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Theorem lmiisolem 28593
Description: Lemma for lmiiso 28594. (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 28583 . . . . . . . . . . . 12 (𝜑 → (𝑀𝐴) ∈ 𝑃)
131, 2, 3, 4, 7, 8, 12midcl 28574 . . . . . . . . . . 11 (𝜑 → (𝐴(midG‘𝐺)(𝑀𝐴)) ∈ 𝑃)
14 lmiiso.2 . . . . . . . . . . . 12 (𝜑𝐵𝑃)
151, 2, 3, 4, 7, 9, 10, 11, 14lmicl 28583 . . . . . . . . . . . 12 (𝜑 → (𝑀𝐵) ∈ 𝑃)
161, 2, 3, 4, 7, 14, 15midcl 28574 . . . . . . . . . . 11 (𝜑 → (𝐵(midG‘𝐺)(𝑀𝐵)) ∈ 𝑃)
171, 2, 3, 4, 7, 13, 16midcl 28574 . . . . . . . . . 10 (𝜑 → ((𝐴(midG‘𝐺)(𝑀𝐴))(midG‘𝐺)(𝐵(midG‘𝐺)(𝑀𝐵))) ∈ 𝑃)
186, 17eqeltrid 2833 . . . . . . . . 9 (𝜑𝑍𝑃)
1918adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝑆𝐴) = 𝑍) → 𝑍𝑃)
20 eqid 2728 . . . . . . . . . 10 (pInvG‘𝐺) = (pInvG‘𝐺)
21 lmiisolem.s . . . . . . . . . 10 𝑆 = ((pInvG‘𝐺)‘𝑍)
221, 2, 3, 10, 20, 4, 18, 21, 8mircl 28458 . . . . . . . . 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 28454 . . . . . . . 8 ((𝜑 ∧ (𝑆𝐴) = 𝑍) → (𝑍 (𝑆𝐴)) = (𝑍 𝐴))
26 simpr 484 . . . . . . . . 9 ((𝜑 ∧ (𝑆𝐴) = 𝑍) → (𝑆𝐴) = 𝑍)
2726eqcomd 2734 . . . . . . . 8 ((𝜑 ∧ (𝑆𝐴) = 𝑍) → 𝑍 = (𝑆𝐴))
281, 2, 3, 5, 19, 23, 19, 24, 25, 27tgcgreq 28279 . . . . . . 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 7430 . . . . . . . . . . . 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 2787 . . . . . . . . . . 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 28578 . . . . . . . . . . 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 2768 . . . . . . . . . 10 ((𝜑 ∧ (𝐴(midG‘𝐺)(𝑀𝐴)) = (𝐵(midG‘𝐺)(𝑀𝐵))) → 𝑍 = (𝐴(midG‘𝐺)(𝑀𝐴)))
37 eqidd 2729 . . . . . . . . . . . . 13 (𝜑 → (𝑀𝐴) = (𝑀𝐴))
381, 2, 3, 4, 7, 9, 10, 11, 8, 12islmib 28584 . . . . . . . . . . . . 13 (𝜑 → ((𝑀𝐴) = (𝑀𝐴) ↔ ((𝐴(midG‘𝐺)(𝑀𝐴)) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿(𝑀𝐴)) ∨ 𝐴 = (𝑀𝐴)))))
