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Theorem lmiisolem 28818
Description: Lemma for lmiiso 28819. (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 480 . . . . . . . 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 28808 . . . . . . . . . . . 12 (𝜑 → (𝑀𝐴) ∈ 𝑃)
131, 2, 3, 4, 7, 8, 12midcl 28799 . . . . . . . . . . 11 (𝜑 → (𝐴(midG‘𝐺)(𝑀𝐴)) ∈ 𝑃)
14 lmiiso.2 . . . . . . . . . . . 12 (𝜑𝐵𝑃)
151, 2, 3, 4, 7, 9, 10, 11, 14lmicl 28808 . . . . . . . . . . . 12 (𝜑 → (𝑀𝐵) ∈ 𝑃)
161, 2, 3, 4, 7, 14, 15midcl 28799 . . . . . . . . . . 11 (𝜑 → (𝐵(midG‘𝐺)(𝑀𝐵)) ∈ 𝑃)
171, 2, 3, 4, 7, 13, 16midcl 28799 . . . . . . . . . 10 (𝜑 → ((𝐴(midG‘𝐺)(𝑀𝐴))(midG‘𝐺)(𝐵(midG‘𝐺)(𝑀𝐵))) ∈ 𝑃)
186, 17eqeltrid 2842 . . . . . . . . 9 (𝜑𝑍𝑃)
1918adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝑆𝐴) = 𝑍) → 𝑍𝑃)
20 eqid 2734 . . . . . . . . . 10 (pInvG‘𝐺) = (pInvG‘𝐺)
21 lmiisolem.s . . . . . . . . . 10 𝑆 = ((pInvG‘𝐺)‘𝑍)
221, 2, 3, 10, 20, 4, 18, 21, 8mircl 28683 . . . . . . . . 9 (𝜑 → (𝑆𝐴) ∈ 𝑃)
2322adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝑆𝐴) = 𝑍) → (𝑆𝐴) ∈ 𝑃)
248adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝑆𝐴) = 𝑍) → 𝐴𝑃)
251, 2, 3, 10, 20, 5, 19, 21, 24mircgr 28679 . . . . . . . 8 ((𝜑 ∧ (𝑆𝐴) = 𝑍) → (𝑍 (𝑆𝐴)) = (𝑍 𝐴))
26 simpr 484 . . . . . . . . 9 ((𝜑 ∧ (𝑆𝐴) = 𝑍) → (𝑆𝐴) = 𝑍)
2726eqcomd 2740 . . . . . . . 8 ((𝜑 ∧ (𝑆𝐴) = 𝑍) → 𝑍 = (𝑆𝐴))
281, 2, 3, 5, 19, 23, 19, 24, 25, 27tgcgreq 28504 . . . . . . 7 ((𝜑 ∧ (𝑆𝐴) = 𝑍) → 𝑍 = 𝐴)
29 simpr 484 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝐴(midG‘𝐺)(𝑀𝐴)) = (𝐵(midG‘𝐺)(𝑀𝐵))) → (𝐴(midG‘𝐺)(𝑀𝐴)) = (𝐵(midG‘𝐺)(𝑀𝐵)))
3029oveq2d 7446 . . . . . . . . . . . 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))))
316, 30eqtr4id 2793 . . . . . . . . . . 11 ((𝜑 ∧ (𝐴(midG‘𝐺)(𝑀𝐴)) = (𝐵(midG‘𝐺)(𝑀𝐵))) → 𝑍 = ((𝐴(midG‘𝐺)(𝑀𝐴))(midG‘𝐺)(𝐴(midG‘𝐺)(𝑀𝐴))))
324adantr 480 . . . . . . . . . . . 12 ((𝜑 ∧ (𝐴(midG‘𝐺)(𝑀𝐴)) = (𝐵(midG‘𝐺)(𝑀𝐵))) → 𝐺 ∈ TarskiG)
337adantr 480 . . . . . . . . . . . 12 ((𝜑 ∧ (𝐴(midG‘𝐺)(𝑀𝐴)) = (𝐵(midG‘𝐺)(𝑀𝐵))) → 𝐺DimTarskiG≥2)
3413adantr 480 . . . . . . . . . . . 12 ((𝜑 ∧ (𝐴(midG‘𝐺)(𝑀𝐴)) = (𝐵(midG‘𝐺)(𝑀𝐵))) → (𝐴(midG‘𝐺)(𝑀𝐴)) ∈ 𝑃)
351, 2, 3, 32, 33, 34, 34midid 28803 . . . . . . . . . . 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 2774 . . . . . . . . . 10 ((𝜑 ∧ (𝐴(midG‘𝐺)(𝑀𝐴)) = (𝐵(midG‘𝐺)(𝑀𝐵))) → 𝑍 = (𝐴(midG‘𝐺)(𝑀𝐴)))
