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Theorem lmiisolem 26161
Description: Lemma for lmiiso 26162. (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 474 . . . . . . . 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 26151 . . . . . . . . . . . 12 (𝜑 → (𝑀𝐴) ∈ 𝑃)
131, 2, 3, 4, 7, 8, 12midcl 26142 . . . . . . . . . . 11 (𝜑 → (𝐴(midG‘𝐺)(𝑀𝐴)) ∈ 𝑃)
14 lmiiso.2 . . . . . . . . . . . 12 (𝜑𝐵𝑃)
151, 2, 3, 4, 7, 9, 10, 11, 14lmicl 26151 . . . . . . . . . . . 12 (𝜑 → (𝑀𝐵) ∈ 𝑃)
161, 2, 3, 4, 7, 14, 15midcl 26142 . . . . . . . . . . 11 (𝜑 → (𝐵(midG‘𝐺)(𝑀𝐵)) ∈ 𝑃)
171, 2, 3, 4, 7, 13, 16midcl 26142 . . . . . . . . . 10 (𝜑 → ((𝐴(midG‘𝐺)(𝑀𝐴))(midG‘𝐺)(𝐵(midG‘𝐺)(𝑀𝐵))) ∈ 𝑃)
186, 17syl5eqel 2863 . . . . . . . . 9 (𝜑𝑍𝑃)
1918adantr 474 . . . . . . . 8 ((𝜑 ∧ (𝑆𝐴) = 𝑍) → 𝑍𝑃)
20 eqid 2778 . . . . . . . . . 10 (pInvG‘𝐺) = (pInvG‘𝐺)
21 lmiisolem.s . . . . . . . . . 10 𝑆 = ((pInvG‘𝐺)‘𝑍)
221, 2, 3, 10, 20, 4, 18, 21, 8mircl 26029 . . . . . . . . 9 (𝜑 → (𝑆𝐴) ∈ 𝑃)
2322adantr 474 . . . . . . . 8 ((𝜑 ∧ (𝑆𝐴) = 𝑍) → (𝑆𝐴) ∈ 𝑃)
248adantr 474 . . . . . . . 8 ((𝜑 ∧ (𝑆𝐴) = 𝑍) → 𝐴𝑃)
251, 2, 3, 10, 20, 5, 19, 21, 24mircgr 26025 . . . . . . . 8 ((𝜑 ∧ (𝑆𝐴) = 𝑍) → (𝑍 (𝑆𝐴)) = (𝑍 𝐴))
26 simpr 479 . . . . . . . . 9 ((𝜑 ∧ (𝑆𝐴) = 𝑍) → (𝑆𝐴) = 𝑍)
2726eqcomd 2784 . . . . . . . 8 ((𝜑 ∧ (𝑆𝐴) = 𝑍) → 𝑍 = (𝑆𝐴))
281, 2, 3, 5, 19, 23, 19, 24, 25, 27tgcgreq 25850 . . . . . . 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 6940 . . . . . . . . . . . 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 2833 . . . . . . . . . . 11 ((𝜑 ∧ (𝐴(midG‘𝐺)(𝑀𝐴)) = (𝐵(midG‘𝐺)(𝑀𝐵))) → 𝑍 = ((𝐴(midG‘𝐺)(𝑀𝐴))(midG‘𝐺)(𝐴(midG‘𝐺)(𝑀𝐴))))
324adantr 474 . . . . . . . . . . . 12 ((𝜑 ∧ (𝐴(midG‘𝐺)(𝑀𝐴)) = (𝐵(midG‘𝐺)(𝑀𝐵))) → 𝐺 ∈ TarskiG)
337adantr 474 . . . . . . . . . . . 12 ((𝜑 ∧ (𝐴(midG‘𝐺)(𝑀𝐴)) = (𝐵(midG‘𝐺)(𝑀𝐵))) → 𝐺DimTarskiG≥2)
3413adantr 474 . . . . . . . . . . . 12 ((𝜑 ∧ (𝐴(midG‘𝐺)(𝑀𝐴)) = (𝐵(midG‘𝐺)(𝑀𝐵))) → (𝐴(midG‘𝐺)(𝑀𝐴)) ∈ 𝑃)
351, 2, 3, 32, 33, 34, 34midid 26146 . . . . . . . . . . 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 2814 . . . . . . . . . 10 ((𝜑 ∧ (𝐴(midG‘𝐺)(𝑀𝐴)) = (𝐵(midG‘𝐺)(𝑀𝐵))) → 𝑍 = (𝐴(midG‘𝐺)(𝑀𝐴)))
37 eqidd 2779 . . . . . . . . . . . . 13 (𝜑 → (𝑀𝐴) = (𝑀𝐴))
381, 2, 3, 4, 7, 9, 10, 11, 8, 12islmib 26152 . . . . . . . . . . . . 13 (𝜑 → ((𝑀𝐴) = (𝑀𝐴) ↔ ((𝐴(midG‘𝐺)(𝑀𝐴)) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿(𝑀𝐴)) ∨ 𝐴 = (𝑀𝐴)))))