3937, 38mpbid 231 . . . . . . . . . . . 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 2829 . . . . . . . . 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 28576 . . . . . . . . . . . . 13 (𝜑 → ((𝐴(midG‘𝐺)(𝑀𝐴))(midG‘𝐺)(𝐵(midG‘𝐺)(𝑀𝐵))) ∈ ((𝐴(midG‘𝐺)(𝑀𝐴))𝐼(𝐵(midG‘𝐺)(𝑀𝐵))))
496, 48eqeltrid 2833 . . . . . . . . . . . 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 28416 . . . . . . . . . 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 2729 . . . . . . . . . . . . . 14 (𝜑 → (𝑀𝐵) = (𝑀𝐵))
551, 2, 3, 4, 7, 9, 10, 11, 14, 15islmib 28584 . . . . . . . . . . . . . 14 (𝜑 → ((𝑀𝐵) = (𝑀𝐵) ↔ ((𝐵(midG‘𝐺)(𝑀𝐵)) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐵𝐿(𝑀𝐵)) ∨ 𝐵 = (𝑀𝐵)))))
5654, 55mpbid 231 . . . . . . . . . . . . 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 28433 . . . . . . . . . 10 ((𝜑 ∧ (𝐴(midG‘𝐺)(𝑀𝐴)) ≠ (𝐵(midG‘𝐺)(𝑀𝐵))) → 𝐷 = ((𝐴(midG‘𝐺)(𝑀𝐴))𝐿(𝐵(midG‘𝐺)(𝑀𝐵))))
6051, 59eleqtrrd 2832 . . . . . . . . 9 ((𝜑 ∧ (𝐴(midG‘𝐺)(𝑀𝐴)) ≠ (𝐵(midG‘𝐺)(𝑀𝐵))) → 𝑍𝐷)
6142, 60pm2.61dane 3025 . . . . . . . 8 (𝜑𝑍𝐷)
6261adantr 480 . . . . . . 7 ((𝜑 ∧ (𝑆𝐴) = 𝑍) → 𝑍𝐷)
6328, 62eqeltrrd 2830 . . . . . 6 ((𝜑 ∧ (𝑆𝐴) = 𝑍) → 𝐴𝐷)
641, 2, 3, 4, 7, 9, 10, 11, 8lmiinv 28589 . . . . . . 7 (𝜑 → ((𝑀𝐴) = 𝐴𝐴𝐷))
6564biimpar 477 . . . . . 6 ((𝜑𝐴𝐷) → (𝑀𝐴) = 𝐴)
6663, 65syldan 590 . . . . 5 ((𝜑 ∧ (𝑆𝐴) = 𝑍) → (𝑀𝐴) = 𝐴)
6766, 28eqtr4d 2771 . . . 4 ((𝜑 ∧ (𝑆𝐴) = 𝑍) → (𝑀𝐴) = 𝑍)
6867oveq1d 7429 . . 3 ((𝜑 ∧ (𝑆𝐴) = 𝑍) → ((𝑀𝐴) (𝑀𝐵)) = (𝑍 (𝑀𝐵)))
69 eqidd 2729 . . . . . . . . 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 28576 . . . . . . . . . . . 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 7430 . . . . . . . . . . 11 ((𝜑𝐵 = (𝑀𝐵)) → (𝐵𝐼𝐵) = (𝐵𝐼(𝑀𝐵)))
7774, 76eleqtrrd 2832 . . . . . . . . . 10 ((𝜑𝐵 = (𝑀𝐵)) → (𝐵(midG‘𝐺)(𝑀𝐵)) ∈ (𝐵𝐼𝐵))
781, 2, 3, 70, 71, 72, 77axtgbtwnid 28263 . . . . . . . . 9 ((𝜑𝐵 = (𝑀𝐵)) → 𝐵 = (𝐵(midG‘𝐺)(𝑀𝐵)))
79 eqidd 2729 . . . . . . . . 9 ((𝜑𝐵 = (𝑀𝐵)) → 𝐵 = 𝐵)