37 eqidd 2735 . . . . . . . . . . . . 13 (𝜑 → (𝑀𝐴) = (𝑀𝐴))
381, 2, 3, 4, 7, 9, 10, 11, 8, 12islmib 28809 . . . . . . . . . . . . 13 (𝜑 → ((𝑀𝐴) = (𝑀𝐴) ↔ ((𝐴(midG‘𝐺)(𝑀𝐴)) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿(𝑀𝐴)) ∨ 𝐴 = (𝑀𝐴)))))
3937, 38mpbid 232 . . . . . . . . . . . 12 (𝜑 → ((𝐴(midG‘𝐺)(𝑀𝐴)) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿(𝑀𝐴)) ∨ 𝐴 = (𝑀𝐴))))
4039simpld 494 . . . . . . . . . . 11 (𝜑 → (𝐴(midG‘𝐺)(𝑀𝐴)) ∈ 𝐷)
4140adantr 480 . . . . . . . . . 10 ((𝜑 ∧ (𝐴(midG‘𝐺)(𝑀𝐴)) = (𝐵(midG‘𝐺)(𝑀𝐵))) → (𝐴(midG‘𝐺)(𝑀𝐴)) ∈ 𝐷)
4236, 41eqeltrd 2838 . . . . . . . . 9 ((𝜑 ∧ (𝐴(midG‘𝐺)(𝑀𝐴)) = (𝐵(midG‘𝐺)(𝑀𝐵))) → 𝑍𝐷)
434adantr 480 . . . . . . . . . . 11 ((𝜑 ∧ (𝐴(midG‘𝐺)(𝑀𝐴)) ≠ (𝐵(midG‘𝐺)(𝑀𝐵))) → 𝐺 ∈ TarskiG)
4413adantr 480 . . . . . . . . . . 11 ((𝜑 ∧ (𝐴(midG‘𝐺)(𝑀𝐴)) ≠ (𝐵(midG‘𝐺)(𝑀𝐵))) → (𝐴(midG‘𝐺)(𝑀𝐴)) ∈ 𝑃)
4516adantr 480 . . . . . . . . . . 11 ((𝜑 ∧ (𝐴(midG‘𝐺)(𝑀𝐴)) ≠ (𝐵(midG‘𝐺)(𝑀𝐵))) → (𝐵(midG‘𝐺)(𝑀𝐵)) ∈ 𝑃)
4618adantr 480 . . . . . . . . . . 11 ((𝜑 ∧ (𝐴(midG‘𝐺)(𝑀𝐴)) ≠ (𝐵(midG‘𝐺)(𝑀𝐵))) → 𝑍𝑃)
47 simpr 484 . . . . . . . . . . 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 28801 . . . . . . . . . . . . 13 (𝜑 → ((𝐴(midG‘𝐺)(𝑀𝐴))(midG‘𝐺)(𝐵(midG‘𝐺)(𝑀𝐵))) ∈ ((𝐴(midG‘𝐺)(𝑀𝐴))𝐼(𝐵(midG‘𝐺)(𝑀𝐵))))
496, 48eqeltrid 2842 . . . . . . . . . . . 12 (𝜑𝑍 ∈ ((𝐴(midG‘𝐺)(𝑀𝐴))𝐼(𝐵(midG‘𝐺)(𝑀𝐵))))
5049adantr 480 . . . . . . . . . . 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 28641 . . . . . . . . . 10 ((𝜑 ∧ (𝐴(midG‘𝐺)(𝑀𝐴)) ≠ (𝐵(midG‘𝐺)(𝑀𝐵))) → 𝑍 ∈ ((𝐴(midG‘𝐺)(𝑀𝐴))𝐿(𝐵(midG‘𝐺)(𝑀𝐵))))
5211adantr 480 . . . . . . . . . . 11 ((𝜑 ∧ (𝐴(midG‘𝐺)(𝑀𝐴)) ≠ (𝐵(midG‘𝐺)(𝑀𝐵))) → 𝐷 ∈ ran 𝐿)
5340adantr 480 . . . . . . . . . . 11 ((𝜑 ∧ (𝐴(midG‘𝐺)(𝑀𝐴)) ≠ (𝐵(midG‘𝐺)(𝑀𝐵))) → (𝐴(midG‘𝐺)(𝑀𝐴)) ∈ 𝐷)
54 eqidd 2735 . . . . . . . . . . . . . 14 (𝜑 → (𝑀𝐵) = (𝑀𝐵))
551, 2, 3, 4, 7, 9, 10, 11, 14, 15islmib 28809 . . . . . . . . . . . . . 14 (𝜑 → ((𝑀𝐵) = (𝑀𝐵) ↔ ((𝐵(midG‘𝐺)(𝑀𝐵)) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐵𝐿(𝑀𝐵)) ∨ 𝐵 = (𝑀𝐵)))))
5654, 55mpbid 232 . . . . . . . . . . . . 13 (𝜑 → ((𝐵(midG‘𝐺)(𝑀𝐵)) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐵𝐿(𝑀𝐵)) ∨ 𝐵 = (𝑀𝐵))))
5756simpld 494 . . . . . . . . . . . 12 (𝜑 → (𝐵(midG‘𝐺)(𝑀𝐵)) ∈ 𝐷)
5857adantr 480 . . . . . . . . . . 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 28658 . . . . . . . . . 10 ((𝜑 ∧ (𝐴(midG‘𝐺)(𝑀𝐴)) ≠ (𝐵(midG‘𝐺)(𝑀𝐵))) → 𝐷 = ((𝐴(midG‘𝐺)(𝑀𝐴))𝐿(𝐵(midG‘𝐺)(𝑀𝐵))))