3937, 38mpbid 224 . . . . . . . . . . . 12 (𝜑 → ((𝐴(midG‘𝐺)(𝑀𝐴)) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿(𝑀𝐴)) ∨ 𝐴 = (𝑀𝐴))))
4039simpld 490 . . . . . . . . . . 11 (𝜑 → (𝐴(midG‘𝐺)(𝑀𝐴)) ∈ 𝐷)
4140adantr 474 . . . . . . . . . 10 ((𝜑 ∧ (𝐴(midG‘𝐺)(𝑀𝐴)) = (𝐵(midG‘𝐺)(𝑀𝐵))) → (𝐴(midG‘𝐺)(𝑀𝐴)) ∈ 𝐷)
4236, 41eqeltrd 2859 . . . . . . . . 9 ((𝜑 ∧ (𝐴(midG‘𝐺)(𝑀𝐴)) = (𝐵(midG‘𝐺)(𝑀𝐵))) → 𝑍𝐷)
434adantr 474 . . . . . . . . . . 11 ((𝜑 ∧ (𝐴(midG‘𝐺)(𝑀𝐴)) ≠ (𝐵(midG‘𝐺)(𝑀𝐵))) → 𝐺 ∈ TarskiG)
4413adantr 474 . . . . . . . . . . 11 ((𝜑 ∧ (𝐴(midG‘𝐺)(𝑀𝐴)) ≠ (𝐵(midG‘𝐺)(𝑀𝐵))) → (𝐴(midG‘𝐺)(𝑀𝐴)) ∈ 𝑃)
4516adantr 474 . . . . . . . . . . 11 ((𝜑 ∧ (𝐴(midG‘𝐺)(𝑀𝐴)) ≠ (𝐵(midG‘𝐺)(𝑀𝐵))) → (𝐵(midG‘𝐺)(𝑀𝐵)) ∈ 𝑃)
4618adantr 474 . . . . . . . . . . 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 26144 . . . . . . . . . . . . 13 (𝜑 → ((𝐴(midG‘𝐺)(𝑀𝐴))(midG‘𝐺)(𝐵(midG‘𝐺)(𝑀𝐵))) ∈ ((𝐴(midG‘𝐺)(𝑀𝐴))𝐼(𝐵(midG‘𝐺)(𝑀𝐵))))
496, 48syl5eqel 2863 . . . . . . . . . . . 12 (𝜑𝑍 ∈ ((𝐴(midG‘𝐺)(𝑀𝐴))𝐼(𝐵(midG‘𝐺)(𝑀𝐵))))
5049adantr 474 . . . . . . . . . . 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 25987 . . . . . . . . . 10 ((𝜑 ∧ (𝐴(midG‘𝐺)(𝑀𝐴)) ≠ (𝐵(midG‘𝐺)(𝑀𝐵))) → 𝑍 ∈ ((𝐴(midG‘𝐺)(𝑀𝐴))𝐿(𝐵(midG‘𝐺)(𝑀𝐵))))
5211adantr 474 . . . . . . . . . . 11 ((𝜑 ∧ (𝐴(midG‘𝐺)(𝑀𝐴)) ≠ (𝐵(midG‘𝐺)(𝑀𝐵))) → 𝐷 ∈ ran 𝐿)
5340adantr 474 . . . . . . . . . . 11 ((𝜑 ∧ (𝐴(midG‘𝐺)(𝑀𝐴)) ≠ (𝐵(midG‘𝐺)(𝑀𝐵))) → (𝐴(midG‘𝐺)(𝑀𝐴)) ∈ 𝐷)
54 eqidd 2779 . . . . . . . . . . . . . 14 (𝜑 → (𝑀𝐵) = (𝑀𝐵))
551, 2, 3, 4, 7, 9, 10, 11, 14, 15islmib 26152 . . . . . . . . . . . . . 14 (𝜑 → ((𝑀𝐵) = (𝑀𝐵) ↔ ((𝐵(midG‘𝐺)(𝑀𝐵)) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐵𝐿(𝑀𝐵)) ∨ 𝐵 = (𝑀𝐵)))))
5654, 55mpbid 224 . . . . . . . . . . . . 13 (𝜑 → ((𝐵(midG‘𝐺)(𝑀𝐵)) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐵𝐿(𝑀𝐵)) ∨ 𝐵 = (𝑀𝐵))))
5756simpld 490 . . . . . . . . . . . 12 (𝜑 → (𝐵(midG‘𝐺)(𝑀𝐵)) ∈ 𝐷)
5857adantr 474 . . . . . . . . . . 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 26004 . . . . . . . . . 10 ((𝜑 ∧ (𝐴(midG‘𝐺)(𝑀𝐴)) ≠ (𝐵(midG‘𝐺)(𝑀𝐵))) → 𝐷 = ((𝐴(midG‘𝐺)(𝑀𝐴))𝐿(𝐵(midG‘𝐺)(𝑀𝐵))))
6051, 59eleqtrrd 2862 . . . . . . . . 9 ((𝜑 ∧ (𝐴(midG‘𝐺)(𝑀𝐴)) ≠ (𝐵(midG‘𝐺)(𝑀𝐵))) → 𝑍𝐷)
6142, 60pm2.61dane 3057 . . . . . . . 8 (𝜑𝑍𝐷)
6261adantr 474 . . . . . . 7 ((𝜑 ∧ (𝑆𝐴) = 𝑍) → 𝑍𝐷)
6328, 62eqeltrrd 2860 . . . . . 6 ((𝜑 ∧ (𝑆𝐴) = 𝑍) → 𝐴𝐷)
641, 2, 3, 4, 7, 9, 10, 11, 8lmiinv 26157 . . . . . . 7 (𝜑 → ((𝑀𝐴) = 𝐴𝐴𝐷))
6564biimpar 471 . . . . . 6 ((𝜑𝐴𝐷) → (𝑀𝐴) = 𝐴)
6663, 65syldan 585 . . . . 5 ((𝜑 ∧ (𝑆𝐴) = 𝑍) → (𝑀𝐴) = 𝐴)
