Proof of Theorem mirbtwnhl
| Step | Hyp | Ref
| Expression |
| 1 | | simpr 484 |
. . . 4
⊢ ((𝜑 ∧ 𝑍 = 𝐴) → 𝑍 = 𝐴) |
| 2 | | mirval.p |
. . . . . 6
⊢ 𝑃 = (Base‘𝐺) |
| 3 | | mirval.i |
. . . . . 6
⊢ 𝐼 = (Itv‘𝐺) |
| 4 | | mirhl.k |
. . . . . 6
⊢ 𝐾 = (hlG‘𝐺) |
| 5 | | mirhl.a |
. . . . . 6
⊢ (𝜑 → 𝐴 ∈ 𝑃) |
| 6 | | mirhl.x |
. . . . . 6
⊢ (𝜑 → 𝑋 ∈ 𝑃) |
| 7 | | mirval.g |
. . . . . 6
⊢ (𝜑 → 𝐺 ∈ TarskiG) |
| 8 | 2, 3, 4, 5, 6, 5, 7 | hleqnid 28616 |
. . . . 5
⊢ (𝜑 → ¬ 𝐴(𝐾‘𝐴)𝑋) |
| 9 | 8 | adantr 480 |
. . . 4
⊢ ((𝜑 ∧ 𝑍 = 𝐴) → ¬ 𝐴(𝐾‘𝐴)𝑋) |
| 10 | 1, 9 | eqnbrtrd 5161 |
. . 3
⊢ ((𝜑 ∧ 𝑍 = 𝐴) → ¬ 𝑍(𝐾‘𝐴)𝑋) |
| 11 | 1 | fveq2d 6910 |
. . . . 5
⊢ ((𝜑 ∧ 𝑍 = 𝐴) → (𝑀‘𝑍) = (𝑀‘𝐴)) |
| 12 | | mirval.d |
. . . . . . 7
⊢ − =
(dist‘𝐺) |
| 13 | | mirval.l |
. . . . . . 7
⊢ 𝐿 = (LineG‘𝐺) |
| 14 | | mirval.s |
. . . . . . 7
⊢ 𝑆 = (pInvG‘𝐺) |
| 15 | | mirhl.m |
. . . . . . 7
⊢ 𝑀 = (𝑆‘𝐴) |
| 16 | 2, 12, 3, 13, 14, 7, 5, 15 | mircinv 28676 |
. . . . . 6
⊢ (𝜑 → (𝑀‘𝐴) = 𝐴) |
| 17 | 16 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ 𝑍 = 𝐴) → (𝑀‘𝐴) = 𝐴) |
| 18 | 11, 17 | eqtrd 2777 |
. . . 4
⊢ ((𝜑 ∧ 𝑍 = 𝐴) → (𝑀‘𝑍) = 𝐴) |
| 19 | | mirhl.y |
. . . . . 6
⊢ (𝜑 → 𝑌 ∈ 𝑃) |
| 20 | 2, 3, 4, 5, 19, 5,
7 | hleqnid 28616 |
. . . . 5
⊢ (𝜑 → ¬ 𝐴(𝐾‘𝐴)𝑌) |
| 21 | 20 | adantr 480 |
. . . 4
⊢ ((𝜑 ∧ 𝑍 = 𝐴) → ¬ 𝐴(𝐾‘𝐴)𝑌) |
| 22 | 18, 21 | eqnbrtrd 5161 |
. . 3
⊢ ((𝜑 ∧ 𝑍 = 𝐴) → ¬ (𝑀‘𝑍)(𝐾‘𝐴)𝑌) |
| 23 | 10, 22 | 2falsed 376 |
. 2
⊢ ((𝜑 ∧ 𝑍 = 𝐴) → (𝑍(𝐾‘𝐴)𝑋 ↔ (𝑀‘𝑍)(𝐾‘𝐴)𝑌)) |
| 24 | | simplr 769 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ 𝑍(𝐾‘𝐴)𝑋) → 𝑍 ≠ 𝐴) |
| 25 | 24 | neneqd 2945 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ 𝑍(𝐾‘𝐴)𝑋) → ¬ 𝑍 = 𝐴) |
| 26 | 7 | ad3antrrr 730 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ 𝑍(𝐾‘𝐴)𝑋) ∧ (𝑀‘𝑍) = 𝐴) → 𝐺 ∈ TarskiG) |
| 27 | 5 | ad3antrrr 730 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ 𝑍(𝐾‘𝐴)𝑋) ∧ (𝑀‘𝑍) = 𝐴) → 𝐴 ∈ 𝑃) |
| 28 | | mirhl.z |
. . . . . . . 8
⊢ (𝜑 → 𝑍 ∈ 𝑃) |
| 29 | 28 | ad3antrrr 730 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ 𝑍(𝐾‘𝐴)𝑋) ∧ (𝑀‘𝑍) = 𝐴) → 𝑍 ∈ 𝑃) |
| 30 | | simpr 484 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ 𝑍(𝐾‘𝐴)𝑋) ∧ (𝑀‘𝑍) = 𝐴) → (𝑀‘𝑍) = 𝐴) |