6051, 59eleqtrrd 2841 . . . . . . . . 9 ((𝜑 ∧ (𝐴(midG‘𝐺)(𝑀𝐴)) ≠ (𝐵(midG‘𝐺)(𝑀𝐵))) → 𝑍𝐷)
6142, 60pm2.61dane 3026 . . . . . . . 8 (𝜑𝑍𝐷)
6261adantr 480 . . . . . . 7 ((𝜑 ∧ (𝑆𝐴) = 𝑍) → 𝑍𝐷)
6328, 62eqeltrrd 2839 . . . . . 6 ((𝜑 ∧ (𝑆𝐴) = 𝑍) → 𝐴𝐷)
641, 2, 3, 4, 7, 9, 10, 11, 8lmiinv 28814 . . . . . . 7 (𝜑 → ((𝑀𝐴) = 𝐴𝐴𝐷))
6564biimpar 477 . . . . . 6 ((𝜑𝐴𝐷) → (𝑀𝐴) = 𝐴)
6663, 65syldan 591 . . . . 5 ((𝜑 ∧ (𝑆𝐴) = 𝑍) → (𝑀𝐴) = 𝐴)
6766, 28eqtr4d 2777 . . . 4 ((𝜑 ∧ (𝑆𝐴) = 𝑍) → (𝑀𝐴) = 𝑍)
6867oveq1d 7445 . . 3 ((𝜑 ∧ (𝑆𝐴) = 𝑍) → ((𝑀𝐴) (𝑀𝐵)) = (𝑍 (𝑀𝐵)))
69 eqidd 2735 . . . . . . . . 9 ((𝜑𝐵 = (𝑀𝐵)) → 𝑍 = 𝑍)
704adantr 480 . . . . . . . . . 10 ((𝜑𝐵 = (𝑀𝐵)) → 𝐺 ∈ TarskiG)
7114adantr 480 . . . . . . . . . 10 ((𝜑𝐵 = (𝑀𝐵)) → 𝐵𝑃)
7216adantr 480 . . . . . . . . . 10 ((𝜑𝐵 = (𝑀𝐵)) → (𝐵(midG‘𝐺)(𝑀𝐵)) ∈ 𝑃)
731, 2, 3, 4, 7, 14, 15midbtwn 28801 . . . . . . . . . . . 12 (𝜑 → (𝐵(midG‘𝐺)(𝑀𝐵)) ∈ (𝐵𝐼(𝑀𝐵)))
7473adantr 480 . . . . . . . . . . 11 ((𝜑𝐵 = (𝑀𝐵)) → (𝐵(midG‘𝐺)(𝑀𝐵)) ∈ (𝐵𝐼(𝑀𝐵)))
75 simpr 484 . . . . . . . . . . . 12 ((𝜑𝐵 = (𝑀𝐵)) → 𝐵 = (𝑀𝐵))
7675oveq2d 7446 . . . . . . . . . . 11 ((𝜑𝐵 = (𝑀𝐵)) → (𝐵𝐼𝐵) = (𝐵𝐼(𝑀𝐵)))
7774, 76eleqtrrd 2841 . . . . . . . . . 10 ((𝜑𝐵 = (𝑀𝐵)) → (𝐵(midG‘𝐺)(𝑀𝐵)) ∈ (𝐵𝐼𝐵))
781, 2, 3, 70, 71, 72, 77axtgbtwnid 28488 . . . . . . . . 9 ((𝜑𝐵 = (𝑀𝐵)) → 𝐵 = (𝐵(midG‘𝐺)(𝑀𝐵)))
79 eqidd 2735 . . . . . . . . 9 ((𝜑𝐵 = (𝑀𝐵)) → 𝐵 = 𝐵)
8069, 78, 79s3eqd 14899 . . . . . . . 8 ((𝜑𝐵 = (𝑀𝐵)) → ⟨“𝑍𝐵𝐵”⟩ = ⟨“𝑍(𝐵(midG‘𝐺)(𝑀𝐵))𝐵”⟩)
811, 2, 3, 10, 20, 4, 18, 14, 14ragtrivb 28724 . . . . . . . . 9 (𝜑 → ⟨“𝑍𝐵𝐵”⟩ ∈ (∟G‘𝐺))
8281adantr 480 . . . . . . . 8 ((𝜑𝐵 = (𝑀𝐵)) → ⟨“𝑍𝐵𝐵”⟩ ∈ (∟G‘𝐺))
8380, 82eqeltrrd 2839 . . . . . . 7 ((𝜑𝐵 = (𝑀𝐵)) → ⟨“𝑍(𝐵(midG‘𝐺)(𝑀𝐵))𝐵”⟩ ∈ (∟G‘𝐺))
844adantr 480 . . . . . . . 8 ((𝜑𝐵 ≠ (𝑀𝐵)) → 𝐺 ∈ TarskiG)
8561adantr 480 . . . . . . . 8 ((𝜑𝐵 ≠ (𝑀𝐵)) → 𝑍𝐷)
8657adantr 480 . . . . . . . 8 ((𝜑𝐵 ≠ (𝑀𝐵)) → (𝐵(midG‘𝐺)(𝑀𝐵)) ∈ 𝐷)
8714adantr 480 . . . . . . . 8 ((𝜑𝐵 ≠ (𝑀𝐵)) → 𝐵𝑃)
88 df-ne 2938 . . . . . . . . . 10 (𝐵 ≠ (𝑀𝐵) ↔ ¬ 𝐵 = (𝑀𝐵))
8956simprd 495 . . . . . . . . . . . 12 (𝜑 → (𝐷(⟂G‘𝐺)(𝐵𝐿(𝑀𝐵)) ∨ 𝐵 = (𝑀𝐵)))
9089orcomd 871 . . . . . . . . . . 11 (𝜑 → (𝐵 = (𝑀𝐵) ∨ 𝐷(⟂G‘𝐺)(𝐵𝐿(𝑀𝐵))))
9190orcanai 1004 . . . . . . . . . 10 ((𝜑 ∧ ¬ 𝐵 = (𝑀𝐵)) → 𝐷(⟂G‘𝐺)(𝐵𝐿(𝑀𝐵)))
9288, 91sylan2b 594 . . . . . . . . 9 ((𝜑𝐵 ≠ (𝑀𝐵)) → 𝐷(⟂G‘𝐺)(𝐵𝐿(𝑀𝐵)))
9315adantr 480 . . . . . . . . . . 11 ((𝜑𝐵 ≠ (𝑀𝐵)) → (𝑀𝐵) ∈ 𝑃)
94 simpr 484 . . . . . . . . . . 11 ((𝜑𝐵 ≠ (𝑀𝐵)) → 𝐵 ≠ (𝑀𝐵))