6766, 28eqtr4d 2817 . . . 4 ((𝜑 ∧ (𝑆𝐴) = 𝑍) → (𝑀𝐴) = 𝑍)
6867oveq1d 6939 . . 3 ((𝜑 ∧ (𝑆𝐴) = 𝑍) → ((𝑀𝐴) (𝑀𝐵)) = (𝑍 (𝑀𝐵)))
69 eqidd 2779 . . . . . . . . 9 ((𝜑𝐵 = (𝑀𝐵)) → 𝑍 = 𝑍)
704adantr 474 . . . . . . . . . 10 ((𝜑𝐵 = (𝑀𝐵)) → 𝐺 ∈ TarskiG)
7114adantr 474 . . . . . . . . . 10 ((𝜑𝐵 = (𝑀𝐵)) → 𝐵𝑃)
7216adantr 474 . . . . . . . . . 10 ((𝜑𝐵 = (𝑀𝐵)) → (𝐵(midG‘𝐺)(𝑀𝐵)) ∈ 𝑃)
731, 2, 3, 4, 7, 14, 15midbtwn 26144 . . . . . . . . . . . 12 (𝜑 → (𝐵(midG‘𝐺)(𝑀𝐵)) ∈ (𝐵𝐼(𝑀𝐵)))
7473adantr 474 . . . . . . . . . . 11 ((𝜑𝐵 = (𝑀𝐵)) → (𝐵(midG‘𝐺)(𝑀𝐵)) ∈ (𝐵𝐼(𝑀𝐵)))
75 simpr 479 . . . . . . . . . . . 12 ((𝜑𝐵 = (𝑀𝐵)) → 𝐵 = (𝑀𝐵))
7675oveq2d 6940 . . . . . . . . . . 11 ((𝜑𝐵 = (𝑀𝐵)) → (𝐵𝐼𝐵) = (𝐵𝐼(𝑀𝐵)))
7774, 76eleqtrrd 2862 . . . . . . . . . 10 ((𝜑𝐵 = (𝑀𝐵)) → (𝐵(midG‘𝐺)(𝑀𝐵)) ∈ (𝐵𝐼𝐵))
781, 2, 3, 70, 71, 72, 77axtgbtwnid 25834 . . . . . . . . 9 ((𝜑𝐵 = (𝑀𝐵)) → 𝐵 = (𝐵(midG‘𝐺)(𝑀𝐵)))
79 eqidd 2779 . . . . . . . . 9 ((𝜑𝐵 = (𝑀𝐵)) → 𝐵 = 𝐵)
8069, 78, 79s3eqd 14021 . . . . . . . 8 ((𝜑𝐵 = (𝑀𝐵)) → ⟨“𝑍𝐵𝐵”⟩ = ⟨“𝑍(𝐵(midG‘𝐺)(𝑀𝐵))𝐵”⟩)
811, 2, 3, 10, 20, 4, 18, 14, 14ragtrivb 26070 . . . . . . . . 9 (𝜑 → ⟨“𝑍𝐵𝐵”⟩ ∈ (∟G‘𝐺))
8281adantr 474 . . . . . . . 8 ((𝜑𝐵 = (𝑀𝐵)) → ⟨“𝑍𝐵𝐵”⟩ ∈ (∟G‘𝐺))
8380, 82eqeltrrd 2860 . . . . . . 7 ((𝜑𝐵 = (𝑀𝐵)) → ⟨“𝑍(𝐵(midG‘𝐺)(𝑀𝐵))𝐵”⟩ ∈ (∟G‘𝐺))
844adantr 474 . . . . . . . 8 ((𝜑𝐵 ≠ (𝑀𝐵)) → 𝐺 ∈ TarskiG)
8561adantr 474 . . . . . . . 8 ((𝜑𝐵 ≠ (𝑀𝐵)) → 𝑍𝐷)
8657adantr 474 . . . . . . . 8 ((𝜑𝐵 ≠ (𝑀𝐵)) → (𝐵(midG‘𝐺)(𝑀𝐵)) ∈ 𝐷)
8714adantr 474 . . . . . . . 8 ((𝜑𝐵 ≠ (𝑀𝐵)) → 𝐵𝑃)
88 df-ne 2970 . . . . . . . . . 10 (𝐵 ≠ (𝑀𝐵) ↔ ¬ 𝐵 = (𝑀𝐵))
8956simprd 491 . . . . . . . . . . . 12 (𝜑 → (𝐷(⟂G‘𝐺)(𝐵𝐿(𝑀𝐵)) ∨ 𝐵 = (𝑀𝐵)))
9089orcomd 860 . . . . . . . . . . 11 (𝜑 → (𝐵 = (𝑀𝐵) ∨ 𝐷(⟂G‘𝐺)(𝐵𝐿(𝑀𝐵))))
9190orcanai 988 . . . . . . . . . 10 ((𝜑 ∧ ¬ 𝐵 = (𝑀𝐵)) → 𝐷(⟂G‘𝐺)(𝐵𝐿(𝑀𝐵)))
9288, 91sylan2b 587 . . . . . . . . 9 ((𝜑𝐵 ≠ (𝑀𝐵)) → 𝐷(⟂G‘𝐺)(𝐵𝐿(𝑀𝐵)))
9315adantr 474 . . . . . . . . . . 11 ((𝜑𝐵 ≠ (𝑀𝐵)) → (𝑀𝐵) ∈ 𝑃)
94 simpr 479 . . . . . . . . . . 11 ((𝜑𝐵 ≠ (𝑀𝐵)) → 𝐵 ≠ (𝑀𝐵))
9516adantr 474 . . . . . . . . . . 11 ((𝜑𝐵 ≠ (𝑀𝐵)) → (𝐵(midG‘𝐺)(𝑀𝐵)) ∈ 𝑃)
964adantr 474 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝐵(midG‘𝐺)(𝑀𝐵)) = 𝐵) → 𝐺 ∈ TarskiG)
9714adantr 474 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝐵(midG‘𝐺)(𝑀𝐵)) = 𝐵) → 𝐵𝑃)
9815adantr 474 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝐵(midG‘𝐺)(𝑀𝐵)) = 𝐵) → (𝑀𝐵) ∈ 𝑃)