| 31 | 16 | ad3antrrr 730 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ 𝑍(𝐾‘𝐴)𝑋) ∧ (𝑀‘𝑍) = 𝐴) → (𝑀‘𝐴) = 𝐴) |
| 32 | 30, 31 | eqtr4d 2780 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ 𝑍(𝐾‘𝐴)𝑋) ∧ (𝑀‘𝑍) = 𝐴) → (𝑀‘𝑍) = (𝑀‘𝐴)) |
| 33 | 2, 12, 3, 13, 14, 26, 27, 15, 29, 27, 32 | mireq 28673 |
. . . . . 6
⊢ ((((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ 𝑍(𝐾‘𝐴)𝑋) ∧ (𝑀‘𝑍) = 𝐴) → 𝑍 = 𝐴) |
| 34 | 25, 33 | mtand 816 |
. . . . 5
⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ 𝑍(𝐾‘𝐴)𝑋) → ¬ (𝑀‘𝑍) = 𝐴) |
| 35 | 34 | neqned 2947 |
. . . 4
⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ 𝑍(𝐾‘𝐴)𝑋) → (𝑀‘𝑍) ≠ 𝐴) |
| 36 | | mirbtwnhl.2 |
. . . . 5
⊢ (𝜑 → 𝑌 ≠ 𝐴) |
| 37 | 36 | ad2antrr 726 |
. . . 4
⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ 𝑍(𝐾‘𝐴)𝑋) → 𝑌 ≠ 𝐴) |
| 38 | 7 | ad2antrr 726 |
. . . . 5
⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ 𝑍(𝐾‘𝐴)𝑋) → 𝐺 ∈ TarskiG) |
| 39 | 6 | ad2antrr 726 |
. . . . 5
⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ 𝑍(𝐾‘𝐴)𝑋) → 𝑋 ∈ 𝑃) |
| 40 | 5 | ad2antrr 726 |
. . . . 5
⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ 𝑍(𝐾‘𝐴)𝑋) → 𝐴 ∈ 𝑃) |
| 41 | 2, 12, 3, 13, 14, 7, 5, 15, 28 | mircl 28669 |
. . . . . 6
⊢ (𝜑 → (𝑀‘𝑍) ∈ 𝑃) |
| 42 | 41 | ad2antrr 726 |
. . . . 5
⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ 𝑍(𝐾‘𝐴)𝑋) → (𝑀‘𝑍) ∈ 𝑃) |
| 43 | 19 | ad2antrr 726 |
. . . . 5
⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ 𝑍(𝐾‘𝐴)𝑋) → 𝑌 ∈ 𝑃) |
| 44 | | mirbtwnhl.1 |
. . . . . 6
⊢ (𝜑 → 𝑋 ≠ 𝐴) |
| 45 | 44 | ad2antrr 726 |
. . . . 5
⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ 𝑍(𝐾‘𝐴)𝑋) → 𝑋 ≠ 𝐴) |
| 46 | 28 | ad2antrr 726 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ 𝑍(𝐾‘𝐴)𝑋) → 𝑍 ∈ 𝑃) |
| 47 | 2, 3, 4, 28, 6, 5,
7 | ishlg 28610 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑍(𝐾‘𝐴)𝑋 ↔ (𝑍 ≠ 𝐴 ∧ 𝑋 ≠ 𝐴 ∧ (𝑍 ∈ (𝐴𝐼𝑋) ∨ 𝑋 ∈ (𝐴𝐼𝑍))))) |
| 48 | 47 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑍 ≠ 𝐴) → (𝑍(𝐾‘𝐴)𝑋 ↔ (𝑍 ≠ 𝐴 ∧ 𝑋 ≠ 𝐴 ∧ (𝑍 ∈ (𝐴𝐼𝑋) ∨ 𝑋 ∈ (𝐴𝐼𝑍))))) |
| 49 | 48 | biimpa 476 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ 𝑍(𝐾‘𝐴)𝑋) → (𝑍 ≠ 𝐴 ∧ 𝑋 ≠ 𝐴 ∧ (𝑍 ∈ (𝐴𝐼𝑋) ∨ 𝑋 ∈ (𝐴𝐼𝑍)))) |