9516adantr 480 . . . . . . . . . . 11 ((𝜑𝐵 ≠ (𝑀𝐵)) → (𝐵(midG‘𝐺)(𝑀𝐵)) ∈ 𝑃)
964adantr 480 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝐵(midG‘𝐺)(𝑀𝐵)) = 𝐵) → 𝐺 ∈ TarskiG)
9714adantr 480 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝐵(midG‘𝐺)(𝑀𝐵)) = 𝐵) → 𝐵𝑃)
9815adantr 480 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝐵(midG‘𝐺)(𝑀𝐵)) = 𝐵) → (𝑀𝐵) ∈ 𝑃)
997adantr 480 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝐵(midG‘𝐺)(𝑀𝐵)) = 𝐵) → 𝐺DimTarskiG≥2)
100 simpr 484 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝐵(midG‘𝐺)(𝑀𝐵)) = 𝐵) → (𝐵(midG‘𝐺)(𝑀𝐵)) = 𝐵)
1011, 2, 3, 96, 99, 97, 98, 100midcgr 28802 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝐵(midG‘𝐺)(𝑀𝐵)) = 𝐵) → (𝐵 𝐵) = (𝐵 (𝑀𝐵)))
102101eqcomd 2740 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝐵(midG‘𝐺)(𝑀𝐵)) = 𝐵) → (𝐵 (𝑀𝐵)) = (𝐵 𝐵))
1031, 2, 3, 96, 97, 98, 97, 102axtgcgrid 28485 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝐵(midG‘𝐺)(𝑀𝐵)) = 𝐵) → 𝐵 = (𝑀𝐵))
104103ex 412 . . . . . . . . . . . . 13 (𝜑 → ((𝐵(midG‘𝐺)(𝑀𝐵)) = 𝐵𝐵 = (𝑀𝐵)))
105104necon3d 2958 . . . . . . . . . . . 12 (𝜑 → (𝐵 ≠ (𝑀𝐵) → (𝐵(midG‘𝐺)(𝑀𝐵)) ≠ 𝐵))
106105imp 406 . . . . . . . . . . 11 ((𝜑𝐵 ≠ (𝑀𝐵)) → (𝐵(midG‘𝐺)(𝑀𝐵)) ≠ 𝐵)
10773adantr 480 . . . . . . . . . . . 12 ((𝜑𝐵 ≠ (𝑀𝐵)) → (𝐵(midG‘𝐺)(𝑀𝐵)) ∈ (𝐵𝐼(𝑀𝐵)))
1081, 3, 10, 84, 87, 93, 95, 94, 107btwnlng1 28641 . . . . . . . . . . 11 ((𝜑𝐵 ≠ (𝑀𝐵)) → (𝐵(midG‘𝐺)(𝑀𝐵)) ∈ (𝐵𝐿(𝑀𝐵)))
1091, 3, 10, 84, 87, 93, 94, 95, 106, 108tglineelsb2 28654 . . . . . . . . . 10 ((𝜑𝐵 ≠ (𝑀𝐵)) → (𝐵𝐿(𝑀𝐵)) = (𝐵𝐿(𝐵(midG‘𝐺)(𝑀𝐵))))
1101, 3, 10, 84, 95, 87, 106tglinecom 28657 . . . . . . . . . 10 ((𝜑𝐵 ≠ (𝑀𝐵)) → ((𝐵(midG‘𝐺)(𝑀𝐵))𝐿𝐵) = (𝐵𝐿(𝐵(midG‘𝐺)(𝑀𝐵))))
111109, 110eqtr4d 2777 . . . . . . . . 9 ((𝜑𝐵 ≠ (𝑀𝐵)) → (𝐵𝐿(𝑀𝐵)) = ((𝐵(midG‘𝐺)(𝑀𝐵))𝐿𝐵))
11292, 111breqtrd 5173 . . . . . . . 8 ((𝜑𝐵 ≠ (𝑀𝐵)) → 𝐷(⟂G‘𝐺)((𝐵(midG‘𝐺)(𝑀𝐵))𝐿𝐵))
1131, 2, 3, 10, 84, 85, 86, 87, 112perpdrag 28750 . . . . . . 7 ((𝜑𝐵 ≠ (𝑀𝐵)) → ⟨“𝑍(𝐵(midG‘𝐺)(𝑀𝐵))𝐵”⟩ ∈ (∟G‘𝐺))
11483, 113pm2.61dane 3026 . . . . . 6 (𝜑 → ⟨“𝑍(𝐵(midG‘𝐺)(𝑀𝐵))𝐵”⟩ ∈ (∟G‘𝐺))
1151, 2, 3, 10, 20, 4, 18, 16, 14israg 28719 . . . . . 6 (𝜑 → (⟨“𝑍(𝐵(midG‘𝐺)(𝑀𝐵))𝐵”⟩ ∈ (∟G‘𝐺) ↔ (𝑍 𝐵) = (𝑍 (((pInvG‘𝐺)‘(𝐵(midG‘𝐺)(𝑀𝐵)))‘𝐵))))
116114, 115mpbid 232 . . . . 5 (𝜑 → (𝑍 𝐵) = (𝑍 (((pInvG‘𝐺)‘(𝐵(midG‘𝐺)(𝑀𝐵)))‘𝐵)))
117 eqidd 2735 . . . . . . 7 (𝜑 → (𝐵(midG‘𝐺)(𝑀𝐵)) = (𝐵(midG‘𝐺)(𝑀𝐵)))