997adantr 474 . . . . . . . . . . . . . . . . 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 26145 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝐵(midG‘𝐺)(𝑀𝐵)) = 𝐵) → (𝐵 𝐵) = (𝐵 (𝑀𝐵)))
102101eqcomd 2784 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝐵(midG‘𝐺)(𝑀𝐵)) = 𝐵) → (𝐵 (𝑀𝐵)) = (𝐵 𝐵))
1031, 2, 3, 96, 97, 98, 97, 102axtgcgrid 25831 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝐵(midG‘𝐺)(𝑀𝐵)) = 𝐵) → 𝐵 = (𝑀𝐵))
104103ex 403 . . . . . . . . . . . . 13 (𝜑 → ((𝐵(midG‘𝐺)(𝑀𝐵)) = 𝐵𝐵 = (𝑀𝐵)))
105104necon3d 2990 . . . . . . . . . . . 12 (𝜑 → (𝐵 ≠ (𝑀𝐵) → (𝐵(midG‘𝐺)(𝑀𝐵)) ≠ 𝐵))
106105imp 397 . . . . . . . . . . 11 ((𝜑𝐵 ≠ (𝑀𝐵)) → (𝐵(midG‘𝐺)(𝑀𝐵)) ≠ 𝐵)
10773adantr 474 . . . . . . . . . . . 12 ((𝜑𝐵 ≠ (𝑀𝐵)) → (𝐵(midG‘𝐺)(𝑀𝐵)) ∈ (𝐵𝐼(𝑀𝐵)))
1081, 3, 10, 84, 87, 93, 95, 94, 107btwnlng1 25987 . . . . . . . . . . 11 ((𝜑𝐵 ≠ (𝑀𝐵)) → (𝐵(midG‘𝐺)(𝑀𝐵)) ∈ (𝐵𝐿(𝑀𝐵)))
1091, 3, 10, 84, 87, 93, 94, 95, 106, 108tglineelsb2 26000 . . . . . . . . . 10 ((𝜑𝐵 ≠ (𝑀𝐵)) → (𝐵𝐿(𝑀𝐵)) = (𝐵𝐿(𝐵(midG‘𝐺)(𝑀𝐵))))
1101, 3, 10, 84, 95, 87, 106tglinecom 26003 . . . . . . . . . 10 ((𝜑𝐵 ≠ (𝑀𝐵)) → ((𝐵(midG‘𝐺)(𝑀𝐵))𝐿𝐵) = (𝐵𝐿(𝐵(midG‘𝐺)(𝑀𝐵))))
111109, 110eqtr4d 2817 . . . . . . . . 9 ((𝜑𝐵 ≠ (𝑀𝐵)) → (𝐵𝐿(𝑀𝐵)) = ((𝐵(midG‘𝐺)(𝑀𝐵))𝐿𝐵))
11292, 111breqtrd 4914 . . . . . . . 8 ((𝜑𝐵 ≠ (𝑀𝐵)) → 𝐷(⟂G‘𝐺)((𝐵(midG‘𝐺)(𝑀𝐵))𝐿𝐵))
1131, 2, 3, 10, 84, 85, 86, 87, 112perpdrag 26093 . . . . . . 7 ((𝜑𝐵 ≠ (𝑀𝐵)) → ⟨“𝑍(𝐵(midG‘𝐺)(𝑀𝐵))𝐵”⟩ ∈ (∟G‘𝐺))
11483, 113pm2.61dane 3057 . . . . . 6 (𝜑 → ⟨“𝑍(𝐵(midG‘𝐺)(𝑀𝐵))𝐵”⟩ ∈ (∟G‘𝐺))
1151, 2, 3, 10, 20, 4, 18, 16, 14israg 26065 . . . . . 6 (𝜑 → (⟨“𝑍(𝐵(midG‘𝐺)(𝑀𝐵))𝐵”⟩ ∈ (∟G‘𝐺) ↔ (𝑍 𝐵) = (𝑍 (((pInvG‘𝐺)‘(𝐵(midG‘𝐺)(𝑀𝐵)))‘𝐵))))
116114, 115mpbid 224 . . . . 5 (𝜑 → (𝑍 𝐵) = (𝑍 (((pInvG‘𝐺)‘(𝐵(midG‘𝐺)(𝑀𝐵)))‘𝐵)))
117 eqidd 2779 . . . . . . 7 (𝜑 → (𝐵(midG‘𝐺)(𝑀𝐵)) = (𝐵(midG‘𝐺)(𝑀𝐵)))
1181, 2, 3, 4, 7, 14, 15, 20, 16ismidb 26143 . . . . . . 7 (𝜑 → ((𝑀𝐵) = (((pInvG‘𝐺)‘(𝐵(midG‘𝐺)(𝑀𝐵)))‘𝐵) ↔ (𝐵(midG‘𝐺)(𝑀𝐵)) = (𝐵(midG‘𝐺)(𝑀𝐵))))
119117, 118mpbird 249 . . . . . 6 (𝜑 → (𝑀𝐵) = (((pInvG‘𝐺)‘(𝐵(midG‘𝐺)(𝑀𝐵)))‘𝐵))
120119oveq2d 6940 . . . . 5 (𝜑 → (𝑍 (𝑀𝐵)) = (𝑍 (((pInvG‘𝐺)‘(𝐵(midG‘𝐺)(𝑀𝐵)))‘𝐵)))
121116, 120eqtr4d 2817 . . . 4 (𝜑 → (𝑍 𝐵) = (𝑍 (𝑀𝐵)))
122121adantr 474 . . 3 ((𝜑 ∧ (𝑆𝐴) = 𝑍) → (𝑍 𝐵) = (𝑍 (𝑀𝐵)))
12328oveq1d 6939 . . 3 ((𝜑 ∧ (𝑆𝐴) = 𝑍) → (𝑍 𝐵) = (𝐴 𝐵))
12468, 122, 1233eqtr2d 2820 . 2 ((𝜑 ∧ (𝑆𝐴) = 𝑍) → ((𝑀𝐴) (𝑀𝐵)) = (𝐴 𝐵))
1254adantr 474 . . . 4 ((𝜑 ∧ (𝑆𝐴) ≠ 𝑍) → 𝐺 ∈ TarskiG)
12622adantr 474 . . . 4 ((𝜑 ∧ (𝑆𝐴) ≠ 𝑍) → (𝑆𝐴) ∈ 𝑃)
12718adantr 474 . . . 4 ((𝜑 ∧ (𝑆𝐴) ≠ 𝑍) → 𝑍𝑃)