8069, 78, 79s3eqd 14841 . . . . . . . 8 ((𝜑𝐵 = (𝑀𝐵)) → ⟨“𝑍𝐵𝐵”⟩ = ⟨“𝑍(𝐵(midG‘𝐺)(𝑀𝐵))𝐵”⟩)
811, 2, 3, 10, 20, 4, 18, 14, 14ragtrivb 28499 . . . . . . . . 9 (𝜑 → ⟨“𝑍𝐵𝐵”⟩ ∈ (∟G‘𝐺))
8281adantr 480 . . . . . . . 8 ((𝜑𝐵 = (𝑀𝐵)) → ⟨“𝑍𝐵𝐵”⟩ ∈ (∟G‘𝐺))
8380, 82eqeltrrd 2830 . . . . . . 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 2937 . . . . . . . . . 10 (𝐵 ≠ (𝑀𝐵) ↔ ¬ 𝐵 = (𝑀𝐵))
8956simprd 495 . . . . . . . . . . . 12 (𝜑 → (𝐷(⟂G‘𝐺)(𝐵𝐿(𝑀𝐵)) ∨ 𝐵 = (𝑀𝐵)))
9089orcomd 870 . . . . . . . . . . 11 (𝜑 → (𝐵 = (𝑀𝐵) ∨ 𝐷(⟂G‘𝐺)(𝐵𝐿(𝑀𝐵))))
9190orcanai 1001 . . . . . . . . . 10 ((𝜑 ∧ ¬ 𝐵 = (𝑀𝐵)) → 𝐷(⟂G‘𝐺)(𝐵𝐿(𝑀𝐵)))
9288, 91sylan2b 593 . . . . . . . . 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 28577 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝐵(midG‘𝐺)(𝑀𝐵)) = 𝐵) → (𝐵 𝐵) = (𝐵 (𝑀𝐵)))
102101eqcomd 2734 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝐵(midG‘𝐺)(𝑀𝐵)) = 𝐵) → (𝐵 (𝑀𝐵)) = (𝐵 𝐵))
1031, 2, 3, 96, 97, 98, 97, 102axtgcgrid 28260 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝐵(midG‘𝐺)(𝑀𝐵)) = 𝐵) → 𝐵 = (𝑀𝐵))
104103ex 412 . . . . . . . . . . . . 13 (𝜑 → ((𝐵(midG‘𝐺)(𝑀𝐵)) = 𝐵𝐵 = (𝑀𝐵)))
105104necon3d 2957 . . . . . . . . . . . 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 28416 . . . . . . . . . . 11 ((𝜑𝐵 ≠ (𝑀𝐵)) → (𝐵(midG‘𝐺)(𝑀𝐵)) ∈ (𝐵𝐿(𝑀𝐵)))
1091, 3, 10, 84, 87, 93, 94, 95, 106, 108tglineelsb2 28429 . . . . . . . . . 10 ((𝜑𝐵 ≠ (𝑀𝐵)) → (𝐵𝐿(𝑀𝐵)) = (𝐵𝐿(𝐵(midG‘𝐺)(𝑀𝐵))))
1101, 3, 10, 84, 95, 87, 106tglinecom 28432 . . . . . . . . . 10 ((𝜑𝐵 ≠ (𝑀𝐵)) → ((𝐵(midG‘𝐺)(𝑀𝐵))𝐿𝐵) = (𝐵𝐿(𝐵(midG‘𝐺)(𝑀𝐵))))
111109, 110eqtr4d 2771 . . . . . . . . 9 ((𝜑𝐵 ≠ (𝑀𝐵)) → (𝐵𝐿(𝑀𝐵)) = ((𝐵(midG‘𝐺)(𝑀𝐵))𝐿𝐵))
11292, 111breqtrd 5168 . . . . . . . 8 ((𝜑𝐵 ≠ (𝑀𝐵)) → 𝐷(⟂G‘𝐺)((𝐵(midG‘𝐺)(𝑀𝐵))𝐿𝐵))
1131, 2, 3, 10, 84, 85, 86, 87, 112perpdrag 28525 . . . . . . 7 ((𝜑𝐵 ≠ (𝑀𝐵)) → ⟨“𝑍(𝐵(midG‘𝐺)(𝑀𝐵))𝐵”⟩ ∈ (∟G‘𝐺))
11483, 113pm2.61dane 3025 . . . . . 6 (𝜑 → ⟨“𝑍(𝐵(midG‘𝐺)(𝑀𝐵))𝐵”⟩ ∈ (∟G‘𝐺))