1181, 2, 3, 4, 7, 14, 15, 20, 16ismidb 28800 . . . . . . 7 (𝜑 → ((𝑀𝐵) = (((pInvG‘𝐺)‘(𝐵(midG‘𝐺)(𝑀𝐵)))‘𝐵) ↔ (𝐵(midG‘𝐺)(𝑀𝐵)) = (𝐵(midG‘𝐺)(𝑀𝐵))))
119117, 118mpbird 257 . . . . . 6 (𝜑 → (𝑀𝐵) = (((pInvG‘𝐺)‘(𝐵(midG‘𝐺)(𝑀𝐵)))‘𝐵))
120119oveq2d 7446 . . . . 5 (𝜑 → (𝑍 (𝑀𝐵)) = (𝑍 (((pInvG‘𝐺)‘(𝐵(midG‘𝐺)(𝑀𝐵)))‘𝐵)))
121116, 120eqtr4d 2777 . . . 4 (𝜑 → (𝑍 𝐵) = (𝑍 (𝑀𝐵)))
122121adantr 480 . . 3 ((𝜑 ∧ (𝑆𝐴) = 𝑍) → (𝑍 𝐵) = (𝑍 (𝑀𝐵)))
12328oveq1d 7445 . . 3 ((𝜑 ∧ (𝑆𝐴) = 𝑍) → (𝑍 𝐵) = (𝐴 𝐵))
12468, 122, 1233eqtr2d 2780 . 2 ((𝜑 ∧ (𝑆𝐴) = 𝑍) → ((𝑀𝐴) (𝑀𝐵)) = (𝐴 𝐵))
1254adantr 480 . . . 4 ((𝜑 ∧ (𝑆𝐴) ≠ 𝑍) → 𝐺 ∈ TarskiG)
12622adantr 480 . . . 4 ((𝜑 ∧ (𝑆𝐴) ≠ 𝑍) → (𝑆𝐴) ∈ 𝑃)
12718adantr 480 . . . 4 ((𝜑 ∧ (𝑆𝐴) ≠ 𝑍) → 𝑍𝑃)
1288adantr 480 . . . 4 ((𝜑 ∧ (𝑆𝐴) ≠ 𝑍) → 𝐴𝑃)
1291, 2, 3, 10, 20, 4, 18, 21, 12mircl 28683 . . . . 5 (𝜑 → (𝑆‘(𝑀𝐴)) ∈ 𝑃)
130129adantr 480 . . . 4 ((𝜑 ∧ (𝑆𝐴) ≠ 𝑍) → (𝑆‘(𝑀𝐴)) ∈ 𝑃)
13112adantr 480 . . . 4 ((𝜑 ∧ (𝑆𝐴) ≠ 𝑍) → (𝑀𝐴) ∈ 𝑃)
13214adantr 480 . . . 4 ((𝜑 ∧ (𝑆𝐴) ≠ 𝑍) → 𝐵𝑃)
13315adantr 480 . . . 4 ((𝜑 ∧ (𝑆𝐴) ≠ 𝑍) → (𝑀𝐵) ∈ 𝑃)
134 simpr 484 . . . 4 ((𝜑 ∧ (𝑆𝐴) ≠ 𝑍) → (𝑆𝐴) ≠ 𝑍)
1351, 2, 3, 10, 20, 125, 127, 21, 128mirbtwn 28680 . . . 4 ((𝜑 ∧ (𝑆𝐴) ≠ 𝑍) → 𝑍 ∈ ((𝑆𝐴)𝐼𝐴))
1361, 2, 3, 10, 20, 125, 127, 21, 131mirbtwn 28680 . . . 4 ((𝜑 ∧ (𝑆𝐴) ≠ 𝑍) → 𝑍 ∈ ((𝑆‘(𝑀𝐴))𝐼(𝑀𝐴)))
137 eqidd 2735 . . . . . . . . . . . 12 ((𝜑𝐴 = (𝑀𝐴)) → 𝑍 = 𝑍)
1384adantr 480 . . . . . . . . . . . . 13 ((𝜑𝐴 = (𝑀𝐴)) → 𝐺 ∈ TarskiG)
1398adantr 480 . . . . . . . . . . . . 13 ((𝜑𝐴 = (𝑀𝐴)) → 𝐴𝑃)
14013adantr 480 . . . . . . . . . . . . 13 ((𝜑𝐴 = (𝑀𝐴)) → (𝐴(midG‘𝐺)(𝑀𝐴)) ∈ 𝑃)
1411, 2, 3, 4, 7, 8, 12midbtwn 28801 . . . . . . . . . . . . . . 15 (𝜑 → (𝐴(midG‘𝐺)(𝑀𝐴)) ∈ (𝐴𝐼(𝑀𝐴)))
142141adantr 480 . . . . . . . . . . . . . 14 ((𝜑𝐴 = (𝑀𝐴)) → (𝐴(midG‘𝐺)(𝑀𝐴)) ∈ (𝐴𝐼(𝑀𝐴)))
143 simpr 484 . . . . . . . . . . . . . . 15 ((𝜑𝐴 = (𝑀𝐴)) → 𝐴 = (𝑀𝐴))
144143oveq2d 7446 . . . . . . . . . . . . . 14 ((𝜑𝐴 = (𝑀𝐴)) → (𝐴𝐼𝐴) = (𝐴𝐼(𝑀𝐴)))
145142, 144eleqtrrd 2841 . . . . . . . . . . . . 13 ((𝜑𝐴 = (𝑀𝐴)) → (𝐴(midG‘𝐺)(𝑀𝐴)) ∈ (𝐴𝐼𝐴))
1461, 2, 3, 138, 139, 140, 145axtgbtwnid 28488 . . . . . . . . . . . 12 ((𝜑𝐴 = (𝑀𝐴)) → 𝐴 = (𝐴(midG‘𝐺)(𝑀𝐴)))
147 eqidd 2735 . . . . . . . . . . . 12 ((𝜑𝐴 = (𝑀𝐴)) → 𝐴 = 𝐴)
148137, 146, 147s3eqd 14899 . . . . . . . . . . 11 ((𝜑𝐴 = (𝑀𝐴)) → ⟨“𝑍𝐴𝐴”⟩ = ⟨“𝑍(𝐴(midG‘𝐺)(𝑀𝐴))𝐴”⟩)
1491, 2, 3, 10, 20, 4, 18, 8, 8ragtrivb 28724 . . . . . . . . . . . 12 (𝜑 → ⟨“𝑍𝐴𝐴”⟩ ∈ (∟G‘𝐺))