1288adantr 474 . . . 4 ((𝜑 ∧ (𝑆𝐴) ≠ 𝑍) → 𝐴𝑃)
1291, 2, 3, 10, 20, 4, 18, 21, 12mircl 26029 . . . . 5 (𝜑 → (𝑆‘(𝑀𝐴)) ∈ 𝑃)
130129adantr 474 . . . 4 ((𝜑 ∧ (𝑆𝐴) ≠ 𝑍) → (𝑆‘(𝑀𝐴)) ∈ 𝑃)
13112adantr 474 . . . 4 ((𝜑 ∧ (𝑆𝐴) ≠ 𝑍) → (𝑀𝐴) ∈ 𝑃)
13214adantr 474 . . . 4 ((𝜑 ∧ (𝑆𝐴) ≠ 𝑍) → 𝐵𝑃)
13315adantr 474 . . . 4 ((𝜑 ∧ (𝑆𝐴) ≠ 𝑍) → (𝑀𝐵) ∈ 𝑃)
134 simpr 479 . . . 4 ((𝜑 ∧ (𝑆𝐴) ≠ 𝑍) → (𝑆𝐴) ≠ 𝑍)
1351, 2, 3, 10, 20, 125, 127, 21, 128mirbtwn 26026 . . . 4 ((𝜑 ∧ (𝑆𝐴) ≠ 𝑍) → 𝑍 ∈ ((𝑆𝐴)𝐼𝐴))
1361, 2, 3, 10, 20, 125, 127, 21, 131mirbtwn 26026 . . . 4 ((𝜑 ∧ (𝑆𝐴) ≠ 𝑍) → 𝑍 ∈ ((𝑆‘(𝑀𝐴))𝐼(𝑀𝐴)))
137 eqidd 2779 . . . . . . . . . . . 12 ((𝜑𝐴 = (𝑀𝐴)) → 𝑍 = 𝑍)
1384adantr 474 . . . . . . . . . . . . 13 ((𝜑𝐴 = (𝑀𝐴)) → 𝐺 ∈ TarskiG)
1398adantr 474 . . . . . . . . . . . . 13 ((𝜑𝐴 = (𝑀𝐴)) → 𝐴𝑃)
14013adantr 474 . . . . . . . . . . . . 13 ((𝜑𝐴 = (𝑀𝐴)) → (𝐴(midG‘𝐺)(𝑀𝐴)) ∈ 𝑃)
1411, 2, 3, 4, 7, 8, 12midbtwn 26144 . . . . . . . . . . . . . . 15 (𝜑 → (𝐴(midG‘𝐺)(𝑀𝐴)) ∈ (𝐴𝐼(𝑀𝐴)))
142141adantr 474 . . . . . . . . . . . . . 14 ((𝜑𝐴 = (𝑀𝐴)) → (𝐴(midG‘𝐺)(𝑀𝐴)) ∈ (𝐴𝐼(𝑀𝐴)))
143 simpr 479 . . . . . . . . . . . . . . 15 ((𝜑𝐴 = (𝑀𝐴)) → 𝐴 = (𝑀𝐴))
144143oveq2d 6940 . . . . . . . . . . . . . 14 ((𝜑𝐴 = (𝑀𝐴)) → (𝐴𝐼𝐴) = (𝐴𝐼(𝑀𝐴)))
145142, 144eleqtrrd 2862 . . . . . . . . . . . . 13 ((𝜑𝐴 = (𝑀𝐴)) → (𝐴(midG‘𝐺)(𝑀𝐴)) ∈ (𝐴𝐼𝐴))
1461, 2, 3, 138, 139, 140, 145axtgbtwnid 25834 . . . . . . . . . . . 12 ((𝜑𝐴 = (𝑀𝐴)) → 𝐴 = (𝐴(midG‘𝐺)(𝑀𝐴)))
147 eqidd 2779 . . . . . . . . . . . 12 ((𝜑𝐴 = (𝑀𝐴)) → 𝐴 = 𝐴)
148137, 146, 147s3eqd 14021 . . . . . . . . . . 11 ((𝜑𝐴 = (𝑀𝐴)) → ⟨“𝑍𝐴𝐴”⟩ = ⟨“𝑍(𝐴(midG‘𝐺)(𝑀𝐴))𝐴”⟩)
1491, 2, 3, 10, 20, 4, 18, 8, 8ragtrivb 26070 . . . . . . . . . . . 12 (𝜑 → ⟨“𝑍𝐴𝐴”⟩ ∈ (∟G‘𝐺))
150149adantr 474 . . . . . . . . . . 11 ((𝜑𝐴 = (𝑀𝐴)) → ⟨“𝑍𝐴𝐴”⟩ ∈ (∟G‘𝐺))
151148, 150eqeltrrd 2860 . . . . . . . . . 10 ((𝜑𝐴 = (𝑀𝐴)) → ⟨“𝑍(𝐴(midG‘𝐺)(𝑀𝐴))𝐴”⟩ ∈ (∟G‘𝐺))
1524adantr 474 . . . . . . . . . . 11 ((𝜑𝐴 ≠ (𝑀𝐴)) → 𝐺 ∈ TarskiG)
15361adantr 474 . . . . . . . . . . 11 ((𝜑𝐴 ≠ (𝑀𝐴)) → 𝑍𝐷)
15440adantr 474 . . . . . . . . . . 11 ((𝜑𝐴 ≠ (𝑀𝐴)) → (𝐴(midG‘𝐺)(𝑀𝐴)) ∈ 𝐷)
1558adantr 474 . . . . . . . . . . 11 ((𝜑𝐴 ≠ (𝑀𝐴)) → 𝐴𝑃)
156 df-ne 2970 . . . . . . . . . . . . 13 (𝐴 ≠ (𝑀𝐴) ↔ ¬ 𝐴 = (𝑀𝐴))
15739simprd 491 . . . . . . . . . . . . . . 15 (𝜑 → (𝐷(⟂G‘𝐺)(𝐴𝐿(𝑀𝐴)) ∨ 𝐴 = (𝑀𝐴)))
158157orcomd 860 . . . . . . . . . . . . . 14 (𝜑 → (𝐴 = (𝑀𝐴) ∨ 𝐷(⟂G‘𝐺)(𝐴𝐿(𝑀𝐴))))
159158orcanai 988 . . . . . . . . . . . . 13 ((𝜑 ∧ ¬ 𝐴 = (𝑀𝐴)) → 𝐷(⟂G‘𝐺)(𝐴𝐿(𝑀𝐴)))