1151, 2, 3, 10, 20, 4, 18, 16, 14israg 28494 . . . . . 6 (𝜑 → (⟨“𝑍(𝐵(midG‘𝐺)(𝑀𝐵))𝐵”⟩ ∈ (∟G‘𝐺) ↔ (𝑍 𝐵) = (𝑍 (((pInvG‘𝐺)‘(𝐵(midG‘𝐺)(𝑀𝐵)))‘𝐵))))
116114, 115mpbid 231 . . . . 5 (𝜑 → (𝑍 𝐵) = (𝑍 (((pInvG‘𝐺)‘(𝐵(midG‘𝐺)(𝑀𝐵)))‘𝐵)))
117 eqidd 2729 . . . . . . 7 (𝜑 → (𝐵(midG‘𝐺)(𝑀𝐵)) = (𝐵(midG‘𝐺)(𝑀𝐵)))
1181, 2, 3, 4, 7, 14, 15, 20, 16ismidb 28575 . . . . . . 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 7430 . . . . 5 (𝜑 → (𝑍 (𝑀𝐵)) = (𝑍 (((pInvG‘𝐺)‘(𝐵(midG‘𝐺)(𝑀𝐵)))‘𝐵)))
121116, 120eqtr4d 2771 . . . 4 (𝜑 → (𝑍 𝐵) = (𝑍 (𝑀𝐵)))
122121adantr 480 . . 3 ((𝜑 ∧ (𝑆𝐴) = 𝑍) → (𝑍 𝐵) = (𝑍 (𝑀𝐵)))
12328oveq1d 7429 . . 3 ((𝜑 ∧ (𝑆𝐴) = 𝑍) → (𝑍 𝐵) = (𝐴 𝐵))
12468, 122, 1233eqtr2d 2774 . 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 28458 . . . . 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 28455 . . . 4 ((𝜑 ∧ (𝑆𝐴) ≠ 𝑍) → 𝑍 ∈ ((𝑆𝐴)𝐼𝐴))
1361, 2, 3, 10, 20, 125, 127, 21, 131mirbtwn 28455 . . . 4 ((𝜑 ∧ (𝑆𝐴) ≠ 𝑍) → 𝑍 ∈ ((𝑆‘(𝑀𝐴))𝐼(𝑀𝐴)))
137 eqidd 2729 . . . . . . . . . . . 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 28576 . . . . . . . . . . . . . . 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 7430 . . . . . . . . . . . . . 14 ((𝜑𝐴 = (𝑀𝐴)) → (𝐴𝐼𝐴) = (𝐴𝐼(𝑀𝐴)))
145142, 144eleqtrrd 2832 . . . . . . . . . . . . 13 ((𝜑𝐴 = (𝑀𝐴)) → (𝐴(midG‘𝐺)(𝑀𝐴)) ∈ (𝐴𝐼𝐴))
1461, 2, 3, 138, 139, 140, 145axtgbtwnid 28263 . . . . . . . . . . . 12 ((𝜑𝐴 = (𝑀𝐴)) → 𝐴 = (𝐴(midG‘𝐺)(𝑀𝐴)))
147 eqidd 2729 . . . . . . . . . . . 12 ((𝜑𝐴 = (𝑀𝐴)) → 𝐴 = 𝐴)
148137, 146, 147s3eqd 14841 . . . . . . . . . . 11 ((𝜑𝐴 = (𝑀𝐴)) → ⟨“𝑍𝐴𝐴”⟩ = ⟨“𝑍(𝐴(midG‘𝐺)(𝑀𝐴))𝐴”⟩)
1491, 2, 3, 10, 20, 4, 18, 8, 8ragtrivb 28499 . . . . . . . . . . . 12 (𝜑 → ⟨“𝑍𝐴𝐴”⟩ ∈ (∟G‘𝐺))
150149adantr 480 . . . . . . . . . . 11 ((𝜑𝐴 = (𝑀𝐴)) → ⟨“𝑍𝐴𝐴”⟩ ∈ (∟G‘𝐺))
151148, 150eqeltrrd 2830 . . . . . . . . . 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 2937 . . . . . . . . . . . . 13 (𝐴 ≠ (𝑀𝐴) ↔ ¬ 𝐴 = (𝑀𝐴))
15739simprd 495 . . . . . . . . . . . . . . 15 (𝜑 → (𝐷(⟂G‘𝐺)(𝐴𝐿(𝑀𝐴)) ∨ 𝐴 = (𝑀𝐴)))