150149adantr 480 . . . . . . . . . . 11 ((𝜑𝐴 = (𝑀𝐴)) → ⟨“𝑍𝐴𝐴”⟩ ∈ (∟G‘𝐺))
151148, 150eqeltrrd 2839 . . . . . . . . . 10 ((𝜑𝐴 = (𝑀𝐴)) → ⟨“𝑍(𝐴(midG‘𝐺)(𝑀𝐴))𝐴”⟩ ∈ (∟G‘𝐺))
1524adantr 480 . . . . . . . . . . 11 ((𝜑𝐴 ≠ (𝑀𝐴)) → 𝐺 ∈ TarskiG)
15361adantr 480 . . . . . . . . . . 11 ((𝜑𝐴 ≠ (𝑀𝐴)) → 𝑍𝐷)
15440adantr 480 . . . . . . . . . . 11 ((𝜑𝐴 ≠ (𝑀𝐴)) → (𝐴(midG‘𝐺)(𝑀𝐴)) ∈ 𝐷)
1558adantr 480 . . . . . . . . . . 11 ((𝜑𝐴 ≠ (𝑀𝐴)) → 𝐴𝑃)
156 df-ne 2938 . . . . . . . . . . . . 13 (𝐴 ≠ (𝑀𝐴) ↔ ¬ 𝐴 = (𝑀𝐴))
15739simprd 495 . . . . . . . . . . . . . . 15 (𝜑 → (𝐷(⟂G‘𝐺)(𝐴𝐿(𝑀𝐴)) ∨ 𝐴 = (𝑀𝐴)))
158157orcomd 871 . . . . . . . . . . . . . 14 (𝜑 → (𝐴 = (𝑀𝐴) ∨ 𝐷(⟂G‘𝐺)(𝐴𝐿(𝑀𝐴))))
159158orcanai 1004 . . . . . . . . . . . . 13 ((𝜑 ∧ ¬ 𝐴 = (𝑀𝐴)) → 𝐷(⟂G‘𝐺)(𝐴𝐿(𝑀𝐴)))
160156, 159sylan2b 594 . . . . . . . . . . . 12 ((𝜑𝐴 ≠ (𝑀𝐴)) → 𝐷(⟂G‘𝐺)(𝐴𝐿(𝑀𝐴)))
16112adantr 480 . . . . . . . . . . . . . 14 ((𝜑𝐴 ≠ (𝑀𝐴)) → (𝑀𝐴) ∈ 𝑃)
162 simpr 484 . . . . . . . . . . . . . 14 ((𝜑𝐴 ≠ (𝑀𝐴)) → 𝐴 ≠ (𝑀𝐴))
16313adantr 480 . . . . . . . . . . . . . 14 ((𝜑𝐴 ≠ (𝑀𝐴)) → (𝐴(midG‘𝐺)(𝑀𝐴)) ∈ 𝑃)
1644adantr 480 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝐴(midG‘𝐺)(𝑀𝐴)) = 𝐴) → 𝐺 ∈ TarskiG)
1658adantr 480 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝐴(midG‘𝐺)(𝑀𝐴)) = 𝐴) → 𝐴𝑃)
16612adantr 480 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝐴(midG‘𝐺)(𝑀𝐴)) = 𝐴) → (𝑀𝐴) ∈ 𝑃)
1677adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝐴(midG‘𝐺)(𝑀𝐴)) = 𝐴) → 𝐺DimTarskiG≥2)
168 simpr 484 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝐴(midG‘𝐺)(𝑀𝐴)) = 𝐴) → (𝐴(midG‘𝐺)(𝑀𝐴)) = 𝐴)
1691, 2, 3, 164, 167, 165, 166, 168midcgr 28802 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝐴(midG‘𝐺)(𝑀𝐴)) = 𝐴) → (𝐴 𝐴) = (𝐴 (𝑀𝐴)))
170169eqcomd 2740 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝐴(midG‘𝐺)(𝑀𝐴)) = 𝐴) → (𝐴 (𝑀𝐴)) = (𝐴 𝐴))
1711, 2, 3, 164, 165, 166, 165, 170axtgcgrid 28485 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝐴(midG‘𝐺)(𝑀𝐴)) = 𝐴) → 𝐴 = (𝑀𝐴))
172171ex 412 . . . . . . . . . . . . . . . 16 (𝜑 → ((𝐴(midG‘𝐺)(𝑀𝐴)) = 𝐴𝐴 = (𝑀𝐴)))
173172necon3d 2958 . . . . . . . . . . . . . . 15 (𝜑 → (𝐴 ≠ (𝑀𝐴) → (𝐴(midG‘𝐺)(𝑀𝐴)) ≠ 𝐴))
174173imp 406 . . . . . . . . . . . . . 14 ((𝜑𝐴 ≠ (𝑀𝐴)) → (𝐴(midG‘𝐺)(𝑀𝐴)) ≠ 𝐴)
175141adantr 480 . . . . . . . . . . . . . . 15 ((𝜑𝐴 ≠ (𝑀𝐴)) → (𝐴(midG‘𝐺)(𝑀𝐴)) ∈ (𝐴𝐼(𝑀𝐴)))
1761, 3, 10, 152, 155, 161, 163, 162, 175btwnlng1 28641 . . . . . . . . . . . . . 14 ((𝜑𝐴 ≠ (𝑀𝐴)) → (𝐴(midG‘𝐺)(𝑀𝐴)) ∈ (𝐴𝐿(𝑀𝐴)))
