𝐴)(midG‘𝐺)(𝑆‘(𝑀𝐴))) = (𝑆‘(𝐴(midG‘𝐺)(𝑀𝐴))))
1986eqcomi 2769 . . . . . . . . . . 11 ((𝐴(midG‘𝐺)(𝑀𝐴))(midG‘𝐺)(𝐵(midG‘𝐺)(𝑀𝐵))) = 𝑍
1991, 2, 3, 4, 7, 13, 16, 20, 18ismidb 25890 . . . . . . . . . . 11 (𝜑 → ((𝐵(midG‘𝐺)(𝑀𝐵)) = (((pInvG‘𝐺)‘𝑍)‘(𝐴(midG‘𝐺)(𝑀𝐴))) ↔ ((𝐴(midG‘𝐺)(𝑀𝐴))(midG‘𝐺)(𝐵(midG‘𝐺)(𝑀𝐵))) = 𝑍))
200198, 199mpbiri 248 . . . . . . . . . 10 (𝜑 → (𝐵(midG‘𝐺)(𝑀𝐵)) = (((pInvG‘𝐺)‘𝑍)‘(𝐴(midG‘𝐺)(𝑀𝐴))))
201196, 197, 2003eqtr4a 2820 . . . . . . . . 9 (𝜑 → ((𝑆𝐴)(midG‘𝐺)(𝑆‘(𝑀𝐴))) = (𝐵(midG‘𝐺)(𝑀𝐵)))
2021, 2, 3, 4, 7, 22, 129, 20, 16ismidb 25890 . . . . . . . . 9 (𝜑 → ((𝑆‘(𝑀𝐴)) = (((pInvG‘𝐺)‘(𝐵(midG‘𝐺)(𝑀𝐵)))‘(𝑆𝐴)) ↔ ((𝑆𝐴)(midG‘𝐺)(𝑆‘(𝑀𝐴))) = (𝐵(midG‘𝐺)(𝑀𝐵))))
203201, 202mpbird 247 . . . . . . . 8 (𝜑 → (𝑆‘(𝑀𝐴)) = (((pInvG‘𝐺)‘(𝐵(midG‘𝐺)(𝑀𝐵)))‘(𝑆𝐴)))
204119, 203oveq12d 6832 . . . . . . 7 (𝜑 → ((𝑀𝐵) (𝑆‘(𝑀𝐴))) = ((((pInvG‘𝐺)‘(𝐵(midG‘𝐺)(𝑀𝐵)))‘𝐵) (((pInvG‘𝐺)‘(𝐵(midG‘𝐺)(𝑀𝐵)))‘(𝑆𝐴))))
205 eqid 2760 . . . . . . . 8 ((pInvG‘𝐺)‘(𝐵(midG‘𝐺)(𝑀𝐵))) = ((pInvG‘𝐺)‘(𝐵(midG‘𝐺)(𝑀𝐵)))
2061, 2, 3, 10, 20, 4, 16, 205, 14, 22miriso 25785 . . . . . . 7 (𝜑 → ((((pInvG‘𝐺)‘(𝐵(midG‘𝐺)(𝑀𝐵)))‘𝐵) (((pInvG‘𝐺)‘(𝐵(midG‘𝐺)(𝑀𝐵)))‘(𝑆𝐴))) = (𝐵 (𝑆𝐴)))
207204, 206eqtr2d 2795 . . . . . 6 (𝜑 → (𝐵 (𝑆𝐴)) = ((𝑀𝐵) (𝑆‘(𝑀𝐴))))
208207adantr 472 . . . . 5 ((𝜑 ∧ (𝑆𝐴) ≠ 𝑍) → (𝐵 (𝑆𝐴)) = ((𝑀𝐵) (𝑆‘(𝑀𝐴))))
2091, 2, 3, 125, 132, 126, 133, 130, 208tgcgrcomlr 25595 . . . 4 ((𝜑 ∧ (𝑆𝐴) ≠ 𝑍) → ((𝑆𝐴) 𝐵) = ((𝑆‘(𝑀𝐴)) (𝑀𝐵)))
210121adantr 472 . . . 4 ((𝜑 ∧ (𝑆𝐴) ≠ 𝑍) → (𝑍 𝐵) = (𝑍 (𝑀𝐵)))
2111, 2, 3, 125, 126, 127, 128, 130, 127, 131, 132, 133, 134, 135, 136, 194, 195, 209, 210axtg5seg 25584 . . 3 ((𝜑 ∧ (𝑆𝐴) ≠ 𝑍) → (𝐴 𝐵) = ((𝑀𝐴) (𝑀𝐵)))
212211eqcomd 2766 . 2 ((𝜑 ∧ (𝑆𝐴) ≠ 𝑍) → ((𝑀𝐴) (𝑀𝐵)) = (𝐴 𝐵))
213124, 212pm2.61dane 3019 1 (𝜑 → ((𝑀𝐴) (𝑀𝐵)) = (𝐴 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 382  wa 383   = wceq 1632  wcel 2139  wne 2932   class class class wbr 4804  ran crn 5267  cfv 6049  (class class class)co 6814  2c2 11282  ⟨“cs3 13807  Basecbs 16079  distcds 16172  TarskiGcstrkg 25549  DimTarskiGcstrkgld 25553  Itvcitv 25555  LineGclng 25556  pInvGcmir 25767  ∟Gcrag 25808  ⟂Gcperpg 25810  midGcmid 25884  lInvGclmi 25885
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-cnex 10204  ax-resscn 10205  ax-1cn 10206  ax-icn 10207  ax-addcl 10208  ax-addrcl 10209  ax-mulcl 10210  ax-mulrcl 10211  ax-mulcom 10212  ax-addass 10213  ax-mulass 10214  ax-distr 10215  ax-i2m1 10216  ax-1ne0 10217  ax-1rid 10218  ax-rnegex 10219  ax-rrecex 10220  ax-cnre 10221  ax-pre-lttri 10222  ax-pre-lttrn 10223  ax-pre-ltadd 10224  ax-pre-mulgt0 10225
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-om 7232  df-1st 7334  df-2nd 7335  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-1o 7730  df-oadd 7734  df-er 7913  df-map 8027  df-pm 8028  df-en 8124  df-dom 8125  df-sdom 8126  df-fin 8127  df-card 8975  df-cda 9202  df-pnf 10288  df-mnf 10289  df-xr 10290  df-ltxr 10291  df-le 10292  df-sub 10480  df-neg 10481  df-nn 11233  df-2 11291  df-3 11292  df-n0 11505  df-xnn0 11576  df-z 11590  df-uz 11900  df-fz 12540  df-fzo 12680  df-hash 13332  df-word 13505  df-concat 13507  df-s1 13508  df-s2 13813  df-s3 13814  df-trkgc 25567  df-trkgb 25568  df-trkgcb 25569  df-trkgld 25571  df-trkg 25572  df-cgrg 25626  df-leg 25698  df-mir 25768  df-rag 25809  df-perpg 25811  df-mid 25886  df-lmi 25887
This theorem is referenced by:  lmiiso  25909
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