158157orcomd 870 . . . . . . . . . . . . . 14 (𝜑 → (𝐴 = (𝑀𝐴) ∨ 𝐷(⟂G‘𝐺)(𝐴𝐿(𝑀𝐴))))
159158orcanai 1001 . . . . . . . . . . . . 13 ((𝜑 ∧ ¬ 𝐴 = (𝑀𝐴)) → 𝐷(⟂G‘𝐺)(𝐴𝐿(𝑀𝐴)))
160156, 159sylan2b 593 . . . . . . . . . . . 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 28577 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝐴(midG‘𝐺)(𝑀𝐴)) = 𝐴) → (𝐴 𝐴) = (𝐴 (𝑀𝐴)))
170169eqcomd 2734 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝐴(midG‘𝐺)(𝑀𝐴)) = 𝐴) → (𝐴 (𝑀𝐴)) = (𝐴 𝐴))
1711, 2, 3, 164, 165, 166, 165, 170axtgcgrid 28260 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝐴(midG‘𝐺)(𝑀𝐴)) = 𝐴) → 𝐴 = (𝑀𝐴))
172171ex 412 . . . . . . . . . . . . . . . 16 (𝜑 → ((𝐴(midG‘𝐺)(𝑀𝐴)) = 𝐴𝐴 = (𝑀𝐴)))
173172necon3d 2957 . . . . . . . . . . . . . . 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 28416 . . . . . . . . . . . . . 14 ((𝜑𝐴 ≠ (𝑀𝐴)) → (𝐴(midG‘𝐺)(𝑀𝐴)) ∈ (𝐴𝐿(𝑀𝐴)))
1771, 3, 10, 152, 155, 161, 162, 163, 174, 176tglineelsb2 28429 . . . . . . . . . . . . 13 ((𝜑𝐴 ≠ (𝑀𝐴)) → (𝐴𝐿(𝑀𝐴)) = (𝐴𝐿(𝐴(midG‘𝐺)(𝑀𝐴))))
1781, 3, 10, 152, 163, 155, 174tglinecom 28432 . . . . . . . . . . . . 13 ((𝜑𝐴 ≠ (𝑀𝐴)) → ((𝐴(midG‘𝐺)(𝑀𝐴))𝐿𝐴) = (𝐴𝐿(𝐴(midG‘𝐺)(𝑀𝐴))))
179177, 178eqtr4d 2771 . . . . . . . . . . . 12 ((𝜑𝐴 ≠ (𝑀𝐴)) → (𝐴𝐿(𝑀𝐴)) = ((𝐴(midG‘𝐺)(𝑀𝐴))𝐿𝐴))
180160, 179breqtrd 5168 . . . . . . . . . . 11 ((𝜑𝐴 ≠ (𝑀𝐴)) → 𝐷(⟂G‘𝐺)((𝐴(midG‘𝐺)(𝑀𝐴))𝐿𝐴))
1811, 2, 3, 10, 152, 153, 154, 155, 180perpdrag 28525 . . . . . . . . . 10 ((𝜑𝐴 ≠ (𝑀𝐴)) → ⟨“𝑍(𝐴(midG‘𝐺)(𝑀𝐴))𝐴”⟩ ∈ (∟G‘𝐺))
182151, 181pm2.61dane 3025 . . . . . . . . 9 (𝜑 → ⟨“𝑍(𝐴(midG‘𝐺)(𝑀𝐴))𝐴”⟩ ∈ (∟G‘𝐺))
1831, 2, 3, 10, 20, 4, 18, 13, 8israg 28494 . . . . . . . . 9 (𝜑 → (⟨“𝑍(𝐴(midG‘𝐺)(𝑀𝐴))𝐴”⟩ ∈ (∟G‘𝐺) ↔ (𝑍 𝐴) = (𝑍 (((pInvG‘𝐺)‘(𝐴(midG‘𝐺)(𝑀𝐴)))‘𝐴))))
184182, 183mpbid 231 . . . . . . . 8 (𝜑 → (𝑍 𝐴) = (𝑍 (((pInvG‘𝐺)‘(𝐴(midG‘𝐺)(𝑀𝐴)))‘𝐴)))
185 eqidd 2729 . . . . . . . . . 10 (𝜑 → (𝐴(midG‘𝐺)(𝑀𝐴)) = (𝐴(midG‘𝐺)(𝑀𝐴)))