1771, 3, 10, 152, 155, 161, 162, 163, 174, 176tglineelsb2 28654 . . . . . . . . . . . . 13 ((𝜑𝐴 ≠ (𝑀𝐴)) → (𝐴𝐿(𝑀𝐴)) = (𝐴𝐿(𝐴(midG‘𝐺)(𝑀𝐴))))
1781, 3, 10, 152, 163, 155, 174tglinecom 28657 . . . . . . . . . . . . 13 ((𝜑𝐴 ≠ (𝑀𝐴)) → ((𝐴(midG‘𝐺)(𝑀𝐴))𝐿𝐴) = (𝐴𝐿(𝐴(midG‘𝐺)(𝑀𝐴))))
179177, 178eqtr4d 2777 . . . . . . . . . . . 12 ((𝜑𝐴 ≠ (𝑀𝐴)) → (𝐴𝐿(𝑀𝐴)) = ((𝐴(midG‘𝐺)(𝑀𝐴))𝐿𝐴))
180160, 179breqtrd 5173 . . . . . . . . . . 11 ((𝜑𝐴 ≠ (𝑀𝐴)) → 𝐷(⟂G‘𝐺)((𝐴(midG‘𝐺)(𝑀𝐴))𝐿𝐴))
1811, 2, 3, 10, 152, 153, 154, 155, 180perpdrag 28750 . . . . . . . . . 10 ((𝜑𝐴 ≠ (𝑀𝐴)) → ⟨“𝑍(𝐴(midG‘𝐺)(𝑀𝐴))𝐴”⟩ ∈ (∟G‘𝐺))
182151, 181pm2.61dane 3026 . . . . . . . . 9 (𝜑 → ⟨“𝑍(𝐴(midG‘𝐺)(𝑀𝐴))𝐴”⟩ ∈ (∟G‘𝐺))
1831, 2, 3, 10, 20, 4, 18, 13, 8israg 28719 . . . . . . . . 9 (𝜑 → (⟨“𝑍(𝐴(midG‘𝐺)(𝑀𝐴))𝐴”⟩ ∈ (∟G‘𝐺) ↔ (𝑍 𝐴) = (𝑍 (((pInvG‘𝐺)‘(𝐴(midG‘𝐺)(𝑀𝐴)))‘𝐴))))
184182, 183mpbid 232 . . . . . . . 8 (𝜑 → (𝑍 𝐴) = (𝑍 (((pInvG‘𝐺)‘(𝐴(midG‘𝐺)(𝑀𝐴)))‘𝐴)))
185 eqidd 2735 . . . . . . . . . 10 (𝜑 → (𝐴(midG‘𝐺)(𝑀𝐴)) = (𝐴(midG‘𝐺)(𝑀𝐴)))
1861, 2, 3, 4, 7, 8, 12, 20, 13ismidb 28800 . . . . . . . . . 10 (𝜑 → ((𝑀𝐴) = (((pInvG‘𝐺)‘(𝐴(midG‘𝐺)(𝑀𝐴)))‘𝐴) ↔ (𝐴(midG‘𝐺)(𝑀𝐴)) = (𝐴(midG‘𝐺)(𝑀𝐴))))
187185, 186mpbird 257 . . . . . . . . 9 (𝜑 → (𝑀𝐴) = (((pInvG‘𝐺)‘(𝐴(midG‘𝐺)(𝑀𝐴)))‘𝐴))
188187oveq2d 7446 . . . . . . . 8 (𝜑 → (𝑍 (𝑀𝐴)) = (𝑍 (((pInvG‘𝐺)‘(𝐴(midG‘𝐺)(𝑀𝐴)))‘𝐴)))
189184, 188eqtr4d 2777 . . . . . . 7 (𝜑 → (𝑍 𝐴) = (𝑍 (𝑀𝐴)))
1901, 2, 3, 10, 20, 4, 18, 21, 8mircgr 28679 . . . . . . 7 (𝜑 → (𝑍 (𝑆𝐴)) = (𝑍 𝐴))
1911, 2, 3, 10, 20, 4, 18, 21, 12mircgr 28679 . . . . . . 7 (𝜑 → (𝑍 (𝑆‘(𝑀𝐴))) = (𝑍 (𝑀𝐴)))
192189, 190, 1913eqtr4d 2784 . . . . . 6 (𝜑 → (𝑍 (𝑆𝐴)) = (𝑍 (𝑆‘(𝑀𝐴))))
193192adantr 480 . . . . 5 ((𝜑 ∧ (𝑆𝐴) ≠ 𝑍) → (𝑍 (𝑆𝐴)) = (𝑍 (𝑆‘(𝑀𝐴))))
1941, 2, 3, 125, 127, 126, 127, 130, 193tgcgrcomlr 28502 . . . 4 ((𝜑 ∧ (𝑆𝐴) ≠ 𝑍) → ((𝑆𝐴) 𝑍) = ((𝑆‘(𝑀𝐴)) 𝑍))
195189adantr 480 . . . 4 ((𝜑 ∧ (𝑆𝐴) ≠ 𝑍) → (𝑍 𝐴) = (𝑍 (𝑀𝐴)))
19621fveq1i 6907 . . . . . . . . . 10 (𝑆‘(𝐴(midG‘𝐺)(𝑀𝐴))) = (((pInvG‘𝐺)‘𝑍)‘(𝐴(midG‘𝐺)(𝑀𝐴)))
1971, 2, 3, 4, 7, 8, 12, 21, 18mirmid 28805 . . . . . . . . . 10 (𝜑 → ((𝑆𝐴)(midG‘𝐺)(𝑆‘(𝑀𝐴))) = (𝑆‘(𝐴(midG‘𝐺)(𝑀𝐴))))
1986eqcomi 2743 . . . . . . . . . . 11 ((𝐴(midG‘𝐺)(𝑀𝐴))(midG‘𝐺)(𝐵(midG‘𝐺)(𝑀𝐵))) = 𝑍
1991, 2, 3, 4, 7, 13, 16, 20, 18ismidb 28800 . . . . . . . . . . 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 258 . . . . . . . . . 10 (𝜑 → (𝐵(midG‘𝐺)(𝑀𝐵)) = (((pInvG‘𝐺)‘𝑍)‘(𝐴(midG‘𝐺)(𝑀𝐴))))