160156, 159sylan2b 587 . . . . . . . . . . . 12 ((𝜑𝐴 ≠ (𝑀𝐴)) → 𝐷(⟂G‘𝐺)(𝐴𝐿(𝑀𝐴)))
16112adantr 474 . . . . . . . . . . . . . 14 ((𝜑𝐴 ≠ (𝑀𝐴)) → (𝑀𝐴) ∈ 𝑃)
162 simpr 479 . . . . . . . . . . . . . 14 ((𝜑𝐴 ≠ (𝑀𝐴)) → 𝐴 ≠ (𝑀𝐴))
16313adantr 474 . . . . . . . . . . . . . 14 ((𝜑𝐴 ≠ (𝑀𝐴)) → (𝐴(midG‘𝐺)(𝑀𝐴)) ∈ 𝑃)
1644adantr 474 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝐴(midG‘𝐺)(𝑀𝐴)) = 𝐴) → 𝐺 ∈ TarskiG)
1658adantr 474 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝐴(midG‘𝐺)(𝑀𝐴)) = 𝐴) → 𝐴𝑃)
16612adantr 474 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝐴(midG‘𝐺)(𝑀𝐴)) = 𝐴) → (𝑀𝐴) ∈ 𝑃)
1677adantr 474 . . . . . . . . . . . . . . . . . . . 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 26145 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝐴(midG‘𝐺)(𝑀𝐴)) = 𝐴) → (𝐴 𝐴) = (𝐴 (𝑀𝐴)))
170169eqcomd 2784 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝐴(midG‘𝐺)(𝑀𝐴)) = 𝐴) → (𝐴 (𝑀𝐴)) = (𝐴 𝐴))
1711, 2, 3, 164, 165, 166, 165, 170axtgcgrid 25831 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝐴(midG‘𝐺)(𝑀𝐴)) = 𝐴) → 𝐴 = (𝑀𝐴))
172171ex 403 . . . . . . . . . . . . . . . 16 (𝜑 → ((𝐴(midG‘𝐺)(𝑀𝐴)) = 𝐴𝐴 = (𝑀𝐴)))
173172necon3d 2990 . . . . . . . . . . . . . . 15 (𝜑 → (𝐴 ≠ (𝑀𝐴) → (𝐴(midG‘𝐺)(𝑀𝐴)) ≠ 𝐴))
174173imp 397 . . . . . . . . . . . . . 14 ((𝜑𝐴 ≠ (𝑀𝐴)) → (𝐴(midG‘𝐺)(𝑀𝐴)) ≠ 𝐴)
175141adantr 474 . . . . . . . . . . . . . . 15 ((𝜑𝐴 ≠ (𝑀𝐴)) → (𝐴(midG‘𝐺)(𝑀𝐴)) ∈ (𝐴𝐼(𝑀𝐴)))
1761, 3, 10, 152, 155, 161, 163, 162, 175btwnlng1 25987 . . . . . . . . . . . . . 14 ((𝜑𝐴 ≠ (𝑀𝐴)) → (𝐴(midG‘𝐺)(𝑀𝐴)) ∈ (𝐴𝐿(𝑀𝐴)))
1771, 3, 10, 152, 155, 161, 162, 163, 174, 176tglineelsb2 26000 . . . . . . . . . . . . 13 ((𝜑𝐴 ≠ (𝑀𝐴)) → (𝐴𝐿(𝑀𝐴)) = (𝐴𝐿(𝐴(midG‘𝐺)(𝑀𝐴))))
1781, 3, 10, 152, 163, 155, 174tglinecom 26003 . . . . . . . . . . . . 13 ((𝜑𝐴 ≠ (𝑀𝐴)) → ((𝐴(midG‘𝐺)(𝑀𝐴))𝐿𝐴) = (𝐴𝐿(𝐴(midG‘𝐺)(𝑀𝐴))))
179177, 178eqtr4d 2817 . . . . . . . . . . . 12 ((𝜑𝐴 ≠ (𝑀𝐴)) → (𝐴𝐿(𝑀𝐴)) = ((𝐴(midG‘𝐺)(𝑀𝐴))𝐿𝐴))
180160, 179breqtrd 4914 . . . . . . . . . . 11 ((𝜑𝐴 ≠ (𝑀𝐴)) → 𝐷(⟂G‘𝐺)((𝐴(midG‘𝐺)(𝑀𝐴))𝐿𝐴))
1811, 2, 3, 10, 152, 153, 154, 155, 180perpdrag 26093 . . . . . . . . . 10 ((𝜑𝐴 ≠ (𝑀𝐴)) → ⟨“𝑍(𝐴(midG‘𝐺)(𝑀𝐴))𝐴”⟩ ∈ (∟G‘𝐺))
182151, 181pm2.61dane 3057 . . . . . . . . 9 (𝜑 → ⟨“𝑍(𝐴(midG‘𝐺)(𝑀𝐴))𝐴”⟩ ∈ (∟G‘𝐺))
1831, 2, 3, 10, 20, 4, 18, 13, 8israg 26065 . . . . . . . . 9 (𝜑 → (⟨“𝑍(𝐴(midG‘𝐺)(𝑀𝐴))𝐴”⟩ ∈ (∟G‘𝐺) ↔ (𝑍 𝐴) = (𝑍 (((pInvG‘𝐺)‘(𝐴(midG‘𝐺)(𝑀𝐴)))‘𝐴))))
184182, 183mpbid 224 . . . . . . . 8 (𝜑 → (𝑍 𝐴) = (𝑍 (((pInvG‘𝐺)‘(𝐴(midG‘𝐺)(𝑀𝐴)))‘𝐴)))