1861, 2, 3, 4, 7, 8, 12, 20, 13ismidb 28575 . . . . . . . . . 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 7430 . . . . . . . 8 (𝜑 → (𝑍 (𝑀𝐴)) = (𝑍 (((pInvG‘𝐺)‘(𝐴(midG‘𝐺)(𝑀𝐴)))‘𝐴)))
189184, 188eqtr4d 2771 . . . . . . 7 (𝜑 → (𝑍 𝐴) = (𝑍 (𝑀𝐴)))
1901, 2, 3, 10, 20, 4, 18, 21, 8mircgr 28454 . . . . . . 7 (𝜑 → (𝑍 (𝑆𝐴)) = (𝑍 𝐴))
1911, 2, 3, 10, 20, 4, 18, 21, 12mircgr 28454 . . . . . . 7 (𝜑 → (𝑍 (𝑆‘(𝑀𝐴))) = (𝑍 (𝑀𝐴)))
192189, 190, 1913eqtr4d 2778 . . . . . 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 28277 . . . 4 ((𝜑 ∧ (𝑆𝐴) ≠ 𝑍) → ((𝑆𝐴) 𝑍) = ((𝑆‘(𝑀𝐴)) 𝑍))
195189adantr 480 . . . 4 ((𝜑 ∧ (𝑆𝐴) ≠ 𝑍) → (𝑍 𝐴) = (𝑍 (𝑀𝐴)))
19621fveq1i 6892 . . . . . . . . . 10 (𝑆‘(𝐴(midG‘𝐺)(𝑀𝐴))) = (((pInvG‘𝐺)‘𝑍)‘(𝐴(midG‘𝐺)(𝑀𝐴)))
1971, 2, 3, 4, 7, 8, 12, 21, 18mirmid 28580 . . . . . . . . . 10 (𝜑 → ((𝑆𝐴)(midG‘𝐺)(𝑆‘(𝑀𝐴))) = (𝑆‘(𝐴(midG‘𝐺)(𝑀𝐴))))
1986eqcomi 2737 . . . . . . . . . . 11 ((𝐴(midG‘𝐺)(𝑀𝐴))(midG‘𝐺)(𝐵(midG‘𝐺)(𝑀𝐵))) = 𝑍
1991, 2, 3, 4, 7, 13, 16, 20, 18ismidb 28575 . . . . . . . . . . 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 2794 . . . . . . . . 9 (𝜑 → ((𝑆𝐴)(midG‘𝐺)(𝑆‘(𝑀𝐴))) = (𝐵(midG‘𝐺)(𝑀𝐵)))
2021, 2, 3, 4, 7, 22, 129, 20, 16ismidb 28575 . . . . . . . . 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 7432 . . . . . . 7 (𝜑 → ((𝑀𝐵) (𝑆‘(𝑀𝐴))) = ((((pInvG‘𝐺)‘(𝐵(midG‘𝐺)(𝑀𝐵)))‘𝐵) (((pInvG‘𝐺)‘(𝐵(midG‘𝐺)(𝑀𝐵)))‘(𝑆𝐴))))
205 eqid 2728 . . . . . . . 8 ((pInvG‘𝐺)‘(𝐵(midG‘𝐺)(𝑀𝐵))) = ((pInvG‘𝐺)‘(𝐵(midG‘𝐺)(𝑀𝐵)))
2061, 2, 3, 10, 20, 4, 16, 205, 14, 22miriso 28467 . . . . . . 7 (𝜑 → ((((pInvG‘𝐺)‘(𝐵(midG‘𝐺)(𝑀𝐵)))‘𝐵) (((pInvG‘𝐺)‘(𝐵(midG‘𝐺)(𝑀𝐵)))‘(𝑆𝐴))) = (𝐵 (𝑆𝐴)))
207204, 206eqtr2d 2769 . . . . . 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 28277 . . . 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 28262 . . 3 ((𝜑 ∧ (𝑆𝐴) ≠ 𝑍) → (𝐴 𝐵) = ((𝑀𝐴) (𝑀𝐵)))
212211eqcomd 2734 . 2 ((𝜑 ∧ (𝑆𝐴) ≠ 𝑍) → ((𝑀𝐴) (𝑀𝐵)) = (𝐴 𝐵))