| 50 | 49 | simp3d 1145 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ 𝑍(𝐾‘𝐴)𝑋) → (𝑍 ∈ (𝐴𝐼𝑋) ∨ 𝑋 ∈ (𝐴𝐼𝑍))) |
| 51 | 50 | orcomd 872 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ 𝑍(𝐾‘𝐴)𝑋) → (𝑋 ∈ (𝐴𝐼𝑍) ∨ 𝑍 ∈ (𝐴𝐼𝑋))) |
| 52 | 2, 12, 3, 13, 14, 38, 15, 40, 39, 46, 51 | mirconn 28686 |
. . . . 5
⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ 𝑍(𝐾‘𝐴)𝑋) → 𝐴 ∈ (𝑋𝐼(𝑀‘𝑍))) |
| 53 | | mirbtwnhl.3 |
. . . . . 6
⊢ (𝜑 → 𝐴 ∈ (𝑋𝐼𝑌)) |
| 54 | 53 | ad2antrr 726 |
. . . . 5
⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ 𝑍(𝐾‘𝐴)𝑋) → 𝐴 ∈ (𝑋𝐼𝑌)) |
| 55 | 2, 3, 38, 39, 40, 42, 43, 45, 52, 54 | tgbtwnconn2 28584 |
. . . 4
⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ 𝑍(𝐾‘𝐴)𝑋) → ((𝑀‘𝑍) ∈ (𝐴𝐼𝑌) ∨ 𝑌 ∈ (𝐴𝐼(𝑀‘𝑍)))) |
| 56 | 2, 3, 4, 41, 19, 5, 7 | ishlg 28610 |
. . . . . 6
⊢ (𝜑 → ((𝑀‘𝑍)(𝐾‘𝐴)𝑌 ↔ ((𝑀‘𝑍) ≠ 𝐴 ∧ 𝑌 ≠ 𝐴 ∧ ((𝑀‘𝑍) ∈ (𝐴𝐼𝑌) ∨ 𝑌 ∈ (𝐴𝐼(𝑀‘𝑍)))))) |
| 57 | 56 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ 𝑍 ≠ 𝐴) → ((𝑀‘𝑍)(𝐾‘𝐴)𝑌 ↔ ((𝑀‘𝑍) ≠ 𝐴 ∧ 𝑌 ≠ 𝐴 ∧ ((𝑀‘𝑍) ∈ (𝐴𝐼𝑌) ∨ 𝑌 ∈ (𝐴𝐼(𝑀‘𝑍)))))) |
| 58 | 57 | adantr 480 |
. . . 4
⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ 𝑍(𝐾‘𝐴)𝑋) → ((𝑀‘𝑍)(𝐾‘𝐴)𝑌 ↔ ((𝑀‘𝑍) ≠ 𝐴 ∧ 𝑌 ≠ 𝐴 ∧ ((𝑀‘𝑍) ∈ (𝐴𝐼𝑌) ∨ 𝑌 ∈ (𝐴𝐼(𝑀‘𝑍)))))) |
| 59 | 35, 37, 55, 58 | mpbir3and 1343 |
. . 3
⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ 𝑍(𝐾‘𝐴)𝑋) → (𝑀‘𝑍)(𝐾‘𝐴)𝑌) |
| 60 | | simplr 769 |
. . . 4
⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ (𝑀‘𝑍)(𝐾‘𝐴)𝑌) → 𝑍 ≠ 𝐴) |
| 61 | 44 | ad2antrr 726 |
. . . 4
⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ (𝑀‘𝑍)(𝐾‘𝐴)𝑌) → 𝑋 ≠ 𝐴) |
| 62 | 7 | ad2antrr 726 |
. . . . 5
⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ (𝑀‘𝑍)(𝐾‘𝐴)𝑌) → 𝐺 ∈ TarskiG) |
| 63 | 19 | ad2antrr 726 |
. . . . 5
⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ (𝑀‘𝑍)(𝐾‘𝐴)𝑌) → 𝑌 ∈ 𝑃) |
| 64 | 5 | ad2antrr 726 |
. . . . 5
⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ (𝑀‘𝑍)(𝐾‘𝐴)𝑌) → 𝐴 ∈ 𝑃) |
| 65 | 28 | ad2antrr 726 |
. . . . 5
⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ (𝑀‘𝑍)(𝐾‘𝐴)𝑌) → 𝑍 ∈ 𝑃) |
| 66 | 6 | ad2antrr 726 |
. . . . 5
⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ (𝑀‘𝑍)(𝐾‘𝐴)𝑌) → 𝑋 ∈ 𝑃) |
| 67 | 36 | ad2antrr 726 |
. . . . 5
⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ (𝑀‘𝑍)(𝐾‘𝐴)𝑌) → 𝑌 ≠ 𝐴) |
| 68 | 16 | ad2antrr 726 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ (𝑀‘𝑍)(𝐾‘𝐴)𝑌) → (𝑀‘𝐴) = 𝐴) |
| 69 | 41 | ad2antrr 726 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ (𝑀‘𝑍)(𝐾‘𝐴)𝑌) → (𝑀‘𝑍) ∈ 𝑃) |
| 70 | 2, 12, 3, 13, 14, 62, 64, 15, 63 | mircl 28669 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ (𝑀‘𝑍)(𝐾‘𝐴)𝑌) → (𝑀‘𝑌) ∈ 𝑃) |
| 71 | 57 | biimpa 476 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ (𝑀‘𝑍)(𝐾‘𝐴)𝑌) → ((𝑀‘𝑍) ≠ 𝐴 ∧ 𝑌 ≠ 𝐴 ∧ ((𝑀‘𝑍) ∈ (𝐴𝐼𝑌) ∨ 𝑌 ∈ (𝐴𝐼(𝑀‘𝑍))))) |
| 72 | 71 | simp3d 1145 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ (𝑀‘𝑍)(𝐾‘𝐴)𝑌) → ((𝑀‘𝑍) ∈ (𝐴𝐼𝑌) ∨ 𝑌 ∈ (𝐴𝐼(𝑀‘𝑍)))) |
| 73 | 2, 12, 3, 13, 14, 62, 15, 64, 69, 63, 72 | mirconn 28686 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ (𝑀‘𝑍)(𝐾‘𝐴)𝑌) → 𝐴 ∈ ((𝑀‘𝑍)𝐼(𝑀‘𝑌))) |
| 74 | 2, 12, 3, 62, 69, 64, 70, 73 | tgbtwncom 28496 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ (𝑀‘𝑍)(𝐾‘𝐴)𝑌) → 𝐴 ∈ ((𝑀‘𝑌)𝐼(𝑀‘𝑍))) |
| 75 | 68, 74 | eqeltrd 2841 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ (𝑀‘𝑍)(𝐾‘𝐴)𝑌) → (𝑀‘𝐴) ∈ ((𝑀‘𝑌)𝐼(𝑀‘𝑍))) |
| 76 | 2, 12, 3, 13, 14, 62, 64, 15, 63, 64, 65 | mirbtwnb 28680 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ (𝑀‘𝑍)(𝐾‘𝐴)𝑌) → (𝐴 ∈ (𝑌𝐼𝑍) ↔ (𝑀‘𝐴) ∈ ((𝑀‘𝑌)𝐼(𝑀‘𝑍)))) |
| 77 | 75, 76 | mpbird 257 |
. . . . 5
⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ (𝑀‘𝑍)(𝐾‘𝐴)𝑌) → 𝐴 ∈ (𝑌𝐼𝑍)) |
| 78 | 2, 12, 3, 7, 6, 5, 19, 53 | tgbtwncom 28496 |
. . . . . 6
⊢ (𝜑 → 𝐴 ∈ (𝑌𝐼𝑋)) |
| 79 | 78 | ad2antrr 726 |
. . . . 5
⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ (𝑀‘𝑍)(𝐾‘𝐴)𝑌) → 𝐴 ∈ (𝑌𝐼𝑋)) |
| 80 | 2, 3, 62, 63, 64, 65, 66, 67, 77, 79 | tgbtwnconn2 28584 |
. . . 4
⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ (𝑀‘𝑍)(𝐾‘𝐴)𝑌) → (𝑍 ∈ (𝐴𝐼𝑋) ∨ 𝑋 ∈ (𝐴𝐼𝑍))) |
| 81 | 48 | adantr 480 |
. . . 4
⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ (𝑀‘𝑍)(𝐾‘𝐴)𝑌) → (𝑍(𝐾‘𝐴)𝑋 ↔ (𝑍 ≠ 𝐴 ∧ 𝑋 ≠ 𝐴 ∧ (𝑍 ∈ (𝐴𝐼𝑋) ∨ 𝑋 ∈ (𝐴𝐼𝑍))))) |
| 82 | 60, 61, 80, 81 | mpbir3and 1343 |
. . 3
⊢ (((𝜑 ∧ 𝑍 ≠ 𝐴) ∧ (𝑀‘𝑍)(𝐾‘𝐴)𝑌) → 𝑍(𝐾‘𝐴)𝑋) |
| 83 | 59, 82 | impbida 801 |
. 2
⊢ ((𝜑 ∧ 𝑍 ≠ 𝐴) → (𝑍(𝐾‘𝐴)𝑋 ↔ (𝑀‘𝑍)(𝐾‘𝐴)𝑌)) |
| 84 | 23, 83 | pm2.61dane 3029 |
1
⊢ (𝜑 → (𝑍(𝐾‘𝐴)𝑋 ↔ (𝑀‘𝑍)(𝐾‘𝐴)𝑌)) |