185 eqidd 2779 . . . . . . . . . 10 (𝜑 → (𝐴(midG‘𝐺)(𝑀𝐴)) = (𝐴(midG‘𝐺)(𝑀𝐴)))
1861, 2, 3, 4, 7, 8, 12, 20, 13ismidb 26143 . . . . . . . . . 10 (𝜑 → ((𝑀𝐴) = (((pInvG‘𝐺)‘(𝐴(midG‘𝐺)(𝑀𝐴)))‘𝐴) ↔ (𝐴(midG‘𝐺)(𝑀𝐴)) = (𝐴(midG‘𝐺)(𝑀𝐴))))
187185, 186mpbird 249 . . . . . . . . 9 (𝜑 → (𝑀𝐴) = (((pInvG‘𝐺)‘(𝐴(midG‘𝐺)(𝑀𝐴)))‘𝐴))
188187oveq2d 6940 . . . . . . . 8 (𝜑 → (𝑍 (𝑀𝐴)) = (𝑍 (((pInvG‘𝐺)‘(𝐴(midG‘𝐺)(𝑀𝐴)))‘𝐴)))
189184, 188eqtr4d 2817 . . . . . . 7 (𝜑 → (𝑍 𝐴) = (𝑍 (𝑀𝐴)))
1901, 2, 3, 10, 20, 4, 18, 21, 8mircgr 26025 . . . . . . 7 (𝜑 → (𝑍 (𝑆𝐴)) = (𝑍 𝐴))
1911, 2, 3, 10, 20, 4, 18, 21, 12mircgr 26025 . . . . . . 7 (𝜑 → (𝑍 (𝑆‘(𝑀𝐴))) = (𝑍 (𝑀𝐴)))
192189, 190, 1913eqtr4d 2824 . . . . . 6 (𝜑 → (𝑍 (𝑆𝐴)) = (𝑍 (𝑆‘(𝑀𝐴))))
193192adantr 474 . . . . 5 ((𝜑 ∧ (𝑆𝐴) ≠ 𝑍) → (𝑍 (𝑆𝐴)) = (𝑍 (𝑆‘(𝑀𝐴))))
1941, 2, 3, 125, 127, 126, 127, 130, 193tgcgrcomlr 25848 . . . 4 ((𝜑 ∧ (𝑆𝐴) ≠ 𝑍) → ((𝑆𝐴) 𝑍) = ((𝑆‘(𝑀𝐴)) 𝑍))
195189adantr 474 . . . 4 ((𝜑 ∧ (𝑆𝐴) ≠ 𝑍) → (𝑍 𝐴) = (𝑍 (𝑀𝐴)))
19621fveq1i 6449 . . . . . . . . . 10 (𝑆‘(𝐴(midG‘𝐺)(𝑀𝐴))) = (((pInvG‘𝐺)‘𝑍)‘(𝐴(midG‘𝐺)(𝑀𝐴)))
1971, 2, 3, 4, 7, 8, 12, 21, 18mirmid 26148 . . . . . . . . . 10 (𝜑 → ((𝑆𝐴)(midG‘𝐺)(𝑆‘(𝑀𝐴))) = (𝑆‘(𝐴(midG‘𝐺)(𝑀𝐴))))
1986eqcomi 2787 . . . . . . . . . . 11 ((𝐴(midG‘𝐺)(𝑀𝐴))(midG‘𝐺)(𝐵(midG‘𝐺)(𝑀𝐵))) = 𝑍
1991, 2, 3, 4, 7, 13, 16, 20, 18ismidb 26143 . . . . . . . . . . 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 250 . . . . . . . . . 10 (𝜑 → (𝐵(midG‘𝐺)(𝑀𝐵)) = (((pInvG‘𝐺)‘𝑍)‘(𝐴(midG‘𝐺)(𝑀𝐴))))
201196, 197, 2003eqtr4a 2840 . . . . . . . . 9 (𝜑 → ((𝑆𝐴)(midG‘𝐺)(𝑆‘(𝑀𝐴))) = (𝐵(midG‘𝐺)(𝑀𝐵)))
2021, 2, 3, 4, 7, 22, 129, 20, 16ismidb 26143 . . . . . . . . 9 (𝜑 → ((𝑆‘(𝑀𝐴)) = (((pInvG‘𝐺)‘(𝐵(midG‘𝐺)(𝑀𝐵)))‘(𝑆𝐴)) ↔ ((𝑆𝐴)(midG‘𝐺)(𝑆‘(𝑀𝐴))) = (𝐵(midG‘𝐺)(𝑀𝐵))))
203201, 202mpbird 249 . . . . . . . 8 (𝜑 → (𝑆‘(𝑀𝐴)) = (((pInvG‘𝐺)‘(𝐵(midG‘𝐺)(𝑀𝐵)))‘(𝑆𝐴)))
204119, 203oveq12d 6942 . . . . . . 7 (𝜑 → ((𝑀𝐵) (𝑆‘(𝑀𝐴))) = ((((pInvG‘𝐺)‘(𝐵(midG‘𝐺)(𝑀𝐵)))‘𝐵) (((pInvG‘𝐺)‘(𝐵(midG‘𝐺)(𝑀𝐵)))‘(𝑆𝐴))))
205 eqid 2778 . . . . . . . 8 ((pInvG‘𝐺)‘(𝐵(midG‘𝐺)(𝑀𝐵))) = ((pInvG‘𝐺)‘(𝐵(midG‘𝐺)(𝑀𝐵)))
2061, 2, 3, 10, 20, 4, 16, 205, 14, 22miriso 26038 . . . . . . 7 (𝜑 → ((((pInvG‘𝐺)‘(𝐵(midG‘𝐺)(𝑀𝐵)))‘𝐵) (((pInvG‘𝐺)‘(𝐵(midG‘𝐺)(𝑀𝐵)))‘(𝑆𝐴))) = (𝐵 (𝑆𝐴)))
207204, 206eqtr2d 2815 . . . . . 6 (𝜑 → (𝐵 (𝑆𝐴)) = ((𝑀𝐵) (𝑆‘(𝑀𝐴))))
208207adantr 474 . . . . 5 ((𝜑 ∧ (𝑆𝐴) ≠ 𝑍) → (𝐵 (𝑆𝐴)) = ((𝑀𝐵) (𝑆‘(𝑀𝐴))))
2091, 2, 3, 125, 132, 126, 133, 130, 208tgcgrcomlr 25848 . . . 4 ((𝜑 ∧ (𝑆𝐴) ≠ 𝑍) → ((𝑆𝐴) 𝐵) = ((𝑆‘(𝑀𝐴)) (𝑀𝐵)))