213124, 212pm2.61dane 3025 1 (𝜑 → ((𝑀𝐴) (𝑀𝐵)) = (𝐴 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wo 846   = wceq 1534  wcel 2099  wne 2936   class class class wbr 5142  ran crn 5673  cfv 6542  (class class class)co 7414  2c2 12291  ⟨“cs3 14819  Basecbs 17173  distcds 17235  TarskiGcstrkg 28224  DimTarskiGcstrkgld 28228  Itvcitv 28230  LineGclng 28231  pInvGcmir 28449  ∟Gcrag 28490  ⟂Gcperpg 28492  midGcmid 28569  lInvGclmi 28570
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-cnex 11188  ax-resscn 11189  ax-1cn 11190  ax-icn 11191  ax-addcl 11192  ax-addrcl 11193  ax-mulcl 11194  ax-mulrcl 11195  ax-mulcom 11196  ax-addass 11197  ax-mulass 11198  ax-distr 11199  ax-i2m1 11200  ax-1ne0 11201  ax-1rid 11202  ax-rnegex 11203  ax-rrecex 11204  ax-cnre 11205  ax-pre-lttri 11206  ax-pre-lttrn 11207  ax-pre-ltadd 11208  ax-pre-mulgt0 11209
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-nel 3043  df-ral 3058  df-rex 3067  df-rmo 3372  df-reu 3373  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3964  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-tp 4629  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7865  df-1st 7987  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-oadd 8484  df-er 8718  df-map 8840  df-pm 8841  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-dju 9918  df-card 9956  df-pnf 11274  df-mnf 11275  df-xr 11276  df-ltxr 11277  df-le 11278  df-sub 11470  df-neg 11471  df-nn 12237  df-2 12299  df-3 12300  df-n0 12497  df-xnn0 12569  df-z 12583  df-uz 12847  df-fz 13511  df-fzo 13654  df-hash 14316  df-word 14491  df-concat 14547  df-s1 14572  df-s2 14825  df-s3 14826  df-trkgc 28245  df-trkgb 28246  df-trkgcb 28247  df-trkgld 28249  df-trkg 28250  df-cgrg 28308  df-leg 28380  df-mir 28450  df-rag 28491  df-perpg 28493  df-mid 28571  df-lmi 28572
This theorem is referenced by:  lmiiso  28594
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