201196, 197, 2003eqtr4a 2800 . . . . . . . . 9 (𝜑 → ((𝑆𝐴)(midG‘𝐺)(𝑆‘(𝑀𝐴))) = (𝐵(midG‘𝐺)(𝑀𝐵)))
2021, 2, 3, 4, 7, 22, 129, 20, 16ismidb 28800 . . . . . . . . 9 (𝜑 → ((𝑆‘(𝑀𝐴)) = (((pInvG‘𝐺)‘(𝐵(midG‘𝐺)(𝑀𝐵)))‘(𝑆𝐴)) ↔ ((𝑆𝐴)(midG‘𝐺)(𝑆‘(𝑀𝐴))) = (𝐵(midG‘𝐺)(𝑀𝐵))))
203201, 202mpbird 257 . . . . . . . 8 (𝜑 → (𝑆‘(𝑀𝐴)) = (((pInvG‘𝐺)‘(𝐵(midG‘𝐺)(𝑀𝐵)))‘(𝑆𝐴)))
204119, 203oveq12d 7448 . . . . . . 7 (𝜑 → ((𝑀𝐵) (𝑆‘(𝑀𝐴))) = ((((pInvG‘𝐺)‘(𝐵(midG‘𝐺)(𝑀𝐵)))‘𝐵) (((pInvG‘𝐺)‘(𝐵(midG‘𝐺)(𝑀𝐵)))‘(𝑆𝐴))))
205 eqid 2734 . . . . . . . 8 ((pInvG‘𝐺)‘(𝐵(midG‘𝐺)(𝑀𝐵))) = ((pInvG‘𝐺)‘(𝐵(midG‘𝐺)(𝑀𝐵)))
2061, 2, 3, 10, 20, 4, 16, 205, 14, 22miriso 28692 . . . . . . 7 (𝜑 → ((((pInvG‘𝐺)‘(𝐵(midG‘𝐺)(𝑀𝐵)))‘𝐵) (((pInvG‘𝐺)‘(𝐵(midG‘𝐺)(𝑀𝐵)))‘(𝑆𝐴))) = (𝐵 (𝑆𝐴)))
207204, 206eqtr2d 2775 . . . . . 6 (𝜑 → (𝐵 (𝑆𝐴)) = ((𝑀𝐵) (𝑆‘(𝑀𝐴))))
208207adantr 480 . . . . 5 ((𝜑 ∧ (𝑆𝐴) ≠ 𝑍) → (𝐵 (𝑆𝐴)) = ((𝑀𝐵) (𝑆‘(𝑀𝐴))))
2091, 2, 3, 125, 132, 126, 133, 130, 208tgcgrcomlr 28502 . . . 4 ((𝜑 ∧ (𝑆𝐴) ≠ 𝑍) → ((𝑆𝐴) 𝐵) = ((𝑆‘(𝑀𝐴)) (𝑀𝐵)))
210121adantr 480 . . . 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 28487 . . 3 ((𝜑 ∧ (𝑆𝐴) ≠ 𝑍) → (𝐴 𝐵) = ((𝑀𝐴) (𝑀𝐵)))
212211eqcomd 2740 . 2 ((𝜑 ∧ (𝑆𝐴) ≠ 𝑍) → ((𝑀𝐴) (𝑀𝐵)) = (𝐴 𝐵))
213124, 212pm2.61dane 3026 1 (𝜑 → ((𝑀𝐴) (𝑀𝐵)) = (𝐴 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wo 847   = wceq 1536  wcel 2105  wne 2937   class class class wbr 5147  ran crn 5689  cfv 6562  (class class class)co 7430  2c2 12318  ⟨“cs3 14877  Basecbs 17244  distcds 17306  TarskiGcstrkg 28449  DimTarskiGcstrkgld 28453  Itvcitv 28455  LineGclng 28456  pInvGcmir 28674  ∟Gcrag 28715  ⟂Gcperpg 28717  midGcmid 28794  lInvGclmi 28795
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-oadd 8508  df-er 8743  df-map 8866  df-pm 8867  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-dju 9938  df-card 9976  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-nn 12264  df-2 12326  df-3 12327  df-n0 12524  df-xnn0 12597  df-z 12611  df-uz 12876  df-fz 13544  df-fzo 13691  df-hash 14366  df-word 14549  df-concat 14605  df-s1 14630  df-s2 14883  df-s3 14884  df-trkgc 28470  df-trkgb 28471  df-trkgcb 28472  df-trkgld 28474  df-trkg 28475  df-cgrg 28533  df-leg 28605  df-mir 28675  df-rag 28716  df-perpg 28718  df-mid 28796  df-lmi 28797
This theorem is referenced by:  lmiiso  28819
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