210121adantr 474 . . . 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 25833 . . 3 ((𝜑 ∧ (𝑆𝐴) ≠ 𝑍) → (𝐴 𝐵) = ((𝑀𝐴) (𝑀𝐵)))
212211eqcomd 2784 . 2 ((𝜑 ∧ (𝑆𝐴) ≠ 𝑍) → ((𝑀𝐴) (𝑀𝐵)) = (𝐴 𝐵))
213124, 212pm2.61dane 3057 1 (𝜑 → ((𝑀𝐴) (𝑀𝐵)) = (𝐴 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 386  wo 836   = wceq 1601  wcel 2107  wne 2969   class class class wbr 4888  ran crn 5358  cfv 6137  (class class class)co 6924  2c2 11435  ⟨“cs3 13999  Basecbs 16266  distcds 16358  TarskiGcstrkg 25798  DimTarskiGcstrkgld 25802  Itvcitv 25804  LineGclng 25805  pInvGcmir 26020  ∟Gcrag 26061  ⟂Gcperpg 26063  midGcmid 26137  lInvGclmi 26138
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5008  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228  ax-cnex 10330  ax-resscn 10331  ax-1cn 10332  ax-icn 10333  ax-addcl 10334  ax-addrcl 10335  ax-mulcl 10336  ax-mulrcl 10337  ax-mulcom 10338  ax-addass 10339  ax-mulass 10340  ax-distr 10341  ax-i2m1 10342  ax-1ne0 10343  ax-1rid 10344  ax-rnegex 10345  ax-rrecex 10346  ax-cnre 10347  ax-pre-lttri 10348  ax-pre-lttrn 10349  ax-pre-ltadd 10350  ax-pre-mulgt0 10351
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4674  df-int 4713  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-tr 4990  df-id 5263  df-eprel 5268  df-po 5276  df-so 5277  df-fr 5316  df-we 5318  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-pred 5935  df-ord 5981  df-on 5982  df-lim 5983  df-suc 5984  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-riota 6885  df-ov 6927  df-oprab 6928  df-mpt2 6929  df-om 7346  df-1st 7447  df-2nd 7448  df-wrecs 7691  df-recs 7753  df-rdg 7791  df-1o 7845  df-oadd 7849  df-er 8028  df-map 8144  df-pm 8145  df-en 8244  df-dom 8245  df-sdom 8246  df-fin 8247  df-card 9100  df-cda 9327  df-pnf 10415  df-mnf 10416  df-xr 10417  df-ltxr 10418  df-le 10419  df-sub 10610  df-neg 10611  df-nn 11380  df-2 11443  df-3 11444  df-n0 11648  df-xnn0 11720  df-z 11734  df-uz 11998  df-fz 12649  df-fzo 12790  df-hash 13442  df-word 13606  df-concat 13667  df-s1 13692  df-s2 14005  df-s3 14006  df-trkgc 25816  df-trkgb 25817  df-trkgcb 25818  df-trkgld 25820  df-trkg 25821  df-cgrg 25879  df-leg 25951  df-mir 26021  df-rag 26062  df-perpg 26064  df-mid 26139  df-lmi 26140
This theorem is referenced by:  lmiiso  26162
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